{"1": {"fulltext": "", "height": "4507", "width": "2991", "jp2-path": "calculuswithappl00haye_0001.jp2"}, "2": {"fulltext": "LIBRARY OF CONGRESS.\\nQA303\\nChap. Copyright No..\\nSheli\\nUNITED STATES OF AMERICA.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0002.jp2"}, "3": {"fulltext": "", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0003.jp2"}, "4": {"fulltext": "", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0004.jp2"}, "5": {"fulltext": "CALCULUS\\nWITH APPLICATIONS\\nAN INTRODUCTION TO THE MATHEMATICAL\\nTREATMENT OF SCIENCE\\nBT\\nELLEN HAYES\\nPROFESSOR OF APPLIED MATHEMATICS IN WELLESLEY COLLEGE\\nBoston\\nALLYN AND BACON\\n1900", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0005.jp2"}, "6": {"fulltext": "__ 70387\\ni-itor^i y of Oonor\u00c2\u00ab9\u00c2\u00ab\\n\\\\*u Cohu Hectiveo\\nNOV 3 1900\\nCopyright entry\\nSECOND COPY.\\nDelivered to\\nORDfcS DIVISION,\\nNOV 20 1900\\nCOPYRIGHT, 19 00,\\nBY ELLEN HAYES.\\nNcrfoooti -Iftes\\nJ. S. Cushing Co. Berwick Smith\\nNorwood Mass. U.S.A.", "height": "4267", "width": "2650", "jp2-path": "calculuswithappl00haye_0006.jp2"}, "7": {"fulltext": "PREFACE.\\nThis little book lias been written for two classes of\\npersons those who wish, for purposes of culture, to know,\\nin as simple and direct a way as possible, what the calculus\\nis and what it is for; and students primarily engaged in\\nwork in chemistry, astronomy, economics, etc., who have\\nnot time or inclination to take long courses in mathematics,\\nyet who would like to know how to use a tool as fine as\\nthe calculus.\\nThe pure mathematician will note the omission of\\nvarious subjects that are important from his point of view;\\nbut for him there are admirable and lengthy treatises on\\npure calculus. Also the student whose experience has\\nled him to conceive of mathematical study as the doing\\nof interminable lists of exercises, will be surprised and,\\npossibly, disax- pointed. This book is a reading lesson in\\napplied mathematics. Fancy exercises have been avoided.\\nThe examples are, for the most part, real problems from\\nmechanics and astronomy. This plan has been pursued in\\nthe conviction that such problems are just as good as\\nmake-believe ones for purposes of discipline, and a good\\ndeal better for purposes of knowledge. The time-honored\\nmethod of presenting calculus is much as if travelers\\nshould be stopped and made to pound stone on the high-\\niii", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0007.jp2"}, "8": {"fulltext": "IV PREFACE.\\nway, so that they never get anywhere or even know what\\nthe road is for. The following pages are a protest against\\nthe conventional method for I am wholly in sympathy\\nwith a remark made by Professor Lester F. Ward, in his\\nOutlines of Sociology There is no more vicions educa-\\ntional practice, and scarcely any more common one, than\\nthat of keeping the student in the dark as to the end and\\npurpose of his work. It breeds indifference, discourage-\\nment, and despair.\\nA chapter on analytic geometry has been introduced, in\\nthe hope that teachers will try the plan of presenting the\\nelements of the calculus and of analytic geometry together.\\nThere is no good reason either for keeping them distinct\\nor for presenting analytic geometry first.\\nTo three works I have to express my deep obligation.\\nThe spirit manifest in them has been my chief encourage-\\nment in preparing this book. I refer to GreenhilPs Differ-\\nential and Integral Calculus, Perry s Calculus for Engineers,\\nand Nernst and Schonnies Einfulirung in die mathematische\\nBehandlung der Naturwissenschaften.\\nWe have in these works, let us hope, an indication of\\nthe role which the calculus is to play in schemes for liberal\\nand scientific education in the not far distant future.\\nELLEN HAYES.\\nWellesley College,\\nSeptember, 1900.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0008.jp2"}, "9": {"fulltext": "CONTENTS.\\nCHAPTER I.\\nDifferentiation and Integration.\\nARTICLES PAGE\\n1- 6. Introduction 1\\n7-13. Differentiation of algebraic functions 8\\n1-4-16. Integration 12\\n17. Exercises 14\\n18-19. Implicit functions. Exercises 16\\n20-22. Differentiation of trigonometric functions 17\\n23-24. Differentiation of exponential and logarithmic func-\\ntions 21\\n25-27. Second derivatives. Partial and total differentials 24\\n28. Taylor s theorem 26\\n29. Maclaurin s theorem 28\\n30-31. Binomial theorem. Converging series 29\\n32-33. Indeterminate forms 31\\n34-35. Exercises. Logarithms 34\\nCHAPTER II.\\nThe Graph.\\n36-37. Cartesian system of coordinates\\ndy\\n38-40. Geometric meaning of Exercises\\nax\\n41-43. Maxima and minima. Exercises\\n44. Examples in maxima and minima\\n45-46. Polar coordinates\\nv\\n41\\n44\\n48\\n50\\n56", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0009.jp2"}, "10": {"fulltext": "VI CONTENTS.\\nCHAPTER III.\\nApplications.\\nARTICLES PAGE\\n47-54. Velocity and acceleration 58\\n55-59. Simple harmonic motion 64\\n60-63. Falling bodies 67\\n64-65. Rectilinear motion 74\\n66-69. Parabolic motion 77\\n70. Motion in a vertical curve 81\\n71-72. Simple pendulum 82\\n73-74. Areas. Examples 85\\n75-77. Mean values 88\\n78-79. Work 91\\n80. Lengths of curves 92\\n81. Volumes and surfaces of revolution 94\\n82. Double and triple integrals 96\\n83. Perfect differential 97\\n84-85. Moment of inertia. Examples 99\\n86-93. Kepler s laws 102\\nCHAPTER IV.\\nAnalytic Geometry.\\n94-99. The equation (x, y) Ill\\n114\\n117\\n117\\n118\\n118\\n119\\n119\\n121\\n100-103. Change of axes\\n104. Condition of parallelism, and of perpendicularity\\n105. Straight line in terms of slope and intercept\\n106. Straight line in terms of two points\\n106. Straight line in terms of one point and slope\\n107. Distance between two points\\n108. Distance from point to line\\n109-116. The ellipse", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0010.jp2"}, "11": {"fulltext": "CONTENTS. vil\\nARTICLES PAGE\\n117-125. The hyperbola 127\\n126-131. The parabola 134\\n132-133. Tangent and normal to a curve 137\\n134. Path of middle point of ellipse-chord 140\\n135. Determination of center and axes of ellipse 142\\n136. Tangent in terms of slope and intercept 143\\n137. Exercises 145\\n138-139. Space coordinates .147\\n140-141. Distance between two points in space 148\\n142. Equation to a plane surface 148\\nCHAPTER Y.\\nFormulas.\\nFundamental integrals 151\\nOther integrals .153\\nMiscellaneous formulas 155\\nIndex 159", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0011.jp2"}, "12": {"fulltext": "Perhaps we should all know hozv to use a tool as fine as the\\ncalculus. J. McKeen Cattell.\\nA man learns to use the calculus as he learns to use the\\nchisel or the file on actual concrete bits of work. John\\nPerry.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0012.jp2"}, "13": {"fulltext": "CALCULUS.\\nCHAPTER I.\\nDIFFERENTIATION AND INTEGRATION.\\n1. In the experiences of every-day life and in the\\noperations of science, people are continually dealing\\nwith things which keep changing in quantity, and with\\nthings so connected that a change in one of them is\\nfollowed by a change in another. For instance, we\\nknow that work varies in amount, and that the amount\\ndone by workmen depends on the time. We find crops\\nnow abundant and now scanty and, other things being\\nequal, the crops are seen to vary with the amount\\nof fertilizer used. We observe that the height of the\\nmercury in a thermometer changes with the temperature.\\nIn these examples and all similar ones there is a\\nrelation of cause and effect, or at least a relation of\\nantecedent and consequent; and Ave say that a quan-\\ntitative change in the cause is accompanied by a\\nquantitative change in the effect.\\nIt is part of the business of science not only to dis-\\ncover relations of cause and effect, but also to try to\\nexpress these relations with precision. When a relation\\nof cause and effect can be stated with exactness, the lan-\\n1", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0013.jp2"}, "14": {"fulltext": "2 CALCULUS.\\nguage of mathematics is the best one to use, because it\\ngives compact and unambiguous expressions, and because\\na further examination of the relation may then be con-\\nducted in that language and results easily reached which\\ncould be arrived at only with much difficulty, if at all,\\nin any ordinary language. For example, the expansive\\nforce of air was a property observed by Guericke (1602-\\n1686), but Boyle (1627-1691) discovered that the vol-\\nume varies inversely as the pressure. That is, if v\\nrepresents the volume of a given quantity of air and p\\nits pressure on unit area of the containing vessel, vac-,\\nP\\nand pv a constant. We shall see later how we may\\nlearn more about this law by using the equation pv c.\\nExperience in balancing bodies of equal or unequal\\nweights no doubt furnished ancient craftsmen with\\nsome vague notions regarding equilibrium but Archi-\\nmedes (287(?)-212 B.C.), from a few assumptions, con-\\ncluded that two bodies suspended from a bar are in\\nequilibrium when their distances from the point of\\nsupport of the bar are inversely proportional to their\\nweights. That is, if I, V are the distances of the bodies\\nwhose weights are w, w respectively, w w V L The\\nprinciple of the lever, as thus stated by Archimedes,\\nwas later fully established. To illustrate further, from\\nearliest times men must have noticed that unsupported\\nbodies fall to the ground but after the investigations\\nmade by Newton (1612-1727), it was possible to state\\nthe law of gravitation with mathematical accuracy\\nThe mutual attraction (stress) between any two bodies\\nvaries directly as the product of their masses and in-\\nversely as the square of their distance from each other,", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0014.jp2"}, "15": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 3\\nThus, if F is the whole attraction between the earth\\nand the moon, for instance, M the mass of the earth, m\\nthe mass of the moon, and r the distance between them,\\nF c These examples go to show that when a\\nr 2\\nprecise quantitative statement can be made in science,\\nmathematics, with its unambiguous symbolic shorthand,\\noffers the most economical way of making it.\\n2. Two modes of quantitative change or variation\\npresent themselves. As an illustration of the first, the\\nnumber of roses in a handful may be varied by adding\\none and another and another, until the number has\\nchanged from a to b or we may add several at a time\\nuntil the number has changed from a to b. But we\\ncannot do less than add one wdiole rose at a time for,\\nin this case, the variation element is a whole unit, that\\nis, a whole rose, and not any fraction of it. Again, we\\nmay measure a day with a minute as a unit of measure,\\nand say that a day contains 1440 minutes but this is\\nonly an artificial convenience. Time does not increase\\na minute at a time,* or even a second at a time, but\\nby elements of time which are immeasurably small frac-\\ntions of a second. This is the second mode of varia-\\ntion a quantitative change not by jumps or finite\\namounts, but by indefinitely small amounts.\\n3. By the term variable we mean a quantity which\\nchanges in the second manner above described. We\\nuse it in speaking of such things as volume, pressure,\\ndistance, etc., when they are conceived as being in a\\nstate of continuous variation. The term function is\\napplied to the quantity which necessarily changes be-", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0015.jp2"}, "16": {"fulltext": "4 CALCULUS.\\ncause of a change in a variable with which it is con-\\nnected. For example, the pressure of steam on the\\npiston of the cylinder is a function of the volume of\\nthe steam. The attraction which the earth exerts on\\nthe moon is a function of the distance of the moon\\nfrom the earth.\\nIf the symbol x stands for the variable and y for the\\nfunction, we briefly express the fact of their connection\\nby the general statement y =f(x). The precise nature\\nof the connection is shown by specializing /(V). For\\nexample, if f(x) is log x, we have the particular\\nstatement y log x.\\nWhen we need to distinguish one function from\\nanother, we use such forms as FQx), (x), u, v, etc.\\nThe nature of these conventional symbols should be\\ncarefully noticed. The parenthesis merely serves to\\nseparate the quantity symbol x from the other symbols\\nF, etc., which are not quantity symbols and hence\\nnot factors. is only algebraic shorthand for the\\nexpression a function of the varying quantity\\nFunctions are classified as algebraic and transcen-\\ndental. An algebraic function is defined as one which\\nimplies only a finite number of the algebraic opera-\\ntions, addition, subtraction, multiplication, division,\\ninvolution, and evolution.\\nThe trigonometric functions sin#, cos^, tanx, etc.,\\nare transcendental; so also are e x log a;, sin 1 etc.\\n4. Let us now suppose a change in pressure, or dis-\\ntance, or time, or whatever quantity we are dealing\\nwith under the symbol x. Let Sx stand for the amount\\nof change. Then the new quantity is represented by", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0016.jp2"}, "17": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 5\\nx r, and f(x) becomes f(x Bx). Subtracting tlie\\nformer value from the latter, we have f(x hx) f(x),\\nthe amount of change in the function occasioned by the\\nchange in the variable. It may be represented by 8y if\\ny=f(x). Then or its equal is\\nox ox\\nthe ratio of the increment of the function to that\\nof the variable. Let Sx now be supposed to become\\nsmaller and smaller until we cannot tell the difference\\nbetween it and zero. We say it has zero for its limit,\\nor it diminishes without limit, and to show that this\\nsupposition has been made we use the symbol dx in\\nplace of 8x, and also use dy in place of 8y for the indefi-\\nnitely small change in the function, dx is called the\\ndifferential of and dy the differential of y. The ratio\\nis called the first differential coefficient oif(x) with\\ndx\\nrespect to x, or briefly, the derivative.\\ndxi\\nWe shall find that the ratio -p is itself, in general,\\nsome function of x hence it is often written f (x).\\nThe symbols \u00e2\u0080\u0094f(x), f r (x) all mean the same\\nthing. It is important to notice that although dy and\\nd v\\ndx, the individual terms of the ratio -j-, are indefinitely\\nsmall, the ratio itself is usually finite.\\nThis dx of the mathematician is suggestive of the atom of\\nthe chemist, the particle of the physicist, and even the cell\\nof the biologist. It is the ultimate element of that with which the\\nmathematician deals, and always implies one property of the quan-\\ntity symbolized by x namely, its continuous variation.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0017.jp2"}, "18": {"fulltext": "b CALCULUS.\\n5. To illustrate the nature of let f (x) ttx 2 the\\ndx\\narea of a circle. Suppose this circular area to be cut\\nout of a thin sheet of metal and to have heat supplied\\nto it in such a way as to cause it to expand, but to re-\\nmain circular. Let the radius increase by the amount\\nSx then\\ny 8y f(x -f- 8x) ir (x 8x) 2\\n7TX 2 2 TTxSx IT (8xJ 2\\nand 8y f(x Sx) -f(x) 2 wxSx it (8x) 2\\nBy\\nhence 2 irx tt8x it (2 x 8x).\\nNow, if 8x be made indefinitely small, the limit of\\n2 x 8x is 2 x, and therefore\\n2\u00e2\u0084\u00a2.\\nThis means that if the quantity of heat used is so.\\nsmall that the increase in the length of the radius is\\nindefinitely small, the ratio of the increment of the\\narea to the increment of the radius is equal to the cir-\\ncumference of the circle, a result which might have\\nbeen guessed beforehand if we had reflected that the\\ngrowth in area is a belt only dx wide around the circle,\\nand dx is next to nothing.\\nAs another illustration, suppose we take the equa-\\ntion 2/ which states the law concerning the mutual\\nx\\ndependence of the volume and pressure of a gas, x repre-", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0018.jp2"}, "19": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 7\\nsenting volume and y representing pressure. Let us\\nlook at pressure as a function of volume then\\nJu\\nand y By f(x z)\\ntherefore By\\nand\\nx 8x\\nc cBx\\nx Bx x x(x Bx)\\nSy g__\\nx(x Bx}\\nhence\\ndy _\\ndx\\nIn other words, the ratio of the increment of the\\npressure to the increment of the volume is inversely\\nas the square of the volume. The minus sign means\\nthat when the volume takes an increment the incre-\\nment of the pressure is negative that is, the pressure-\\nincrement is really a decrement. This agrees with what\\nis said by the equation itself namely, that the press-\\nure decreases as the volume increases, and vice versa.\\nIf the student will follow the thought in a few con-\\ncrete examples like the two just given, he will gain a\\nbetter insight into the nature and purpose of the cal-\\nculus than he can acquire from the mechanical working\\nof a great number of meaningless exercises.\\n6. The importance of the ratio is soon realized\\ndx\\nby the student of the mathematical side of any of the\\nsciences which admit of mathematical treatment, such\\nas astronomy, thermodynamics, electricity, chemistry,", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0019.jp2"}, "20": {"fulltext": "8 CALCULUS.\\neconomics, etc. Granting its importance, we should\\nknow how to find, by the most direct method, the\\nderivative of any ordinary algebraic or transcendental\\nfunction and, what is even more essential, we should\\nbe able to perform the reverse operation that is, hav-\\ning -f- =z f (x) to find /(a;). In the following articles,\\nax\\ntheorems are established by means of which derivatives\\nmay be directly written, so that we need not take any\\nintermediate steps as in Art. 5.\\nThis operation of finding derivatives is called differ-\\nentiation. We begin with a function which is itself\\nthe sum of two functions.\\n7. Let u some function of x, and v some other func-\\ntion of x, and let\\ny=f(x)=u v.\\nThen, if x takes the increment r,\\ny By f(x Bx) u Bu v Si\\nand By Su Sv\\nBy Bu Bv\\nhence -f-\\nox ox ox\\nand in the limit -f- (1)\\nax ax ax\\nIt is evident that the same proof applies to any number\\nof functions connected by plus and minus signs.\\nA constant quantity, because it is a constant or un-\\nvarying quantity, has no increment and if we attempt\\nto express its derivative, we have nothing to divide by\\ndx. This amounts to saying that the derivative of a\\nconstant is zero.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0020.jp2"}, "21": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 9\\n8. Let the given function consist of the product of\\ntwo functions as expressed by\\ny=f(x)=uv.\\nThen\\ny Sy f(x r) (it Sii) (y hv)\\nuv vSu uSv SuSv\\nhence\\nSy vSu uSv SuSv\\ntherefore\\nSy Su Sv Sv\\n-f- v y- u Su\\nox ox ox ox\\nand\\ndy _ du dv\\ndx dx dx\\n(2)\\nAM UJU\\nThe last term Su is disposed of by observing that\\nox\\nas Su diminishes without limit, any quantity (except oc)\\nmultiplied by Su diminishes without limit, and is there-\\nfore dropped.\\nSimilarly, if y uvw where u, v, w, etc., are func-\\ntions of x,\\ndy ^du A dv ^dtv\\n_\u00c2\u00a3 (w (ww z; )-r--l etc.\\nax ax ax ax\\n9. Taking the last expression of the preceding article,\\nsuppose v u, w u, etc., so that uvw becomes (u) n\\nn being the number of functions of x. Then y u n\\nand the expression\\ndy ^du r A dv N a 1\\n-f- (vw -\u00e2\u0080\u00a20^ Quw---)\u00e2\u0080\u0094 (uv etc.,\\nax ax ax ax\\nax ax 6?x", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0021.jp2"}, "22": {"fulltext": "10 CALCULUS.\\nThis is a polynomial of n terms hence we can write,\\n(3)\\ndy n _ Y du\\n-2- nu n l\\ndx dx\\nAs a special case, if\\nu x, v x, iv x, etc.,\\ny x n\\nand formula (3) becomes\\nnx n l\\ndx\\np\\n10. Let y =f(x)== u q in which p and q are constant\\nquantities, positive and integral.\\nThen\\ny q =u*,\\nand by (3),\\ni dy n ,du\\nhence\\ndy p u p 1 du\\ndx q y q l dx\\nEliminating y from this expression by means of the\\nV\\ngiven expression y u q we have\\ndx q dx\\n11. Let y =f(x) u~ n ,n being integral and positive.\\nThen y and so yu n 1. Using the formula for\\nu n", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0022.jp2"}, "23": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 11\\nthe derivative of the product of two functions, and\\nwriting zero for the derivative of unity,\\nU n _M_ _j_ y nu n-\\\\ __ _ Q\\ndx dx\\ndy ynu n l du nu n l du\\nthat is, -f- _ _,\\ndx u n ax u m dx\\nor -f- m^ w x (5)\\n12. Comparing formulas (3), (4), (5), it is seen that\\nif y u n in which w is a function of x and is any\\nconstant,\\ndy \u00e2\u0080\u009e_i c?m\\ncfe dx\\nThe translation of this formula gives, therefore, the\\nonly rule that is needed for finding the derivative of\\na function affected with any constant exponent. It\\nshould be noticed, however, that since this expression\\ncl ii all f\\nfor the first derivative, contains as a factor, we\\ndx dx\\nmay require various other rules if we are to find the\\nexpression for which is the symbol. For example,\\nsuppose y (log 3 We now know that 3 (log x) 2\\nmultiplied by the derivative of log x, whatever it is.\\nWhat it is we shall learn in a subsequent article. At\\npresent we can only write\\n-Jl 3 (log x) 2 log x.\\ndx dx", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0023.jp2"}, "24": {"fulltext": "12 CALCULUS.\\n13. For the ratio of the indefinitely small increment\\nof the function to the increment of the variable we may\\nof course use f(x), as well as the symbol It\\ndx dx\\nshould be carefully noticed that the d standing above\\ndx is here, as everywhere, a symbol of operation and not\\nof quantity, signifying the differential oif(x). Notice\\nalso that an indicated operation counts for the same as a\\nsy\u00c2\u00a3i ry*\\nperformed one. For example, in an operation\\na x\\n/y* /y*a\\nis indicated, and the expression has everywhere\\nthe same value as a x, the result obtained by actually\\nd\\nperforming the division. So, for example, irx 2 has\\nthe same value as 2 ttx.\\n14. Differentiation is seen to be a tearing down\\nprocess, whereby we reach an ultimate element of quan-\\ntity. The reverse operation, one of building up, is\\nknown as integration; its symbol is (long s).\\nAs already shown, rules are established for the\\ndifferentiation of functions, but integration is largely\\na matter of guesswork and experiment. The test of\\nthe correctness of any integration is this differentiate\\nthe result; we should get the given differential form.\\nTables of integrals enable the student to write\\ndirectly the expression corresponding to an indicated\\nintegration, so that he need not go through the process\\nof discovering the required expression.\\nIn accordance with what has just been said, we have\\nJ dy y, the symbols 6?, J neutralizing each other.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0024.jp2"}, "25": {"fulltext": "DIFFERENTIATION AND INTEGRATION.\\nAlso,\\n,if\\ndy _ du dv\\ndx dx dx\\ndy du (7y\\nand\\nw v\\nIf\\ndy _ cfe c/^\\ne?# dx dx\\n13\\ny (ydu wtfo;)\\ntw.\\nIf\\nm 1\\n15. The symbol \\\\f(x)dx is known as a general or\\nindefinite integral. After discovering a function, say\\nwhich differentiated will give f(x), we ought to\\nwrite\\nff(x)dx cf (x)+C,\\nin which (7 is a quantity primarily undetermined, and\\nknown as the constant of integration. Since C is a\\nconstant,\\n-f O (x) \u00e2\u0082\u00acT\\\\ (x) -f (7=/\\nand as a constant term may thus exist in connection\\nwith the original function we give the integral the\\nbenefit of the doubt and write as stated, cf (x) 0.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0025.jp2"}, "26": {"fulltext": "14 CALCULUS.\\n16. The symbol I f(x)dx is known as a definite\\nintegral. Its meaning is this Find the general inte-\\ngral, which will be some function of and substitute b\\nfor x then substitute a for and subtract the latter\\nexpression from the former. To state the process sym-\\nbolically, let\\n)f(x)dx 4 (x)\\\\\\nj\\nthen I f(x)dx 4 (x)\\nK\u00c2\u00ab\\na and b are called the limits of the integral. The\\nconstant O disappears since\\nC]-[ k\u00c2\u00ab) C] K\u00c2\u00ab).\\nFor further discussion of definite integrals, see Art. 73,\\nand for other methods of finding the value of (7, see\\nArts. 60, 61, etc.\\nExercises.\\n17. 1. If y nu, n Prove this in two ways:\\ndx dx\\n(1) by beginning y 8y n (u 8u) (2) by regarding nu\\nas a special case of the product of two functions in which\\none of the functions is a constant (a function by courtesy).\\n2. If dy nf(x)dx, show that\\ny J nf(x) dx n I f(x) dx\\nthat is, a constant factor under the integral sign may be\\nremoved and written as a coefficient of the integral. Notice\\nthat if dy nf(x) dx, -1= f( x dx.\\nn w", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0026.jp2"}, "27": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 15\\nu\\n3. If y show that\\nJ v\\ndie do\\nV u\\ndy __ dx dx\\ndx v 2\\nTranslate this formula into a theorem.\\n4. Prove that if y m being a constant.\\nm dx m dx\\nDo this in two ways: (1) by observing that the result in\\nexercise 1 holds when w is a fraction (2) by making a\\nspecial case under exercise 3.