LIBRARY OF CONGRESS. QA303 Chap. Copyright No.. Sheli UNITED STATES OF AMERICA. CALCULUS WITH APPLICATIONS AN INTRODUCTION TO THE MATHEMATICAL TREATMENT OF SCIENCE BT ELLEN HAYES PROFESSOR OF APPLIED MATHEMATICS IN WELLESLEY COLLEGE Boston ALLYN AND BACON 1900 __ 70387 i-itor^i y of Oonor«9« '\*u Cohu Hectiveo NOV 3 1900 Copyright entry SECOND COPY. Delivered to ORDfcS DIVISION, NOV 20 1900 COPYRIGHT, 19 00, BY ELLEN HAYES. Ncrfoooti -Iftes J. S. Cushing & Co. — Berwick & Smith Norwood Mass. U.S.A. PREFACE. This little book lias been written for two classes of persons : those who wish, for purposes of culture, to know, in as simple and direct a way as possible, what the calculus is and what it is for; and students primarily engaged in work in chemistry, astronomy, economics, etc., who have not time or inclination to take long courses in mathematics, yet who would like " to know how to use a tool as fine as the calculus. " The 'pure' mathematician will note the omission of various subjects that are important from his point of view; but for him there are admirable and lengthy treatises on pure calculus. Also the student whose experience has led him to conceive of mathematical study as the doing of interminable lists of exercises, will be surprised and, possibly, disax->pointed. This book is a reading lesson in applied mathematics. Fancy exercises have been avoided. The examples are, for the most part, real problems from mechanics and astronomy. This plan has been pursued in the conviction that such problems are just as good as make-believe ones for purposes of discipline, and a good deal better for purposes of knowledge. The time-honored method of presenting calculus is much as if travelers should be stopped and made to pound stone on the high- iii IV PREFACE. way, so that they never get anywhere or even know what the road is for. The following pages are a protest against the conventional method ; for I am wholly in sympathy with a remark made by Professor Lester F. Ward, in his Outlines of Sociology : "There is no more vicions educa- tional practice, and scarcely any more common one, than that of keeping the student in the dark as to the end and purpose of his work. It breeds indifference, discourage- ment, and despair." A chapter on analytic geometry has been introduced, in the hope that teachers will try the plan of presenting the elements of the calculus and of analytic geometry together. There is no good reason either for keeping them distinct or for presenting analytic geometry first. To three works I have to express my deep obligation. The spirit manifest in them has been my chief encourage- ment in preparing this book. I refer to GreenhilPs Differ- ential and Integral Calculus, Perry's Calculus for Engineers, and Nernst and Schonnies' Einfulirung in die mathematische Behandlung der Naturwissenschaften. We have in these works, let us hope, an indication of the role which the calculus is to play in schemes for liberal and scientific education in the not far distant future. ELLEN HAYES. Wellesley College, September, 1900. CONTENTS. CHAPTER I. Differentiation and Integration. ARTICLES PAGE 1- 6. Introduction 1 7-13. Differentiation of algebraic functions ... 8 1-4-16. Integration 12 17. Exercises 14 18-19. Implicit functions. Exercises 16 20-22. Differentiation of trigonometric functions ... 17 23-24. Differentiation of exponential and logarithmic func- tions 21 25-27. Second derivatives. Partial and total differentials . 24 28. Taylor's theorem 26 29. Maclaurin's theorem ....... 28 30-31. Binomial theorem. Converging series ... 29 32-33. Indeterminate forms 31 34-35. Exercises. Logarithms 34 CHAPTER II. The Graph. 36-37. Cartesian system of coordinates . dy 38-40. Geometric meaning of ~ Exercises ax 41-43. Maxima and minima. Exercises 44. Examples in maxima and minima 45-46. Polar coordinates .... v 41 44 48 50 56 VI CONTENTS. CHAPTER III. Applications. ARTICLES PAGE 47-54. Velocity and acceleration 58 55-59. Simple harmonic motion 64 60-63. Falling bodies 67 64-65. Rectilinear motion ....... 74 66-69. Parabolic motion 77 70. Motion in a vertical curve 81 71-72. Simple pendulum 82 73-74. Areas. Examples 85 75-77. Mean values 88 78-79. Work 91 80. Lengths of curves 92 81. Volumes and surfaces of revolution ... 94 82. Double and triple integrals 96 83. Perfect differential 97 84-85. Moment of inertia. Examples .... 99 86-93. Kepler's laws 102 CHAPTER IV. Analytic Geometry. 94-99. The equation / (x, y) = Ill 114 117 117 118 118 119 119 121 100-103. Change of axes 104. Condition of parallelism, and of perpendicularity 105. Straight line in terms of slope and intercept . 106. Straight line in terms of two points 106. Straight line in terms of one point and slope . 107. Distance between two points .... 108. Distance from point to line .... 109-116. The ellipse CONTENTS. vil ARTICLES PAGE 117-125. The hyperbola 127 126-131. The parabola 134 132-133. Tangent and normal to a curve .... 137 134. Path of middle point of ellipse-chord . . . 140 135. Determination of center and axes of ellipse . . 142 136. Tangent in terms of slope and intercept . . . 143 137. Exercises 145 138-139. Space coordinates .147 140-141. Distance between two points in space . . . 148 142. Equation to a plane surface 148 CHAPTER Y. Formulas. Fundamental integrals . 151 Other integrals . . . . . . . . .153 Miscellaneous formulas 155 Index 159 "Perhaps we should all know hozv to use a tool as fine as the calculus." — J. McKeen Cattell. "A man learns to use the calculus as he learns to use the chisel or the file on actual concrete bits of work." — John Perry. CALCULUS. CHAPTER I. DIFFERENTIATION AND INTEGRATION. 1. In the experiences of every-day life and in the operations of science, people are continually dealing with things which keep changing in quantity, and with things so connected that a change in one of them is followed by a change in another. For instance, we know that work varies in amount, and that the amount done by workmen depends on the time. We find crops now abundant and now scanty ; and, other things being equal, the crops are seen to vary with the amount of fertilizer used. We observe that the height of the mercury in a thermometer changes with the temperature. In these examples and all similar ones there is a relation of cause and effect, or at least a relation of antecedent and consequent; and Ave say that a quan- titative change in the cause is accompanied by a quantitative change in the effect. It is part of the business of science not only to dis- cover relations of cause and effect, but also to try to express these relations with precision. When a relation of cause and effect can be stated with exactness, the lan- 1 2 CALCULUS. guage of mathematics is the best one to use, because it gives compact and unambiguous expressions, and because a further examination of the relation may then be con- ducted in that language and results easily reached which could be arrived at only with much difficulty, if at all, in any ordinary language. For example, the expansive force of air was a property observed by Guericke (1602- 1686), but Boyle (1627-1691) discovered that the vol- ume varies inversely as the pressure. That is, if v represents the volume of a given quantity of air and p its pressure on unit area of the containing vessel, vac-, P and pv — a constant. We shall see later how we may learn more about this law by using the equation pv = c. Experience in balancing bodies of equal or unequal weights no doubt furnished ancient craftsmen with some vague notions regarding equilibrium ; but Archi- medes (287(?)-212 B.C.), from a few assumptions, con- cluded that two bodies suspended from a bar are in equilibrium when their distances from the point of support of the bar are inversely proportional to their weights. That is, if I, V are the distances of the bodies whose weights are w, w', respectively, w : w' : : V : L The principle of the lever, as thus stated by Archimedes, was later fully established. To illustrate further, from earliest times men must have noticed that unsupported bodies fall to the ground ; but after the investigations made by Newton (1612-1727), it was possible to state the law of gravitation with mathematical accuracy : The mutual attraction (stress) between any two bodies varies directly as the product of their masses and in- versely as the square of their distance from each other, DIFFERENTIATION AND INTEGRATION. 