\\n5. Prove that if y\\ndu\\nm\\ndy dx\\ndx u 2\\n6. Prove that the derivative of the square root of any\\nfunction is the derivative of the function divided by twice\\nthe square root of the function. That is, if y Vw,\\ndu\\ndy dx\\ndx 2 V\\na\\nf(x) dx f f(x) dx that is, the limits\\nmay be reversed if the sign of the integral is changed.\\n8. If y is the surface of a sphere whose radius is x,\\ndx\\n9. Show that the ratio of the differential of the circum-\\nference of a circle is to the differential of its radius as 2 ir 1.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0027.jp2"}, "28": {"fulltext": "16 CALCULUS.\\n18. Thus far y has been in each instance an explicit\\nfunction of x. Suppose now that x and y are so com-\\nbined in any expression that y is only implicitly a\\nfunction of x for example, as in the expression\\nax by c 0. This statement is the equivalent of\\nthe explicit statement y A little algebraic\\nconsideration will show that it is unnecessary to solve\\nfor y before proceeding to find We may differ-\\ned 7\\nentiate immediately, and then solve for -f-. Thus, if\\ndx\\nax by e 0, differentiating term by term with x as\\nthe fundamental variable, we have a b and\\n7 dx\\nhence -f- which is the same result that we get\\ndx b _\\nby differentiating y\\nExercises.\\n19\\nl.\u00c2\u00a3 \u00c2\u00a3=l 5 showthat^=\u00c2\u00b1^.\\na 2 b 2 dx a -y/tf\\nDifferentiating immediately,\\n2 x dx\\n-T+-T2\u00e2\u0080\u0094\\ndy dy __ b 2 x\\ndx dx a 2 y\\nsolving for -f-\\ny may now be eliminated by using its value\\nb\\n-ya 2 x 2\\na\\nobtained from the given equation.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0028.jp2"}, "29": {"fulltext": "DIFFERENTIA TION\\nAND\\nINTEGRATION,\\n2.\\n8 y 2,\\n3 axy fin\\ndx\\n3.\\nx s y 3\\n3 axy a 3\\nfind\\ndy\\ndx\\n4.\\npv c\\nshow that -J-\\ndv\\nc -p\\nif J)\\nBy\\nformula\\n,(2), *f +i\\nhence\\nJ\\ndp _\\ndv\\ni\\nG\\n~V 2\\n17\\n5. 27v k c show that\\nHere, and also in exercise 4 it is just as well to solve\\nfor p before beginning to differentiate. We have p so\\nthat vk\\ndp _ _ kc v k l _ _ kc\\ndv v 2k v k+1\\n6. F= c in which c, M, m are constants show that\\nc]F _2GMm m\\ndr r 3\\nIt is often desirable to use other letters besides x and y to denote\\nthe variable quantities. The student should therefore accustom him-\\nself at the outset to such symbols as those given in exercises 4, 5, 6.\\n20. If f(x) is a varying angle, it is clear that any\\ntrigonometric function of the angle must also vary.\\nWe have now to find -j- when y represents each one\\nof the trigonometric functions in turn.\\nSuppose y sin/(^). Let f(x) u then y sin u,\\n1 Sy _ sin (u S-u) sin u\\n(XllKX\\nOX ox", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0029.jp2"}, "30": {"fulltext": "18 CALCULUS.\\nPut u Su a, and u {3\\nthen, since sin a sin /3 2 sin J (a cos J (cc /3),\\nSy sin (w Su sin w sin a sin /3\\n2 sin Sw cos \\\\(2 u hu)\\n2 cos (u I Sw) sin 1 cm\\nS?/ N sin Sw S^\\nhence cos (u A- ou) j-g -k\u00e2\u0080\u0094\\nox A J \\\\ou Sx\\nBut when an angle diminishes without limit, we may\\nwrite the angle itself for its sine so we now have\\ndy d du sn\\\\\\nsin u cos u (b)\\ndx dx dx\\nLet y cos w sin u J\\n,i dy d (it (it d (tt\\nthen sm u cos u) u\\nV2 y V2 Jdx\\\\2 J 9\\ni n ay a au ^tn\\nthereiore \u00e2\u0080\u0094oosu= smu\u00e2\u0080\u0094\u00e2\u0080\u0094 (7)\\n\\\\AiJU \\\\XlJU \\\\AlJU\\nsinw\\nLet V tan u\\ncosw\\nUsing exercise 3, Art. 17, together with formulas\\n(6) and (7),\\ndy cos 2 u sin 2 u du 9 du\\nJ 2 1~ SeC M TT 5\\ndx cos A u ax ax\\ntherefore -M. tan u sec 2 u (8)", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0030.jp2"}, "31": {"fulltext": "BIFFEBENTIATION AND INTEGRATION. 19\\nT COS u\\nLet y cot u\\nsin u\\ndu sin 2 u cos 2 u du 9 du\\n(9)\\ndx\\nsin 2 u dx\\ntherefore\\ndy d\\ncot u\\n\\\\XiJu LI il/\\n9 C?7\u00c2\u00a3\\ncosec 5 m\\ndx\\nLet\\n1\\ny sec u\\nu COS 76\\nthen, using\\nexercise 5, Art. 17,\\ndy\\ndu\\nsin u\\ndx\\ntan ser\\ndu\\nu\\nand Ave have\\ndy d du /1AX\\n-\u00e2\u0080\u0094sec 7^= tan u secu (lu)\\ndx dx dx\\nLet y cosec u\\nSUlTi\\ncos u\\ndy dx t du\\ncot u cosec u\\nax sur u dx\\nand therefore\\ndy d du\\n-f- cosec u= cot u cosec w (11)\\n21. In formulas (6) to (11) inclusive we have the\\nratio of the differential of the trigonometric function to\\nthe differential of the angle x. These formulas, which\\nare remarkable for their simplicity, should be translated\\nand committed to memory. We may next regard the\\ntrigonometric function as varying by equal increments,\\nand thereby causing a change in the angle. From this", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0031.jp2"}, "32": {"fulltext": "20 CALCULUS.\\npoint of view we have to find the ratio of the differen-\\ntial of the angle to the differential of the trigonometric\\nfunction.\\n22. Let y sin -1 u then u siny.\\nHence, by formula (6),\\ndu dy\\ncos y -j-\\ndx dx\\ndu\\ndy dx\\nthat is,\\ndx cos y\\nBut cos y Vl u 2\\ndu\\ntherefore \u00c2\u00a7y _ n -i u (12)\\ncfa cte Vl-i( 2\\nLet 2/ cos -1 1\u00c2\u00a3 then w cos y, and proceeding as\\nbefore, Ave find\\ndu\\ndy d tfe ox\\n-f- -\u00e2\u0080\u0094cos 1 (lo)\\na# a# VI _ ^2\\nLet 2/ tan -1 u then m tan y,\\n6?W\\n6?w dy dy dx\\nsec z that is, 5-\\na^ a^ a# sec z 2/\\ndu\\nand sec 2 y that is,\\ntherefore _^ tan -i M _^ (14)\\ndx ax 1 w\\nsimilarly, if y= cot 1 2/,\\ndy d dx sh r x\\n-j- cot -1 w q 5- (15;\\ndx dx 1 ?r", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0032.jp2"}, "33": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 21\\nLet y sec 2 u. u sec y,\\ndu dy\\nand -7- tan y sec v -7 1\\ndx dx\\ndu\\nthat is, dx\\ndx tan y secy\\nBnt since u sec y, tan y V^ 2 1 and we have\\ndu\\ndy d dfe ^i^n\\n-f sec- 1 (Id)\\ndx dx U -y/y? _ l\\nFinally, if y cosec -1 we find that\\ndu\\ndy d _- 6fe\\ncosec\\ndx dx u V^ 2 1\\n(17)\\nAll these operations of differentiating may be reversed,\\nso that if\\ndy du\\ndx\\ndu r\\ncos u -j-, y I cos udu sm u, etc.\\n23. It remains to find -!f- when 2/ e x a x log log 1 w,\\ne w a* u being, as before, equal to f(x).\\nFrom algebra* we have\\n[2 [3 |4 |n_\\nin which g is the number 2.7182818284 forming the\\nbase of the Napierian system of logarithms. If y e x\\nc li A e A_(i x\\ndx dx dx\\\\ 1 2 [3\\n*See Hall and Knight s Elementary Algebra, Art. 537. Edition of", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0033.jp2"}, "34": {"fulltext": "22 CALCULUS.\\nPerforming the operation indicated, that is, differen-\\ntiating this series term by term,\\nbut this result is the original series which e x equals\\ntherefore\\n(18)\\ndx dx\\nThis result is unique, being the only case known in\\nwhich the derivative of a function is the function itself.\\nWe have also from algebra\\na x =1 +xlog e a+ v 7/\\ntherefore\\na x log e a\\ndx\\n1 log e a v e\\n2 (log e a) 2\\nlog e a\\nHence, if y a x\\n\u00e2\u0080\u0094-a x a x log*, a. (1 9)\\n24. Let log e x then e y and if we regard x\\nas a function of y, we have by formula (18),\\ndx t dy 1 1\\n=e y that is, -f-.=\\ndy ax e y x\\ntherefore -y- t 1\u00c2\u00b0^ (20)\\ndx dx x", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0034.jp2"}, "35": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 23\\ndy\\nTo find -j- when y log c /(x) \\\\og e u, we notice that\\n\u00c2\u00abF ^r- -s- and in the limit\\nOX 0^ OX\\n6?^/ dy du\\ndx du dx\\nNow if y log e u, by formula (20),\\nCtU U\\nand therefore -\u00e2\u0080\u0094\\\\og e u (21)\\ndx ax u dx y\\nIf y e u log e y u\\nand -r= -r- log y by formula (21),\\ndx dx y dx J v J\\n6?V 6?M 6?W\\nhence y 5\\nax ax ax\\ndy d du\\nso that we have e 11 e ll -j-.\\ndx dx dx\\nFinally, if y a u log y u log a, and\\n(22)\\ndu Id 1 dy\\ndx log a dx y log a c?x\\ndy du du\\nhence ~r y -=r~ lose a M log a,\\nc/x dx h dx b\\nand therefore -=-a M a w T log a. (23)\\nax dx dx\\nFormulas (1) to (23), together with the correspond-\\ning integration formulas, are collected in Chapter V.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0035.jp2"}, "36": {"fulltext": "24 CALCULUS.\\n25. Since first derivatives are themselves generally\\nfunctions of the fundamental variable, their first deriva-\\ntives may in turn be found. These last are called second\\nderivatives of the original functions. Second deriva-\\ntives are indicated by the symbols f n (x), o-\\ndx\\\\dxj dx 2\\nIt is to be carefully noticed that i 2 like d, is not a\\nsymbol of quantity, but of operation. The d 2 should\\nnever be read u d square, but second d Since\\nalways means it is best to use the latter\\ndx 2 dx\\\\dxj\\nform until there is no danger of misunderstanding the\\nd u\\nsymbol The derivative of the second derivative is\\n(XX 70\\nd V\\ncalled the third derivative and is written the next\\nPy\\nis written and so on.\\nax*\\n26. In what has preceded we have assumed one\\nfundamental variable but reference to common ex-\\namples shows us that we may have a function of two or\\nmore independent variables. For instance, crops vary\\nnot only with the amount of fertilizer used, but also with\\nthe amount of sunshine and moisture.\\nIf z is a function of two independent variables x and\\ny, expressed by writing u f(x, y), we may differen-\\ntiate, supposing x to vary and y to remain constant, or\\nAve may suppose y to vary and x to remain constant.\\nIn the former case we have and in the latter,\\ndx dy\\nThese ratios are known as partial differential coefficients,\\nand to indicate this we may use the parenthesis, writing", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0036.jp2"}, "37": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 25\\nj and t V When we suppose x and y to vary\\nsimultaneously the corresponding change in the func-\\ntion is called the total differential. As an example\\nof partial differentials suppose we have pv nt, where\\np and v are pressure and volume as before and t is\\nthe absolute temperature of a gas. Suppose t varies\\nwhile v remains constant then f-^ J Again, let v\\nvary while t remains constant, I -J-] as in exer-\\na a j. -tr\\\\ \\\\dvj V A\\ncise 4, Art. 19. x y\\n27. Having z f(x, if), let us differentiate z with\\nrespect to x and then differentiate the result with respect\\nto y. The order of the steps is indicated by\\nd fdz\\\\ d 2 z\\nor\\ndy \\\\dxj dydx\\nThe reverse order is indicated by\\nd f dz\\\\ d 2 z\\nor\\ndx\\\\dyj dxdy\\nWe proceed to show that\\nd f dz\\\\ __ d f dz\\\\\\ndy \\\\dx) dx \\\\dyj\\nthat is, we get the same result in whichever order we\\nproceed.\\nLet x take the increment Sx while y remains constant\\nthen g -/(^y)\\nSx Sx", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0037.jp2"}, "38": {"fulltext": "26 CALCULUS.\\nNow let y in this result take the increment Sy while x\\nremains constant.\\nBy \\\\8xJ\\nf(x e, y 8y -/(a?, y 8y) -f(x 8a;, y) y)\\nSySa;\\nReversing the order, we have\\n8z f(x, y 8y) -/(a;, y)\\n8y 8y\\n8_(8z y\\n8x\\\\8yj\\n_ f(x 8x,y 8y) -f(x k, y -/(a;, y Sy +/(s, y\\nSxSy\\nHence _\u00e2\u0080\u0094=\u00e2\u0080\u0094(\u00e2\u0080\u0094;\\noy\\\\oxj cx\\\\oyJ\\nand in the limit\\nd fdz\\\\ _ d fdz\\\\ m d 2 z _ _ d 2 z\\ndy\\\\dxj dx\\\\dyj dydx dxdy\\nIn any scientific investigation in which the calculus\\nis used the context must show what and how many\\nvariables are involved, and what partial differential\\ncoefficients will occur. For example, Carnot s Prin-\\nciple, with its applications as presented in thermo-\\ndynamics, affords an abundance of cases of these partial\\ndifferential coefficients.\\n28. Ordinary text-books in algebra and trigonometry\\nusually give methods for expanding (a x) m e x a x\\nlog(l x), sin a;, etc., into series in ascending powers", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0038.jp2"}, "39": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 27\\nof x. We can now establish one general theorem by\\nmeans of which these functions and all similar ones\\nmay be expanded.\\nWe first notice that if y =f(z x) and we differ-\\nentiate regarding x as a variable and z as a constant,\\nthe result is just the same as if we should differentiate\\nwith z for the variable and x as a constant. That is, if\\ny =f(z x), This is obviously true if\\nwe consider that it makes no difference whether we\\nchange the function by changing z or x.\\nSuppose\\nf(z x)= A Bx Ox 9 Dx* E (a)\\nin which A, B, C, etc., are functions of z and not of x.\\nLet us now differentiate successively the first member\\nof equation (a) with respect to g, and the second mem-\\nber with respect to and put x after each differ-\\nentiation.\\nThen, since f(z x) f(z x),\\ndz ax\\ndz\\nB 2Cx 3Dx 2 4:I]x 3\\nand f(z)=B.\\nf (z x) 2 2 3 Dx 3 4 fix 2\\nand (s)=2 7.\\nf (z x)=2-SD 2-3-4Hx+\\nand 2 -8 2", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0039.jp2"}, "40": {"fulltext": "28 CALCULUS.\\nAlso, if x in equation (a), f(z) A.\\nWe now have\\nA B=fXz), 0=\\\\f (z), etc.\\nPutting these values into the assumed series (V),\\naO=/ V (3 2\\n[n\\nThis formula is Taylor s theorem. It enables us to\\nexpand functions of the sum of two variables in ascend-\\ning powers of one of the variables, combined with finite\\ncoefficients depending on the other variable.\\n29. Suppose we have a function of one variable and\\nwish to expand it into a series. Following the method\\nof the preceding article, assume\\nf(x) A Bx Ox 2 Dx 3 Eat (5)\\nDifferentiating successively and putting x after\\neach differentiation, we have\\nf(x) B 2Cx 2,Bx i 4Bx z\\nand f(*)\\\\ B.\\nf (x)= 2C 2 3 Dx 3 4 Bx 2\\nand o 2a\\nf (x)= 2-3D 2-3-4\\n(z)] o 2.3D.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0040.jp2"}, "41": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 29\\nAlso, f{x-)\\\\ A.\\nThe assumed coefficients A, B^ etc., are thus deter-\\nmined, for we have\\nA =/(V)] or, as it is usually written, /(0);\\n(*)]o=/ (0);\\n[3 [3\\nPutting these values into the assumed series (5),\\n(0)z 2 (O)s 8\\nf(x)=f(P)+f(0-)x\\nj^CQ)^\\nThis formula is Maclaurin s theorem.\\nIt will be observed that if z is made equal to zero in\\nTaylor s theorem, we have Maclaurin s theorem. The\\nlatter may therefore be regarded as a special case under\\nTaylor s theorem.\\n30. Suppose f(x) (a-\\\\-x) m\\nlet us expand this function according to Maclaurin s\\ntheorem.\\nIf x 0, the function becomes a m", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0041.jp2"}, "42": {"fulltext": "30 CALCULUS.\\nFurther, (x) rn(a x) m\\nand (0) m O m_1 ma m l\\nf n (x) =m(m I) (a x) m 2\\nf (0) m (m 1) (a w 2 m (m 1) a\u00e2\u0084\u00a2 2\\nf (x)=m(m l)(a m 3\\nand hence\\nCO) m(m l)(w 2)(a x) m 3\\nm (m 1) (m 2) a m 3\\nTherefore Ave have\\n(a m a m ma m l x 4\\nm Cm 1) a m 2 2\\n[2\\nm(m\u00e2\u0080\u0094 1) Cm 2) a\u00e2\u0084\u00a2 V\\nm(m 1) (m -?i 2)a w (w -V- 1\\n|w 1\\nThis formula will be recognized as the binomial\\ntheorem. It provides for the expansion of a binomial\\naffected with any constant exponent.\\n31. A series must be known to be a converging series\\nbefore any practical use can be made of it. The sim-\\nplest tests for convergency are given in algebra text-\\nbooks.* If a series is found to be diverging, it is\\nSee Hall and Knight s Elementary Algebra, Arts. 470-477.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0042.jp2"}, "43": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 31\\nrejected for such values of the variable as make it\\ndiverging; or it is transformed into a series which\\nconverges and, if possible, into one which converges\\nrapidly, in order that only a few terms need be used.\\n32. If one function is divided by another, as A\\nit sometimes happens that the functions are of such\\na nature that upon evaluating them for some particu-\\nlar quantity each function reduces to zero, so that we\\nhave The question arises what does this expres-\\nsion mean, and what is its value\\nStudents often say that must be unity sometimes\\nthey are inclined to think it is zero. In some instances\\nthe first view is correct in others, the second in others\\nstill, a value will be found which is neither unity nor\\nzero. Now it is evident that if we can find a limit\\nf(x)\\nwhich the ratio J is approaching as x approaches\\nnearer and nearer to that value which makes f(x) and\\nj (x) each equal to zero, we have caught the correct\\nvalue of We proceed to find a general expression\\nfor this limit.\\nSuppose x to take the increment Bx then by Taylor s\\ntheorem,", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0043.jp2"}, "44": {"fulltext": "32 CALCULUS.\\nLet a be the quantity that makes both f(x) and f (x)\\nequal to zero. Substituting a for x,\\n/0, fa) HA.)i.^W t\\nrv J r y e) 2\\nDividing both numerator and denominator of the\\nsecond member of this expression by 8x,\\nFinally, when r becomes c?:z,\\nf(a) f (a)\\nHence, if becomes x when evaluated for any\\nquantity as a, the value of this indeterminate form is\\no.)\\nevaluated for a.\\nIf it should happen that ^j\\nwe divide both\\nf f (x)\\\\\\nnumerator and denominator of equation (2) by Sx again\\nand have\\n/O) f (a)\\n(a)\\n/CO\\nIf becomes when evaluated for any quantity", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0044.jp2"}, "45": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 33\\nas a, this expression may be determined as in the first\\ncase by observing that if\\n/O) oo Kx) _ o\\n4 (x) l o\\nIf f(x) j (x oo when evaluated for some quantity,\\nf (x)\\nT~~~ wn cn ma y e treated as above.\\nToo\\nIf f(x)\u00e2\u0080\u0094 oo \u00e2\u0080\u0094go for some quantity, it should\\nbe transformed into a fraction which takes the form\\nand then determined.\\n33. If y \\\\_f(x)^ x log y j (z)logf(z); and this\\nis indeterminate whenever one of the factors becomes\\nzero and the other infinite for the same value of x.\\n(1) Suppose $(x)= 0, and \\\\ogf(x) oo then\\nf(x) cc or 0. Consequently [/(V)]^ becomes inde-\\nterminate when for some value of x it takes the form\\n0\u00c2\u00b0 or oo\u00c2\u00b0.\\n(2) Suppose \u00c2\u00a3(V)=\u00c2\u00b1ao, and logf(x)= then\\nf(x) l, and \\\\_f(x)~Y ){x) gives the indeterminate forms\\n1 and 1~\u00c2\u00b0\u00c2\u00b0.\\nHence, if we have any of the indeterminate forms 0\u00c2\u00b0,\\noo\u00c2\u00b0, 1~\u00c2\u00b0\u00c2\u00b0, as the result of evaluating [/(V)]^ f\u00c2\u00b0 r some\\nquantity, we change the exponential function to the\\ncorresponding logarithmic function, and then reduce to\\nthe form which is dealt with under the first case.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0045.jp2"}, "46": {"fulltext": "34 CALCULUS.\\nExercises.\\n34. 1. Show that cotit= cosec 2 through the\\niXJu (XX\\nrelation cot u\\ntan ii\\\\\\nv2\\n2. If y [/(a)]* (x) v?, show that\\n-JL VU V 1 1- (log U) U v\\ndx dx dx\\nTake the logarithmic form, logy=vlogu, and differentiate.\\n3. y x x of (1 log x).\\ndx\\ndy 1\\n4. y e x\\ndx\\na-V\\n6. N= e\\\\ tan ,P log tan (45\u00c2\u00b0 F), in which the varia-\\nbles are N and F. Show that\\nc^Y^ A^-cosi^ 7\\ndF cos 2 F\\nWatson s Theoretical Astronomy, p. 69.\\n7. Verify the following expansions by means of Mac-\\nlaurin s theorem\\n/y\u00c2\u00bb\u00c2\u00a3 /y\u00c2\u00bb0 /Vl4 /yffl\\n(ii) a^ l a! log e a ^f^- 3\\n(in)Iog(l a\u00c2\u00bb) f-J+-(-ir 1 f-.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0046.jp2"}, "47": {"fulltext": "DIFFERENTIATION AND INTEGRATION.\\n35\\n/y\u00c2\u00bb3 /v\u00c2\u00bb5\\n(v) C psa! l--\\nShow that (v) might be derived directly from (iv), since\\nsin x cos x.\\ndx\\n8. Show that 1.\\nX\\n-\u00e2\u0080\u00a2o\\nIn this case f(x) cos x and (V) 1 ^L?\\n_\\n9. Show that\\nX\\n-log-\\n1.\\n10. Show that x l\u00c2\u00b0g( 1+a;\\n1 cos as\\n2.\\nWe have\\n^)_ l0 1 ^(iT^)\\nsma;\\nbut this expression evaluated for is as before. Hence\\nwe proceed to the second derivatives and have\\n\\\\-x\\n1\\ncos#\\nand this equals 2 when evaluated for 0.\\n11. Show that\\n1 sin x 2 sin 2\\n1 3 sin x 2 sin 2 x\\n3 when a? 30\u00c2\u00b0.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0047.jp2"}, "48": {"fulltext": "36 CALCULUS.\\n12. Expand sin -1 a;. Using Maclanrin s theorem,\\nfix) sin 1 x /(0) (0) x .M x* tUlp. x 3\\n/(O) 0; -J=J (O) 1.\\nVI x-\\nf (x) ^\u00e2\u0080\u0094;f (0) 0.\\n_ (1 a 2 -I x\\nii-xy _\\nTherefore sin -1 sc a;\\n1.\\n13. Show that 2r sin 1 a(l+-^-\\n2r V 24r 2\\nThomson and Tait s Nat. Phil., Vol. 1, Art. 131.\\nSubstituting for x in the expansion of sin -1 a?, and\\n2 r\\nmultiplying by 2 r, we have the result\\n1\\n24 r 2\\n14. Expand sin 2 to the second power term inclusive.\\n2 02\\nAns. -j\\n15. Given lajlogojeZa?; perform the operation indicated.\\nWe have d(uv) udv vdu (formula 2)\\nhence, m; I \\\\idv j i cfa^ (Art. 14)\\nor, I udv uv I i", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0048.jp2"}, "49": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 37\\nIn the present case let\\nu log x then dv xdx, and du\\nx\\nX X\\nIntegrating dv, v and uv log x\\ntherefore x log xdx log x\\nJ 2 J 2 x\\n\\\\ogx\\n2 4\\nThis process is known as integration by parts.\\n16. Use the -method of example 15 in the following\\nexamples\\n(i) I x cos x dx x sin x cos x.\\n(ii) I e ax x dx (x\\n(iii) I sin -1 x dx x sin -1 x Vl x 2\\n(iv) I log x dx x log x x.\\n17. Given the following integrals, to find their values:\\n(i) I (a x) n dx.\\n(ii) I t nxdx.\\nLet cos x,= u; then du sin x dx, and\\n/tan xdx dx log u log cos x.\\nJ cosx J u", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0049.jp2"}, "50": {"fulltext": "38 CALCULUS.\\n(iii) I sin x cos x dx.\\nThis may be written I i sin 2 x dx, which equals \u00e2\u0080\u0094J cos 2x.\\n(iv) I x Va 2 x? dx.\\nLet u V a 2 or then I a; Va 2 a; 2 da; m 2 dw\\n_^ _(a 2 -ar\u00c2\u00b0)j\\n3 3\\n(v) I Va 2 a; 2 c?x.\\nLet a; ct sin u then I Va 2 a; 2 da; a 2 I cos 2 ^dw\\nf (1 cos 2 u) cfa -(it i sin 2 u)\\nsin x H V a- or sm A Vcr x\\n2 V a a 2 7 2 a 2\\n(vi) I da;.\\nThis may be written\\nJ a 2 __ a 2 _ #2\\\\\\nv dXj which equals\\nVa 2 x 2\\ndx I Va 2 x?dx.\\nVa 2 a? 2\\nTherefore\\n^2^ a ^2 a 2\\n/Oy t o _i a; I eh i x x 9 i\\ndx a 2 sin (-sin L Va- a;\\nVa 2", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0050.jp2"}, "51": {"fulltext": "DIFFERENTIATION AND INTEGRATION. 39\\ndx\\n(vi)\\nVa 2 x 2\\nLet u x Va 2 x 2\\nthen I x I log i\u00c2\u00a3 log (a; Va 2 -f x 2\\nJ ^tf x 2 u\\n(viii) I Va 2 x 2 dx.\\nIntegrate by parts, letting u Va 2 x 2\\nX LLJu\\nthen dv dx* v a;, du wu x Va 2 2\\nand the formula I wcfa; uv I v^w\\nbecomes\\nJ Va 2 2 dx x -Va 2 x 2 I\\nVa 2 a 8\\ncto\\na; Va 2 x 2 I\\n-dx\\nVa 2 or 5\\nx Va 2 x 2 fs/a 2 x 2 dx+ f\\nJ J Va 2 x 2\\nTransposing the middle term and dividing by 2,\\nI Va 2 x*dx V x 2 x 2 I\\ndx.\\ndx\\nVa 2 x 2\\n|V^T^+| log (a? V^?T^ 2\\n35. It should be remembered that the system of\\nlogarithms used in this chapter is the Napierian.\\nWhenever differentiation or integration gives rise to", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0051.jp2"}, "52": {"fulltext": "40 CALCULUS.\\nan expression in which a logarithmic factor occurs, the\\nequation containing this factor must be multiplied\\nthrough by 0.43429448 the modulus of the common\\nsystem, before it can be used in computations involving\\ncommon logarithms. For example, in developing the\\ntheory for determining the place of a comet moving in\\na hyperbolic orbit we encounter the equation\\nk~\\\\/p dt a 2 tan yjr\\nl\u00c2\u00abKH\\nin which t and a are the variables. This is to be in-\\ntegrated between the limits T and t so we have\\n*VpJ[*-/rftant[l.(l l)-g4r\\na 2 tan^r C-e(\\\\ +~\\\\da C-da\\nhence JcVp (T\u00e2\u0080\u0094 t) a 2 tan yfr\\\\ -^e(cr j log e r\\nCommon logarithms cannot be used in this equation\\nor in any modified form of it without first intro-\\nducing the modulus of the common system as a factor\\nthroughout.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0052.jp2"}, "53": {"fulltext": "CHAPTER II.\\nTHE GRAPH.\\n36. The changes in a function corresponding to\\nchanges in its variable may be graphically shown in\\nthe following way\\nDraw a horizontal line with another line at right\\nangles to it. Call the horizontal line XX or the\\nx-axis the vertical line YY or the y-axis and their\\npoint of intersection or the origin.\\nIf y is some specified function of x, give a number of\\nconvenient values to x and find the accompanying values\\nfor y. Beginning at as the zero point, lay off with\\nany convenient unit of length the positive values of x\\nto the right on the a axis, and the negative values to\\nthe left on this axis. At the end, remote from 0, of\\nthis line, which represents a value of x, draw a perpen-\\ndicular (using the same unit of length) to represent\\nthe corresponding value of y. The perpendicular is\\nto be drawn upward from the #-axis in case y is posi-\\ntive, and downward when y is negative.\\nIn this way locate a point for each pair of values of\\nx and y. If many values be given to x, any two con-\\nsecutive values differing but little from each other,\\nwe shall have a correspondingly large number of points\\nwith small distances separating them. Connecting all\\nthe points in order, we have a continuous line, straight\\nor curved, called a graph or locus. The values of x are\\n41", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0053.jp2"}, "54": {"fulltext": "42\\nCALCULUS,\\ncalled abscissas, and the values of y ordinates. The\\ntwo together are known as coordinates.