3 Thus, if F is the whole attraction between the earth and the moon, for instance, M the mass of the earth, m the mass of the moon, and r the distance between them, F = c — — . These examples go to show that when a r 2 precise quantitative statement can be made in science, mathematics, with its unambiguous symbolic shorthand, offers the most economical way of making it. 2. Two modes of quantitative change or variation present themselves. As an illustration of the first, the number of roses in a handful may be varied by adding one and another and another, until the number has changed from a to b ; or we may add several at a time until the number has changed from a to b. But we cannot do less than add one wdiole rose at a time ; for, in this case, the variation element is a whole unit, that is, a whole rose, and not any fraction of it. Again, we may measure a day with a minute as a unit of measure, and say that a day contains 1440 minutes ; but this is only an artificial convenience. Time does not increase a minute " at a time,*' or even a second at a time, but by elements of time which are immeasurably small frac- tions of a second. This is the second mode of varia- tion : a quantitative change not by jumps or finite amounts, but by indefinitely small amounts. 3. By the term variable we mean a quantity which changes in the second manner above described. We use it in speaking of such things as volume, pressure, distance, etc., when they are conceived as being in a state of continuous variation. The term function is applied to the quantity which necessarily changes be- 4 CALCULUS. cause of a change in a variable with which it is con- nected. For example, the pressure of steam on the piston of the cylinder is a function of the volume of the steam. The attraction which the earth exerts on the moon is a function of the distance of the moon from the earth. If the symbol x stands for the variable and y for the function, we briefly express the fact of their connection by the general statement y =f(x). The precise nature of the connection is shown by specializing /(V). For example, if f(x) is log x, we have the particular statement y = log x. When we need to distinguish one function from another, we use such forms as FQx), <$>(x), u, v, etc. The nature of these conventional symbols should be carefully noticed. The parenthesis merely serves to separate the quantity symbol x from the other symbols /, F, etc., which are not quantity symbols and hence not factors. /(#) is only algebraic shorthand for the expression "a function of the varying quantity #." Functions are classified as algebraic and transcen- dental. An algebraic function is defined as one which implies only a finite number of the algebraic opera- tions, addition, subtraction, multiplication, division, involution, and evolution. The trigonometric functions sin#, cos^, tanx, etc., are transcendental; so also are e x , log a;, sin" 1 #, etc. 4. Let us now suppose a change in pressure, or dis- tance, or time, or whatever quantity we are dealing with under the symbol x. Let Sx stand for the amount of change. Then the new quantity is represented by DIFFERENTIATION AND INTEGRATION. 5 x + &r, and f(x) becomes f(x + Bx). Subtracting tlie former value from the latter, we have f(x + hx) — f(x), the amount of change in the function occasioned by the change in the variable. It may be represented by 8y if y=f(x). Then ^* ^/ — , or its equal -^, is ox ox the ratio of the increment of the function to that of the variable. Let Sx now be supposed to become smaller and smaller until we cannot tell the difference between it and zero. We say it " has zero for its limit," or it " diminishes without limit," and to show that this supposition has been made we use the symbol dx in place of 8x, and also use dy in place of 8y for the indefi- nitely small change in the function, dx is called the differential of #, and dy the differential of y. The ratio -^ is called the first differential coefficient oif(x) with dx respect to x, or briefly, the derivative. dxi We shall find that the ratio -p is itself, in general, some function of x ; hence it is often written f (x). The symbols —f(x), -^, f r (x) all mean the same thing. It is important to notice that although dy and d v dx, the individual terms of the ratio -j-, are indefinitely small, the ratio itself is usually finite. This dx of the mathematician is suggestive of the " atom " of the chemist, the " particle " of the physicist, and even the " cell " of the biologist. It is the ultimate element of that with which the mathematician deals, and always implies one property of the quan- tity symbolized by x ; namely, its continuous variation. b CALCULUS. 5. To illustrate the nature of -~ let f (x) — ttx 2 , the dx area of a circle. Suppose this circular area to be cut out of a thin sheet of metal and to have heat supplied to it in such a way as to cause it to expand, but to re- main circular. Let the radius increase by the amount Sx ; then y + 8y — f(x -f- 8x) = ir (x + 8x) 2 = 7TX 2 + 2 TTxSx + IT (8xJ 2 , and 8y = f(x + Sx) -f(x) = 2 wxSx + it (8x) 2 ; By hence # = 2 irx + tt8x = it (2 x + 8x). Now, if 8x be made indefinitely small, the limit of 2 x + 8x is 2 x, and therefore ^ = 2™. This means that if the quantity of heat used is so. small that the increase in the length of the radius is indefinitely small, the ratio of the increment of the area to the increment of the radius is equal to the cir- cumference of the circle, a result which might have been guessed beforehand if we had reflected that the growth in area is a belt only dx wide around the circle, and dx is "next to nothing." As another illustration, suppose we take the equa- tion 2/ = -, which states the law concerning the mutual x dependence of the volume and pressure of a gas, x repre- DIFFERENTIATION AND INTEGRATION. 7 senting volume and y representing pressure. Let us look at pressure as a function of volume ; then Ju and y + By = f(x + &z) = therefore By = and x + 8x c cBx x + Bx x x(x + Bx) Sy = g__ . & x(x + Bx} hence dy _ dx In other words, the ratio of the increment of the pressure to the increment of the volume is inversely as the square of the volume. The minus sign means that when the volume takes an increment the incre- ment of the pressure is negative ; that is, the pressure- increment is really a decrement. This agrees with what is said by the equation itself ; namely, that the press- ure decreases as the volume increases, and vice versa. If the student will follow the thought in a few con- crete examples like the two just given, he will gain a better insight into the nature and purpose of the cal- culus than he can acquire from the mechanical working of a great number of meaningless exercises. 6. The importance of the ratio -^ is soon realized dx by the student of the mathematical side of any of the sciences which admit of mathematical treatment, such as astronomy, thermodynamics, electricity, chemistry, 8 CALCULUS. economics, etc. Granting its importance, we should know how to find, by the most direct method, the derivative of any ordinary algebraic or transcendental function ; and, what is even more essential, we should be able to perform the reverse operation ; that is, hav- ing -f- =z f'(x) to find /(a;). In the following articles, ax theorems are established by means of which derivatives may be directly written, so that we need not take any intermediate steps as in Art. 5. This operation of finding derivatives is called differ- entiation. We begin with a function which is itself the sum of two functions. 7. Let u = some function of x, and v some other func- tion of x, and let y=f(x)=u + v. Then, if x takes the increment &r, y + By = f(x + Bx) = u + Bu + v + Si>, and By = Su + Sv ; , By Bu , Bv hence -f- = — + — > ox ox ox and in the limit -f- = — + — — (1) ax ax ax It is evident that the same proof applies to any number of functions connected by plus and minus signs. A constant quantity, because it is a constant or un- varying quantity, has no increment ; and if we attempt to express its derivative, we have nothing to divide by dx. This amounts to saying that the derivative of a constant is zero. DIFFERENTIATION AND INTEGRATION. 