\\nTo illustrate, suppose y x 2. When 0, y 2\\nwhen x 1, y 3 when a; 2, etc. When\\n0, Ave have no distance to measure off on the #-axis,\\nand since y 2 we measure upward two units, thus\\nlocating the point P v Measuring one unit to the\\nright and three upward, Ave have the point P 2 Lo-\\ncating a number of points in this, way and then con-\\nnecting them, the result looks like a straight line. At\\nany rate Ave have not been able to get any apparent\\nbends or corners provided the plotting has been\\naccurately done. This line presents to the eye the\\nway y changes as x changes when y x 2. The\\nvertical lines representing the values of y seem to\\nget steadily longer as x increases.\\nX\\nX\\nY\\nFig. 1.\\nAs another illustration, let us take the isotherm\\nc\\nequation y (see Art. 5). Suppose c is unity, so\\n1 X\\nthat y When x 1, y 1 when x 2, y=\\\\\\\\\\nx\\nwhen x y 2 etc. Locating these points and", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0054.jp2"}, "55": {"fulltext": "THE GRAPH. 43\\ndrawing a smooth curve through them, the graph\\nappears as in Fig. 1. Two things are clear in regard\\nto this graph: (1) it is related to the ^/-axis precisely\\nas it is to the #-axis (2) as x increases without limit,\\ny diminishes without limit, so that the points are nearer\\nand nearer to the a;-axis. The graph therefore shows\\nwhat the equation says namely, that as the volume\\nbecomes indefinitely great the pressure becomes indefi-\\nnitely small; and conversely, if the volume could be\\ndiminished without limit, the pressure would be indefi-\\nnitely great.\\nWe further observe that when x is negative, y is negative and\\nthus the complete graph includes a branch in the diagonally oppo-\\nsite corner X OY (Art. 123). But this second branch represents\\nno actual pressures and volumes, because pressures and volumes\\nare positive. We shall find numerous instances of equations in\\nwhich the variables, abstractly viewed, have a wider range of\\nvalues than the values possible for the concrete quantities under\\nconsideration.\\n37. If we like, we may think of a graph as the path\\nof a looint which moves from one determined point to\\nthe next one, and thence to the next one in order.\\nThe equation y x 2, for example, merely says that\\nthe point moves so that its ordinate is all the time equal\\nto its abscissa increased by the constant 2. When\\nthe graph is thus looked upon as the path of a moving\\npoint, the variable coordinates x and y are called cur-\\nrent coordinates. Any equation in two variables may be\\nsaid to express the laiv of the point s motion in the plane.\\nFor brevity we shall speak of the curve y=f(x)\\ninstead of saying the curve which the equation\\ny=f(x~) represents.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0055.jp2"}, "56": {"fulltext": "44\\nCALCULUS.\\nFig. 2.\\n38. Suppose the moving point describes the arc CO 1\\nof the graph or curve y =f(x). Let P be any point\\nin the path and Q\\nanother point. As\\nthe moving point\\ngoes from P to Q,\\nits abscissa changes\\nfrom x to x 8x,\\nand its ordinate\\nfrom y to y S?/.\\nDraw PL paral-\\nlel to the ;r-axis.\\nThen PL and\\nLQ 8y. Let ZT\\nbe the chord (produced) passing through the points\\nP, Q. \u00e2\u0080\u0094jy is the tangent of the angle which the line\\nTT 1 makes with the #-axis. Now suppose 8x and Sy\\nto become indefinitely small. P and Q must approach\\nindefinitely near to each other, the chord becomes a\\ntangent, and\\nPL ax\\nWe now have a geometric meaning for the first\\nel v\\nderivative: If y =f(x), -j- is the tangent of the angle\\nwhich the tangent to the curve makes with the x-axis.\\nThe direction of the tangent determines the direc-\\ntion of the curve at the point of tangency. The value\\nof -j- at any particular point on the curve gives us,\\ntherefore, the slope or gradient of the curve at that\\npoint.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0056.jp2"}, "57": {"fulltext": "THE GRAPH. 45\\nIf a tan -1 we also have, when P and Q are\\nTO 7\\nindefinitely near to each other, -7^ sin a, and\\nPL dx P ds\\n~57] 1T cosa ds being the elementary arc PQ.\\n39. The student will at once perceive that the first\\nderivative must be of great use in searching for special\\nfeatures of any graph. For one important applica-\\ntion, let us see what it can tell us about the graph of\\nax by c 0. Differentiating this expression,\\na bf 0,\\nax\\nand therefore\\ndy _ a\\ndx b\\nWe have here a constant value for the tangent of the\\nangle which the graph of ax by c makes with\\nthe #-axis. Accordingly, the slope is constant and the\\ngraph can have no bends for a bend means change\\nof slope. Therefore ax by c must be a straight\\nline and its own tangent. But ax by c is the\\ngeneral equation of the first degree, and any property\\nproved for it holds for any and every particular equa-\\ntion of the first degree. For instance, the graph of\\ny x 2, which seemed in Art. 36 to be a straight\\nline, we now know to be a straight line. Further,\\ndy\\nfrom y x 2 w r e have -j- 1. Since 1 is the gra-\\ndient of this particular line, we know that it makes an\\nangle of 45\u00c2\u00b0 with the horizontal axis.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0057.jp2"}, "58": {"fulltext": "46 CALCULUS.\\nAgain, in the curve y jr 2 Here tana\\nvaries inversely as the square of the abscissa, and is all\\nthe time negative. It follows that at every point the\\ntangent to the curve makes an obtuse angle with the\\n#-axis.\\nThe angle a is always measured from the #-axis on\\nthe right-hand side of the origin, counter-clockwise\\naround to the line which, with the x-axis, forms the\\nangle.\\nExercises.\\n40. 1. A point moves in a circle around the origin as a\\ncenter, with a radius r.\\n(1) The equation to the circle must be x 2 2 r 2 for\\nthe abscissa and ordinate are all the while the sides of a\\nright triangle.\\ndy _ x\\n(2) Show that\\nux\\n(3) Find the coordinates of the point or points where the\\ncircle has a slope of 1.\\n2. Find the point of tangency when the tangent to y\\nx\\nmakes equal angles with the axes of reference.\\nPut tan 135\u00c2\u00b0 and solve for x.\\ndx\\n3. Show that the curve y goes through the\\n1 x 2\\norigin. Find its slope at the origin.\\n4. Construct the curve y 2 4 x. Find the point of tan-\\ngency when the tangent to the curve makes an angle of 45\u00c2\u00b0\\nwith the a axis.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0058.jp2"}, "59": {"fulltext": "THE GRAPH. 47\\n5. Construct the curve y sin x, making as much use as\\npossible of to determine the slope at various points.\\nThe x-axis must here be regarded the circumference of a\\ncircle whose radius is unity, straightened to a right line\\nwith the origin marked 0\u00c2\u00b0 We easily obtain a num-\\nber of points on the curve by using the pairs of coordi-\\nnates 0\u00c2\u00b0, 0; 45\u00c2\u00b0, iV2; 90\u00c2\u00b0, 1 135\u00c2\u00b0, i-V2; 180\u00c2\u00b0, 225\u00c2\u00b0,\\n|-V2, etc. Hence the curve passes through the origin,\\nhas a maximum ordinate at 90\u00c2\u00b0, and crosses the x-axis again\\nat 180\u00c2\u00b0. In order to measure off the abscissas, the angles\\n45\u00c2\u00b0, 90\u00c2\u00b0, 135\u00c2\u00b0, etc., must be expressed in radians. We have\\nthe radian 57\u00c2\u00b0.2958 for the unit of distance. The abscissa\\nindicated by 45\u00c2\u00b0, for instance, is approxi-\\nJ 57.2958 191 Xi\\nmately. The distance from the origin to the second point\\n1 80\\nof crossing is 3.14159 and the maximum ordi-\\nnate therefore meets the o:-axis at a distance 1(3.14159\\nfrom the origin.\\nFrom 180\u00c2\u00b0 to 360\u00c2\u00b0 the values of the sines are a repetition\\nof the values for the first semi-circumference, except that\\nthey are now all negative. Hence this portion of the curve\\nis in every respect like the portion from 0\u00c2\u00b0 to 180\u00c2\u00b0 but it\\nlies below the x-axis, and the direction of its convexity is\\nreversed.\\nSince sin(?i7r x) sin a?, n being even and positive, it is\\nseen that the curve keeps its sinuous character, crossing the\\na axis at regular intervals an unlimited number of times.\\nOn account of the repetition over and over again of the\\nseries of values of sin x, the function is called a periodic\\nfunction. The curve itself is known as the sinusoid.\\n6. Construct the curve y cos a;.\\nIt is obvious in advance that this curve, which might be\\ncalled the co-sinusoid, must be precisely like the sine curve", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0059.jp2"}, "60": {"fulltext": "48 CALCULUS.\\nor sinusoid and that we shall have it in its proper position\\nif we suppose the sine curve moved a distance of 90\u00c2\u00b0 to the\\nleft along the .T-axis.\\n7. Find the first point to the right of the ?/-axis where\\ny sin x and y cos x cross each other (see Art. 95). Show\\nthat the angle at which they cross is 180\u00c2\u00b0 2 tan -1 ^-V2.\\n8. Construct y m sin nx.\\n9. Construct y m cos nx.\\nAssign numerical values to m and n\\\\ then give a series\\nof values to x, as in the first case. If a negative value is\\ngiven to m, the effect is to rotate the curve on the a axis so\\nthat portions which were above are now below, and vice versa.\\n41. If for some value of x, a hence, to\\nfind whether the point describing a curve is anywhere\\ndv\\nmoving parallel to the a axis, we must put equal to\\nCtOu\\nzero. Let .r x represent one root of the equation 0.\\nIf a value of x a little less than this root makes\\ndx\\npositive, and a value a little greater makes it negative,\\nthe tangent to the curve must make an acute angle with\\nthe #-axis, then become parallel to it, then make an\\nobtuse angle with it and the curve must have a bend,\\nbeing convex upward. The ordinate of the highest\\npoint, corresponding to x v is a maximum. So we\\ndefine a maximum value of a function as a value greater\\nthan the value just before it and also than the one just\\nafter it. (PB, Fig. 3.)\\nOn the other hand, if changes from to in\\npassing through zero, the curve is concave upward, and", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0060.jp2"}, "61": {"fulltext": "THE GRAPH.\\n49\\nthe lowest point is the end of a minimum ordinate that\\nis, the value of the function is less than the value just\\nbefore it and the one just after it. (P C, Fig. 3.)\\ndv\\nIt is evident that if -j- is changing from to\\nand if JL is changing from\\ndx\\nd [dy\\ndx \\\\dx.\\nd (dy\\ndx\\\\dx\\nis\\nto\\nis\\n42. A third case arises\\nIf does not change siorn\\ndx h h\\nin passing through zero, there is neither a maximum\\nnor a minimum but the point after reaching P or P f\\ntakes the path indi-\\ncated by the dot-\\nted line. The point\\nwhere -f- is then\\ndx\\ncalled a point of in-\\nflexion. In this case\\ndx\\\\dxj\\nEvery one is famil-\\niar with the point of\\ninflexion as a feature\\nin railroads, when\\nthe track is concave, say with respect to the fields on\\nthe right, and then changes so as to be concave to the\\nfields on the left. Curves containing points of inflexion\\nare very common in architectural forms. Such a curve\\nis then known as an ogee.\\nThe same curve may of course have several maximum\\npoints and several minimum points, and also points of\\nFig. 3.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0061.jp2"}, "62": {"fulltext": "50 CALCULUS.\\ninflexion. Maximum and minimum points must evi-\\ndently alternate.\\nExercises.\\n43. 1. Consider the meaning of the statement cc.\\nax\\nExamine the two cases (a) when changes sign in pass-\\ncioc\\n(h/\\ning through an infinite value (b) when -p does not change\\nsign in passing through such a value.\\n2. Examine the following curves for maxima and minima:\\n(i) y T^\u00e2\u0080\u0094 (iii) y ^ogx.\\nJL X\\n(ii) 1- (iv) y 2px.\\nor 0~\\n3. Draw the curve e x showing that it lies wholly\\nabove the as-axis, crosses the ?/-axis at an angle of 45\u00c2\u00b0, and\\nhas no maximum or minimum points for any finite value\\nof x.\\n44. To illustrate the use of the principles established\\nin Art. 41, suppose Ave know the slant height a of a\\nright cone and wish to find the radius of its base when\\nthe volume is a maximum. Let y be the volume and x\\nthe base then\\nTTX*\\nVa 2 x 2\\nJ 3\\nx and y, being mutually dependent variables, must ad-\\nmit of graphical representation the abscissa of the\\npoint tracing the curve or graph is the varying radius,\\nand the ordinate is the varying volume. Hence, if we\\nput equal to zero and solve the equation so formed,", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0062.jp2"}, "63": {"fulltext": "THE GRAPH. 51\\nthe value of x obtained will be the radius which gives\\nthe maximum volume. Differentiating\\ny Vr- z z\\nU 3\\ndx\\n\u00e2\u0096\u00a0K\\n3\\nIT\\n8\\n2 Va 2 a; 2 x 2\\n-2x\\nv2Va 2 2\\n2 (fa 2 x 2 x s\\nVa 2 2\\nputting this expression equal to zero,\\n2x(a*-x*)-x s =0;\\nhence x V-| a.\\nThat is, the volume of the cone w T ill be greatest when\\nthe radius of the base is Vf a.\\nIn a case like this it is unnecessary to inquire whether\\n-j- changes sign, and whether the change is from to\\nor from to For the volume of a cone of given\\nslant height evidently varies from no volume when the\\nradius is zero, through finite values to no volume again,\\nwhen the radius is equal to the slant height that is,\\nfrom a cone that is all height and no base to one that\\nis all base and no height. Somewhere between these\\ntwo extreme cases there must be a cone of ordinary\\nshape whose volume is the greatest possible. It is well\\noccasionally to supplement mathematics with common\\nsense rather than to rely mechanically and invariably\\non some rule or formula.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0063.jp2"}, "64": {"fulltext": "52 CALCULUS.\\nExamples.\\n1. Find the altitude of the right cylinder of greatest\\nvolume inscribed in a sphere whose radius is r.\\nAlt. -I!.\\nV3\\n2. Given a point on the axis of the parabola y^\u00e2\u0080\u0094Xfx^\\nat the distance I from the vertex, rind the abscissa of\\nthe point of the curve nearest to it. x I 2 p.\\n3. Find the maximum rectangle that can be inscribed\\nin the ellipse whose axes are a and b.\\nThe sides are aV2 and 5V2.\\n4. A talus resting on a horizontal plane has a slope\\nof 30\u00c2\u00b0; at the top of the talus is a series of strata 5 ft.\\nthick the entire height of the ledge is 30 ft. How\\nfar must one stand from the foot of the talus to get\\nthe best view of the strata\\nThe angle at the observer s eye, formed by lines\\ndrawn to the bottom and to the top of the strata, must\\nbe a maximum. Let this angle be the angle sub-\\ntended by the talus, /3, and the angle subtended by\\nboth talus and strata, 7. Also, let x be the horizontal\\ndistance from the observer to a point directly beneath\\n25 30\\nthe strata. Then tan /3 tan 7 and\\nx\\ntherefore tan a\\n30_25\\nxx 5x\\n750 x 2 750\\nd 5 (a? 750) -10 a?\\nand tan a\\ndx (x 2 750) 2", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0064.jp2"}, "65": {"fulltext": "THE GRAPH. 53\\nEquating this derivative to zero, x 5V30, and\\nfinally the distance sought is\\n5 V30 5 V3 VTO 5).\\ntan 30\u00c2\u00b0 v J\\n5. The strength of a rectangular beam of given\\nlength, loaded and supported in any particular way,\\nis proportional to the breadth of the section multiplied\\nby the square of the depth. If the diameter a is given\\nof a cylindric tree, what is the strongest beam which\\nmay be cut from it\\nLet x be the beam s breadth then Va 2 x 2 must be\\nits depth. Hence, if y =x(a 2 x 2 the strength is a\\nmaximum when y is a maximum.\\na 2 3 x 2 0, and therefore x\\ndx V3\\nIn the same way find the stiffest beam which may be\\ncut from the tree by making the breadth multiplied by\\nthe cube of the depth a maximum. We now have\\ny x(a 2 x 2\\na 2 _ 2.2)1 z x Q a 2 _ ^)i(_ 2 x) 0, and x\\ndx 2\\nPerry s Calculus for Engineers.\\n6. The volume of a circular cylindric cistern being\\ngiven (no cover), show that its surface is a minimum\\nwhen the radius of the base is equal to the height of\\nthe cistern.\\nLet x be the radius and y the height then the\\nvolume is irx 2 y, which equals a constant, say a. If S\\nis the surface,\\nS 7rx 2 2 irxy irx 2 since y 5\\nx irx 1", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0065.jp2"}, "66": {"fulltext": "54 CALCULUS.\\nd S\\nFinding and putting it equal to zero, we have\\nax\\nTTX 1 1 XT T\\nand x y. How do we know that tins\\nIT\\nmakes the surface a minimum rather than a maximum\\nx* and x y\\nIT\\n7. Determine the speed most economical in fuel to\\nsteam against a tide, supposing the resistance to vary\\nas the nth power of the velocity through the water.\\nLet a denote the velocity of the tide, x the velocity\\nof the steamer through the water then x a will he\\nthe velocity of the steamer relatively to the bank.\\nThe power required, and therefore the coal burnt per\\nhour, will vary as the product of the resistance and the\\nspeed that is, as af +1 and therefore the coal burnt per\\nx i+1\\nmile will vary as This is to be a minimum,\\ni t x a\\nhence we have\\nd_( x n+1 \\\\(n l)x n (x -a)- x n+1 _ m\\ndx \\\\x a J (x a) 2\\nx _, 1 x a 1\\nand or\\na n an\\nThus if the resistance is taken to vary as the square\\nof the velocity, the speed past the bank should be half\\nthe velocity of the current.\\nGreenhilPs Differential and Integral Calculus.\\n8. Let A and B be two point-sources of heat. It is\\nrequired to find the point M on the straight line AB,\\nwhich is at the lowest temperature, the intensity of the\\nradiation of heat varying inversely as the square of the\\ndistance from the source of heat. Let a be the distance", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0066.jp2"}, "67": {"fulltext": "THE GRAPH. 55\\nbetween the points A and B, and x the distance from A\\nof the point M on the straight line then\\nAM x, and BM= a x.\\nLet the intensities of heat at unit distance from the\\nsources of heat be denoted bj^ a and /3 respectively.\\nThen the total intensity of heat co at the point M\\nwill be\\na /3\\n~n\\nx 2 (a x) 2\\nFor a maximum or minimum,\\ndco\\ndx\\nthat is,\\nand\\n_2a\\n3\\n2/3\\n2 x) z\\n(a\\n2:) 3\\n/8\\na?\\nI\\na\\nX\\n^8\\no,\\nV\\nThe distances i?ikf and Jl71T have, therefore, the same\\nratio as the cube roots of the corresponding heat\\nintensities.\\nSolving for x,\\n_ aV\u00c2\u00ab\\nIn this case it is necessary to see whether the value\\nfound corresponds to a maximum or a minimum. Dif-\\nferentiating the expression for we have\\ndx\\nd 2 co__2-3a 2-3/3\\ndx 2 x^ (a 4", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0067.jp2"}, "68": {"fulltext": "56 CALCULUS.\\nwhich is positive for all values of .r, including the value\\na-fy\\n/a+ /{3\\nay is therefore a minimum.*\\n45. Suppose a line AB through the origin to revolve\\naround counter-clockwise, making the variable angle\\nwith the .T-axis. Let AB pass through a point\\nP(x, and let the distance of P from be denoted\\nby r. r and are called polar coordinates; Ois the pole.\\nProjecting OP onto the x and y axes respectively, we\\nhave x rco$0 and y rsin0. Through these rela-\\ntions F (a?, y) becomes .F(rcos0, rsin0)==O. For\\nexample, the equation oft y 2 2ay becomes in\\npolar coordinates r 2 cos 2 ,2 sin 2 2a(rsin#) 0;\\nthat is, r 1 a sin 0. This is readily seen to be a circle\\nto which the x-axis is tangent, the point of tangency\\nbeing the origin, or pole.\\nWhenever any value of makes r negative, we meas-\\nure from the origin away from that end of the line AB\\nwhich is tracing the arc that measures 0. If we im-\\nagine an arrow to lie in the line AB and rotate with\\nit, the barb may be regarded as tracing the arc that\\nmeasures while the feather-end is negative.\\n46. Let PP f be an arc 8s of a curve f(r, 0) 0, PQ\\nthe arc of a circle whose radius is r and let the angle\\nPOP 80. In the limit PQP is a right triangle,\\nPQ r d0, QP ^dr, and PP ds. Let j be the\\nNernst and Schonflies EinfuKrung in die mathematische Beliand-\\nlung der Naturwissenschafien.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0068.jp2"}, "69": {"fulltext": "THE GRAPH. 57\\nangle made by the radius vector OP and the tangent\\nto the curve then tan 6\\nQP dr\\nWhenever the radius vector r is a maximum or mini-\\nmum, the tangent at\\nits extremity must\\nbe at right angles to\\nit that is,\\nrdO dr A\\ndr rdd\\nPoints for which\\nr is a maximum or\\nminimum are called o A\\napsides. To find, FlG 4\\ntherefore, whether\\nI u r\\na given curve has an apsis, we must put and\\nsolve this equation.\\nFor example, let us take the polar equation to the\\nellipse, the pole Tbeing at the right hand focus (see\\nArt. 115).\\n_ a (1 2 -j dr __ a (1 6 2 e sin 6 u\\n1 e cos ff an Id (1 ^cos^) 2\\n1 dr sm\\nthen 77T q n\\\\ and equating this to zero,\\nr dO 1 e cos 6\\nsin 6 0. Hence, the apsidal values of are 0\u00c2\u00b0 and\\n180\u00c2\u00b0. These results agree with what we observe in an\\nexamination of the given equation to the ellipse r is a\\nmaximum, a(l g), when 0=180\u00c2\u00b0, and a minimum,\\na(l -O, when 0\u00c2\u00b0.\\nThe student who is unacquainted with the formal analytic\\ngeometry of the straight line and the conic section is advised to\\nread Chapter IV before beginning the next chapter.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0069.jp2"}, "70": {"fulltext": "CHAPTER III.\\nAPPLICATIONS.\\n47. In the mathematical sciences one of the most\\ncommon of fundamental variables is time and when the\\nfunction of time is the space passed over by a body, the\\nfirst and second derivatives and are of great\\ntit dt\\\\dt h\\nimportance. N\\nSuppose a body moves over equal spaces in equal\\ntimes. The space divided by the time gives the speed\\nor velocity of the body. That is, if s is the space passed\\nover in the time t, is the velocity of the body.\\nWhile the camels were being loaded, T measured my first base-\\nline of 400 metres. Boghra (my riding camel) walked it in five\\nand one-half minutes. This was a daily recurring task, for the\\ncontours of the ground varied a good deal, and the depth of the\\nsand made a very appreciable difference in the time the camels\\ntook to do the same distance.\\nSven Hedin s Through Asia, Vol. I, p. 482.\\nIn this illustrative case, the speed of the\\nt 51\\ncamel expressed in metres per minute. Assuming that\\no\\nwas a constant during each day, the distance travelled\\nu\\non any given day by Hedin s caravan was known by\\nmultiplying the speed by the time spent in travel.\\n58", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0070.jp2"}, "71": {"fulltext": "VELOCITY. 59\\n48. If the motion is variable so that the body does not\\nmove over equal spaces in equal times, we may obtain\\nan expression for velocity by taking the time so short\\nthat during that time the motion must be uniform. So\\nif dt be an indefinitely short time and ds the indefinitely\\nsmall space passed over in that time, is the velocity\\nand is measured by the space that would have been\\npassed over in a unit of time if the body had kept on\\nmoving for a whole unit with the velocity which it had\\nat the instant considered. For instance, if we say that\\na train is running at the rate of 30 miles an hour, we\\nmean that if it were to run for a whole hour with the\\nsame speed which it has at this instant it would pass\\nover a distance of 30 miles. As a matter of fact it may\\nstop in a few minutes that has nothing to do with its\\nspeed at this instant. But 30 miles per hour is the same\\nas 1 mile in 2 minutes, or 4.4 feet in .1 of a second, and\\nso on. Evidently the rate remains the same so long as\\nthe ratio of the space to the time is the same, however\\nsmall the space and the time may be individually.\\nHence, in this case, 30 miles per hour.\\ndt L\\nIf we know the whole space passed over by a body\\nand know also the time taken, the space divided by the\\ntime is the average velocity: it must not be confused\\nwith the velocity proper, which may have varied during\\nthe time. For example, the first mail cartridge sent by\\ncompressed air from the Boston post office to the North\\nUnion Station (Dec. 17, 1897) required 1 minute and\\n2 seconds to pass from one place to the other, a distance\\nof 4500 feet. The average velocity was 72.58+ feet\\nper second.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0071.jp2"}, "72": {"fulltext": "60 CALCULUS.\\n49. If a body is moving in a northeasterly direction,\\nit plainly has a motion eastward and a motion northward.\\nFor instance, if it is moving due northeast with a ve-\\nlocity of 20 miles per hour, it is getting eastward at\\nthe rate of 20 cos 45\u00c2\u00b0 miles per hour, and northward\\nat the same rate. If it is moving east 30\u00c2\u00b0 north at the\\nrate of 20 miles per hour, it is moving east at the rate\\nof 20 cos 30\u00c2\u00b0 miles per hour, and north at the rate of\\n20 cos 60\u00c2\u00b0 miles per hour.\\nIn general, if a body is moving with a velocity v along\\na line which makes with the .r-axis an angle of a degrees,\\nits component velocity parallel to the #-axis is v cos a,\\nand its component velocity parallel to the y-axis is\\nv sin a. We have already seen (Art. 38) that cos a\\n7 7 ds\\n(1 1J Ci S\\nand sin a. Hence, if is the velocity of a body at\\nds at\\nCIS (XX CtX CtX ,i c\\nany instant, v cos a and is theretore\\ndt ds dt at\\nthe component velocity parallel to the #-axis. Similarly,\\nv sin a the component velocity parallel\\ndt ds dt r J L\\nto the y-axis.