9 8. Let the given function consist of the product of two functions as expressed by y=f(x)=uv. Then y + Sy = f(x + &r) = (it + Sii) (y + hv) = uv + vSu + uSv + SuSv ; hence Sy = vSu + uSv + SuSv ; therefore Sy Su , Sv , ^ Sv -f- = v y- + u — + Su — , ox ox ox ox and dy _ du dv dx dx dx (2) AM UJU The last term Su — is disposed of by observing that ox as Su diminishes without limit, any quantity (except oc) multiplied by Su diminishes without limit, and is there- fore dropped. Similarly, if y = uvw ■•- where u, v, w, etc., are func- tions of x, dy , ^du . , A dv , . ^dtv , , _£ = (w • ••) — + (ww •••)^- + (^ z; -- # )-r--l ^ etc. ax ax ax ax 9. Taking the last expression of the preceding article, suppose v = u, w = u, etc., so that uvw ••• becomes (u) n , n being the number of functions of x. Then y = u n , and the expression dy , ^du . r A dv . , N a 1 ^ . , -f- = (vw -•0^ + Quw---)— + (uv -..) — + ••, etc., ax ax ax ax ax ax ' 6?x 10 CALCULUS. This is a polynomial of n terms ; hence we can write, (3) dy n _ Y du -2- = nu n l — dx dx As a special case, if u = x, v = x, iv = x, etc., y = x n , and formula (3) becomes — £ = nx n ~ l . dx p 10. Let y =f(x)== u q in which p and q are constant quantities, positive and integral. Then y q =u*, and by (3), „ i dy n ,du hence dy p u p ~ 1 du dx q y q ~ l dx Eliminating y from this expression by means of the V given expression y = u q , we have dx q dx 11. Let y =f(x) = u~ n ,n being integral and positive. Then y = — , and so yu n = 1. Using the formula for u n DIFFERENTIATION AND INTEGRATION. 11 the derivative of the product of two functions, and writing zero for the derivative of unity, U n _M_ _j_ y nu n-\ __ _ Q . dx dx ,, , ■ dy ynu n ~ l du nu n ~ l du that is, -f- = — ^ _ = _, dx u n ax u m dx or -f- = — m^ w x — (5) 12. Comparing formulas (3), (4), (5), it is seen that if y — u n , in which w is a function of x and % is any constant, dy „_i c?m cfe dx The translation of this formula gives, therefore, the only rule that is needed for finding the derivative of a function affected with any constant exponent. It should be noticed, however, that since this expression cl ii all f for the first derivative, -^-, contains — as a factor, we dx dx may require various other rules if we are to find the expression for which — is the symbol. For example, suppose y — (log #) 3 . We now know that -^- = 3 (log x) 2 multiplied by the derivative of log x, whatever it is. What it is we shall learn in a subsequent article. At present we can only write -Jl = 3 (log x) 2 — log x. dx dx 12 CALCULUS. 13. For the ratio of the indefinitely small increment of the function to the increment of the variable we may of course use — f(x), as well as the symbol -^. It dx dx should be carefully noticed that the d standing above dx is here, as everywhere, a symbol of operation and not of quantity, signifying the differential oif(x). Notice also that an indicated operation counts for the same as a sy£i ry*& performed one. For example, in an operation a ~\~ x /y* /y*a is indicated, and the expression has everywhere the same value as a — x, the result obtained by actually d performing the division. So, for example, — irx 2 has the same value as 2 ttx. 14. Differentiation is seen to be a tearing down process, whereby we reach an ultimate element of quan- tity. The reverse operation, one of building up, is known as integration; its symbol is ( (long s). As already shown, rules are established for the differentiation of functions, but integration is largely a matter of guesswork and experiment. The test of the correctness of any integration is this : differentiate the result; we should get the given differential form. Tables of integrals enable the student to write directly the expression corresponding to an indicated integration, so that he need not go through the process of discovering the required expression. In accordance with what has just been said, we have J dy = y, the symbols 6?, J , neutralizing each other. DIFFERENTIATION AND INTEGRATION. Also, ,if dy _ du dv dx dx dx dy = du + (7y + ..., and # = w + v + .... If dy _ cfe c/^ e?# dx dx 13 y — \ (ydu + wtfo;) tw. If # m + 1 15. The symbol \f(x)dx is known as a general or indefinite integral. After discovering a function, say (/>(#), which differentiated will give f(x), we ought to write ff(x)dx = cf>(x)+C, in which (7 is a quantity primarily undetermined, and known as the constant of integration. Since C is a constant, -f O (x) + €T\ = ±+ (x) + -f (7=/<», and as a constant term may thus exist in connection with the original function we give the integral the benefit of the doubt and write as stated, cf> (x) + 0. 14 CALCULUS. 16. The symbol I f(x)dx is known as a definite integral. Its meaning is this : Find the general inte- gral, which will be some function of #, and substitute b for x ; then substitute a for #, and subtract the latter expression from the former. To state the process sym- bolically, let )f(x)dx = 4>(x)\ j> then I f(x)dx = 4>(x) = £(*)- a and b are called the limits of the integral. The constant O disappears since [>(&) + C]-[(£)- , y may now be eliminated by using its value b ± - -ya 2 — x 2 , a obtained from the given equation. DIFFERENTIA TION AND INTEGRATION, 2. # 8 + y 2, - - 3 axy = ; fin dx 3. x s + y 3 - - 3 axy + a 3 = ; find dy dx 4. pv = c ; show that -J- = dv c -p : - - if J) = = /(*)■ By formula ,(2), *f +i > = '•> hence 'J dp _ dv i ; = - G ~V 2 ' 17 5. 27v k = c'] show that — = -• Here, and also in exercise 4 ? it is just as well to solve for p before beginning to differentiate. We have p = — , so that vk dp _ _ kc'v k ~ l _ _ kc' dv v 2k v k+1 6. F= c — — ? in which c, M, m are constants ; show that c]F = _2GMm m dr r 3 It is often desirable to use other letters besides x and y to denote the variable quantities. The student should therefore accustom him- self at the outset to such symbols as those given in exercises 4, 5, 6. 20. If f(x) is a varying angle, it is clear that any trigonometric function of the angle must also vary. We have now to find -j- , when y represents each one of the trigonometric functions in turn. Suppose y = sin/(^). Let f(x) = u ; then y = sin u, 1 Sy _ sin (u + S-u) — sin u (XllKX — — ~ OX ox 18 CALCULUS. Put u + Su = a, and u = {3 ; then, since sin a — sin /3 = 2 sin J (a — £) cos J (cc + /3), Sy = sin (w + Su') — sin w = sin a — sin /3 = 2 sin | Sw cos \(2>u + hu) = 2 cos (u + I Sw) sin 1 cm ; , S?/ ., ^ N sin ^- Sw S^ hence ^- = cos (u + A- ou) — j-g — • -k— ox A J \ou Sx But when an angle diminishes without limit, we may write the angle itself for its sine ; so we now have dy d . du sn\ -~= — sin u = cos u — — (b) dx dx dx Let y = cos w = sin ( — — u J ; ,i dy d . (it \ (it \ d (tt \ then -^ = — sm — — u )= cos — — u) — — — u ; ^ ^ V2 y V2 Jdx\2 J 9 . i n ay a • au ^tn thereiore -^- = —oosu= — smu—— (7) \AiJU \XlJU \AlJU sinw Let V = tan u = cosw Using exercise 3, Art. 17, together with formulas (6) and (7), dy cos 2 u + sin 2 u du 9 du J = 2 1~ = SeC M TT 5 dx cos A u ax ax therefore -M. = — tan u = sec 2 u — • (8) BIFFEBENTIATION AND INTEGRATION. 19 T . COS u Let y = cot u = — sin u du — sin 2 u — cos 2 u du 9 du (9) dx sin 2 u dx therefore dy d -£- = — cot u = — \XiJu LI il/ 9 C?7£ cosec" 5 m — dx Let 1 y = sec u = : u COS 76 then, using exercise 5, Art. 17, dy du sin u — dx = = = tan ?/ ser du *, u — % and Ave have dy d , du /1AX -^- = -—sec 7^= tan u secu — (lu) dx dx dx Let y = cosec u SUlTi — cos u — dy dx t du -^- = — } = — cot u cosec u -^-, ax sur u dx and therefore dy d , du .---,. -f- = — cosec u= — cot u cosec w — (11) 21. In formulas (6) to (11) inclusive we have the ratio of the differential of the trigonometric function to the differential of the angle x. These formulas, which are remarkable for their simplicity, should be translated and committed to memory. We may next regard the trigonometric function as varying by equal increments, and thereby causing a change in the angle. From this 20 CALCULUS. point of view we have to find the ratio of the differen- tial of the angle to the differential of the trigonometric function. 