\\nEvidently a velocity parallel to any line furnishes\\na component velocity parallel to any other line if it\\nbe multiplied by the cosine of the angle between the\\nlines.\\n50. Suppose a particle is moving in a plane curve and\\nwe wish to know its component velocities at any instant\\n(1) along the radius vector, and (2) perpendicular to\\nthe radius vector.\\nWe have x r cos 6 and y r sin 0, in which x, y, r,", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0072.jp2"}, "73": {"fulltext": "VELOCITY.\\n61\\nand 6 depend upon the time t. Differentiating with t\\nas the fundamental variable,\\ndx _ dr\\ndi~~dt\\ncos i\\nsin 6\\ndd\\ndt\\ndy dr a add\\n-2- sin u r cos u\\ndt dt dt\\ndx\\nO)\\ndx\\nAccording to the preceding article, is the velocity\\ndx\\nparallel to the #-axis and cos 6 is the component\\nwhich it furnishes along the radius vector. Similarly,\\n-2 sin 6 is the component which -2 furnishes along the\\nradius vector. The sum of these components is the\\nFig. 5.\\nwhole velocity along the radius vector. From equa-\\ntions (a) and (b) we have\\ndt dt at\\nO)", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0073.jp2"}, "74": {"fulltext": "^COS0-\\ndx n dO\\nsm v r\\ndt\\ndt dt\\n62 CALCULUS.\\nAgain, resolving along a line perpendicular to the\\nradius vector and combining the parts,\\noo\\nThe reason for the minus sign in the first member\\nof equation (Y?) should be noticed. The velocities\\n-f- cos 6 and sin 6 are oppositely directed (see\\ndt dt\\nFig. 5); hence, when combined, their difference must\\nbe expressed.\\n51. -t\\\\-t\\\\ the rate of change of a variable velocity,\\nis called acceleration.\\nand are the accelerations parallel to\\ndt\\\\dt) dt\\\\dtj l\\nthe #-axis and ?/-axis respectively and we can now find\\nthe component accelerations (1) along the radius vector,\\nand (2) perpendicular to the radius vector.\\nDifferentiating equations (a) and (J) of the preced-\\ning article,\\nd 2 x\\nIt 2\\nd 2 y\\nIt?\\n--r cos0 2 +r\u00e2\u0080\u0094 sin (9, (e)\\n_dt 2 \\\\dt) J dt dt dt 2\\nr sm 0- 2- r cos0.\\n_dt 2 \\\\dtj\\ndt dt dt 2\\nMultiplying equation by sin and equation (e)\\nby cos 6 and adding, we have\\nfor the acceleration along the radius vector.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0074.jp2"}, "75": {"fulltext": "ANGULAR VELOCITY. 63\\nAgain, multiplying by cos (9, and (e) by sin\\nand subtracting the latter product from the former, we\\nd 2 y d 2 x n dr d6 d 2 7\\n\u00e2\u0080\u0094f cos sin 2 r (h)\\ndt l dt 2 dt dt dt 2\\nfor the acceleration perpendicular to the radius vector.\\nIt is to be noticed that the second member of equa-\\ntion (li) may be written r 2\\nJ r dt\\\\ dt)\\n52. Angular velocity is defined as the ratio of the\\nangle differential, d0, to the time differential, dt. This\\nd0\\nratio, may be a constant or a variable. For ex-\\ncic\\nample, the earth rotates on her axis with constant\\nangular velocity, and but she moves in her\\ns J dt 2-i h\\norbit around the sun with a variable angular velocity.\\n(See Art. 88.)\\n53. As an important application of the results given\\nin equations and (K) above, suppose a particle is\\nmoving in a circle with constant angular velocity.\\nThen, since r is a constant, =0 and 0.\\ndt dt\\\\dtj\\nTherefore the acceleration along the radius vector re-\\nduces to ri\u00e2\u0080\u0094 Also, in equation (A), since 0,\\nthe term 2 is zero; and since is a constant,\\n0, and the term r is zero; and therefore\\ndAdtJ dt 2\\nthe acceleration perpendicular to the radius vector is\\nzero. This conclusion is what we might have expected", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0075.jp2"}, "76": {"fulltext": "64\\nCALCULUS.\\nfrom the premise that the particle moves in a circle\\nwith constant angular velocity.\\nf ao\\\\*\\n54. The above expression, r for the accelera-\\ntion along the radius vector when a particle is moving in\\na circle with constant angular velocity, may be written\\nr \\\\dt\\nor\\n1 (rd0\\\\*\\nr\\\\dt)\\nbut since rd0 is the length of the arc corresponding to\\nthe angle dd, is the linear velocity v of the particle.\\nLit o\\nHence we have as a simple form for the accelera-\\nr\\ntion along the radius vector when the particle or body\\nmoves in a circle with constant angular velocity.\\n55. Suppose a point Q moves with constant angular\\nd0\\nvelocity or co in a circle AQA f whose radius is r.\\nTake the center as or-\\nigin, and let QP be\\nthe perpendicular from\\nQ to the j/-axis OA.\\nAs the angle AOQ\\nincreases, the line QP\\nincreases from to r\\nand then decreases.\\nThe changes in the\\nratios etc\\ndue to changes in the\\nangle AOQ, have already been discussed (Art. 20).\\nWe shall now consider the motion of the point P as\\nFig. 6.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0076.jp2"}, "77": {"fulltext": "SIMPLE HARMONIC MOTION. 65\\nQ describes the circle. OP is the ordinate of P at\\nany instant, and if Q has taken the time t to move\\nfrom A to Q, the angle AOQ cot hence 2/ r cos o\\nIf starts at some point and is the time required\\nto move from Q 1 to A, the angle OA cot hence,\\ncounting the time from the start at Q the angle Q OQ\\ncot and the angle A OQ cot cot and therefore\\ny r cos (jot o)\u00c2\u00a3 r cos (o e)\\nif we write e for the constant, cot\\n56. In regard to the motion of P, we notice at once\\nthat it must cross the circle on the diameter AA f and\\nreturn to A in the time that Q is describing the circum-\\nference so its motion is vibratory. It starts with zero\\nvelocity, and must be going with its greatest velocity\\nwhen at the center; for its direction of motion is then\\nparallel to that of Q.\\nTo get a more precise knowledge of the motion of P,\\nlet us take e 0, so that y r cos cot. By doing this\\nthe equation gains in simplicity and the motion remains\\nthe same, but the time is counted from the instant when\\nQ is at A instead of Q f\\nWe now have y r cos cot, (a)\\n-Jl rco sin cot. (b~)\\ndt V J\\n-JL rco 2 cos cot. (c)\\ndt\\\\\\nEquation (6) shows that the velocity of the point is\\ngreatest when cot 90\u00c2\u00b0 that is, when P is at the cen-", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0077.jp2"}, "78": {"fulltext": "66 CALCULUS.\\nter. Equation (V) shows that the acceleration is great-\\nest when cot 0\u00c2\u00b0 and 180\u00c2\u00b0 that is, at the start and at\\nA! Also, the acceleration is least when cot 90\u00c2\u00b0.\\n57. The variation in the ordinate OP may be best\\nappreciated by noticing the identity of the equation\\ny r cos cot with the equation y m eosnx given in\\nexercise 9, Art. 40. Equation (a) accordingly repre-\\nsents a cosine curve. Further, if the velocity equation\\n(6) be graphically shown, its curve must be the sinus-\\noid. And finally, the acceleration equation is\\nanother cosine curve, differing from the first, how-\\never, on account of the coefficient r 2 which has\\nreplaced the coefficient r in equation (a).\\nIt is w^ell worth the student s while to construct care-\\nfully the graphs for the three equations (a), (V),\\nusing the same unit of length for all three. The usual\\n#-axis now becomes a time axis in each case, sii\\\\ce the\\nabscissas are times. The y-axis for (a) is a displace-\\nment axis for (J) it is a velocity axis and for (c) an\\nacceleration axis.\\n58. We are now familiar with the geometrical mean-\\ning of when y If y is analogous\\n7 (XX Ct L\\nto and must have the same geometrical meaning.\\ndx\\nThat is, viewed geometrically rather than kinematically,\\nis the tangent of the angle which the tangent to the\\nCtv\\ncurve y f (t) makes with the \u00c2\u00a3-axis. Accordingly,\\nequation (6), Art. 56, might be called the curve of the\\ntangent to (a) for any ordinate (with the abscissa t f", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0078.jp2"}, "79": {"fulltext": "SIMPLE HARMONIC MOTION. 67\\nin the graph of (5) represents the magnitude of the\\nslope of (a) at the point whose abscissa is t Evidently\\nthe curve of (c) is related to (6) just as is to (a).\\n59. The point P, vibrating back and forth across the\\ncircle (Fig. 6), is said to have simple harmonic motion.\\nIt is such motion as this that Jupiter s satellites seem\\nto have as we look at his orbit edge on.\\nThe range OA or OA! on one side or the other of\\nthe middle point is called the amplitude and the ordi-\\nnate OP is the displacement. The period of a simple\\nharmonic motion is the time which elapses from any\\ninstant until the point moves again in the same direc-\\ntion through the same position that is, the time\\nrequired by P to move from P to A! thence back to J.,\\nand finally to the initial position P f is the period. The\\nphase is the fraction of the period of vibration.\\n2 7T\\nThe epoch is the angle e.\\nThis expression y r cos (wt e) is to be found, perhaps more\\nfrequently than any other, in all branches of mathematical physics.\\nIt is in terms, or series of terms, of this form that every periodic\\nphenomenon can be described mathematically. From the expres-\\nsions for the longitude and radius vector of a planet or satellite to\\nthose of the most complex undulations, whether in water, in air,\\nor in the luminiferous medium, all are alike dependent upon it.\\nTait s Dynamics.\\nExample. Find an expression for the up and down\\nmotion of the connecting-rod of a locomotive.\\n60. The downward fall of an unsupported bocly is\\ndue to the accelerating force exerted by the earth and\\nknown as gravity. At small distances above the earth s", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0079.jp2"}, "80": {"fulltext": "68 CALCULUS.\\nsurface this force is practically constant the accel-\\neration caused by it is denoted by g. When g is de-\\ntermined at different places on the earth, it is found\\nto vary within narrow limits. This variation is due to\\nseveral causes, the chief one being the rotation of the\\nearth on its axis, g has its least value at the equator\\nand its greatest value at the poles. At Washington,\\nD.C., g is 980.098 dynes that is, the observed accel-\\neration due to gravity is, at that point on the earth s\\nsurface, 32.155 feet per second.\\nTaking the origin at the point from which a body\\nfalls, with the positive end of the ^-axis downward, we\\nnow have\\na fdy\\\\_\\nJt\\\\dtJ 9\\ntherefore, after integrating,\\ncl l v gt+ C. (Art. 15.)\\nat\\nIf the body falls from rest, v when t there-\\nfore (7=0, and the equation becomes\\nw\\nMultiplying by dt and integrating again,\\ny \u00c2\u00b1Cjt*+C l\\nSince y when t 0, O f\\ntherefore y gt 2 (5)\\nU. S. Coast and Geodetic Survey.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0080.jp2"}, "81": {"fulltext": "FALLING BODIES. 69\\nCombining equations (a) and (b) so as to eliminate t,\\nEquation (e) enables us to find the velocity with\\nwhich a body is moving when it has fallen through a\\ngiven space. For example, the monument at Washing-\\nton is 555 feet high if a ball is dropped from the top,\\nwhat is its velocity upon reaching the ground We\\nmay take g 32, a value sufficiently accurate in this\\nexample and similar ones. Then v 8V555 188 feet\\nper second, approximately.\\n61. If the body is projected directly upward,\\ndt\\\\dt)\\nbecause the acceleration is now a retardation tending\\nto diminish y. Integrating as before,\\nIf the body is projected with the velocity V, V\\nwhen t therefore C FJ and the equation becomes\\nMultiplying by dt and integrating again,\\nand since y when t 0, (7 and we have", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0081.jp2"}, "82": {"fulltext": "70 CALCULUS.\\nCombining equations (J) and (e) so as to eliminate t,\\n62. By means of the equations of the two preceding\\narticles we can readily show that if a body is projected\\nvertically upward, it takes the same time to come down\\nthat it does to go up also, upon reaching the point\\nfrom which it was projected, it has the same velocity as\\nthat with which it was projected.\\nFrom (d),\\nwhen\\ndt g\\ntime up\\nfrom CO,\\nwhen\\ndt V 2g\\nspace up\\nfrom (5),\\nwhen\\n-0 9\\ntime down\\nfrom (e),\\nwhen\\nV 2\\n2 9\\nIn the derivation of formulas (a) to no account\\nhas been taken of the resistance offered by the air to\\nthe fall or rise of a body. The formulas are strictly\\ntrue only on the supposition that the acceleration is\\nconstant, and that the motion takes place in a vacuum.\\n63. We may now consider the case when the height\\nis so great that the acceleration cannot be regarded as\\nconstant. What is the velocity of the body on reach-\\ning the earth\\nA homogeneous sphere, or a sphere composed of con-\\ncentric layers with the density varying only from one\\nlayer to another, attracts an external body with an", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0082.jp2"}, "83": {"fulltext": "FALLING BODIES. 71\\nintensity varying inversely as the square of the dis-\\ntance of the bocty from the center of the sphere. Let\\ng be the acceleration due to the earth when the body is\\nat the earth s surface, and/ the acceleration at the dis-\\ntance y from the center. (Notice that the center thus\\nbecomes the origin.)\\nThen, if R is the earth s radius,\\ngB?\\n9\\nR2 +1 f\\nk that is.\\ny 1\\nand therefore\\nwe\\nhave\\nd d y\\\\ _\\ndt\\\\dt)\\nV 2,\\nThe minus sign is taken because y is diminishing as\\nthe body falls that is, dy is negative, and since is\\nincreasing numerically, -q-\\\\-j~) must also be negative.\\nIf we multiply by dt, as in the previous articles, and\\nattempt to integrate, we have\\ndt~J y* dt\\nan indicated operation which cannot be performed un-\\nless we know what function y is of t and this we do\\nnot know in advance. But multiplying by dy instead\\nof dt,\\nd fdy\\\\ __ dy fdy\\\\ __ gR 2\\nd n\\\\tt) iti d Kdtr-^ cly\\nThe first member is immediately integrated by ob-\\nserving that it is of the form xdx, and that\\nJU CI Jb c)", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0083.jp2"}, "84": {"fulltext": "72- CALCULUS.\\nWe have, therefore,\\ngjr\\nIf the body falls from the height h above the earth s\\n(Jb II (1 RP*\\nsurface so that y R h when =0, C r\\nat R h\\nand the equation becomes\\nyw= qm L_\\n2\\\\dtJ y \\\\y R h\\nThe same result is reached by writing a definite\\nintegral (Art. 16) whose limits are R h and y. We\\nthen have\\n1W p _/z^\\nX\\n2\\\\efc/ \u00c2\u00ab^+a 2\\ndy\\n1\\n2\\n2\\n-B A.\\n\\\\y R h\\nSuppose that\\ndy_\\ndt\\nv l when y R\\\\ that is, v 1 is the\\nvelocity which the body has when it reaches the earth s\\nsurface. Then\\niw=#-\u00e2\u0080\u009e-\\nR R h\\ngR\\nR h\\ngh\\nR\\nR h", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0084.jp2"}, "85": {"fulltext": "FALLING BODIES. 73\\nand therefore\\n^V^-i+I-\\nr m\\nIf h R, the series within the parenthesis is converg-\\ning and if h is very small in comparison with 72, we\\nmay drop all terms of the series after the first term\\nwe then have v f V2^A, which is identical with for-\\nmula (Y), Art. 60.\\nIf A _R, the series is diverging; the formula con-\\ntaining it cannot therefore be used, and we return to\\none of the other expressions. For example, suppose a\\nbody falls from an indefinitely great distance what\\nwill be its velocity on reaching the surface of the\\nearth, all forces besides the earth s attraction being\\ndisregarded\\nWe have V) 2 2 A\\nor v r ^J2gR,\\nwhen h is indefinitely great.\\nIf R 3960 x 5280 feet and g 32.155,\\nv V2(32.155)(3960)(5280) secQ\\n5280 L\\n7 miles per second, approximately.\\nBy great heights we may mean such various\\nheights as those attained by the kites flown at the\\nBlue Hill Observatory (8000 ft,), or by Andrews bal-\\nloon; or the height of a meteorite when it first becomes\\nvisible. In the practical consideration of the velocities", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0085.jp2"}, "86": {"fulltext": "74 CALCULUS.\\nof bodies falling from such heights, the resistance of\\nthe air must be taken into account. For a discussion\\nof the vertical motion of a body in a resisting medium,\\nsee Greenhill s Calculus, Art. 7(5.\\n64. Let a body free to move be subjected to an\\nattractive force that varies directly as the distance of\\nthe body from the point where the force is located.\\nIf we take this point as origin, with a line passing\\nthrough the body for the ^-axis, we have\\nd fdx\\\\\\nThe coefficient /jl is seen to be the value of the accel-\\neration when x=l that is, when the body is at a unit s\\ndistance from the origin. The minus sign is used for\\nthe same reason that was given in Art. 63.\\nMultiplying by dx and integrating,\\nlA7.r\\\\ 2 __^ 2 c\\n2\\\\dt) 2 X +C\\nIf when x a, C\\ndt 2\\ntherefore j =-(a? x 2\\nWriting this equation so that dt shall stand by itself,\\ndt=- 1 dx\\nV//, Va 2 x 2\\nAfter extracting the square root only the negative\\nsign is retained because dt is positive, and dx is nega-\\ntive when the body is moving toward the origin.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0086.jp2"}, "87": {"fulltext": "BECTILINEAR MOTION. 75\\nIntegrating again,\\nV/x a\\nIf t when x a, O r\\n1\\ntherefore cos -1\\nV\\nthat is, a cos V/x\u00c2\u00a3.\\nComparing this result with equation (a), Art. 56, we\\nconclude that a body subjected to an attractive force\\nvarying directly as the distance will move with simple\\nharmonic motion.\\n65. Suppose the body is driven away from the origin\\nby a force varying directly as the distance of the body.\\nand proceeding as before,\\nI J /jlx 2 C fji (x 2 a 2 (a)\\nthat is, Vfidt\\n-Vx 2 a 2\\nIntegrating again, we have\\ntVJi+ C r log (x Vx 2 a 2\\nNotice that the constant of integration m y be written\\nin either member of the equation as suits our conven-\\nience. Heretofore it has been written in the right-hand\\nmember.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0087.jp2"}, "88": {"fulltext": "76 CALCULUS.\\nNow suppose that x a when t\\nthen C f log a,\\nand tVfi log a log (x V^ 2 a 2\\ni (x Va 2 a 2\\n^V/x log\\na\\nFrom this expression we have\\nLt\\nx V.t 2 a 2, ae\\nFurther, since (x Vx 2 a 2 V# 2 a 2 a 2\\na; V.r 2 a 2 ae y//x\\nAdding the expressions for\\nx VV 2 a 2, and .r V x l a\\\\\\n2x tuft* ae\\nthat is, x a (e^ f e\\nIf we now differentiate this expression, we shall have\\nthe velocity a function of the time instead of a function\\nof the distance as in equation (a) for\\ndt 2 K J\\n2\\nEquations (5) and (c) show that as t increases, the\\nbody is driven farther and farther from the origin with\\never increasing velocity. These equations involve the", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0088.jp2"}, "89": {"fulltext": "RECTILINEAR MOTION. 77\\nsupposition that the initial velocity is zero. Let us now\\nsuppose that the initial velocity is aVft. Resuming\\nthe equation 7 xo\\nsince [_p\\\\== aVa when x a. (7=0:\\n\\\\dt)\\nand the equation becomes\\nfdx^ 2\\ndx\\nor V udt,\\nx\\nthe minus sign being used because the motion is toward\\nthe origin.\\nWe now have log x Vfit G\\\\\\nand since x a when t 0, C f log a\\ntherefore V/jit log\\nand x ae 1\\nThis equation shows that with the initial velocity\\na V/x the body constantly approaches the origin, but\\nnever reaches it.\\n66. Suppose that a body instead of being projected\\nvertically, is projected in a direction making the angle a\\nwith the horizontal plane, V being the velocity of pro-\\njection. The body thus has a vertical velocity and a\\nhorizontal velocity. The horizontal velocity is evi-\\ndently unaccelerated, whilst the vertical velocity is\\nbeing retarded by gravity. That is, taking the hori-", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0089.jp2"}, "90": {"fulltext": "78 CALCULUS.\\nzontal side of the angle a for the #-axis, and taking the\\ny-axis vertical and positive upward with the point\\nfrom which the body is projected as origin,\\nd fdx\\\\ _ n d fdy\\\\ _\\ndt\\\\dtJ~ It\\\\di)~ 9\\nThese two statements are the equations of motion\\nof the body. Examples of such equations have already\\noccurred in preceding articles. Integrating the first\\n(1 v\\none, J^cos a, the constant horizontal velocity. In-\\ntegrating again,\\nx tv cos a, {a)\\nthe constant of integration being zero if t when\\nx= Q.\\nIntegrating the second equation of motion,\\nWhen t 0, the time of projection, is the vertical\\n1 J dt\\ncomponent of the velocity for the same instant. This\\ninitial vertical velocity being F~sin a, we have\\nat F sin\\ndt J\\nand integrating again,\\ny \\\\gt 2 t V sin a. (J)\\nEquations (a) and (b) give the coordinates of the body\\nat any time t. Eliminating we have\\ny x tan a f x\\\\\\n2 F 2 cos 2 a\\nthe equation to the path of the body.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0090.jp2"}, "91": {"fulltext": "PARABOLIC MOTION. 79\\n67. If we transform equation (e) by passing to a\\nnew pair of axes parallel to the first with\\nV 2 sin a cos a V 2 sin 2 a\\n9 2 9\\nfor the coordinates of the new origin, we have (Art. 100),\\nY^sin 2 a f F 2 sin a cos a\\ny H tan a[ x\\n2 9 V g\\ng_ f F 2 sinacos^ 2\\n2 V 2 cos 2 a\\\\ g\\nAfter reduction this becomes\\n2 F 2 cos 2 a\\nx 2 y,\\n9\\nwhich is seen to be a parabola convex upward with its\\nvertex at the origin of coordinates. (Art. 129.)\\nThis curve is approximately shown in a stream of\\nwater issuing from a hose. It may also be traced by\\nwatching a tennis-ball or base-ball as the ball moves\\nthrough the air.\\n68. To find the horizontal range, we put y in\\nequation then x This value is great-\\nest when sin 2 a is greatest; that is, when the angle\\nof projection, is 45\u00c2\u00b0.\\nIt may be noticed that persons skilled in throwing have learned\\nfrom experience that in order to throw as far as possible the ball\\nor stone must be thrown in a direction about half way between\\nhorizontal and straight up.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0091.jp2"}, "92": {"fulltext": "80 CALCULUS.\\n69. To find the range on an inclined plane let the\\nstraight line y x tan /3 express the slope of the plane.\\nWe have then to find where the line y x tan /3 cuts\\nthe parabola\\ny x tan a x\\\\\\nI V* cosset\\nEliminating y, we obtain\\n_ 2 V 2 cos a sin (a /3)\\ng cosp\\nthe abscissa of the point of intersection. The distance\\nfrom the point of projection to this point of intersec-\\ntion is therefore\\nQ t 2V 2 cos a sin (a ff)\\n#sec/3, which equals\\ng cos 2 ft\\nTo find the particular value of a that will make this\\ndistance a maximum, we must view this expression as\\na function of a and equate the first derivative to zero\\nthat is, if R is the range, is the condition for a\\nda\\nmaximum (or minimum). (Art. 41.)\\nWe have then\\ndR d n\\nx sec p\\nda da\\n2 V 2 sin a sin (a /3) 2V 2 cos a cos (a\u00e2\u0080\u0094 /3)\\ng cos 2 ft\\nEquating this to zero and reducing,\\ncos a cos /3) sin a sin (a /3)\\nthat is, cos /3)] 0,\\nand hence 2\u00c2\u00ab-/3= 90\u00c2\u00b0,\\n90\u00c2\u00b0+/3\\na\\n\u00c2\u00a3-1-1(90\u00c2\u00b0-/?).", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0092.jp2"}, "93": {"fulltext": "MOTION IX A VERTICAL CURVE. 81\\nTherefore the direction of projection which secures\\nthe greatest range on a given inclined plane bisects the\\nangle between the vertical and the inclined plane.\\nThe student should of course raise the question:\\nHow do we know that the above equation of condition\\ngives a value of a that secures a maximum range instead\\nof a minimum\\n70. Suppose that a piece of smooth Avire or small-bore\\ntubing, smooth on the inside, is bent into the shape of\\nsome plane curve and hung up vertically. Further,\\nsuppose that a bead* is strung on the wire, or a small\\nball dropped into the tube. The body, say the ball,\\nwill slide downward under the action of gravity, but\\nit will be obliged to follow a certain path. What will\\nits velocity be at any point P\\nDraw the usual axes in the vertical plane in which\\nthe curve lies. Let A be the position of the body when\\n\u00c2\u00a3=0; P its position (x, y) at any time t\\\\ and let\\narc^LP s. If a is the angle which the tangent at the\\npoint P makes with the #-axis, g sin a is the acceleration\\nalong the curve at P. But sin a f- hence\\nas\\nd fds\\\\ _ _ cly\\nJt\\\\dtJ~ 9 Ts\\nMultiplying by 2 ds and integrating, we obtain\\nr ds^ 2\\ndtj\\nIf we call the ordinate of the point A y and the\\nvelocity at A v we have\\nand therefore v 2 v* 2 g (y g)", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0093.jp2"}, "94": {"fulltext": "82 CALCULUS.\\nNow let the ordinate t/ be produced upward to a\\npoint B, making AB A, the height from which the\\nbody falling freely would have to fall in order to ac-\\nquire the velocity v Draw BN a line parallel to the\\n#-axis. Let C be the point where the ordinate y pro-\\nduced meets the line BN. v 2 =2 /h (Art. 60\\nSubstituting this value of v 2 in the equation above,\\nv*= 2gh Lg{y- y 2g(h y -y-) 2g PC.\\nHence the velocity at any point P is the same as the\\nvelocity that would have been acquired had the body\\nfallen directly from the line BN to P.