22. Let y = sin -1 u \ then u = siny. Hence, by formula (6), du dy — = cos y -j- ; dx ° dx du dy dx that is, dx cos y But cos y = Vl — u 2 ; du therefore §y _ —^ n -i u = ^ . (12) cfa cte Vl-i( 2 Let 2/ = cos -1 1£ ; then w = cos y, and proceeding as before, Ave find du dy d - tfe ^-, ox -f- = -—cos" 1 ^= — =— (lo) a# a# VI _ ^2 Let 2/ — tan -1 u ; then m = tan y, 6?W 6?w dy ,, , . dy dx — = sec z ^ -~ ; that is, -^ = — 5- ■ a^ a^ a# sec z 2/ ' du and -^ = sec 2 y -~^ ; that is, -^ therefore ^ = _^ tan -i M = _^ ; (14) dx ax 1 + w similarly, if y= cot" 1 2/, dy d ^ . dx sh r x -# = -j- cot -1 w = - q 5- (15; dx dx 1 + ?r DIFFERENTIATION AND INTEGRATION. 21 Let y = sec 2 u. u = sec y, , du dy and -7- = tan y sec v -7 1 : dx * * dx du that is, % = dx dx tan y secy Bnt since u = sec y, tan y = V^ 2 — 1 ; and we have du dy d ., dfe ^i^n -f =-— sec- 1 ^^ , (Id) dx dx U -y/y? _ l Finally, if y = cosec -1 % we find that du dy d _- 6fe = ^- cosec * % dx dx u V^ 2 — 1 (17) All these operations of differentiating may be reversed, so that if dy du dx du r , . , cos u -j-, y = I cos udu = sm u, etc. 23. It remains to find -!f- when 2/ = e x , a x , log #, log 1 w, e w , a* u being, as before, equal to f(x). From algebra* we have [2 [3 |4 |n_ in which g is the number 2.7182818284 . . . forming the base of the Napierian system of logarithms. If y = e x , c li = A e * = A_(i + x + ^ + ^ + dx dx dx\ 1 2 [3 *See Hall and Knight's Elementary Algebra, Art. 537. Edition of 22 CALCULUS. Performing the operation indicated, that is, differen- tiating this series term by term, but this result is the original series which e x equals ; therefore *£ =*-«*=«• (18) dx dx This result is unique, being the only case known in which the derivative of a function is the function itself. We have also from algebra a x =1 +xlog e a+ v + 7/ +•••; therefore — a x = log e a dx 1 + # log e a + - v ,^ e # 2 (log e a) 2 = ^ log e a Hence, if y = a x , -^- = —-a x = a x log*, a. (1 9) 24. Let ^/ = log e x ; then # = e y ; and if we regard x as a function of y, we have by formula (18), dx t . , . dy 1 1 -=- =e y ; that is, -f-.= — = -; dy ■ ax e y x therefore -y- = t"1°^ » = — (20) dx dx ° x DIFFERENTIATION AND INTEGRATION. 23 dy To find -j- when y = log c /(x) = \og e u, we notice that «F = ^r- • -s- and in the limit OX 0^ OX 6?^/ dy du dx du dx Now if y — log e u, -=^ = -, by formula (20), CtU U and therefore -^- = -—\og e u = — =-• (21) dx ax u dx y If y = e u , log e y = u and -r= - = -r- log y = - -¥-, by formula (21), dx dx ** y dx J v J " . 6?V 6?M 6?W hence ^ = y — = and so on. ax* 26. In what has preceded we have assumed one fundamental variable ; but reference to common ex- amples shows us that we may have a function of two or more independent variables. For instance, crops vary not only with the amount of fertilizer used, but also with the amount of sunshine and moisture. If z is a function of two independent variables x and y, expressed by writing u = f(x, y), we may differen- tiate, supposing x to vary and y to remain constant, or Ave may suppose y to vary and x to remain constant. In the former case we have — -; and in the latter, — . dx dy These ratios are known as partial differential coefficients, and to indicate this we may use the parenthesis, writing DIFFERENTIATION AND INTEGRATION. 25 ( — j and t — V When we suppose x and y to vary simultaneously the corresponding change in the func- tion is called the total differential. As an example of partial differentials suppose we have pv = nt, where p and v are pressure and volume as before and t is the absolute temperature of a gas. Suppose t varies while v remains constant ; then f-^ J == — Again, let v vary while t remains constant, I -J-] — . — — , as in exer- a a j. -tr\ \dvj V A cise 4, Art. 19. x y 27. Having z = f(x, if), let us differentiate z with respect to x and then differentiate the result with respect to y. The order of the steps is indicated by d fdz\ d 2 z ■ — — or dy \dxj dydx The reverse order is indicated by d f dz\ d 2 z ■ — — or dx\dyj dxdy We proceed to show that d f dz\ __ d f dz\ . dy \dx) dx \dyj that is, we get the same result in whichever order we proceed. Let x take the increment Sx while y remains constant ; then — = /<> + g ^?/) -/(^y) , Sx Sx 26 CALCULUS. Now let y in this result take the increment Sy while x remains constant. By \8xJ = f(x + &e, y + 8y ) -/(a?, y + 8y) -f(x + 8a;, y) +/(>, y) SySa; Reversing the order, we have 8z = f(x, y + 8y) -/(a;, y) 8y 8y 8_(8z y 8x\8yj _ f(x + 8x,y + 8y) -f(x + &k, y ) -/(a;, y + Sy ) +/(s, y ) ^ SxSy Hence _—=—(—; oy\oxj cx\oyJ and in the limit » d fdz\ _ d fdz\ m d 2 z _ _ d 2 z dy\dxj dx\dyj ' dydx dxdy In any scientific investigation in which the calculus is used the context must show what and how many variables are involved, and what partial differential coefficients will occur. For example, Carnot's Prin- ciple, with its applications as presented in thermo- dynamics, affords an abundance of cases of these partial differential coefficients. 28. Ordinary text-books in algebra and trigonometry usually give methods for expanding (a + x) m , e x , a x , log(l + x), sin a;, etc., into series in ascending powers DIFFERENTIATION AND INTEGRATION. 27 of x. We can now establish one general theorem by means of which these functions and all similar ones may be expanded. We first notice that if y =f(z + x) and we differ- entiate regarding x as a variable and z as a constant, the result is just the same as if we should differentiate with z for the variable and x as a constant. That is, if y =f(z + x), (-¥- )= ( — )• This is obviously true if we consider that it makes no difference whether we change the function by changing z or x. Suppose f(z + x)= A + Bx + Ox 9 - + Dx* + E& + ..., (a) in which A, B, C, etc., are functions of z and not of x. Let us now differentiate successively the first member of equation (a) with respect to g, and the second mem- ber with respect to #, and put x = after each differ- entiation. Then, since — f(z + x) = — f(z + x), dz ax dz = B + 2Cx + 3Dx 2 + 4:I]x 3 + ..., and f(z)=B. f"(z + x) = 2 + 2 • 3 Dx + 3 • 4 fix 2 + ..., and /"(s)=2<7. f"(z + x)=2-SD + 2-3-4Hx+ -, and /"(*) = 2 -8 2>. 28 CALCULUS. Also, if x = in equation (a), f(z) = A. We now have A =/<», B=fXz), 0=\f"(z), etc. Putting these values into the assumed series (V), /<> + aO=/+/'(3> + -^> 2 + /^! + [n This formula is Taylor's theorem. It enables us to expand functions of the sum of two variables in ascend- ing powers of one of the variables, combined with finite coefficients depending on the other variable. 29. Suppose we have a function of one variable and wish to expand it into a series. Following the method of the preceding article, assume f(x) = A + Bx + Ox 2 + Dx 3 + Eat ■ •-. (5) Differentiating successively and putting x = after each differentiation, we have f(x) = B + 2Cx + 2,Bx i + 4Bx z + ..., and f(*)\ = B. f"(x)= 2C + 2 ■ 3 Dx + 3 ■ 4 Bx 2 + ...,. and /"<»] o = 2a f'"(x)= 2-3D + 2-3-4& + -, /'"(z)] o = 2.3D. DIFFERENTIATION AND INTEGRATION. 29 Also, f{x-)\ = A. The assumed coefficients A, B^ etc., are thus deter- mined, for we have A =/(V)] , or, as it is usually written, /(0); £=/'(*)]o=/'(0); "" ~ [3 " [3 ' Putting these values into the assumed series (5), /'(0)z 2 /'"(O)s 8 f(x)=f(P)+f(0-)x + ^-^ + "—^ - + , j^CQ)^ , This formula is Maclaurin's theorem. It will be observed that if z is made equal to zero in Taylor's theorem, we have Maclaurin's theorem. The latter may therefore be regarded as a special case under Taylor's theorem. 30. Suppose f(x) = (a-\-x) m \ let us expand this function according to Maclaurin's theorem. If x = 0, the function becomes a m . 30 CALCULUS. Further, /' (x) = rn(a + x) m ~\ and /'(0) = m O + ^) m_1 ] = ma m ~ l . f n (x) =m(m — I) (a + x) m ~ 2 , f" (0) = m (m - 1) (a + ^) w ~ 2 ] = m (m - 1) a™" 2 . f"(x)=m(m — l)(a + ^) m " 3 , and hence /'"CO) = m(m - l)(w - 2)(a + x) m - 3 ] = m (m — 1) (m — 2) a m ~ 3 . Therefore Ave have (a + ^) m = a m + ma m ~ l x 4 m Cm — 1) a m 2 # 2 [2 m(m— 1) Cm — 2) a™ V m(m - 1) - (m -?i + 2)a w - (w -V- 1 |w — 1 This formula will be recognized as the binomial theorem. It provides for the expansion of a binomial affected with any constant exponent. 31. A series must be known to be a converging series before any practical use can be made of it. The sim- plest tests for convergency are given in algebra text- books.* If a series is found to be diverging, it is * See Hall and Knight's Elementary Algebra, Arts. 470-477. DIFFERENTIATION AND INTEGRATION. 31 rejected for such values of the variable as make it diverging; or it is transformed into a series which converges and, if possible, into one which converges rapidly, in order that only a few terms need be used. 32. If one function is divided by another, as *. A it sometimes happens that the functions are of such a nature that upon evaluating them for some particu- lar quantity each function reduces to zero, so that we have -• The question arises : what does this expres- sion mean, and what is its value ? Students often say that - must be unity ; sometimes they are inclined to think it is zero. In some instances the first view is correct ; in others, the second ; in others still, a value will be found which is neither unity nor zero. Now it is evident that if we can find a limit f(x) which the ratio \) J is approaching as x approaches nearer and nearer to that value which makes f(x) and (x) each equal to zero, we have caught the correct value of — We proceed to find a general expression for this limit. Suppose x to take the increment Bx ; then by Taylor's theorem, 32 CALCULUS. Let a be the quantity that makes both f(x) and (x) equal to zero. Substituting a for x, /0, + fa) HA.)i.^W t ... rv J + <£'<>)&* + r ^ y (&e) 2 + ... Dividing both numerator and denominator of the second member of this expression by 8x, Finally, when &r becomes c?:z, f(a) = f'(a) /"<» Hence, if ^ becomes x when evaluated for any quantity as a, the value of this indeterminate form is <£'o.) evaluated for a. If it should happen that ^j = -, we divide both f (x)\ numerator and denominator of equation (2) by Sx again and have /O) f"(a) (a) <£"<» /CO If , ) ; becomes ^ when evaluated for any quantity DIFFERENTIATION AND INTEGRATION. 33 as a, this expression may be determined as in the first case by observing that if /O) = oo (x) °°' l o" If f(x)(x') = oo . when evaluated for some quantity, (x) — T~~~ = (z)logf(z); and this is indeterminate whenever one of the factors becomes zero and the other infinite for the same value of x. (1) Suppose $(x)= 0, and \ogf(x) = ± oo ; then f(x) = cc or 0. Consequently [/(V)]^ becomes inde- terminate when for some value of x it takes the form 0° or oo°. (2) Suppose <£(V)=±ao, and logf(x)= ; then f(x) = l, and \_f(x)~Y ){x) gives the indeterminate forms 1" and 1~°°. Hence, if we have any of the indeterminate forms 0°, oo°, 1~°°, as the result of evaluating [/(V)]^ f° r some quantity, we change the exponential function to the corresponding logarithmic function, and then reduce to the form -, which is dealt with under the first case. 34 CALCULUS. Exercises. 34. 1. Show that — cotit= — cosec 2 ^ — through the iXJu (XX relation cot u = tan \- — ii\ v2 ; 2. If y = [/(a)]* (x) = v?, show that -JL = VU V 1 1- (log U) U v dx dx dx Take the logarithmic form, logy=vlogu, and differentiate. 3. y = x x ; -^ = of (1 + log x). dx -- dy 1 4. y = e x ; -^ = dx - a-V 6. N= e\ tan ,P — log tan (45° + \ F), in which the varia- bles are N and F. Show that c^Y^ A^-cosi^ 7 ) dF cos 2 F — Watson's Theoretical Astronomy, p. 69. 7. Verify the following expansions by means of Mac- laurin's theorem : /y»£ /y»0 /Vl4 /yffl (ii) a^ = l + a! log e a + ^^ + ^f^- 3 +"-. (in)Iog(l + a») = »-| + f-J+-(-ir 1 f-. DIFFERENTIATION AND INTEGRATION. 35 /y»3 /v»5 (v) C psa! = l-- + - . Show that (v) might be derived directly from (iv), since — sin x = cos x. dx 8. Show that • = 1. X -•o In this case f(x) = cos x and <£'(V) = 1 ; ^L? _ 9. Show that X -log- = 1. 10. Show that x l°g( 1+a; ) " 1 — cos as = 2. We have / ^)_ l0 ^ 1 + ^) + ^(iT^) *'(*) sma; . but this expression evaluated for is - as before. Hence we proceed to the second derivatives and have ■ + ^— \-x 1 <£"(#) cos# and this equals 2 when evaluated for 0. 11. Show that 1 — sin x — 2 sin 2 # 1 — 3 sin x -\- 2 sin 2 x = 3 when a? = 30°. 36 CALCULUS. 12. Expand sin -1 a;. Using Maclanrin's theorem, fix) = sin" 1 x = /(0) + /'(0) x + .M x* + tUlp. x 3 + - /(O) - 0; /'(*) = -J=J /'(O) = 1. VI — x- f"(x) = ^—;f'(0) = 0. _ (1 - a 2 )* - -I x ii-xy _ Therefore sin -1 sc = a; + = 1. 13. Show that 2r sin" 1 — = a(l+-^- +...). 2r V 24r 2 ; — Thomson and Tait's Nat. Phil., Vol. 1, Art. 131. Substituting — for x in the expansion of sin -1 a?, and 2 r multiplying by 2 r, we have the result 1 + — + 24 r 2 14. Expand sin 2 - to the second power term inclusive. 2 02 Ans. -j' 15. Given lajlogojeZa?; perform the operation indicated. We have d(uv) = udv + vdu ; (formula 2) hence, m; = I \idv + j i>cfa^ (Art. 14) or, I udv — uv — I i>(^. DIFFERENTIATION AND INTEGRATION. 37 In the present case let u = log x ; then dv = xdx, and du — -- x X X Integrating dv, v = — , and uv = — log x ; therefore ( x log xdx = — log x — ( — — J 2 J 2 x — — \ogx 2 & 4 This process is known as integration by parts. 16. Use the -method of example 15 in the following examples : (i) I x cos x dx = x sin x + cos x. (ii) I e ax x dx = — (x ] • (iii) I sin -1 x dx = x sin -1 x + Vl — x 2 - (iv) I log x dx = x log x — x. 17. Given the following integrals, to find their values: (i) I (a — x) n dx. (ii) I t&nxdx. Let cos x,= u; then du = — sin x dx, and /tan xdx = | dx = — | — = — log u = — log cos x. J cosx J u 38 CALCULUS. (iii) I sin x cos x dx. This may be written I i sin 2 x dx, which equals —J cos 2x. (iv) I x Va 2 — x? dx. Let u == V a 2 — or ; then I a; Va 2 — a; 2 da; = — | m 2 dw = _^ = _(a 2 -ar°)j 3 3 (v) I Va 2 — a; 2 c?x. Let a; = ct sin u ; then I Va 2 — a; 2 da; = a 2 I cos 2 ^dw = - f (1 + cos 2 u) cfa = -(it + i sin 2 u) = — sin x - H — - V a- — or = — sm A - + - Vcr — x . 2 V a a 2 7 2 a 2 (vi) I da;. This may be written J a 2 __ ( a 2 _ #2\ v ' dXj which equals Va 2 — x 2 — dx — I Va 2 — x?dx. Va 2 — a? 2 ^ Therefore ^2^" a ^2 " a ' 2 /Oy t o • _i a; I eh • i x , x / 9 i — dx = a 2 sin * — (-sin L - + - Va- — a; Va 2 - DIFFERENTIATION AND INTEGRATION. 39 dx (vi) /. Va 2 + x 2 Let u — x = Va 2 + x 2 ; then I — x = I — - = log i£ = log (a; + Va 2 -f x 2 ). J ^tf + x 2 ^ u (viii) I Va 2 + x 2 dx. Integrate by parts, letting u = Va 2 + x 2 ; X LLJu then dv = dx* v = a;, du = — , wu = x Va 2 + # 2 , and the formula I wcfa; = uv — I v^w becomes J Va 2 + # 2 dx = x -Va 2 + x 2 — I - Va 2 + a 8 cto : a; Va 2 + x 2 — I - -dx Va 2 + or 5 = x Va 2 + x 2 - fs/a 2 + x 2 dx+ f J J Va 2 + x 2 Transposing the middle term and dividing by 2, I Va 2 + x*dx = - Vaxis, and the negative values to the left on this axis. At the end, remote from 0, of this line, which represents a value of x, draw a perpen- dicular (using the same unit of length) to represent the corresponding value of y. The perpendicular is to be drawn upward from the #-axis in case y is posi- tive, and downward when y is negative. In this way locate a point for each pair of values of x and y. If many values be given to x, — any two con- secutive values differing but little from each other, — we shall have a correspondingly large number of points with small distances separating them. Connecting all the points in order, we have a continuous line, straight or curved, called a graph or locus. The values of x are 41 42 CALCULUS, called abscissas, and the values of y ordinates. The two together are known as coordinates. To illustrate, suppose y = x + 2. When # = 0, y = 2 ; when x = 1, y = 3 \ when a; = — 2, ?