\\n71. Let us now limit the case to motion in a vertical\\ncircle. Instead of having the ball slide in a circular\\ntube we can just as well\\nsecure circular motion by\\nattaching the ball to the\\nend of a string whose other\\nend is fastened at the cen-\\nter of the circle. We now\\nhave a pendulum. Let C\\nbe the center of the circle\\nFig. 7. its lowest point OX the\\n#-axis, and OY the y-axis.\\nLet A be the starting point of the ball then at A t\\nand v Q 0. Let P be its position and v its velocity at\\nany time t. Also, let 6= angle P CO and a angle A 00\\ns arc AP and I PC, the length of the string.\\nBy the preceding article,\\nfd*\\\\ 2 v 2g PJST= 2#Z(cos cos", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0094.jp2"}, "95": {"fulltext": "SIMPLE PENDULUM. 83\\nbut (*Y-Pf^\\n\\\\dtj \\\\dt.\\nhence -3- (cos cos a)\\n4gf 2 a -off\\nsm z snr\\nI 2 2\\nand therefore 2\\\\k Jsin 2 sin 2\\ndt l* 2 2\\nNotice that after extracting the square root only the\\nminus sign is used, because dt is positive and d0 is\\nnegative.\\nWe now have\\nVf^=-\\ndd\\n\\\\|sin 2 sin\\nthat is,\\nf\u00e2\u0080\u0094 r\\n2\\ni\\\\/sin 2\\n_:_\u00c2\u00bb0\\nThe expression here presented for integration looks\\nquite simple, but it cannot be expressed in finite terms by\\nmeans of the ordinary algebraic or trigonometric func-\\ntions. If, however, we expand sin 2 by Maelaurin s\\n2\\ntheorem (Art. 34, ex. 14), and then take a so small\\nthat we may neglect powers of a (and 0) beyond the\\nsecond, we shall have\\n4j sin 2 |-sin 2 2 2", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0095.jp2"}, "96": {"fulltext": "84 CALCULUS.\\nThe above integral then becomes\\nand this is integrated by formula ll r Chap. V, so that\\nwe obtain\\nC\\nVf-\\na\\ncos 1 cos -1 cos L\\na a a a\\nSolving for 0, a cos y k\\nWhen (9=0, -^2* cos^O\\nhence f the time from J. to (9, is\\n2 V\\nIf T 7 be the time of an oscillation from J. to A 1 (on\\nthe other side of 0),\\nThis result is true only when a is small, as above\\nshown. It is independent of and therefore the time\\nof an oscillation is the same for all small arcs in the\\nsame circle. That is, if a and a! are two small but\\nunequal arcs, the times of oscillation for the same\\npendulum are equal.\\n72. It will be noticed that the equation\\n6 a cos\\\\f- t", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0096.jp2"}, "97": {"fulltext": "AREAS.\\n85\\nis of the form of the equation expressing simple har-\\nmonic motion therefore the pendulum-bob. has simple\\nharmonic motion in a circle which lies in a plane pass-\\ning through OX perpendicular to OF. The radius of\\nthis circle is and the displacement at any time t is 6.\\nIf a is given in degrees, it must be divided by the\\nradian (57\u00c2\u00b0. 295779 For instance, if I 50 inches\\nand a 1\u00c2\u00b0, the radius of the circle across which the\\nharmonic motion takes place is i- inches,\\nL 57.\u00c2\u00b0+ 57\\napproximately.\\n73. Areas. Let PS be a portion of the curve\\ny =f(x) and let it be required to find the area\\nbounded by this arc, the ordinates PM and SN, and the\\n#-axis.\\nFig. 8.\\nLet OM=a, ON=b, OT=z, and OV=z 8p;\\nthen QT=y, and RV=y 8y. If the area OLQT,\\nany varying portion of the area OLSN, equals ^4, area\\nOLR V=A 8 A, and 8A TQR V. Now, if the short", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0097.jp2"}, "98": {"fulltext": "86 CALCULUS.\\narc QR were a straight line, the area TQRS would be\\na trapezoid, and we should have\\n8A Sx\u00c2\u00b1(QT RV)= Sx(y \u00c2\u00b18i,)\\nand y Si/,\\nox\\nIn the limit QR becomes a straight line, and\\ndA\\nthat is, dA ydx f(x)dx\\nand this is a representative strip taken anywhere in the\\narea OLSN.\\nSuppose \\\\f(x)dx f (x) 0;\\nthen A 4 (x)+C.\\nSince we are measuring areas from the y-axis,\\nwhen x 0, A\\nwhen x a, JL area OLPM;\\nwhen 6, -4 area OLSN;\\ntherefore area OLSN= \u00c2\u00a3(J) (7,\\narea OLPM= t (a) O.\\nSubtracting this last expression from the one pre-\\nceding it,\\narea OLSN- area OLPM= mmMPSN\\nf(x)dx.\\na", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0098.jp2"}, "99": {"fulltext": "ABE AS. 87\\nWe have, then, for the area between the ordinates,\\nwhose distances from the ?/-axis are a and b respec-\\ntively, the definite integral\\nr\\nf(x)dx.\\n74. For example, suppose we wish to find the area\\nbounded by the parabola y 2 \\\\px, the a axis, and any\\nordinate y r (the accompanying abscissa being x\\nA= Cydx C x 2p i x*dx:\\n4:p 2 X 2\\n3\\n4 /V)2/v^2\\nWe notice that the rectangle x y f 2p 2 x 2 hence\\nthe area in question equals two-thirds the circumscribed\\nrectangle.\\nExamples.\\n1. Find the area of the upper right-hand quarter of\\nthe ellipse 1.\\nIn this case\\no ydv=J ~^Ja 2 -x l dx\\nh x r* 2 a2 -l\\n_ 7raS\\nThe area of the entire ellipse is therefore it ah. ira 2\\nthe area of the circle x 2 y 2 a 2 may now be viewed\\nas a special case of the ellipse in which b a.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0099.jp2"}, "100": {"fulltext": "88 CALCULUS.\\n2. Find the area between the isotherm pv c, the\\nv-axis, and the two ordinates whose distances from the\\n#-axis are a and b respectively. Aits, c log\\na\\n3. Find the area bounded by the #-axis and the\\ncurve y sin x, from x 0\u00c2\u00b0 to x 180\u00c2\u00b0.\\nAns. 2..\\n75. Mean values. The mean or average value of n\\nquantities is the nt\\\\\\\\ part of their sum. If the quanti-\\nties to be averaged are successive values of a function\\nof some variable, their magnitudes depend not only on\\nthe nature of the function, but also on the law of varia-\\ntion of the fundamental. Thus, suppose we have the\\nisotherm pv c and wish to know the average pressure\\nbetween the volumes i\\\\ and v 2 It is necessary to make\\nsome assumption in regard to the variation of v. (1) If\\nits increments are supposed equal, we understand by the\\nmean value of the pressure the average of the press-\\nures corresponding to the arithmetic series v, v civ,\\nv 2 civ, etc. (2) If the volume is assumed to depend\\non some other variable in such a manner that the\\nabscissa increments are not equal, the mean value will\\nnow be the average of a new series of pressure ordinates\\ncorresponding to the new series of values of v arising\\nunder the second assumption. Evidently the two means\\nwill, in general, be unequal but one is just as properly\\nthe average as the other. An important illustration is\\nafforded if we ask what is the mean distance of a\\nplanet from the sun If a planet moved in its elliptic\\norbit in such a way that the radius vector described\\nequal angles in equal times, that is, if its angular\\nat", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0100.jp2"}, "101": {"fulltext": "MEAN VALUES. 89\\nvelocity, were a constant, the mean length of its radius\\nvector could be shown to be aVl e 2 a being the semi-\\nmajor axis, and e the eccentricity of its orbit. But we\\nknow that the law of gravitation requires that the areal\\nvelocity shall be a constant that is, the radius vector\\ndescribes equal areas, instead of equal angles, in equal\\ntimes (Art. 88). In one case is constant; in the\\nother, A r 2 is constant. A little consideration will\\n2 dt\\nshow that the mean value of r cannot be the same in\\nthe two cases.\\n76. If y=f(x) and all of the dx s are equal, the\\naverage length of y between x a and x b is at once\\nfound by dividing the area I f(x)dx by b a; for\\nreturning to Fig. 8, if the area MPSN be divided by\\nits base b a, the quotient is the altitude of an equiva-\\nlent rectangle of base b a] and the altitude of the\\nrectangle is the average altitude of the strips repre-\\nsented by TQRV\\\\ that is, of the ys.\\nExamples.\\n1. Find the average length of the ordinates of a\\nsemicircle, supposing the series taken equidistant.\\nWe have x 2 y 2 r 2 or, y Vr 2 x 2 therefore\\nM= I Vr 2 x 2 dx =+7rr.\\nFrom this result it appears that the average ordinate\\nequals the length of an arc of 45\u00c2\u00b0.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0101.jp2"}, "102": {"fulltext": "90 CALCULUS.\\n2. Find the average length of the ordinates, sup-\\nposing they are drawn through equidistant points on\\nthe circumference.\\nIn this case\\n2r\\n1 C n\\nI r si\\nM=- rsinddO\\nIT\\n3. Given pv c show that the mean pressure\\nbetween the volumes v x and v\u00c2\u00bb is log- 2 v chang-\\ning by equal increments. 211\\n4. A particle has simple harmonic motion. Find its\\nmean velocity as it passes from the extremity of the\\nradius to the center of the circle.\\n77. The above geometric conception of mean values\\nmay be adopted when a function is expressed in polar\\ncoordinates.\\nIf r let x be written for 6, and y for r, so\\nthat we have y=f(x). This equation furnishes a\\ncurve which sustains peculiar relations to the original\\npolar curve. The radii vectores lose their fan-shaped\\narrangement, and are placed parallel and equidistant\\n(if 6 is an equicrescent variable) with their extremities\\n011 a common line, the :r-axis. The pole may be viewed\\nas developing into this axis, just as if a draw-string\\nwere let out, while a circle of unit radius with the\\npole as center develops into a straight line parallel to\\nthe #-axis, the radii vectores keeping their position\\nof perpendicularity with respect to the circumference of\\nthe circle. The mean value of the radius vector then\\n1 r b\\nbecomes I f(x) dx, as before.\\nb aJ*", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0102.jp2"}, "103": {"fulltext": "WOBK. 91\\nFor example, to find r the mean length of the radius\\na _ e 2\\\\\\nvector of the ellipse r 6 being an equi-\\nF 1 e cos 6 H\\ncrescent variable, we have, using one-half of the ellipse,\\n1 r\u00c2\u00aba(l e 2 7 r A o\\nr I \u00e2\u0080\u0094ax a VI e*.\\nirJol e cos\\nThe radii vectores, now in the role of ordinates, are\\ndistributed at equal intervals through an area A whose\\nbase is 7r.\\nExample. Find the average length of the radius\\nvector of the cardioid r a(l cos\\nM=- a (1 cos cfe\\n7T\u00c2\u00abyo 7T\\nsmo;\\na.\\n78. Work. If a force _F acts on a body of mass m,\\ngiving it an acceleration\\nXT d 2 s\\nF=m dT*\\nMultiplying by ds,\\nFds m f )ds.\\ndt\\\\dtj\\nIntegrating between the limits v and V, V being the\\nvelocity when s 0, and v the velocity when s s,\\nIf F= 0, fFds=\u00c2\u00b1mv 2", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0103.jp2"}, "104": {"fulltext": "92 CALCULUS.\\nFds is defined as the work done on the body m as it\\nis moved through the space ds. I Fds is the work\\ndone in moving the body over the arc s.\\nJ mv 2 is defined as the kinetic energy which the body\\npossesses because work has been expended upon it, the\\nkinetic energy representing the work stored up in the\\nbody. In order to perform the operation indicated by\\nI Fds we of course need to know what function F is\\nof s in case F is a variable depending on s. Suppose\\nthat F= p(s and W represents the work then\\nW\\nFrom this it appears that work can be represented by\\nan area referred to a space axis (x-axis), and a force\\naxis (y-axis).\\n79. In fact, the integral for area is seen to be repre-\\nsentative of all definite single integrals, these integrals\\ntaking the general form I f(x)dx. The primary or\\nhorizontal axis is named for the quantity which x de-\\nnotes, and the secondary or vertical axis is named for\\nthe function of x. The integral itself is then repre-\\nsented by the area MPSN. (Fig. 8.)\\n80. Lengths of curves. Referring to Fig. 2, Art. 38,\\nit is seen that\\nds Vcte 2 dy* =\\\\1+ f^ 2 dx =yjl (^ffdyi", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0104.jp2"}, "105": {"fulltext": "LENGTHS OF CURVES. 93\\nhence, if s is the length of an arc from the point (x 1\\nto the point (x n y n\\nConvenience must decide which of the formulas we\\nshall use in any given example.\\nExamples.\\nc f\\n1. Find the length of the catenary y -I ec e\\nthe curve in which a uniform chain hangs.\\ndy_li\\ndx 2 x\u00e2\u0082\u00ac\\ntherefore ^l+l-^-j -(e c e\\nand s I -le c e c \\\\dx -le\\n2. Find the circumference of the circle x 2 y 2 r 2\\nWe have -f- then, if AB is the first quad-\\ndx y\\nrantal arc of the circle,\\nAB =SS l+ (t) 2dx r i\\ndXJ *^0 y2\\ndx\\n^X~\\\\ V ITT\\nr\\\\ sin\\nL Jo\\nTherefore the circumference of the circle, 4 .Ai?, 2 7rr.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0105.jp2"}, "106": {"fulltext": "94 CALCULUS.\\n81. Volumes of revolution areas of surfaces of\\nrevolution.\\nIf a plane curve revolves around any line in its plane\\nas an axis, it is evident that tlie figure generated is\\nsuch that any cross-section of it by a plane at right\\nangles to the axis is a circle. The volume may be found\\nb)^ taking the axis of revolution as the #-axis and add-\\ning together layers dx in thickness. The area of any\\ncross-section is Try 2 If V represents volume, we have\\nthen b\\nV= I Try 2 dx,\\nin which the equation to the generating curve is\\nv\\nSimilarly, the surface may be found by noticing that\\nthe arc 8s generates the frustum of a cone whose sur-\\nface is known from elementary geometry to be\\nor, in the limit, 2iryJs\\\\ so that, if S is the area of the\\nsurface, 7 N0\\ndy\\\\ 2\\nH W 1+ @D*\\nExamples.\\n0-2 n,2\\n1. The ellipse f- l revolves about its major\\naxis. What is the volume generated\\n1 V= iry 2 dx it (a 2 x 2 dx I (a 2 2 dx\\nJo Jo a 1 a 1 Jo\\nirb 2 2", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0106.jp2"}, "107": {"fulltext": "VOLUMES AND SURFACES.\\n95\\nThe entire volume is therefore irab 2\\nThe volume of the sphere, 7m 3 is a special case, in\\nwhich b a.\\n2. Find the area of the surface generated as the\\nellipse revolves about its major axis.\\n%s\\n+ffi*\\n2irb\\nCV -(a 2 -b 2 )x 2 fdx\\na 2 Jo\\nirb\\\\ b\\nw\\n^J a 2 b 2\\ntherefore the whole surface is\\nw\\n_iVa 2 b 2\\nsin\\na\\n2irb\\n2irb\\nb\\nb\\n_iVa 2 b 2\\n_sm\\na 2 _ ^2 a\\na 2 1 6\\ncos\\nVa 2 b 2 a.\\n3. Find the area of the surface of a sphere whose\\nradius is a.\\nIf we make b a, we have, from the result in the\\npreceding example, for the area of the surface of the\\nsphere,\\n2 ira\\na a 2\\ncos\\na\\n~Va 2 a 2 _\\nWe must now find the value of\\n2ira\\na a 2\\nVa 2 b 2\\nwhen b a.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0107.jp2"}, "108": {"fulltext": "96\\nCALCULUS.\\nTreating b as a variable and applying the principle\\nof Art. 32,\\n1\\nb\\nCOS\\na\\n,4( C0S\\n,7\\n\u00e2\u0096\u00a031\\nVa 2 b*J\\nb=a Va 2\\ndb\\nw 2\\n^-5 n i\\nb\\nVa 2 -6 2\\nTherefore, 2 7ra a a 2 4 ira 2\\nThis result for the area of the surface of a sphere\\nagrees, of course, with the one obtained by the method\\nof elementary geometry.\\n82. The area integral I f(x) dx represents the sum\\n%sa\\nof strips whose height is y and breadth dx. We may\\nreach the same result by starting with the elementary\\nrectangle dxdy and using two integral signs, one to\\nindicate that we add such rectangles together to make\\na strip y in height, and a second to indicate that the\\nstrips are to be added together, making the area from\\na to h (Fig. 8). For example, the area of the ellipse\\nmay be found by adding together the areas dxdy from\\nthe major axis to the curve itself then adding together\\nthe strips from the minor axis to the end of the major\\naxis. To indicate this double operation, we write\\nI I dxdy,\\nusing the right-hand integral sign with dy.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0108.jp2"}, "109": {"fulltext": "DOUBLE INTEGRALS.\\n97\\nPerforming the first operation,\\nI dxdy I ydi\\n*^0\\nThe remaining part of the work is the same as in\\nArt. 74, example 2.\\nThe above procedure in finding areas involves what\\nis known as a double integral. Similarly, three succes-\\nsive indicated integrations constitute a triple integral.\\nExamples of double integrals will occur in subsequent\\narticles.\\n83.* Suppose a point to travel once round the closed\\noval area J., an indicator diagram, for instance, so as\\nalways to have the interior of the curve on the left\\nFig. 9\\nhand. Let B be the minimum point, and C the maxi-\\nmum point with respect to the #-axis D the minimum\\npoint, and E the maximum point with respect to the\\n^/-axis.\\nGreenhill s Differential and Integral Calculus.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0109.jp2"}, "110": {"fulltext": "98 CALCULUS.\\nThen A J dxdy I xdy,\\ntaken round the perimeter of the curve.\\nFrom B to O along BPC, dy is positive, and\\nfxdy area MBP ON.\\nFrom (7 to B along C(?Z?, tf is negative, and\\nj*xdy area 3IBQCJST;\\nso that, taken round the curve,\\nfxdy area MBPCN- area 3IBQCN= A,\\nthe area of the closed curve.\\nBut 1 1 dxdy ycfo\\nand from E to D along EPD, dx is negative, so that\\nJ^/.r area LEQJDK;\\nand from i) to JE along DBE, dx is positive, so that\\nfydx area LBBBK;\\nand therefore, taken round the curve,\\nI ydx A.\\nTherefore taken round the curve,\\nj (ydx xdy)\\nand ydx xdy d (xy) is called a perfect differential.\\nIts integral between two limits is independent of the\\nintermediate values of x and y and of the path described", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0110.jp2"}, "111": {"fulltext": "MOMENT OF INERTIA. 99\\nbetween the limits; so that, taken round any closed\\npath, the integral is zero.\\nWhen Fig. 9 represents an indicator diagram, and\\nKL the reduced stroke of the piston, while the ordinate\\ny represents the pressure of the steam, the pencil will\\ndescribe the contour with the area to the left, when the\\nsteam pressure is urging the piston from L to K. The\\ndiagram taken on the return stroke from the other end\\nof the cylinder will be described in the opposite sense,\\nwith the area on the right hand of the describing pencil.\\n84. Moment of inertia. When a rigid body rotates\\nabout an axis, the linear velocity of any particle of the\\nboc is ds rae\\nv r\\ndt dt\\na) being the angular velocity of the particle, and r its\\ndistance from the axis. Its kinetic energy of rotation is\\ntherefore mv 2 mr 2 co 2 m being the mass of the par-\\nticle and the kinetic energy of rotation of the whole\\nbody is\\n1 mv i _|_ i m V 2 m n v ,!2\\n2 mv 2 2 mco 2 r 2 ft) 2 Smr 2\\nthat is, one-half the product of the square of the angular\\nvelocity and 2mr 2\\nThe symbol 2 is used to indicate a pol} r nomial in\\nwhich the terms are similarly constituted, as in the case\\nbefore us. Since such an expression as 2 J mv 2 is in\\nreality a polynomial, only common factors can be re-\\nmoved and placed before 2, the symbol of summation.\\nThus in 2 mco 2 r 2 is, of course, a common factor co 2 is a\\ncommon factor because the rotating body is supposed to\\nLofC.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0111.jp2"}, "112": {"fulltext": "100\\nCALCULUS.\\nbe rigid, and consequently all of its parts have the same\\nangular velocity but m is not a common factor because\\nit is not supposed that all of the particles have equal\\nmasses neither is r a common factor, for the particles\\nare at different distances from the axis of rotation.\\nThe quantity 2 mr 2 is called the moment of inertia of\\nthe body with respect to the axis, and is seen to be the\\nsum of the products obtained by multiplying the mass of\\neach particle by the square of its distance from the axis.\\nIf a body rotates with a given angular velocity about\\ndifferent axes, the kinetic energy of rotation with respect\\nto any axis must be proportional to 2 mr 2 consequently,\\nthe moment of inertia measures the capacity of a body\\nto store up kinetic energy during rotation about the axis\\nwith respect to which the moment of inertia is taken.\\nExamples.\\n85. l. A sheet of metal, rectangular in shape and\\nof uniform density, is made to rotate about an axis\\ncoinciding with one end.\\nWhat is its moment of\\ninertia\\nTake the axis of rota-\\ntion for the #-axis with\\nthe origin at the left-hand\\ncorner of the rectangle.\\nLet b be the breadth and\\n-2 d the height of the rec-\\ntangle. If p is the density\\nof the metal, pdydx is\\nthe mass of the indefi-\\nnitely small rectangle dy dx cut anywhere from the\\ni\\nFig. 10.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0112.jp2"}, "113": {"fulltext": "MOMENT OF INERTIA. 101\\nsheet and (pdydx^y 2 is the moment of inertia of\\nthis small piece. Hence the moment of inertia of the\\nentire sheet becomes\\ni Jo f al yUydx\\nphj*fcly f*f.\\nFrom this example it is plain that in all cases in which\\nthe density is constant throughout the body, the density\\nfactor may as well be set aside until the integration is\\ncompleted. If, however, the density varies from point\\nto point, so that p is some specified function of x and y,\\nit must be kept under the sign of integration and be\\ntaken account of in the process of integrating.\\n2. A straight slender rod of length Z, whose density\\nvaries directly as the distance from one end, rotates\\nabout an axis perpendicular to it and passing through\\nthe end having the least density. What is the moment\\nof inertia with reference to this axis\\nTake the given axis as the #-axis, with the origin at\\nthe end of the rod.\\npccy therefore p ky if Ar is the density at a unit s\\ndistance from the end. Then the moment of inertia is\\n\u00c2\u00a3py 2 dy JkyHy\\n3. Find the moment of inertia of a circle with refer-\\nence to an axis through its center and perpendicular to\\nit, p being a constant.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0113.jp2"}, "114": {"fulltext": "102 CALCULUS.\\nLet R be the radius of the circle, r the distance of\\nany particle from the axis, and 6 the variable angle\\nmeasured from some chosen radius. Consider an ele-\\nmentary portion bounded by the circles whose radii are\\nr and r dr, and by the radii forming the angle d0.\\nIn the limit this bit of area becomes the rectangle\\n(rd0)dr; hence, the integral is\\nC C \\\\\\\\rdrd6) 2tt Crhlr=-W;\\nand therefore the moment of inertia is\\n2\\n86. Kepler s laws. It is shown in works on the\\ndetermination of orbits* that the equations for the un-\\ndisturbed motion of a planet or comet relative to the\\nsun are\\ng F(l 0, (1)\\ng F(l m)5 0, (2)\\ng l Bl 0, (3)\\nin which x, y, z are the coordinates of the heavenly\\nbody referred to the sun as origin, f, are\\ndt l dt l dt z\\nthe accelerations parallel to the three axes of reference,\\nr is the distance of the body from the sun, k 2 is the mass\\nof the sun, and m the ratio of the mass of the body to\\nthe mass of the sun. Having these three equations, we\\nWatson, Theoretical Astronomy; Dziobek, Planeten-Bewe-\\ngungen Tisserand, Determination des Orbites.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0114.jp2"}, "115": {"fulltext": "KEPLER S LAWS. 103\\ncan at once establish Kepler s laws. Arts. 87, 88, 89,\\n90, 93 are taken, with slight changes, from Watson s\\nTheoretical Astronomy.\\n87. If we multiply equation (1) of the preceding\\narticle by y, and equation (2) by x, and subtract the\\nlast product from the first, we shall have, after inte-\\ngrating the result,\\nxdy ydx _ n\\nJt =C\\nC being the constant of integration.\\nIn a similar manner we obtain\\nxdz zdx _ pi ydz zdy _\\ndt dt\\nIf we multiply these three equations respectively by\\n2, y, and and add the products,\\nCz-C f y C x=0.\\nThis is the equation to a plane passing through the\\norigin of coordinates (Art. 142). Since x, y, z are the\\ncoordinates of the heavenly body, it must remain in this\\nplane. The path of the heavenly body relative to the\\nsun is therefore a plane curve, and the plane of the orbit\\npasses through the center of the sun.\\n88. If we multiply equations (1), (2), and (3) re-\\nspectively by 2 dx, 2 dy, and 2 fe, take the sum and\\nintegrate, we have\\ndx 2 dy 2 dz 2 y Q m C xdx y d y zdz p.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0115.jp2"}, "116": {"fulltext": "104 CALCULUS.\\nBut r 2 x 2 y 2 z\\\\\\ntherefore rdr xdx ydy zdz.\\nIntroducing this value of xdx zdz into equa-\\ntion (4) and performing the integration indicated, we\\nhave\\nr (5)\\nh being the constant of integration.\\nIf we add together the squares of the expressions for\\nC, C, 0 and put C 2 C 2 C 2 4/ 2 we shall have\\n(x 2 y 2 z 2 (d.r 2 di/ 2 ch 2 _ (xdx ydy zdz) 2 ifi\\ndt 2 dt 2 f\\no dx 2 d// 2 dz 2 r 2 dr 2 a\\nthat is, r 2 JL. ____ 4/ 2. (6\\nIf we now represent by dv the infinitely small angle\\ncontained between two consecutive radii-vectores r and\\nr dr, since dx 2 dy 2 dz 2 is the square of ds, the ele-\\nment of path described by the body, we shall have\\ndx 2 dy 2 dz 2 dr 2 r 2 dv 2\\nSubstituting this value of dx 2 dy 2 dz 2 in equa-\\ntion (6),\\nr 2 dv=2fdt. (7)\\nThe quantity r 2 dv is double the area included by the\\nelement of path described in the element of time dt,\\nand by the radii-vectores r and r dr. See Fig. 4,\\nArt. 46. The area of the triangle POP area POQ\\narea PQP 1 but in the limit area PQP f vanishes,", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0116.jp2"}, "117": {"fulltext": "KEPLER S LAWS. 105\\nand area POQ=\\\\OP(PQ)= r (n20) W riting equa-\\ntion (7) in the form\\n\\\\fdv_ f\\ndt 7\\n1 r 2 c J v\\nthe quantity -2\u00e2\u0080\u0094 is the area described by the radius-\\nvector in the time dt, divided by the time, and is defined\\nas the areal velocity. Since/ is a constant, we conclude\\nthat the radius-vector of a planet or comet describes equal\\nareas in equal intervals of time. (Kepler s second law.)\\n89. Combining equations (5) and (6) so as to elimi-\\nnate and solving for dt, we have\\ndt 2\\ndt (8)\\nV2 rk 2 (1 m) hr 2 4 J*\\nSubstituting this value of dt in equation (7),\\ndr _ r V2 rk 2 (1 ni) hr 2 4 f 2\\ndv~ 2/\\n(9)\\nWe have seen (Art. 46) that the condition that r\\nshall be a maximum or minimum is 0. With\\nrd9\\nthe notation of the present article, v 6 some con-\\nstant therefore dv dd, and we have, in order to find\\nthe maximum and minimum values of r,\\nV2 rk 2 (l m)- lir 2 -if 2 Q\\n2/\\nthat is, 2 rk 2 (1 m) hr 2 4 f 2 0.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0117.jp2"}, "118": {"fulltext": "106 CALCULUS.\\nIf 1\\\\ and r 2 represent the two roots of this quadratic\\nequation,\\n_ F(l m) 4/\u00c2\u00bb ^(lT^7\\n_ F(l w) J 4 p /H (1+w 2\\nSince the equation of condition yields only two values\\nof r, the orbit cannot have more than two apsidal points.\\nIf it is a closed curve and not a circle, it must evidently\\nhave two, rather than one, such points. The point\\ncorresponding to r v the maximum value of r, is called\\nthe aphelion, and the point corresponding to r 2 is the\\nperihelion.\\n90. If we put\\nga+ia y-4g i p+-y. i +0\\nh it A 2\\nand add the two expressions, we have\\na\\nAlso, taking the difference of the two expressions and\\nsubstituting the value of h just found,\\n4/ 2 ak\\\\\\\\ m)(l e 2 *\u00c2\u00bbp(l m)\\nif p be written for a(l e 2", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0118.jp2"}, "119": {"fulltext": "KEPLER S LAWS. 107\\nSubstituting these values of h and 4/ 2 in equation (9)\\nit becomes\\n7 \\\\f~vdr\\nav\\nid\\ni\\n^r-l^-p \\\\ll-( l A\\na \\\\e r e)\\nthe integral of which gives\\nv cos 1 G),\\nco being the constant of integration; and, therefore,\\nwe have -f 1 cos\\nthat is, solving for r,\\nr r (10)\\n1+6 cos (v CO) v y\\nThis expression is seen to be the polar equation to a\\nconic section (Art. 115), the pole being at the focus, p\\nbeing the semi-latus rectum, e the eccentricity, and co\\nthe angle at the focus between the major axis and a\\nfixed line in the plane of the orbit. the vectorial angle,\\nis measured from this latter line.\\nIf co 0, equation (10) becomes\\nr i a(1 2) (11)\\n1 e cos v 1 e cos v\\nIn this case v is called the true anomaly. We now\\nconclude that the orbit of a heavenly body revolving\\naround the sun is a conic section with the sun in one of\\nthe foci. (Kepler s first law.)", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0119.jp2"}, "120": {"fulltext": "108 CALCULUS.\\n91. The planets revolve around the sun in ellipses,\\nand these ellipses are, as a rule, characterized by small\\neccentricities. Thus the eccentricity of the earth s orbit\\nis at present 0.01G77. Of all the major planets Mercury\\nhas the most elliptic orbit, its eccentricity being 0.2056.\\nThe orbits of comets, on the other hand, may be\\ndescribed as parabolic, by which we mean that they are\\neither ellipses of great eccentricity (almost unity), or\\nhyperbolas whose eccentricity differs but little from\\nunity. In many cases the eccentricity cannot be found\\nto differ from unity the orbit is then of course described\\nas a parabola. Of the periodic comets which have been\\nobserved at more than one perihelion passage, Tempel s\\ncomet has the least eccentricity, namely 0.4051.\\n92. In putting r x and r 2 equal to a(l e) and a(l e)\\nrespectively, the argument has much the air of begging\\nthe question, seeming to assume that the orbit is a conic\\nsection and then using the assumption in the proof.\\nBut it is to be noted that when we adopt the expres-\\nsions a(l and a(\\\\ e), e does not mean eccentri-\\ncity, neither does a mean semi-major axis. They do not\\nbear these meanings until in equation (10) we identify\\nthem with the constants in the polar equation of\\ncoordinate geometry.\\n93. If the values of h and 4/ 2 as found above, are\\nintroduced into equation (8), we have\\n7 Va rdr\\nat-.\\nk Vl m VaV (a r) 2", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0120.jp2"}, "121": {"fulltext": "KEPLER S LAWS.\\nwhich may be written\\n109\\ndt\\nae\\nV1\\nm\\nVW^Y\\nae\\nor\\ndt\\nkVl\\n-d\\na r\\nae\\na r -.fa r\\na\\nVi-\\na r\\nae\\ne\\nae\\nae\\nv\u00c2\u00bb\\nr\\\\2.\\na r\\nae\\nthe integration of which gives\\n3\\na 2\\nt\\nVl\\nm\\ncos\\n-\u00c2\u00abJl-\\nae\\nae\\nC. (12)\\nWhen the heavenly body is in perihelion, r a(l e),\\nand the integral reduces to t 0; therefore, if we\\ndenote the time from perihelion by t we have\\nt n\\n*Vi~T\\nill\\nCOS\\na i\\nae\\n-e\\\\l\\na r\\\\-\\nae\\n(13)\\nWe here integrate between the limits t and with\\nt t t\\nIn aphelion r a (1 e) putting this value of r\\ninto equation (13), and denoting by \\\\t the time from\\nperihelion to aphelion, we have\\n1 t\\n2 T\\nVl\\n7T.\\n(14)\\nW2\\nAccording to Kepler s second law, the time from\\naphelion to perihelion must equal the time from peri-", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0121.jp2"}, "122": {"fulltext": "110 CALCULUS.\\nhelion to aphelion therefore r is the time of a com-\\nplete revolution.\\nThe time of a complete revolution is termed the\\nperiodic time.\\nFrom equation (14)\\nt 2 4tt 2 i aS (15)\\n2 (l m) v J\\nand for a second planet,\\nt 2 4tt 2 t f S t (16)\\nA 2 (1 m f V y\\nComparing equations (15) and (16), we see that\\n(1 111) T 2 a 3 ,.rj,\\n(l m )r 2 a 8\\nIf the masses of the two planets are very nearly the\\nsame, we may take 1 m 1 ?n and hence, in this\\ncase, it follows that the squares of the periodic times of\\ntivo planets are to each other as the cubes of the semi-\\nmajor axes. (Kepler s third law.)", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0122.jp2"}, "123": {"fulltext": "CHAPTER IV.\\nANALYTIC GEOMETRY.\\n94. In this chapter it is proposed to present the\\nelementary principles of analytic (coordinate) geometry,\\nwith especial reference to conic sections.\\nThe Cartesian system* of coordinates has already\\nbeen explained in Art. 36. Here, as there, we shall\\nspeak of the curve F(x, y) 0, the curve y =f(x the\\nline ax by c 0, etc., instead of saying the curve\\nwhich the equation F(x, y) represents, etc.\\n95. If the equations\\ny 0O), (6)\\nare treated as simultaneous, the x and y of equation (a)\\nmust mean the same as the x and y of equation (6);\\nconsequently, as coordinates they are restricted to the\\npoint or points common to the two curves (a) and (5).\\nIf equations (a) and (6) have been so combined as\\nto eliminate one of the coordinates, say y, the x of the\\nresulting equation is the abscissa of the point of inter-\\nsection of the two curves, and the curves intersect in\\nas many real points as there are real roots of this new\\nequation.\\nCalled the Cartesian system, after Rene Descartes (1596-\\n1650), the inventor of coordinate geometry.\\nIll", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0123.jp2"}, "124": {"fulltext": "112 CALCULUS.\\nFor example, if y be eliminated between the two\\nequations x 2 y 2 4 and x y 1 0, we have\\nx 2 x f 0. V7 are therefore the abscissas of\\nthe points where x 2 y 2 4 and x y 1\\nintersect.\\n96. If, however, equations (a) and (5) have been so\\ncombined that neither x nor y is eliminated, the x and\\nnow refer primarily to the points common to the curves\\nof (a) and but we may treat them as a new x and\\ny, the current coordinates of a point describing a\\nnew curve, which passes through the intersections of\\nthe curves (a) and (J).\\nFor example, suppose we have the equations y=2x 2\\nand 2y x 2. Adding them, y x, a straight line\\ndistinct from the given lines, but passing through their\\npoint of intersection.\\n97. If the coordinates of a given point satisfy a\\ngiven equation, the point evidently lies on the curve\\nwhich the equation represents. Conversely, if a point\\nis on a curve, its coordinates will satisfy the equation\\nto the curve.\\nIf an equation F(x, y) can be written\\nf(x, y) y)=o,\\nthe curve of the given equation is made up of the com-\\nbined curves of f(x, y} and $(x, y)= for any\\npoint whose coordinates cause m f(x, y) to vanish, thus\\nsatisfying the equation f(x,y )=0, will also cause\\ny} (x, y} to vanish. Hence, all points 011/(2;, y^}\\nare also points on f(x, y^)$(x, y) 0. Similarly, all\\npoints on $(2;, are points on f(x, y^)$(x, 0.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0124.jp2"}, "125": {"fulltext": "ANALYTIC GEOMETRY. 113\\nFurther, there are no other points on f(x, y^)c\\\\ (x, y) 0,\\nbecause f(x, y) cannot vanish except by the\\nvanishing of either f(x, y) or y).\\nFor example,\\nx 2 2 y 2 3 xy x y (x y) (x 2 y 1)\\nhence, the curve which x 2 2 y 2 3 xy x y\\nrepresents is made up of the straight lines x y\\nand x 2y 1 0.\\n(The term curve is here used as inclusive of\\nstraight lines.)\\n98. If we have the equations y =f(x) and y\\nthe equation y =f(x) f (x) represents a curve whose\\nordinate for any abscissa x is the product of the orcli-\\nnates corresponding to x in the two primary curves.\\nFor example, if y x and y log x, y x log x is\\na third curve whose ordinate at any point equals the\\nproduct of the corresponding ordinates.\\nIn drawing such a set of curves to the same axes of reference\\nit is well to use colored pencils or crayons. For instance, if the\\nstraight line y x is drawn in red, the logarithmic curve in yellow,\\nand the curve y x log x in blue, the resulting diagram appeals to\\nthe eye much more forcibly than if all were done in black or white.\\n99. If y =f(x) and y (x), the equation\\nrepresents a curve whose ordinate at any point is the\\nsum of the corresponding ordinates of the given curves.\\nFor example, the so-called equation of time is\\nmade up of two parts one due to the eccentricity of", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0125.jp2"}, "126": {"fulltext": "114 CALCULUS.\\nthe earth s orbit, the other to the obliquity of the\\necliptic. If E x and E 2 represent these two parts re-\\nspectively, E, the whole equation of time, equals E x E v\\nWith a scale of dates one year long for the x-axis, and\\na scale marked to minutes for the ?/-axis, we may con-\\nstruct the curve of E 1 and also the curve of E v A\\nthird .curve, whose ordinate for any date is the sum of\\nthe ordinates of the first two curves for that date, then\\nrepresents E. See Young s General Astronomy, Fig. 64,\\nedition of 1898.\\nThe principles of this section and the preceding one\\nf(x)\\ncan evidently be extended to such forms as y\\ny K*) et\u00e2\u0082\u00ac. w\\n100. If we move the #-axis parallel to itself through\\nthe distance y\\\\ every ordinate is changed by the amount\\ny Similarly, if the y-axis is moved parallel to itself\\nthrough the distance x every abscissa is changed by\\nthe amount x 1 So, if X and Y are the new current\\ncoordinates, x X x and y Y y\\\\ in which x r and\\ny are the coordinates of the new origin referred to the\\nold axes. Hence, if in any equation F(x, y) 0, we\\nwrite x x for x, and y y for y, so that the equation\\nbecomes F(x x y y r 0, the geometric result se-\\ncured is a change of origin to a new point (x y f with\\nnew axes parallel to the old ones.\\nThe new current coordinates may be written x, y,\\ninstead of X, I 7 since they do not occur in connection\\nwith the old coordinates and therefore cannot be con-\\nfused with them.\\nFor example, x 2 y 2 r 2 being the equation to a circle\\nwith its center at the origin, (x a) 2 (y b) 2 r 2 is", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0126.jp2"}, "127": {"fulltext": "ANALYTIC GEOMETRY,\\n115\\nthe same circle with its center at (a, 5). The coordi-\\nnates of the new origin referred to the old axes are\\na, h.\\n101. Suppose the axes to rotate around the origin\\nthrough the angle The new coordinates of any\\npoint P are\\nX= 0B f Y=PB f\\nFig. 11\\nNow\\nOB 1 cosa 0C=0B BC=x BC=z PB f sin\\nTherefore x OB cos a PB f sin\\nthat is, x X cos a Y sin a.\\nSimilarly, Xsin a -f- 1 cos a,", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0127.jp2"}, "128": {"fulltext": "116 CALCULUS.\\nHence, if in any equation F(x, y)=0 we write\\nx cos a y sin a for x and sin a y cos a for y, the\\ngeometric result is the rotation of the axes through\\nthe angle a, in which a may have any value and be\\npositive or negative.\\n102. In the formulas just derived, the old coordi-\\nnates x and y are explicit functions of the new coordi-\\nnates X and Y. If we multiply the first formula by\\ncos a and the second by sin a, and add the products, we\\nhave\\nX x cos a y sin a.\\nSimilarly, Y x sin a -f- y cos a.\\nThe new coordinates are now explicit functions of\\nthe old ones.\\n103. The formulas derived in the three preceding\\narticles are indispensable in astronomy. As an\\nexample of the use of the two in Art. 102, suppose a\\nplanet is referred to the line in which the plane of its\\norbit cuts the ecliptic as the a axis, with a line at right\\nangles to it in the ecliptic as the ?/-axis. The planet\\nmay be referred to a new #-axis having its positive end\\ndirected toward the vernal equinox, with a correspond-\\ning new ?/-axis, if we use the relations\\nX x cos Q y sin Q\\nY x sin Q y cos Q\\nThe axes are here moved backward, that is, in the\\nnegative direction, through the angle (the longitude", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0128.jp2"}, "129": {"fulltext": "ANALYTIC GEOMETRY. 117\\nof the ascending node), Q being the angle between the\\nvernal end of the equinoctial line and the line passing\\nthrough the sun and the point through which the\\nplanet moves in going from the south to the north side\\nof the ecliptic.\\n104. We have seen (Art. 39) that ax by c\\nrepresents a straight line because a constant.\\ndx b\\nIt follows that if in any two equations\\ny mx ?i,\\ny m x n f\\nm m, the lines are parallel, for they have the same\\nslope.\\nAlso, if m 1 the lines are at right angles to\\nm\\neach other for m and m are now the tangents of\\na and 90\u00c2\u00b0 a since tan (90\u00c2\u00b0 cot a.\\n105. If x is put equal to zero in the equation F(x, y) 0,\\nthe resulting value of y must be the ordinate of the\\npoint where the curve crosses the y-axis. Now if x\\nin ax by c 0, y but is the constant\\nb b\\nterm when the equation is written in the form\\na g\\ny x It follows that if any linear equation\\nb b\\nbe written in the form y mx n, the constant term is\\nthe distance from the origin to the point where the line\\ncrosses the y-axis.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0129.jp2"}, "130": {"fulltext": "118\\nCALCULUS.\\n106. Suppose that P 1 (V, P (x y ,f are any two\\npoints on a straight line, while P(x, y) is the moving\\npoint. From similar\\ntriangles (Fig. 12),\\nPQ r _P D.\\nF Q\\nPD\\nthat is,\\ny\\n_y -y\\nx x\\nx x\\nThis equation we\\ndescribe as the equa-\\ntion to the straight\\nline in terms of the coordinates of two points through\\nwhich it passes.\\nFor example, the points (\u00e2\u0080\u00942, 1), (3, 4) determine\\na line whose equation is\\ny\\n1\\n4-1\\nz-(-2) 3-(-2y\\nwhich becomes, after reducing,\\nx y 1 0.\\nIf P and P n are indefinitely near each other,\\ny n y f becomes dy, and x n x becomes dx. Hence\\ny y f ^f^\\nwhich is the equation to the straight line in terms of\\nits gradient (or slope) and one point through which it\\npasses.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0130.jp2"}, "131": {"fulltext": "ANALYTIC GEOMETRY. 119\\n107, Referring to Fig. 12, Ave see that if D is the\\ndistance between any two points,\\nj) v y x r y 2 y 2\\nAlso, if P n is midway between P and P\\n10 *0 and y Ky y 5\\nthat is, the coordinates of a point bisecting the distance\\nbetween two points are the averages of the abscissas\\nand of the ordinates, respectively, of the given points.\\n108. To find the perpendicular distance from a given\\npoint (V, 2/) to a given line\\ny mx n, (1)\\nwe write y y f m(x (2)\\na line passing through (V, y f parallel to (1). The\\nintercept of (1) on the y-axis is w, the intercept of (2)\\non the y-axis is y r mx 1 and the difference of these\\nintercepts is m# n. It is evident that if this\\ndifference of intercepts be multiplied by cos (tan -1\\nthat is, by we shall have the perpendicular\\nVl m 2\\ndistance between the two lines, and hence the distance\\nfrom (V y f to y w?^ w. The formula for the dis-\\ntance from any point (V, y f to any line y mx n is\\ntherefore\\ny W22/ w\\nVl m 2\\nOr, if the equation to the line be in the form\\nax by c 0,", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0131.jp2"}, "132": {"fulltext": "120\\nthe formula is\\nCALCULUS.\\nr a f c\\nwhich equals\\nax f by c\\nVa 2 b 2\\nIt appears, then, that we have simply to evaluate the\\nfunction ax by c for the coordinates of the given\\npoint and divide by the square root of the sum of the\\nsquares of the coefficients of x and y.\\nExercises.\\n1. Given the equation ax by c 0, show that if it\\nbe written in the form\\nc c\\nthe quantities standing beneath x and y are the intercepts\\non the axes of x and y respectively.\\n2. Write the formula for the distance from the origin to\\nthe line ax by c 0.\\n3. Find the equation to a straight line which passes\\nthrough a given point (_p, g) and makes equal angles with\\nthe axes.\\n4. Find the length of the perpendicular from the origin\\non the line a (x a) b (y b) 0. Also, find the portion\\nof this line intercepted by the axes.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0132.jp2"}, "133": {"fulltext": "ANALYTIC GEOMETRY. 121\\n5. Write the equation to a line passing through\\nthe origin and making an angle of 120\u00c2\u00b0 with the\\na axis.\\n6. The coordinates of the vertices of a triangle are\\n(1, 2), (\u00e2\u0080\u00943, i) 3 (4, a/2); write the equations to its\\nsides.\\n7. Find the equation to a line which passes through\\nthe intersection of the lines x a, x +y a 0, and\\nthrough the origin.\\n8. Show that the lines y 2 x 3, y 3 x 4,\\ny 4 5 all pass through one point.\\n9. Find the slope of the line y mx 3 in order\\nthat it may pass through the intersection of the lines\\ny x 1 and y 2 x 2.\\n10. Find the equation to the straight line which is\\nequidistant from the two lines y mx n ?i f\\n11. If a is the angle between the lines y mx+n\\nand y m x n show that tan a\\n1 wim\\n109. The ellipse. Suppose that the circle x 2 y 2 r 2\\nis in a plane JfeT which makes an angle 7 with another\\nplane If. Let the #-axis to which the circle is referred\\nbe parallel to plane If. If perpendiculars to jV are\\ndropped from the extremities of the ordinates of the\\ncircle, the feet of these perpendiculars will form in If\\na new curve which is the projection of the circle on\\nthe plane If; and any ordinate y f of the circle be-\\ncomes the ordinate y f cosy in the new curve. Hence, if\\nwe write the circle-equation in the form y \u00c2\u00b1Va 2 z 2", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0133.jp2"}, "134": {"fulltext": "122 CALCULUS.\\nthe new curve, called the ellipse, has for its\\nequation\\ny cos 7 Va 2 x 2\\na cos 7 is a constant, and is evidently that line of the\\nellipse which replaces that radius of the circle which is\\nat right angles to the #-axis. Put acosy b; then\\ncos 7 and the ellipse-equation becomes\\ny Va 2\\n~2.\\na\\n.2 2\\nthat is, 1.\\na 1 b z\\na is called the semi-major axis, and b the semi-minor\\naxis of the ellipse.\\nThe student should distinguish carefully between the\\naxis of reference (a?-axis) and the major axis. An axis\\nof reference is a mere convenience, and we might study\\nthe ellipse with such an axis occupying some other posi-\\ntion in relation to the curve, or even without any such\\naxis but the major axis is an essential line of the\\nellipse, occupying a special position within it. The\\nsame is true of the minor axis.\\n110. Now let the ellipse and the circle be drawn in\\nthe same plane, the ellipse inside the circle with the\\nmajor axis coinciding with that diameter of the circle\\nwhich lies in the #-axis.\\nThe ellipse has the appearance of a circle flattened in\\nthe direction YY f AA! 2 a is the major axis, and\\nBB 26 is the minor axis. A and A 1 the extremi-\\nties of the major axis, are called the vertices.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0134.jp2"}, "135": {"fulltext": "ANALYTIC GEOMETRY\\n123\\nWith jB, one extremity of the minor axis, as a center\\nand with a radius equal to a, strike an arc. This arc\\nwill cut AA! at points F and F equally distant from 0,\\nthe common center of the circle and the ellipse. Evi-\\ndently the flatter the ellipse is, the farther these points\\nwill be from hence, if we know the ratio of OF to\\nOA, we know how flat the ellipse is, compared with the\\ncircumscribed circle.\\nBF=a \u00c2\u00a30 b; therefore OF Va 2 6 2 and\\nOF\\nOA\\nVa 2 b 2\\nFig. 13.\\nThis important ratio is called the eccentricity of the\\nellipse, and is denoted by e. The somewhat similar\\nQ\\nratio, is called the ellipticity of the ellipse.\\nct\\nThe eccentricity is evidently a proper fraction. The\\npoints F and F are called foci and the double ordi-\\nnate through either focus is known as the latus rectum.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0135.jp2"}, "136": {"fulltext": "!4\\nSince\\nOJ 7\\nC4Z,CDX?7,S.\\n111.\\nV\u00c2\u00ab 2\\na(l\\n6 s\\nVa 2\\nF\\nFA a Va 2\\nSimilarly,\\n-6 2\\na(l-e).\\nFrom the way in which the points F and F f Avere\\nfound we see that the sum of the distances BF and BF\\nis 2 a. It may now be shown that the sum of the dis-\\ntances from any point P(x\\\\ on the ellipse to the foci\\nis 2 a.\\nThe coordinates of F, the right-hand focus, are ae,\\nhence, by Art. 107,\\n(FF) 2 O aey ij 2\\nx 12 2 aex 1 a 2 e 2 y 2\\nSince P is on the ellipse, its coordinates must satisfy\\nthe equation to the ellipse, and we have\\nb 2\\ny 2 (a 2 x 12\\n(l-e 2 )(a 2 -x f2\\nSubstituting this value of y 12 in the expression for\\n(Fpy,\\n(FP) 2 x ,2 -2 aex f a 2 e 2 (1 e 2 )(a 2 x /2\\na 2 2 aex f e 2 x 2\\ntherefore FP a ex 1\\nNotice that the other root, (a\u00e2\u0080\u0094 ea/), is rejected\\nbecause a ex 1 and FP is positive. Repeating the", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0136.jp2"}, "137": {"fulltext": "ANALYTIC GEOMETRY. 125\\nargument for the distance F P, the coordinates of F\\nbeing ae, 0, F P a ex 1 hence FP F f P 2 a.\\n112. The proposition just established affords a way\\nof mechanically constructing an ellipse. Fasten one\\nend of a string at a point F on the blackboard or paper,\\nand the other at a point F taking the distance FF f\\nsomewhat less than the length of the string. Pass the\\nstring around a pencil and move the point of the pencil\\nover the paper, keeping the string taut. An ellipse\\nwill be described.\\nIt is clear now that we might define an ellipse as the\\npath of a point which moves so that the sum of its dis-\\ntances from two fixed points is a constant.\\n113. If b a in *f- 1, the equation returns to\\nthe circle-equation x 2 y 2 a 2 also, e 0.\\nIt thus appears that a circle is merely an ellipse with\\nequal axes and eccentricity equal to zero.\\n114. Let a line DP f (Fig. 14), be drawn parallel\\nto the minor axis. If the equation to the ellipse is\\n1, the minor axis lies in the ?/-axis, and\\na 2 o 2\\nhence the line DD f is parallel to the ?/-axis. If its dis-\\ntance from the axis is its equation is x the\\ne e\\nequation affirming that whatever may be the ordinate\\nof the point tracing the line, the abscissa is constantly\\nNow the distance from any point P(x y 1 on the\\ne", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0137.jp2"}, "138": {"fulltext": "126\\nCALCULUS.\\ncurve to this line is minus the distance of the point\\nfrom the y-axis that is, x r or\\na ex\\nWe have\\nalready seen (Art. Ill) that the distance from P to the\\nfocus is a ex 9 Hence the distance from any point on\\nthe ellipse to the focus and the distance from the point\\nto the line x are in the ratio\\ne\\na ex f\\na ex\\nthat is, e. Accordingly the ellipse may be defined as\\nthe path of a point which moves so that its distance from a\\ngiven jived point and its distance from a given fixed line\\nhave a constant ratio less than unity.\\nThe line x is called the directrix.\\ne\\n115. In Fig. 14 let FP r and angle EFP 6.\\nY\\nD\\nP\\nC\\nE X,\\nL F J\\nD\\nFig. 14.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0138.jp2"}, "139": {"fulltext": "ANALYTIC GEOMETRY. 127\\nWe have\\nFF=--ae;\\ne\\njPi rcos(18O\u00c2\u00b0-0)\\nr cos 6,\\nand also\\nFP\\nPC\\nthen FP\\n:ePC=e(FF+FL);\\nthat is,\\nfa\\nr el ae r cos\\n\\\\e\\na ae 2 er cos i\\nSolving for r,\\n1 e cos\\nThis is the equation to the ellipse in polar coordi-\\nnates with the pole at the right-hand focus. From the\\npoint of view of astronomy it is the most important of\\nall forms of ellipse-equations. See Art. 90.\\n116. It is of interest, logically, to note that the\\ntheory of the ellipse may be developed from the defini-\\ntion given in Art. 112, or the one in Art. Ill, or,\\nindeed, from any fundamental property. In the present\\ninstance we have chosen to begin by viewing the ellipse\\nas the projection of a circle on a plane making a given\\nangle with the plane of the circle.\\n-A\\nir\\n117. The hyperbola. About the ellipse 1\\ncircumscribe a rectangle with its sides parallel to the\\naxes of the ellipse. Draw the diagonals of the rect-", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0139.jp2"}, "140": {"fulltext": "128\\nCALCULUS.