/ = ; etc. When # = 0, Ave have no distance to measure off on the #-axis, and since y = + 2 we measure upward two units, thus locating the point P v Measuring one unit to the right and three upward, Ave have the point P 2 . Lo- cating a number of points in this, way and then con- necting them, the result looks like a straight line. At any rate Ave have not been able to get any apparent bends or corners — provided the plotting has been accurately done. This line presents to the eye the way y changes as x changes when y = x + 2. The vertical lines representing the values of y seem to get steadily longer as x increases. X' X Y Fig. 1. As another illustration, let us take the isotherm c equation y = ~ (see Art. 5). Suppose c is unity, so 1 X that y = — When x = 1, y = 1 ; when x = 2, y=\\ x when x = £, y = 2 ; etc. Locating these points and THE GRAPH. 43 drawing a smooth curve through them, the graph appears as in Fig. 1. Two things are clear in regard to this graph: (1) it is related to the ^/-axis precisely as it is to the #-axis ; (2) as x increases without limit, y diminishes without limit, so that the points are nearer and nearer to the a;-axis. The graph therefore shows what the equation says ; namely, that as the volume becomes indefinitely great the pressure becomes indefi- nitely small; and conversely, if the volume could be diminished without limit, the pressure would be indefi- nitely great. We further observe that when x is negative, y is negative ; and thus the complete graph includes a branch in the diagonally oppo- site corner X'OY' (Art. 123). But this second branch represents no actual pressures and volumes, because pressures and volumes are positive. We shall find numerous instances of equations in which the variables, abstractly viewed, have a wider range of values than the values possible for the concrete quantities under consideration. 37. If we like, we may think of a graph as the path of a looint which moves from one determined point to the next one, and thence to the next one in order. The equation y = x + 2, for example, merely says that the point moves so that its ordinate is all the time equal to its abscissa increased by the constant 2. When the graph is thus looked upon as the path of a moving point, the variable coordinates x and y are called cur- rent coordinates. Any equation in two variables may be said to express the laiv of the point's motion in the plane. For brevity we shall speak of "the curve y=f(x)" instead of saying " the curve which the equation y=f(x~) represents." 44 CALCULUS. Fig. 2. 38. Suppose the moving point describes the arc CO 1 of the graph or curve y =f(x). Let P be any point in the path and Q another point. As the moving point goes from P to Q, its abscissa changes from x to x + 8x, and its ordinate from y to y + S?/. Draw PL paral- lel to the ;r-axis. Then PL = &£ and LQ = 8y. Let ZT' be the chord (produced) passing through the points P, Q. —jy is the tangent of the angle which the line TT 1 makes with the #-axis. Now suppose 8x and Sy to become indefinitely small. P and Q must approach indefinitely near to each other, the chord becomes a tangent, and -~~ = -&- PL ax We now have a geometric meaning for the first el v derivative: If y =f(x), -j- is the tangent of the angle which the tangent to the curve makes with the x-axis. The direction of the tangent determines the direc- tion of the curve at the point of tangency. The value of -j- at any particular point on the curve gives us, therefore, the slope or gradient of the curve at that point. THE GRAPH. 45 If a = tan -1 -^, we also have, when P and Q are TO 7 indefinitely near to each other, -7^ == -^ = sin a, and PL dx P $ ds ~57] = 1T = cosa > ds being the elementary arc PQ. 39. The student will at once perceive that the first derivative must be of great use in searching for special features of any graph. For one important applica- tion, let us see what it can tell us about the graph of ax + by + c = 0. Differentiating this expression, a + bf = 0, ax and therefore dy _ a dx b We have here a constant value for the tangent of the angle which the graph of ax + by + c = makes with the #-axis. Accordingly, the slope is constant and the graph can have no bends ; for a bend means change of slope. Therefore ax + by + c = must be a straight line and its own tangent. But ax + by + c = is the general equation of the first degree, and any property proved for it holds for any and every particular equa- tion of the first degree. For instance, the graph of y = x + 2, which seemed in Art. 36 to be a straight line, we now know to be a straight line. Further, dy from y = x + 2 w r e have -j- = 1. Since 1 is the gra- dient of this particular line, we know that it makes an angle of 45° with the horizontal axis. 46 CALCULUS. Again, in the curve y = -, jr = 2' Here tana varies inversely as the square of the abscissa, and is all the time negative. It follows that at every point the tangent to the curve makes an obtuse angle with the #-axis. The angle a is always measured from the #-axis on the right-hand side of the origin, counter-clockwise around to the line which, with the x-axis, forms the angle. Exercises. 40. 1. A point moves in a circle around the origin as a center, with a radius r. (1) The equation to the circle must be x 2 + ?/ 2 = r 2 ; for the abscissa and ordinate are all the while the sides of a right triangle. dy _ x (2) Show that ux -\/?~ — (3) Find the coordinates of the point or points where the circle has a slope of 1. 2. Find the point of tangency when the tangent to y = - x makes equal angles with the axes of reference. Put -¥- = tan 135° and solve for x. dx 3. Show that the curve y = goes through the 1 + x 2 origin. Find its slope at the origin. 4. Construct the curve y 2 = 4 x. Find the point of tan- gency when the tangent to the curve makes an angle of 45° with the a>axis. THE GRAPH. 47 5. Construct the curve y = sin x, making as much use as possible of -^ to determine the slope at various points. The x-axis must here be regarded the circumference of a circle whose radius is unity, straightened to a right line with the origin marked 0° . We easily obtain a num- ber of points on the curve by using the pairs of coordi- nates : 0°, 0; 45°, iV2; 90°, 1 ; 135°, i-V2; 180°, ; 225°, — |-V2, etc. Hence the curve passes through the origin, has a maximum ordinate at 90°, and crosses the x-axis again at 180°. In order to measure off the abscissas, the angles 45°, 90°, 135°, etc., must be expressed in radians. We have the radian 57°.2958 ••• for the unit of distance. The abscissa indicated by 45°, for instance, is = — approxi- J ' ' 57.2958 191 Xi mately. The distance from the origin to the second point 1 80 of crossing is — ^ = 3.14159 •••, and the maximum ordi- nate therefore meets the o:-axis at a distance 1(3.14159 •••) from the origin. From 180° to 360° the values of the sines are a repetition of the values for the first semi-circumference, except that they are now all negative. Hence this portion of the curve is in every respect like the portion from 0° to 180° ; but it lies below the x-axis, and the direction of its convexity is reversed. Since sin(?i7r + x) = sin a?, n being even and positive, it is seen that the curve keeps its sinuous character, crossing the a>axis at regular intervals an unlimited number of times. On account of the repetition over and over again of the series of values of sin x, the function is called a periodic function. The curve itself is known as the sinusoid. 6. Construct the curve y = cos a;. It is obvious in advance that this curve, which might be called the co-sinusoid, must be precisely like the sine curve 48 CALCULUS. or sinusoid ; and that we shall have it in its proper position if we suppose the sine curve moved a distance of 90° to the left along the .T-axis. 7. Find the first point to the right of the ?/-axis where y = sin x and y = cos x cross each other (see Art. 95). Show that the angle at which they cross is 180° — 2 tan -1 ^-V2. 8. Construct y — m sin nx. 9. Construct y = m cos nx. Assign numerical values to m and n\ then give a series of values to x, as in the first case. If a negative value is given to m, the effect is to rotate the curve on the a>axis so that portions which were above are now below, and vice versa. 