\\nangle. Half of one of the diagonals is Va 2 b 2\\nWith a radius of this length and with the center\\nof the ellipse for center, strike an arc cutting the major\\naxis produced in the points F and F f Now take\\n-vi\\nb 2\\nand draw a line DD f (Fig. 15) whose\\nFig. 15.\\nequation is x This line will cut the ellipse, because\\na\\na.\\ne\\nFollowing the analogy of the ellipse, we proceed to\\nfind the path of a point whose distance from the fixed\\npoint F is to its distance from the fixed line DD r in", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0140.jp2"}, "141": {"fulltext": "ANALYTIC GEOMETRY. 129\\na constant ratio, this ratio e being here defined as\\nVa 2 b 2\\nand hence greater than unity.\\na\\nLet P be the moving point whose coordinates are\\nx PN, y PL. PN cuts DD f at (7, and DD f cuts\\nthe #-axis at P.\\ne; that is, PF=ePC=e(x-^\\nPC e\\nAlso, (PFy (Pi) 2 (PL) 2\\nEquating the two expressions for (PP) 2\\ne 2 ix j y 2 (x ae) 2\\nExpanding and reducing,\\n_ 1) _ y 2 a 2Q e 2 _ ly 9\\na 2 a 2 (e 2 1)\\nthat is, 1.\\n118. The curve whose equation we have now found\\nis the hyperbola. Although closely related to the\\nellipse, it differs from that curve in various important\\nrespects\\n1. a and b being the same in magnitude and position\\nfor the two curves, no portion of the hyperbola lies\\nwithin the area occupied by the ellipse for as soon as\\nx a, y is imaginary.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0141.jp2"}, "142": {"fulltext": "130 CALCULUS.\\n2. The hyperbola has two parts or branches sym-\\nmetrically placed with respect to the axial line in which\\nBB lies for if we assign values x\\\\ x n x ,n etc.,\\nto we obtain the same values for y that are obtained\\nwhen x x n x etc., are the values assigned.\\n3. Values indefinitely large may be assigned to x\\nwithout making y imaginary. The curve, therefore,\\nextends to infinity.\\n119. The equation to the hyperbola may be written\\nb\\ny a\\nVx 2 a 2\\nor, expanding (x 2 a 2 1 by the binomial theorem,\\nbf a 2 a*\\nThe equations to the two diagonals (produced) of the\\nrectangle (Fig. 15) are seen to be\\ny \u00c2\u00b1-x. (5)\\nct\\nComparing equations (a) and (6), we observe that\\nany ordinate of the hyperbola is less than the corre-\\nsponding ordinate of the lines but we also notice that\\nas x becomes larger and larger, the ordinates, according\\nto equations (a) and (J), approach equality.\\nWhenever such a relation exists between a line and a\\ncurve, the distance between them becoming indefi-\\nnitely small as the points describing them recede to\\ninfinity, the straight line is called an asymptote.\\nThe hyperbola 1 has therefore the asymp-\\n7 ct o\\ntotes y x.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0142.jp2"}, "143": {"fulltext": "ANALYTIC GEOMETRY. 131\\n120. The line AA! (Fig. 15), 2 a in length, is called\\nthe transverse axis of the hyperbola and BB f 2 b in\\nlength, is the conjugate axis.\\nIf we consider the equation\\nb 2 a 2\\nwe find it situated with respect to the y-axis just as\\nthe first hyperbola is with respect to the #-axis. This\\ncurve, 2- 5=1, is known as the conjugate hyperbola\\nb 2 a 2\\nin distinction from the primary or transverse hyperbola.\\nFollowing the method of Art. 119, we find that the\\nlines y x are asymptotes of the conjugate\\nhyperbola also.\\n121. The points A and A! where the transverse axis\\nmeets the curve are its vertices. Similarly, B and B r\\nare the vertices of the secondary or conjugate\\nhyperbola.\\nThe transverse and conjugate axes, the asymptotes,\\nand the directrix are all essential lines of the hyperbola,\\nand sustain a fixed geometric relation to it like a rigid\\nframework, so that if the position of the hyperbola is\\nchanged with respect to the x- and ^-axes, these lines\\ngo with it.\\n122. If b a, the rectangle (Fig. 15) becomes a\\nsquare the asymptotes become y x, the two lines\\nnow crossing each other at right angles (Art. 104)\\nthe equation to the hyperbola itself becomes x 2 y 2 a 2\\nand is known as the equilateral hyperbola. It is evi-", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0143.jp2"}, "144": {"fulltext": "132 CALCULUS.\\ndently the hyperbola that would appear in plane M,\\nArt. 109, in connection with the circle x 2 y 2 a 2 and\\nthe circumscribed square.\\n123. If we rotate the axes in the negative direction\\nthrough the angle 45\u00c2\u00b0, we shall have these axes coin-\\nciding with the asymptotes of the hyperbola x 2 y 2 a 2\\nTo do this we use the formulas\\nx x cos a y sin tf,\\ny x sin a -f y cos a, (Art. 101)\\nwhich become, for a 45\u00c2\u00b0,\\nx x %V2 y 1V2 |V2( y),\\ny -xW2 yW2 W 7 2(y x).\\nSubstituting these values for x and y in the equation\\nx 2 y 2 a 2 we have\\nthat is, i .r y) 2 (y x) 2 a 2\\nwhich becomes, after reduction,\\na 2\\nThis is the equation to an equilateral hyperbola\\nreferred to its asymptotes.\\nThe isotherm pv c (Art. 36) is an important illus-\\ntration. (See Maxwell s Theory of Heat, Chap. VI.)\\n124. Following the method of Art. Ill, and using\\nFig. 15, it is found that\\n(FP) 2 e 2 z 2 2 aex a 2", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0144.jp2"}, "145": {"fulltext": "ANALYTIC GEOMETRY. 133\\nTaking FP positive, and noticing that ex a, we\\nhave\\nFP ex f a.\\nSimilarly, FP ex 1 a\\ntherefore FP -FP 2a,\\nand the hyperbola may be defined as the path of a point\\nmoving so that the difference of its distances from two\\nfixed points is a constant.\\n125. The polar equation to the hyperbola may be\\nreadily obtained from Fig. 15.\\nLet FP r, FP r and the angle LFP 6. We\\nhave also r f r 2 a and F f F= 2 ae.\\nThen r 12 r 2 4 ar 4 a 2\\nand since r f is one side of the triangle F PF,\\nr !2 r 2 4 ^2 _ 4 r CQS 180 o _ ffy^\\nEquating these two values of r /2\\nr 2 4 ar 4 a 2 r 2 4 a 2 6 2 4 a#r cos\\na(e 2 -l)\\nhence r 4-\\n1 e cos\\nThis equation may be described as the right-hand\\nfocal polar equation to the hyperbola.\\nWhen 0=0, r a(l e), and the feather-end of\\nthe arrow (Art. 45) gives the vertex of the left-hand\\nbranch of the curve. As the radius vector continues\\nto revolve in the positive direction, we continue to get\\nnegative values of r and points on the left-hand branch", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0145.jp2"}, "146": {"fulltext": "134\\nCALCULUS.\\nuntil 6 cos l r is then infinite, and the radius\\ne j\\nvector is parallel to the asymptote y=-x, because\\n-il -\\\\b\\ncos i tan L\\n6 a 1 1\\nFrom cos to 6 360\u00c2\u00b0 cos -1 r is positive,\\nand the right-hand branch is being traced. Finally,\\nwhen 6 changes from 300\u00c2\u00b0 cos -1 to 360\u00c2\u00b0, the re-\\nmaining part of the left-hand branch is traced.\\n126. The parabola. It remains to inquire what kind\\nof a curve we have when a point moves so that its dis-\\ntance from a fixed point is to its distance from a fixed\\nline in the constant ratio unity.\\nD\\nN\\nY\\nc p^-\\nB\\nX\\ni\\nD\\n\\\\F J\\nF]\\nr\\n[G. 16.\\nLet DD\\\\ Fig. 16, be the fixed line, and F the fixed\\npoint. Let 2p be the length of FB, the distance from\\nF to DD f Take a line through F perpendicular to", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0146.jp2"}, "147": {"fulltext": "ANALYTIC GEOMETRY. 13\\noo\\nDD f for the x-axis, and a line parallel to DD\\\\ bisecting\\nFB, for the ^-axis. Let P be the moving point with\\nthe coordinates OL and PL.\\nThen,\\nFP PN=x+p;\\nalso,\\n(FPy (PL) 2 (PX) 2\\ny 2 (x p) 2\\nHence,\\n(x ff y 2 (x p) 2\\nthat is,\\ny 2 4 px.\\nThis curve is called the parabola.\\nSince it is the path of a point whose distance from\\nthe fixed point is to its distance from the fixed line in\\nthe ratio unity, it must be regarded as the transition\\ncurve between the ellipse and the hyperbola. Its\\neccentricity 0, being the ratio of the two distances, is\\nof course unity.\\n127. Considering the equation y 2 4px, we see that\\nthe parabola has a line of symmetry which has been\\nused as the a axis for if any value be assigned to y\\nhas two values numerically equal and with opposite\\nsigns so that if the area above the #-axis were folded\\nover, the part of the curve in the upper area would\\nexactly fit the part in the lower. This line of sym-\\nmetry is called the axis of the curve, and the point\\nwhere it meets the curve is the vertex. The point F is\\nthe focus, and the double ordinate through the focus\\nis the latus rectum, as in the case of the, ellipse.\\nWhen x\u00e2\u0080\u0094p, y 2p; therefore the semi-latus\\nrectum is twice the distance of the focus from the\\nvertex.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0147.jp2"}, "148": {"fulltext": "136 CALCULUS.\\n128. If p is positive in the equation y 2 \\\\px, posi-\\ntive values may be assigned to x without making y\\nimaginary. Hence, the parabola like the hyperbola\\nextends to infinity. It differs from the hyperbola,\\nhowever, in this important respect it has only one\\nreal branch, because negative values of x make y\\nimaginary.\\n129. If p is negative in the equation y 2 \\\\px, we\\nhave the same law as before, governing the motion of\\nthe point P but the path is now wholly on the nega-\\ntive side of the ^/-axis, for only negative values can\\nnow be assigned to x.\\nSimilarly, x 2 2 py is a parabola above the z-axis,\\nwith the ?/-axis for its line of symmetry, if p is positive\\nwhile x 2 2py is the same curve below the #-axis if p\\nis negative.\\n130. The polar equation to the parabola, the focus\\nbeing pole, is obtained from Fig. 16. FP is r, and the\\nangle XFP is 6. Then\\nFL r cos\\nbut r PJV=FL 2p;\\ntherefore, r r cos 9 2 p\\nthat is, r 2\u00e2\u0080\u0094\\ncos 6\\n131. The ellipse, parabola, and hyperbola are known\\nas conic sections for it can be shown that if a cone of\\nrevolution is cut by a plane in any manner whatever,\\nthe cross-section is one of these three curves. See", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0148.jp2"}, "149": {"fulltext": "ANALYTIC GEOMETRY. 137\\nPuckle s Conic Sections, Arts. 323-325, together with\\nChap. VIII of that treatise.\\n132. A straight line becomes a tangent to a curve at\\nany point (x y if (1) it passes through that point,\\nand (2) if it has the same slope as the curve at that\\npoint. We have already seen (Art. 38) that if\\ny=zf(\u00c2\u00a3) is the equation to a curve, gives its slope\\ndx\\nor gradient at each point. It has also been shown\\n(Art. 107) that\\ny y O *0\\nis the equation to a straight line passing through the\\npoint (z f y with the slope It follows that if\\n(x y) is a point P on the curve y =f(x) and if -r~\\nis specialized for that point, becoming or -JL-,\\ny y ^7 O\\nis the equation to the tangent at P r\\nFor example, let us find the equation to the tangent\\nat the upper extremity of the latus rectum of the par-\\n(a 1J 2 77\\nabola y 2 4 px. Differentiating, we have -f-\\nax y\\nThe coordinates of the upper extremity of the latus\\nrectum are 2 p. Specializing for this point.\\ndx\\ndf] i", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0149.jp2"}, "150": {"fulltext": "138 CALCULUS.\\nand the general equation gives\\ny 2p x p;\\nthat is, y x p,\\nwhich is the equation to the tangent in question. We\\nnotice that this particular tangent makes an angle of\\n45\u00c2\u00b0 with the axis of the curve, which is here the #-axis,\\nand cuts the axis produced where the directrix DD f\\ncuts it. (See Fig. 10.)\\nExample. Find the general equation to the tangent\\nto the ellipse 1. Differentiating and solving\\nfor we have\\n(XX 7 70\\nay _ _ b l x\\ndx a 2 y\\nand the general equation to the tangent becomes, for\\nthe ellipse, 72\\ny y 2 x -v)-\\na A y\\nFor instance, the coordinates of the upper extremity of\\nty\\nthe left-hand latus rectum are ae, bVl e 2 ~ir\\ntherefore becomes\\nb 2, ae\\nthat is, e\\na 2 bVl- e*\\nand the tangent at the point named is\\ny b Vl e 2 e(x -f ae).\\nIt will be noticed that this particular tangent makes\\nwith the major axis an angle whose tangent is the\\neccentricity.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0150.jp2"}, "151": {"fulltext": "ANALYTIC GEOMETRY. 139\\nThe line perpendicular to the tangent at the point\\nof tangency is called the normal. Its equation is\\nevidently\\ny y -j-j( x x\\n133. The general equation to the tangent to the\\nparabola y 2 \\\\px, in terms of the coordinates of the\\npoint of tangency, is seen to be\\ny y -6- O\\nthat is, yy 1 2px y 2 2px\\nor, since y f2 i px f (x\\\\ y being on the parabola,\\nW 1 2jp(a? aO, (1)\\nand y== ^(aj (2)\\nWriting the equation to the line passing through the\\nfocus and P(x y), we have, after reducing,\\nIf -A is the point where the tangent cuts the axis pro-\\nduced, FPA is an isosceles triangle. To prove this,\\n2 p\\nlet tan a= 4-, the coefficient of in (2). Then\\ny\\ntan 2 a 4j?y 4 _J^_\\n1 4p 2 y f2 4: p 2 ipx \u00e2\u0080\u0094ip 2 x f p\\n1 ~y\u00c2\u00a5", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0151.jp2"}, "152": {"fulltext": "140 CALCULUS.\\nBut this is the coefficient of x in (3).\\nHence, angle PAF angle FPA.\\nIt immediately follows that if a line Pi is drawn on\\nthe concave side of the parabola, parallel to its axis, FP\\nand PL make equal angles with the tangent.*\\nAdvantage is taken of this property of the parabola\\nin the construction of reflectors. Since the angle of\\nreflection of a ray of heat or light equals the angle of\\nincidence, if a light be placed at the focus of a para-\\nbolic reflector, the light is reflected in a system of\\n(approximately) parallel rays. Illumination of a rail-\\nroad track for a long distance in front of the locomotive\\nis secured by means of such a reflector. Conversely,\\nif rays of heat or light, parallel to the axis of a para-\\nbolic reflector, fall upon its concave surface, they will\\nconverge at the focus.\\n134. It is required to find the locus (path) of the\\nmiddle point of any ellipse-chord moving parallel to\\nitself.\\nLet C(x n y rf and G\\\\x\\\\ y f be the points where\\n9 9\\nthe chord meets the ellipse 1, and let M(x, y)\\nbisect the chord CC f\\nHertz, in the first of his celebrated experiments on the propa-\\ngation of electric rays, made use of this property of parabolic surfaces.\\nHe employed large reflectors of sheet zinc bent into the form of para-\\nbolic cylinders, in whose focal line the transmitter and the receiver of\\nthe electric waves were placed. The electric rays passed from the\\ntransmitter to the first parabolic reflector, were there reflected so as to\\nbecome parallel, and were then reflected from the second reflector to\\nthe receiver placed at its focus. Young and Linebarger s Calculus.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0152.jp2"}, "153": {"fulltext": "ANALYTIC GEOMETRY. Ill\\nThen x J(V x fr and y ^(y f y ,f\\nthat is, x 2 x z rf and y f 2y y lf\\nSince (V, y and (V 2/ are each on the ellipse,\\n(1)\\nT V= L 2\\nSubstituting in (1) the values just noticed for x r\\nand y f we have\\n(2x-x C^y-y^y _\\na? b*\\n4x 2 -\u00c2\u00b1xx x 2 4y 2 -4yy n y n H\\nor I 1\\na 2 a? b* b*\\nIntroducing relation (2),\\nx(x x y(y y _\\na 2 62 u\\nand therefore\\n6 2\\na; x n a 2 y\\nNow let y y n m(x x ff\\nbe the equation to the chord OC f then\\nKi m,\\nand this is true, of course, when y), the point tracing\\nthe line (7(7 is restricted to the point M.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0153.jp2"}, "154": {"fulltext": "142 CALCULUS.\\nEquating the two values of l rr\\nx x n\\nb 2 x\\na 2 y\\nb 2\\nthat is, y x.\\narm\\nThe path of the middle point of any chord of slope\\nm, moving parallel to itself across the ellipse 1\\na 2 b 2\\nis therefore a straight line passing through the center\\nof the ellipse.\\nIf m 0, the equation becomes x 0, the equation\\nto the minor axis and if m oc, y 0, the equation to\\nthe major axis.\\nWriting b 2 for 2 we have\\nb 2\\nthe corresponding equation in relation to the hyperbola.\\n135. By means of the result in the preceding article,\\nwe are now able to find the center and construct the\\naxes of any ellipse.\\nDraw any two par?dlel chords and bisect them. The\\nchord passing through the points of bisection must\\npass through the center of ellipse and the point of\\nbisection of this third chord is the center. With the\\ncenter now found and a radius of any convenient length,\\nstrike a circle cutting the ellipse. Draw one chord\\ncommon to both ellipse and circle, and finally draw\\nan ellipse-chord perpendicular to the preceding chord", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0154.jp2"}, "155": {"fulltext": "ANALYTIC GEOMETRY. 143\\nat its middle point. It will be the major (or minor)\\naxis of the ellipse.\\nHaving given the hyperbola with its accompanying\\nconjugate hyperbola, the construction of the axes is\\nthe same as for the ellipse.\\n136. The tangent-equation, Art. 132, involves the\\ncoordinates of the point of tangency. It is desirable\\nto obtain a form in which these coordinates do not\\nappear. What conditions must be imposed on the line\\ny mx n so that it shall keep the slope m and yet\\nbe a tangent to the ellipse\\nEliminating y between the equations\\ny mx n, (1)\\nx 2 y 2\\na* P 1 2\\nthe resulting equation,\\nx 2 {mx w) 2 _ i\\na 2 b 2\\nhas for its roots the abscissas of the points of intersec-\\ntion of (1) and (2). These roots are\\nmn m 2 n 2 n 2 b 2\\nb 2 b* b 2\\n1 m 2 I 1 m 2 2 1 m 2\\na? \\\\\\\\a 2 \u00c2\u00a5J ~a 2 T 2\\nThus far the straight line is merely a secant (real\\nor imaginary) of the ellipse. If it is to become a", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0155.jp2"}, "156": {"fulltext": "144 CALCULUS.\\ntangent, the two points of its intersection with the\\nellipse must be indefinitely near to each other; that\\nis, the two abscissas must be equal. Hence, the radi-\\ncal, which now makes them unequal, must vanish, and\\nwe have\\nm 2 n 2 n 2 b 2\\nf m* Y 1 m 2\\nCo 2 W ~a?\\nFrom this equation of condition we obtain\\nn Va 2 m 2 i 2\\nwhich is therefore the relation which must hold between\\nn and a, m and b in order that (1) shall be tangent to\\n(2), and we have\\ny mx ^fa 2 m 2 b 2 (3)\\nThe double sign in (3) plainly means two tangents\\nparallel to each other, one cutting the ^/-axis at the\\ndistance VoW P above the origin, and the other at\\nthe same distance below it.\\nThe corresponding equation for the tangent to the\\nhyperbola may be obtained at once by writing b 2 for\\nb 2 in (3), and we have\\ny mx Va 2 m 2 b 2 (4)\\nIf b a, so that the ellipse becomes a circle, (3)\\nbecomes\\ny mx aVm 2 1. (5)\\nSimilarly, if the hyperbola is equilateral, (4) becomes\\ny mx a^Jm 2 1. (6)", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0156.jp2"}, "157": {"fulltext": "ANALYTIC GEOMETRY 145\\nExercises.\\n137. 1. Construct the ellipse\\nL\\n4 9\\nWhat is its eccentricity How must the formula for e be\\nwritten in this case\\n2. Find the points of intersection of the ellipse and\\nhyperbola whose equations are\\naj 2^ l, 3x 2 -6f l,\\nand show that at each of these points the tangent to the\\nellipse is the normal to the hyperbola. (Puckle s Conic\\nSections.)\\n3. Find the equations to the asymptotes of the hyperbola\\n3 x 2 6 y 2 1.\\n4. Find the distance between the right-hand focus of\\nx 2 2 y 2 1 and the right-hand focus of 3xr 6y 2 l.\\n5. Write the equation to a circle which shall have its\\ncenter coincident with the focus of the parabola y 2 Ap x,\\nand shall be tangent to the parabola.\\n6. Find an expression for the perpendicular distance\\n9 9\\nXT 1/~\\nfrom the right-hand focus of the ellipse 4-^=1 to\\nthe tangent y mx Va 2 m 2 b 2\\n7. Find an expression for the perpendicular distance\\nfrom the focus of the parabola y 2 \u00c2\u00a3px to any normal.\\n8. Show that the line y mx n becomes a tangent to\\nthe parabola y 2 4^px if n\\n9. Show that the path of the middle point of any parab-\\nola-chord moving parallel to itself is a line parallel to the\\naxis of the parabola.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0157.jp2"}, "158": {"fulltext": "146 CALCULUS.\\n10. Given a parabola, find its axis and focus.\\n11. The locus of the foot of the perpendicular from the\\ncenter of the equilateral hyperbola x 2 y 2 a 2 is the lem-\\nniscate (x 2 y 2 2 a 2 (as 2 y 2\\nUse y mx a Vm 2 1.\\nThe line perpendicular to it passing through the center\\nis y x. It is required to find the path of the inter-\\nsection of these two lines as m passes through all values.\\nEliminate m.\\n12. Show that the locus of the foot of the perpendicular\\ndropped from the focus of the parabola on its tangent is\\nthe tangent at the vertex.\\n13. If a source of light or heat is placed in one focus of\\nan ellipse, the rays will be reflected so as to meet in the\\nother focus.\\n14. A. planet at P is moving in the direction PQ. Its\\ndistance PS from the sun at S (one focus of its elliptic\\norbit) is J its major axis. Construct the orbit.\\n15. Given one focus and any point P and the length of\\nthe major axis of an ellipse; show that the eccentricity\\ndepends on the direction of the tangent at P. Construct\\nthe major and minor axes of ellipses corresponding to\\nvarious tangents through P.\\n16. The tangent at any point of a hyperbola is produced\\nto meet the asymptotes show that the triangle cut off is of\\nconstant area\\n17. Find the equation to the path of the center of a circle\\nwhich is tangent to two given circles.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0158.jp2"}, "159": {"fulltext": "ANALYTIC GEOMETRY.\\n147\\n138. Just as a point in a plane may be determined by\\nreferring it to two lines at right angles to each other,\\nso a point in space may be determined by referring it\\nto three planes, each plane intersecting the other two\\nat right angles. The point common to the three planes\\nz\\nM\\n7/^\\nP\\nX\\n2\\nf\\nFig. 17.\\nis called the origin, and the lines of intersection of the\\nplanes are known as the axes of x, y, and z. The posi-\\ntive directions of the axes are usually taken to be repre-\\nsented in the figure by OX, OY, and 0Z\\\\ the negative\\ndirections are then OX OY 0Z r\\nIf P (Fig. 17) is any point in space, then PL, PM,\\nPN, its perpendicular distances from the planes YOZ,\\nZOX, and XOY respectively are the coordinates x, y,\\nand z respectively.\\n139. The three planes evidently divide the space\\naround the origin into eight equal triedral angles. If\\na point is in the upper front right-hand angle, its", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0159.jp2"}, "160": {"fulltext": "148 CALCULUS.\\ncoordinates are all positive, because each one is meas-\\nured parallel to its own axis and in the positive direc-\\ntion. Again, if a point is in the upper front left-hand\\nangle, the y and z coordinates are positive, but the\\nx coordinate is negative because measured in the nega-\\ntive direction parallel to OX In like manner we are\\nable to state the character of each coordinate for points\\nsituated in each one of the other six angles.\\n140. OP, the distance of P from the origin, is the\\ndiagonal of the rectangular parallelopiped, three of\\nwhose edges are PZ, PM, PN. Therefore,\\n(OPf (Pi) 2 (PJ/) 2 (PX) 2\\na? f z i.\\n141. Suppose we have any two points P (x y z\\nand P (x y z Let planes be passed through P\\nand P parallel to the three planes of reference.\\nThere is thus formed a rectangular parallelopiped\\nwhose diagonal is the line P f P and three of whose\\nedges are x x\\\\ y \u00e2\u0080\u0094y f z z Therefore,\\n(jp p y o x r 2 o y y (z z y.\\nThis formula for the distance between two points in\\nspace should be compared with the formula in Art. 107.\\n142. We have seen (Art. 39) that the general equa-\\ntion of the first degree in two variables represents a\\nstraight line. It may now be asked, What is repre-\\nsented by\\nAx By Cz D 0, (1)\\nthe general equation of the first degree in three\\nvariables", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0160.jp2"}, "161": {"fulltext": "AXALYTIC GEOMETRY. 149\\n1. It represents a surface and not a solid. For let\\n(a, 6) be a point in the plane YOX, and suppose that a\\nstraight line be drawn through this point parallel to\\nthe z-axis to meet the locus of equation (1), whatever\\nkind of locus it may be. We now have\\n-Aa-Bh-D\\nz a\\ntherefore the straight line meets the locus in one\\ndefinite point at the distance from the\\nplane YOX, and consequently the locus cannot be\\nmade up of layers either adjacent to one another or\\noccurring at intervals.\\n2. The surface is a plane. For suppose that a\\npoint moving in the surface be so restricted that it\\nmust remain at a constant distance from the plane\\nYOX that is, let z have a constant value, say c.\\nEquation (1) is now reduced to Ax By Cc D 0.\\nTherefore the point moving in the surface and at a\\nconstant distance from the plane YOX is moving in a\\nstraight line. In other words, any section of the sur-\\nface made by a plane parallel to the plane YOX is a\\nstraight line. Hence, if Ax By 4- Oz D is not\\na plane surface, it must be a wavy surface, something\\nlike a corrugated tin roof with the corrugation lines\\nparallel to the plane YOX. But repeating the argu-\\nment, making y a constant, we find that all sections\\nmade by planes parallel to the plane ZOX are straight\\nlines. The surface in question must therefore be a\\nplane.\\nIn case the constant term D is zero, the coordinates", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0161.jp2"}, "162": {"fulltext": "150 CALCULUS.\\nof the origin (0, 0, 0) satisfy the equation, and the\\nplane passes through the origin.\\nThus the equation, Cz -C i/ C x (p. 103),\\nrepresents a plane passing through the origin and\\nany moving point whose coordinates satisfy this equa-\\ntion at each instant, must be moving in the plane and\\nhence in a plane curve.\\nArts. 138-1-12 have been introduced for the sake of Arts. 86-88.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0162.jp2"}, "163": {"fulltext": "dx dx doc\\nCHAPTER V.\\nFORMULAS.\\nli- ^dx ^dx) u v.\\nJ \\\\dx doc J\\n2. J^uv v^+u dv a M*\u00e2\u0080\u0094*\u00c2\u00bb\\ndoc doc doc i J doc doc J\\n3. jL u n nu n 1 3,. (u tn \u00e2\u0080\u0094dx= u +1\\ndx dx J dx m 1\\n4. -4^- sin m cos w 4,. I cos dx sin if.\\ndx dx J dx\\n5. cos w sin 5,. sin w dx cos u\\ndx dx J dx\\n6. tan sec 2 u 6,. f sec 2 if tan u\\ndx dx J dx\\n7 cot u cosec 2 m 7, f cosec 2 u dac=-cotu.\\ndx dx J dx\\n8. -!^-secw=tanwsecw\u00e2\u0080\u0094 8,. {t2LMi eeu^dx=secu.\\ndx dx J dx\\n9. -7- cosec u cot u cosec if\\nax dx\\nd?f\\n9, 1 cot u cosec tf dx cosec ti.\\ndtf du\\nd dx _, f dx 1\\n10. -Sin-l^rr: 10^ J dX Sin 1 if.\\nax Vl _ ^2 vi _\\ndtf dw\\n11. -^-cos 1 Hi. f -^=dx cos 1 w.\\ndx Vi_^2 J Vf^ 2\\n151", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0163.jp2"}, "164": {"fulltext": "152 CALCULUS.\\ndu du\\n12. tan -1 u 12 L J L c dx tan -1 u.\\ndx 1 w a J w 2\\n7?^ dw\\nd d ^y* i d nr\\n13. -cot _1 t^=-- 3\u00c2\u00ab 13x. J da? cot 1 if.\\nda? 1 u 2 1 if 2\\nd?f du\\nia d 1 c e f das t\\n14 -j\u00e2\u0080\u0094 sec _1 i^ 14 x J doc sec n.\\nd 1 f/.r\\n15. cosec~ m\\nrca? Mvie- 1\\nd?\u00c2\u00a3\\n15\\nJ 1 dx cosec -1 u.\\nd ^a? _ ^ac\\n16. ^e x e x 16\\nda?\\n17. ^-a r a x log e 17,. f x log e ada? a x\\ndx J\\n18. -*Le u e u 18,. fe*^da; e M\\nda? da? J da?\\n19 L a ioge a 19l f f\u00c2\u00ab rfx ^_.\\ndx dx J dx log e a\\n20. l0ge 05 X 20:. f loge X.\\ndx x J x\\ndu du\\noi d dx oi f da?\\n21. \u00e2\u0080\u0094\u00e2\u0080\u0094\\\\0geU 1 1 \u00e2\u0080\u0094-dx \\\\0geU.\\ndx u J u\\ndu du\\n22. JLvi 22 1 .J_^^ VS.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0164.jp2"}, "165": {"fulltext": "FORMULAS. 153\\n23 f Vtf^tf dx sin 1 Vr?^?.\\nJ 2 a 2\\n24. fVx 2 \u00c2\u00b1cfdx [xVx 2 a? a 2 log (x V# 2 a 2\\n25. f __^_ log (aj Va; 2 a 2\\nV# 2 a 2\\n26. f X dx Va 2 4- a 2\\nJ Va 2 a 2\\n27. f -^V?3g ^L sin- 1\\nJ Va 2 z 2 2 ^2 a\\n28. f dx ^Vx T Td 2 T log (a? Va^dT^ 2\\nV# 2 a 2\\n29. fxWa^ti 2 dx ^(2x 2 a 2 V^^ sin 1\\n7= lo g\\n#r-\\\\/r/ 2 -4- o^ 2 C6\\nf-\\nlog g\\n\u00c2\u00abVa 2 2 a a Va 2 a; 2\\ncfa _ Va 2 a? 2\\n32, Va V*W Va2 2 -sm-^.\\nJ x 2 x a\\n/X X i\\ndx a vers -1 V2 ax x 2\\n\u00e2\u0096\u00a0\\\\/9, r/cr. cr? a\\n34.\\nV2 ax x 2\\ndx V2 ax x 2\\nxa/2 ax x 2 ax\\nV 2 ax x 2 dx vers -1 V 2 ax x 2\\n2 a 2\\n36. I dec a vers -1 V2 ax x 2\\nJ x a", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0165.jp2"}, "166": {"fulltext": "154 CALCULUS.\\n37 I tan x dx log cos x. 38. j cot x dx log sin x.\\n39. f-^ log tan-- 40. f-^-=logtanf-\\nJ sin. i- 2 J cos a; \\\\2 4\\n41. I cot x. 42. I tan x.\\nJ suras J cos L\\nC 7 i o sin 2 a?\\n43. I sin a; cos a? ao? J cos 2 a?\\n44. I log tan x.\\nJ sin x cos x\\n45. I x sin .r cfo sin x x cos x.\\n46. I ar sin x dx as 2 cos a; 2 x sin a; 2 cos x.\\n47 J sin 2 x dx sin 2 a: x.\\n48. J cos 2 x dx sin 2x x.\\n49 I log x dx a; log a; a:.\\n50. I sin -1 x dx x sin -1 x Vl a?.\\n5 1 f tan 1 a; c?aj a; tan 1 x log (1 x 2\\nJ sec -1 x dx x sec -1 a? log (x Va? 1)\\ni fa 6 f f tan-^J^tan|\\nJ Ja+6cosx Va 2 -6 2 V Xa 2\\n52\\nFor other integrals see Peirce s Short Table of Integrals.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0166.jp2"}, "167": {"fulltext": "FORMULAS. 155\\n54. f(z x)=f{z)+f(z)x+f^\\n55. f(x) =/(Q) (O) X \u00c2\u00a3^j^-\\n56. (a rt) a ma- x !il!izll^\\nm (m 1) (m 2) a 1 3 a^\\n57. e ^l\\n58. a- l g log e a x2(log a)2 g:3(1 a)3\\n[2 [3\\n/y\u00c2\u00bb2 /ytO /y\u00c2\u00bb4\\n59. log(l a; x-|\\n60. sm^ h, 61 cos# l K\\n[3 16 |2 li\\n62. sin cos a. 63. cos a sin a.\\n64. tan cot a. 65. cot (5 a] tan a.\\n66. sin a) sin a cos a) cos a.\\n67. tan a) tan cc cot cot a.\\no o -i sin a sin a\\n68. sm- cos 2 a 1. 69. tan a\\ncos a Vl sin 2 a\\n70. sin (a sin a cos /2 cos a sin /3.\\n71. cos (a cos a cos /J sin a sin\\n72. sin (a sin a cos cos sin\\n73. cos (a j8) cos a cos sin a sin (3.\\n74. sin 2 a 2 sin a cos a.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0167.jp2"}, "168": {"fulltext": "156 CALCULUS.\\n75. cos 2\u00c2\u00ab cos 2 a sin 2 a 2 cos 2 a 1 1 2 sin 2 a.\\nm tan ft tan\\n76. tan (a\\n1 tan tan\\nn tan tan B\\n77. tan (ft S)\\n1 tan ft tan /3\\n78. tan2\u00c2\u00ab= 2tan 79. cot 2 a COt2 1\\n1 tan- ft 2 cot\\nft /l COS ft ft /l\\n80. Sm =yj- 81. COS 2\\nft /l COS ft on ft /1.+ COS ft\\n82. tan-=\\\\- 83. cot-=\\\\/\u00e2\u0080\u0094\\n2 1 cos 2 1 cos ft\\n84. sin sin /3 2 sin i (ft -f- /5) cos i (ft\\n85. sin sin f3 2 cos -J (ft sin J (ft /3).\\n86. cos cos 2 cos (ft /J) cos (ft /J).\\n87. cos ft cos (3 2 sin J (ft /J) sin (a\\n88. log afr log a log 6. 89. log log a log b.\\nb\\n1 i\\n90. log a n n log a. 91. loga n -loga.\\nn\\n92. If asc 2 bx c 0,\\nF 2 c 1 m r\\n2 a \\\\4 a 2 a 2a 2a\\n93. l-2?3 -4 n. 94. log 1=0.\\n95. Iog0=\u00e2\u0080\u0094 oo. 96. log a a=l.\\n97. e 2.7182818284 98. log 10 e 0.43429448\\n99. 7T 3.14159265-... 100. log 10 7r 0.49714987", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0168.jp2"}, "169": {"fulltext": "FORMULAS. 157\\n101. 57\u00c2\u00b0.2957795\\n102. R 180 60 60 206264 .8\\n103. log R\u00c2\u00b0 1.75812263\\n104. log R 5.31442513\\n105. 30\u00c2\u00b0 45\u00c2\u00b0 60\u00c2\u00b0\\nsin, iV2 |V3\\ncos, iV3 |V2 i\\n106. Base of right triangle cos y alt. h sin y.\\nQi hypotlienuse y angle at base.)\\n107. Area of sector of circle r(rO) r*9.\\n108. Area of ellipse =wab.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0169.jp2"}, "170": {"fulltext": "", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0170.jp2"}, "171": {"fulltext": "INDEX.\\nThe references are to pages.\\nAbscissa, 42.\\nAcceleration, 62.\\nAmplitude, 67.\\nAnalytic geometry, 111.\\nAnomaly, true, 107.\\nAphelion, 106.\\nApsides, 57.\\nArchimedes, 2.\\nAreal velocity, 105.\\nAreas, 85.\\nof surfaces of revolution, 94.\\nAsymptote, 130.\\nAsymptotes of hyperbola, 130.\\nAttraction of homogeneous sphere,\\n70.\\nAxes of coordinates, 41.\\nAxes, change of, 115.\\nmajor and minor, 122.\\ntransverse and conjugate, 131.\\nAxis of parabola, 135.\\nof symmetry, 135.\\nBinomial theorem, 30.\\nBodies, falling, 67.\\nBoyle s law, 2.\\nCartesian coordinates, 41, 111.\\nCatenary, equation to, 93.\\nCircle, equation to, 46.\\nCircular functions, 20.\\nComets, orbits of, 108.\\nConcavity of curves, 48.\\nConic sections, 136.\\nexercises on, 145.\\nConjugate hyperbola, 131.\\nConstant, derivative of, 8.\\nof integration, 13.\\nConvergence of series, 30.\\nCoordinates, 42.\\ncurrent, 43.\\npolar, 56.\\ntransformation of, 56,\\n115.\\n114,\\nDefinite integrals, 14.\\nDerivative, 5.\\nof constant, 8.\\nof product, 9.\\nof quotient, 15.\\nof sum, 8.\\nof cos x and sin x, 18.\\nof x n 10.\\nDerivatives, second and higher,\\n24.\\npartial, 24.\\nDescartes, 111.\\nDifferential, 5.\\nperfect, 98.\\ntotal, 25.\\n159", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0171.jp2"}, "172": {"fulltext": "160\\nINDEX.\\nDifferentiation, 8.\\nDirectrix of ellipse, 126.\\nof hyperbola, 129.\\nof parabola, 138.\\nDisplacement, 67.\\nDistance between two points, 119.\\nfrom point to line, 119.\\nDouble integrals, 97.\\nEccentricity of ellipse, 123.\\nof hyperbola, 129.\\nof parabola, 135.\\nEllipse, 121.\\narea of, 87.\\naxes of, 122.\\nconstruction of, 125.\\ndetermination of center and\\naxes of, 142.\\ndirectrix of, 126.\\neccentricity of, 123.\\nellipticity of, 123.\\nequation to, 122.\\npath of middle point of chord\\nof, 140.\\npolar equation to, 127.\\nfoci of, 123.\\nvertices of, 122.\\nEnergy, kinetic, 92.\\nEpoch. 67.\\nEquation to a curve, 111.\\nax by c 0, 45.\\nAx By Cz D 0, 148.\\nEquations of motion, 78.\\nEquilateral hyperbola, 131.\\nFalling bodies, 67.\\nFoci of ellipse, 123.\\nof hyperbola, 128.\\nFocus of parabola, 135.\\nFormulas, collection of, 155.\\nFunction, algebraic, 4.\\nexplicit and implicit, 16.\\nperiodic, 47.\\nFunction, transcendental, 4.\\nof several variables, 24.\\nGradient, 44.\\nGraph, 41.\\nGravitation, law of, 2.\\nGravity, 67.\\nHarmonic motion, 67.\\nHeat, minimum intensity of, 54.\\nHertz, 140.\\nHorizontal range, 79.\\nHyperbola, 127.\\nasymptotes of, 130.\\naxes of, 131.\\ndirectrix of, 129.\\neccentricity of, 129.\\nequation to, 129.\\npolar equation to, 133.\\nfoci of, 128.\\nHyperbola, conjugate, 131.\\nequilateral, 131.\\ntransverse, 131.\\nIndeterminate forms, 31-33.\\nIndicator diagram, 97.\\nInertia, moment of, 100.\\nInflexion, point of, 49.\\nIntegrals, double and triple, 97.\\ndefinite and indefinite, 13.\\ntable of, 151.\\nIntegration, 12.\\nconstant of, 13.\\nby parts, 37.\\nIntercept, 119.\\nIsotherm, 132.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0172.jp2"}, "173": {"fulltext": "INDEX.\\n161\\nKepler s laws, 102.\\nKinetic energy, 92.\\nof rotation, 99.\\nLatus rectum, 123.\\nLaw, Kepler s first, 107.\\nKepler s second, 105.\\nKepler s third, 110.\\nLengths of curves, 92.\\nLimits of definite integrals, 14.\\nLocus, 41.\\nLogarithmic differentiation, 21.\\nLogarithms, common and Napier-\\nian, 39.\\nMaclaurin s theorem, 29.\\nMaxima and minima, 48.\\nexercises in, 52-55.\\nMean values, 88.\\nMoment of inertia, 100.\\nMotion, equations of, 78.\\nof falling body, 68.\\nof rising body, 69.\\nin a parabola, 77.\\npendulum, 82.\\nrectilinear, 74.\\nsimple harmonic, 67.\\nin a vertical curve, 81.\\nNewton, 2.\\nNode, longitude of ascending, 116.\\nNormal, 139.\\nOgee, 49.\\nOperation, indicated, 12.\\nOrbits, eccentricity of, 108.\\nOrdinate, 42.\\nOrigin of coordinates, 41.\\nchange of, 114.\\nParabola, 134.\\naxis of, 135.\\ndirectrix of, 138.\\neccentricity of, 135.\\nfocus of, 135.\\nlatus rectum of, 135.\\nequation to, 135.\\npolar equation to, 136.\\nvertex of, 135.\\nParabolic reflector, 140.\\nParallelism, condition of, 117.\\nPartial differential coefficients, 24.\\nPendulum, 82.\\ntime of oscillation of, 84.\\nPerfect differential, 98.\\nPerihelion, 106.\\nPeriod, 67.\\nPeriodic function, 47.\\ntime, 110.\\nPerpendicularity, condition of,\\n117.\\nPhase, 67.\\nPoint of inflexion, 49.\\nPolar coordinates, 56.\\nPole, m.\\nProjectile, path of, 78.\\nRadian, 47, 85.\\nRadius vector, 57.\\nRange, horizontal, 79.\\non an incline, 80.\\nRevolution, areas of surfaces of,\\n94.\\nvolumes of, 94.\\nSinusoid, 47.\\nSlope, 44.\\nStraight line, 117.\\nexercises in, 120.\\nSymmetry, line of, 135.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0173.jp2"}, "174": {"fulltext": "162\\nINDEX.\\nTable of formulas, 155.\\nof integrals, 151.\\nTangent in terms of slope and\\nintercept, 143.\\nto curve, 137.\\nTaylor s theorem, 28.\\nTransformation of coordinates,\\n56, 114, 115.\\nTransverse hyperbola, 131.\\nTriple integrals, 97.\\nTrue anomaly, 107.\\nVariable, 3.\\nVelocity, areal, 105.\\nangular, 63.\\ncomponent, 60.\\nlinear, 58, 64.\\nVertices of ellipse, 122.\\nVolumes of revolution, 94.\\nWork, 91.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0174.jp2"}, "175": {"fulltext": "SCIENCE. 51\\nPhysics for University Students.\\nBy Professor HENRY S. CARHART, University of Michigan.\\nParti. Mechanics, Sound, and Light. With 154 Illustrations. i2mo,\\ncloth, 330 pages. Price, $1.50.\\nPart II. Heat, Electricity, and Magnetism. With 224 Illustrations.\\ni2mo, cloth, 446 pages. Price, $1.50.\\nTHESE volumes, the outgrowth of long experience in teach-\\ning, offer a full course in University Physics. In preparing\\nthe work, the author has kept constantly in view the actual needs\\nof the class-room. The result is a fresh, practical text-book, and\\nnot a cyclopaedia of physics.\\nParticular attention has been given to the arrangement of\\ntopics, so as to secure a natural and logical sequence. In many\\ndemonstrations the method of the Calculus is used without its\\nformal symbols and, in general, mathematics is called into ser-\\nvice, not for its own sake, but wholly for the purpose of establish-\\ning the relations of physical quantities. At the same time the\\ncourse in Physics represented by this book is supposed to pre-\\ncede the study of calculus, and its methods will in a general way\\nprepare the student for the study of higher mathematics.\\nProfessor W. LeConte Stevens, Rensselaer Polytech?tic Institute, Troy, N. Y.\\nAfter an examination of Carhart s University Physics, I have unhesitat-\\ningly decided to use it with my next class. The book is admirably\\narranged, clearly expressed, and bears the unmistakable mark of the\\nwork of a successful teacher.\\nProfessor Florian Cajori, Colorado College The strong features of his Uni-\\nversity Physics appear to me to be conciseness and accuracy of statement,\\nthe emphasis laid on the more important topics by the exclusion of minor\\ndetails, the embodiment of recent researches whenever possible.\\nProfessor A. A. Atkinson, Ohio University, Athens, O. I am very much\\npleased with the book. The important principles of physics and the\\nessentials of energy are so well set forth for the student for which the book\\nis designed, that it at once commends itself to the teacher.\\nProfessor A. E. Frost, Western University, Allegheny, Pa. I think that it\\ncomes nearer meeting my special needs than any book I have examined,\\nbeing far enough above the High School book to justify its name, and\\nyet not so far above it as to be a discouragement to the average student.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0175.jp2"}, "176": {"fulltext": "52 SCIENCE.\\nPrimary Batteries.\\nBy Professor HENRY S. CARHART, University of Michigan. Sixty-\\nseven Illustrations. i2mo, cloth, 202 pages. Price, $1.50.\\nTHIS is the only book on this subject in English, except a\\ntranslation. It is a thoroughly scientific and systematic\\naccount of the construction, operation, and theory of all the best\\nbatteries. An entire chapter is devoted to a description of stand-\\nards of electromotive force for electrical measurements. An ac-\\ncount of battery tests, with results expressed graphically, occujDies\\nforty pages of this book. This chapter forms an excellent outline\\nguide for laboratory purposes. The tests have all been performed\\neither by the author himself or under his immediate supervision.\\nThey are free from bias and exhibit some facts not heretofore ac-\\ncessible to the public.\\nThe battery as a device for the transformation of energy is;\\nkept constantly in view from first to last and the final chapter-\\non Thermal Relations concludes with the method of calculating\\nelectromotive force from thermal data.\\nProfessor John Trowbridge, Harvard University: I have found it of the\\ngreatest use, and it seems to me to supply a much needed want in the\\nliterature of the subject.\\nProfessor Eli W. Blake, Drown University The book is very opportune,\\nas putting on record, in clear and concise form, what is well worth know-\\ning, but not always easily gotten.\\nProfessor George F. Barker, University of Pennsylvania I have read it\\nwith a great deal of interest, and congratulate you upon the admirable\\nway in which you have put the facts concerning this subject. The latter\\nportion of the book will be especially valuable for students, and I shall\\nbe glad to avail myself of it for that purpose.\\nProfessor John E. Davies, University of Wisconsin I am so much pleased\\nwith it that I have asked all the electrical students to provide themselves\\nwith a copy of it. 1 have assured them that if it is small in size, it\\nis, nevertheless, very solid, and they will do well to study and work over\\nit very carefully.\\nProfessor Alex. Macfarlane, University of Texas Allow me to congratu-\\nlate you on producing a work which contains a great deal of information\\nwhich cannot be obtained readily and compactly elsewhere.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0176.jp2"}, "177": {"fulltext": "SCIENCE. 53\\nElectrical Measurements.\\nBy Professor Henry S. Carhart and Asst. Professor G. W. PATTER-\\nSON, University of Michigan. i2mo, cloth, 344 pages. Price, $2.00.\\nIN this book are presented a graded series of experiments for\\nthe use of classes in electrical measurements. Quantitative\\nexperiments only have been introduced, and these have been\\nselected with the object of illustrating general methods rather\\nthan applications to specific departments of technical work.\\nThe several chapters have been introduced in what the authors\\nbelieve to be the order of their difficulty involved. Explana-\\ntions or demonstrations of the principles involved have been\\ngiven, as well as descriptions of the methods employed.\\nThe Electrical Engineer, New York We can recommend this book very\\nhighly to all teachers in elementary laboratory work.\\nThe Electrical Journal, Chicago: This is a very well-arranged text-book\\nand an excellent laboratory guide.\\nExercises in Physical Measurement\\nBy Louis W. Austin, Ph.D., and Charles B. Thwing, Ph.D.,\\nUniversity of Wisconsin. i2mo, cloth, 198 pages. Price, $1.50.\\nTHIS book puts in compact and convenient form such direc-\\ntions for work and such data as are required by a student\\nin his first year in the physical laboratory.\\nThe exercises in Part I. are essentially those included in the\\nPraciicwn of the best German universities. They are exclu-\\nsively quantitative, and the apparatus required is inexpensive.\\nPart II. contains such suggestions regarding computations and\\nimportant physical manipulations as will make unnecessary the\\npurchase of a second laboratory manual.\\nPart III. contains in tabular form such data as will be needed\\nby the student in making computations and verifying results.\\nProfessor Sara.li F. Whiting, Wellesley College It comprises very nearly\\nthe list of exercises which I have found practical in a first-year college\\ncourse in Physics. I note that while the directions are brief, skill is\\nshown in seizing the very points which need to be emphasized. The\\nIntroduction with Part II. gives a very clear presentation of the essential\\nthings in Measurements, and of the treatment of errors.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0177.jp2"}, "178": {"fulltext": "58 MATHEMATICS.\\nElements of Algebra.\\nBy Professor James M. Taylor, Colgate University, Hamilton, N.Y.\\nAt Press.\\nIN this book Professor Taylor aims primarily at simplicity in\\nmethod and statement, and at a natural and logical sequence\\nin the series of steps which lead the pupil from his arithmetic\\nthrough his algebra.\\nAn introductory chapter explains the meaning and object of\\nliteral notation, and illustrates the use of the equation in solving\\narithmetical problems. This is followed by a drill on particular\\nnumbers before the pupil is introduced to the use of letters to\\nrepresent general algebraic numbers. General principles are\\nbrought out by induction from particular cases, and proofs are\\ngiven in their natural places where the pupil will be unlikely to\\nmemorize without comprehending them. Nomenclature has been\\nlooked to carefully. Many of the misleading terms of the older\\ntext-books have been discarded and others more useful and help-\\nful have been applied.\\nThe methods of working examples have been chosen for their\\nsimplicity and the scope of their application. Suggestions as to\\nmethod of attack are given, but formal rules are stated but rarely.\\nPositive and negative numbers are so explained and defined as to\\ngive clear and true concepts, such as lead naturally to still broader\\nviews of numbers. Factoring is made a fundamental principle in\\nthe solution of quadratic and higher equations. Particular atten-\\ntion is given to the theory of equivalent equations. In illustrating\\nthe meaning of numbers, equations, and systems of equations, the\\ngraphic method is used. In general the aim is to render as clear\\nas possible to the pupil all fundamental processes and to simplify\\nthe statement of rules.\\nThe book is particularly adapted to beginners, and is intended\\nat the same time to prepare for any college or scientific school,\\nas each subject is so treated that the pupil will have nothing to\\nunlearn as he advances in mathematics.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0178.jp2"}, "179": {"fulltext": "MA THE MA TICS. 59\\nAn Academic Algebra.\\nBy Professor J. M. Taylor, Colgate University, Hamilton, N.Y. i6mo,\\ncloth, 348 pages. Price, $1.00.\\nTHIS book is adapted to beginners of any age and covers\\nsufficient ground for admission to any American college or\\nuniversity. In it the fundamental laws of number, the literal\\nnotation, and the method of solving and using the simpler\\nforms of equations, are made familiar before the idea of alge-\\nbraic number is introduced. The theory of equivalent equa-\\ntions and systems of equations is fully and clearly presented.\\nFactoring is made fundamental in the study and solution of\\nequations. Fractions, ratios, and exponents are concisely and\\nscientifically treated, and the theory of limits is briefly and\\nclearly presented.\\nProfessor C. H. Judson, Fur?7ian University, Greenville, S.C. I regard\\nthis and his college treatise as among the very best books on the subject,\\nand shall take pleasure in commending the Academic Algebra to the\\nschools of this State.\\nProfessor E. P. Thompson, Miami University, Oxford, O. The book is\\ncompact, well printed, presenting just the subjects needed in preparation\\nfor college, and in just about the right proportion, and simpiy presented.\\nI like the treatment of the theory of limits, and think the student should\\nbe introduced early to it. I am more pleased with the book the more\\nI examine it.\\nLogarithmic and Other Mathematical Tables.\\nBy William J. Hussey, Professor of Astronomy in the Leland Stan-\\nford Junior University, California. 8vo, cloth, 148 pages. Price, $1.00.\\nIN compiling this book the needs of computers and of students\\nhave been kept in view. Auxiliary tables of proportional\\nparts accompany the logarithmic portions of the book, and all\\nneeded helps are given for facilitating interpolation.\\nVarious mechanical devices make this work specially easy to\\nconsult and the large, clear, open page enables one readily to\\nfind the numbers sought. It commends itself at once to the eye\\nas a piece of careful and successful book making.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0179.jp2"}, "180": {"fulltext": "60 MA THE MA TICS.\\nA College Algebra.\\nBy Professor J. M. TAYLOR, Colgate University, Hamilton, N.Y.\\ni6mo, cloth, 326 pages. Price, $1.50.\\nA VI GO ROUS and scientific method characterizes this book.\\nIn it equations and systems of equations are treated as\\nsuch, and not as equalities simply.\\nA strong feature is the clearness and conciseness in the state-\\nment and proof of general principles, which are always followed\\nby illustrative examples. Only a few examples are contained in\\nthe First Part, which is designed for reference or review. The\\nSecond Part contains numerous and well selected examples.\\nDifferentiation, and the subjects usually treated in university\\nalgebras, are brought within such limits that they can be success-\\nfully pursued in the time allowed in classical courses.\\nEach chapter is as nearly as possible complete in itself so\\nthat the order of their succession can be varied at the discretion\\nof the teachers.\\nProfessor W. P. Durfee, Hobart College, Geneva, N, Y. It seems to me a\\nlogical and modern treatment of the subject. I have no hesitation in pro-\\nnouncing it, in my judgment, the best text-book on algebra published in\\nthis country.\\nProfessor George C. Edwards, University of California: It certainly is a\\nmost excellent book, and is to be commended for its consistent conciseness\\nand clearness, together with the excellent quality of the mechanical work\\nand material used.\\nProfessor Thomas E. Bo3/ce, Middlebury College, Vt. I have examined\\nwith considerable care and interest Taylor s College Algebra, and can say\\nthat I am much pleased with it. I like the author s concise presentation\\nof the subject, and the compact form of the work.\\nProfessor H. M. Perkins, Ohio Wesleyan University I think it is an\\nexcellent work, both as to the selection of subjects, and the clear and\\nconcise method of treatment.\\nS.J. Brown, Formerly of University of Wis cons i?i I am free to say that\\nit is an ideal work for elementary college classes. I like particularly the\\nintroduction into pure algebra, elementary problems in Calculus, and ana-\\nlytical growth. Of course, no book can replace the clear-sighted teacher\\nfor him, however, it is full of suggestion.", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0180.jp2"}, "181": {"fulltext": "", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0181.jp2"}, "182": {"fulltext": "10V 3 1900", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0182.jp2"}, "183": {"fulltext": "", "height": "4150", "width": "2650", "jp2-path": "calculuswithappl00haye_0183.jp2"}, "184": {"fulltext": "", "height": "4429", "width": "3025", "jp2-path": "calculuswithappl00haye_0184.jp2"}}