41. If ' = for some value of x, a = ; hence, to find whether the point describing a curve is anywhere dv moving parallel to the a>axis, we must put -~ equal to CtOu .. zero. Let .r x represent one root of the equation -^- = 0. If a value of x a little less than this root makes ~^- dx positive, and a value a little greater makes it negative, the tangent to the curve must make an acute angle with the #-axis, then become parallel to it, then make an obtuse angle with it ; and the curve must have a bend, being convex upward. The ordinate of the highest point, corresponding to x v is a maximum. So we define a maximum value of a function as a value greater than the value just before it and also than the one just after it. (PB, Fig. 3.) On the other hand, if -~ changes from — to + in passing through zero, the curve is concave upward, and THE GRAPH. 49 the lowest point is the end of a minimum ordinate ; that is, the value of the function is less than the value just before it and the one just after it. (P ; C, Fig. 3.) dv It is evident that if -j- is changing from + to — , — ; and if JL is changing from dx d [dy dx \dx. d (dy dx\dx is to +, is +. 42. A third case arises : If -~ does not change siorn dx h h in passing through zero, there is neither a maximum nor a minimum ; but the point after reaching P or P f takes the path indi- cated by the dot- ted line. The point where -f- = is then dx called a point of in- flexion. In this case dx\dxj Every one is famil- iar with the point of inflexion as a feature in railroads, when the track is concave, say with respect to the fields on the right, and then changes so as to be concave to the fields on the left. Curves containing points of inflexion are very common in architectural forms. Such a curve is then known as an ogee. The same curve may of course have several maximum points and several minimum points, and also points of Fig. 3. 50 CALCULUS. inflexion. Maximum and minimum points must evi- dently alternate. Exercises. 43. 1. Consider the meaning of the statement ~ = cc. ax Examine the two cases : (a) when -— changes sign in pass- cioc (h/ ing through an infinite value ; (b) when -p does not change sign in passing through such a value. 2. Examine the following curves for maxima and minima: (i) y = T^—>' (iii) y = ^ogx. JL - (- X (ii) -, + ^ = 1- (iv) y = 2px. or 0~ 3. Draw the curve ?/ = e x , showing that it lies wholly above the as-axis, crosses the ?/-axis at an angle of 45°, and has no maximum or minimum points for any finite value of x. 44. To illustrate the use of the principles established in Art. 41, suppose Ave know the slant height a of a right cone and wish to find the radius of its base when the volume is a maximum. Let y be the volume and x the base ; then TTX* — Va 2 — x 2 . J 3 x and y, being mutually dependent variables, must ad- mit of graphical representation ; the abscissa of the point tracing the curve or graph is the varying radius, and the ordinate is the varying volume. Hence, if we put -~ equal to zero and solve the equation so formed, THE GRAPH. 51 the value of x obtained will be the radius which gives the maximum volume. Differentiating y = Vr- z z U 3 dx ■K = 3 IT "8 - 2 # Va 2 — a; 2 + x 2 -2x v2Va 2 — # 2 2 # (fa 2 — x 2 ) — x s Va 2 — # 2 putting this expression equal to zero, 2x(a*-x*)-x s =0; hence x = V-| a. That is, the volume of the cone w T ill be greatest when the radius of the base is Vf a. In a case like this it is unnecessary to inquire whether -j- changes sign, and whether the change is from + to — or from — to + . For the volume of a cone of given slant height evidently varies from no volume when the radius is zero, through finite values to no volume again, when the radius is equal to the slant height ; that is, from a cone that is all height and no base to one that is all base and no height. Somewhere between these two extreme cases there must be a cone of ordinary shape whose volume is the greatest possible. It is well occasionally to supplement mathematics with common sense rather than to rely mechanically and invariably on some rule or formula. 52 CALCULUS. Examples. 1. ' Find the altitude of the right cylinder of greatest volume inscribed in a sphere whose radius is r. Alt. -I!. V3 2. Given a point on the axis of the parabola y^—Xfx^ at the distance I from the vertex, rind the abscissa of the point of the curve nearest to it. x = I — 2 p. 3. Find the maximum rectangle that can be inscribed in the ellipse whose axes are a and b. The sides are aV2 and 5V2. 4. A talus resting on a horizontal plane has a slope of 30°; at the top of the talus is a series of strata 5 ft. thick ; the entire height of the ledge is 30 ft. How far must one stand from the foot of the talus to get the best view of the strata ? The angle at the observer's eye, formed by lines drawn to the bottom and to the top of the strata, must be a maximum. Let this angle be «; the angle sub- tended by the talus, /3, and the angle subtended by both talus and strata, 7. Also, let x be the horizontal distance from the observer to a point directly beneath 25 30 the strata. Then tan /3 = — ; tan 7 = — ; and x therefore tan a = 30_25 xx 5x 750 x 2 + 750 ' d 5 (a? + 750) -10 a? and — tan a = — ^ — -+- ^— — dx (x 2 + 750) 2 THE GRAPH. 53 Equating this derivative to zero, x = 5V30, and finally the distance sought is 5 V30 ^|— = 5 V3 ( VTO - 5). tan 30° v J 5. The strength of a rectangular beam of given length, loaded and supported in any particular way, is proportional to the breadth of the section multiplied by the square of the depth. If the diameter a is given of a cylindric tree, what is the strongest beam which may be cut from it ? Let x be the beam's breadth ; then Va 2 — x 2 must be its depth. Hence, if y =x(a 2 — x 2 ), the strength is a maximum when y is a maximum. — = a 2 — 3 x 2 = 0, and therefore x = — — dx V3 In the same way find the stiffest beam which may be cut from the tree by making the breadth multiplied by the cube of the depth a maximum. We now have y = x(a 2 — x 2 )* ; ^ = ( a 2 _ 2.2)1 + z x Q a 2 _ ^)i(_ 2 x) = 0, and x = -. dx 2 — Perry's Calculus for Engineers. 6. The volume of a circular cylindric cistern being given (no cover), show that its surface is a minimum when the radius of the base is equal to the height of the cistern. Let x be the radius and y the height ; then the volume is irx 2 y, which equals a constant, say a. If S is the surface, S = 7rx 2 + 2 irxy = irx 2 -\ , since y = - — 5 - x irx 1 54 CALCULUS. d S Finding — ■ and putting it equal to zero, we have ax TTX 1 *! 1 XT T , • -, and x — y. How do we know that tins IT makes the surface a minimum rather than a maximum ? x* — — = -, and x = y IT 7. Determine the speed most economical in fuel to steam against a tide, supposing the resistance to vary as the nth power of the velocity through the water. Let a denote the velocity of the tide, x the velocity of the steamer through the water ; then x — a will he the velocity of the steamer relatively to the bank. The power required, and therefore the coal burnt per hour, will vary as the product of the resistance and the speed ; that is, as af +1 , and therefore the coal burnt per x' i+1 mile will vary as This is to be a minimum, i t x — a hence we have d_( x n+1 \(n + l)x n (x -a)- x n+1 _ m dx \x — a J (x — a) 2 , x _, 1 x — a 1 and -=!+-, or = — a n an Thus if the resistance is taken to vary as the square of the velocity, the speed past the bank should be half the velocity of the current. — GreenhilPs Differential and Integral Calculus. 8. Let A and B be two point-sources of heat. It is required to find the point M on the straight line AB, which is at the lowest temperature, the intensity of the radiation of heat varying inversely as the square of the distance from the source of heat. Let a be the distance THE GRAPH. 55 between the points A and B, and x the distance from A of the point M on the straight line ; then AM = x, and BM= a — x. Let the intensities of heat at unit distance from the sources of heat be denoted bj^ a and /3 respectively. Then the total intensity of heat co at the point M will be a /3 *> = ~n + x 2 ' (a — x) 2 For a maximum or minimum, dco dx that is, and _2a £ 3 + - 2/3 2 — x) z (a — 2:) 3 /8 a? I « a — X ^8 o, V # The distances i?ikf and Jl71T have, therefore, the same ratio as the cube roots of the corresponding heat intensities. Solving for x, _ aV« In this case it is necessary to see whether the value found corresponds to a maximum or a minimum. Dif- ferentiating the expression for — , we have dx d 2 co__2-3a 2-3/3 dx 2 x^ (a — #) 4 56 CALCULUS. which is positive for all values of .r, including the value a-fy" be the * Nernst and Schonflies' EinfuKrung in die mathematische Beliand- lung der Naturwissenschafien. THE GRAPH. 57 angle made by the radius vector OP and the tangent to the curve ; then tan 6 = ~^~ = -=— . QP' dr Whenever the radius vector r is a maximum or mini- mum, the tangent at its extremity must be at right angles to it ; that is, rdO dr A dr rdd Points for which r is a maximum or minimum are called o A apsides. To find, / FlG 4 therefore, whether I u r a given curve has an apsis, we must put = and solve this equation. For example, let us take the polar equation to the ellipse, the pole Tbeing at the right hand focus (see Art. 115). _ a (1 — £ 2 ) -j dr __ a (1 — 6 2 ) e sin 6 u 1 + e cos ff an Id ~ (1 + ^cos^) 2 ; 1 dr & sm then — 77T = q n\ and equating this to zero, r dO 1 + e cos ^ 6 sin 6 = 0. Hence, the apsidal values of are 0° and 180°. These results agree with what we observe in an examination of the given equation to the ellipse : r is a maximum, a(l + g), when 0=180°, and a minimum, a(l -O, when = 0°. The student who is unacquainted with the formal analytic geometry of the straight line and the conic section is advised to read Chapter IV before beginning the next chapter. CHAPTER III. APPLICATIONS. 47. In the mathematical sciences one of the most common of fundamental variables is time ; and when the function of time is the space passed over by a body, the first and second derivatives — and — ( — - ) are of great , tit dt\dt h importance. N Suppose a body moves over equal spaces in equal times. The space divided by the time gives the speed or velocity of the body. That is, if s is the space passed over in the time t, - is the velocity of the body. " While the camels were being loaded, T measured my first base- line of 400 metres. Boghra (my riding camel) walked it in five and one-half minutes. This was a daily recurring task, for the contours of the ground varied a good deal, and the depth of the sand made a very appreciable difference in the time the camels took to do the same distance." — Sven Hedin's Through Asia, Vol. I, p. 482. In this illustrative case, - = — — = the speed of the t 51 camel expressed in metres per minute. Assuming that o - was a constant during each day, the distance travelled u on any given day by Hedin's caravan was known by multiplying the speed by the time spent in travel. 58 VELOCITY. 59 48. If the motion is variable so that the body does not move over equal spaces in equal times, we may obtain an expression for velocity by taking the time so short that during that time the motion must be uniform. So if dt be an indefinitely short time and ds the indefinitely small space passed over in that time, — is the velocity and is measured by the space that would have been passed over in a unit of time if the body had kept on moving for a whole unit with the velocity which it had at the instant considered. For instance, if we say that a train is running at the rate of 30 miles an hour, we mean that if it were to run for a whole hour with the same speed which it has at this instant it would pass over a distance of 30 miles. As a matter of fact it may stop in a few minutes ; that has nothing to do with its speed at this instant. But 30 miles per hour is the same as 1 mile in 2 minutes, or 4.4 feet in .1 of a second, and so on. Evidently the rate remains the same so long as the ratio of the space to the time is the same, however small the space and the time may be individually. Hence, in this case, — = 30 miles per hour. dt L If we know the whole space passed over by a body and know also the time taken, the space divided by the time is the average velocity: it must not be confused with the velocity proper, which may have varied during the time. For example, the first mail cartridge sent by compressed air from the Boston post office to the North Union Station (Dec. 17, 1897) required 1 minute and 2 seconds to pass from one place to the other, a distance of 4500 feet. The average velocity was 72.58+ feet per second. 60 CALCULUS. 49. If a body is moving in a northeasterly direction, it plainly has a motion eastward and a motion northward. For instance, if it is moving due northeast with a ve- locity of 20 miles per hour, it is getting eastward at the rate of 20 cos 45° miles per hour, and northward at the same rate. If it is moving east 30° north at the rate of 20 miles per hour, it is moving east at the rate of 20 cos 30° miles per hour, and north at the rate of 20 cos 60° miles per hour. In general, if a body is moving with a velocity v along a line which makes with the .r-axis an angle of a degrees, its component velocity parallel to the #-axis is v cos a, and its component velocity parallel to the y-axis is v sin a. We have already seen (Art. 38) that — = cos a 7 7 ds (1 1J Ci S and — ^ = sin a. Hence, if -— is the velocity of a body at ds at , . CIS (XX CtX -\ CtX ■ ,i c any instant, v cos a = — . — - = — ; and — - is theretore dt ds dt at the component velocity parallel to the #-axis. Similarly, v sin a = — . — ^ = -^ = the component velocity parallel dt ds dt r J L to the y-axis. Evidently a velocity parallel to any line furnishes a component velocity parallel to any other line if it be multiplied by the cosine of the angle between the lines. 50. Suppose a particle is moving in a plane curve and we wish to know its component velocities at any instant (1) along the radius vector, and (2) perpendicular to the radius vector. We have x = r cos 6 and y = r sin 0, in which x, y, r, VELOCITY. 61 and 6 depend upon the time t. Differentiating with t as the fundamental variable, dx _ dr di~~dt cos i sin 6 dd dt dy dr . a , add -2- = — sin u + r cos u — dt dt dt dx O) dx According to the preceding article, — is the velocity dx parallel to the #-axis and ^ cos 6 is the component which it furnishes along the radius vector. Similarly, -2 sin 6 is the component which -2 furnishes along the radius vector. The sum of these components is the Fig. 5. whole velocity along the radius vector. From equa- tions (a) and (b) we have dt dt at O) ^COS0- dx . n dO sm v = r — dt dt dt 62 CALCULUS. Again, resolving along a line perpendicular to the radius vector and combining the parts, oo The reason for the minus sign in the first member of equation (Y?) should be noticed. The velocities -f- cos 6 and — sin 6 are oppositely directed (see dt dt Fig. 5); hence, when combined, their difference must be expressed. 51. -t\-t\ the rate of change of a variable velocity, is called acceleration. — ( — ] and — ( — ) are the accelerations parallel to dt\dt) dt\dtj l the #-axis and ?/-axis respectively ; and we can now find the component accelerations (1) along the radius vector, and (2) perpendicular to the radius vector. Differentiating equations (a) and (J) of the preced- ing article, d 2 x It 2 '' d 2 y It? — --r — - cos0 — 2 — — +r— — sin (9, (e) _dt 2 \dt) J \ dt dt dt 2 ) r ( sm 0- 2- — + r — cos0. (/) _dt 2 \dtj dt dt dt 2 ) Multiplying equation (/) by sin #, and equation (e) by cos 6 and adding, we have for the acceleration along the radius vector. ANGULAR VELOCITY. 63 Again, multiplying (/) by cos (9, and (e) by sin and subtracting the latter product from the former, we d 2 y « d 2 x . n dr d6 , d 2 , 7 , —f cos - — ■ sin = 2 — — + r — - (h) dt l dt 2 dt dt dt 2 for the acceleration perpendicular to the radius vector. It is to be noticed that the second member of equa- tion (li) may be written - • — ( r 2 — )• J r dt\ dt) 52. Angular velocity is defined as the ratio of the angle differential, d0, to the time differential, dt. This d0 ratio, — , may be a constant or a variable. For ex- cic ample, the earth rotates on her axis with constant angular velocity, and — = — — ; but she moves in her s J dt 2-i h orbit around the sun with a variable angular velocity. (See Art. 88.) 53. As an important application of the results given in equations (£. If $ starts at some point #', and £ is the time required to move from Q 1 to A, the angle #' OA = cot ; hence, counting the time from the start at Q', the angle Q' OQ = cot and the angle A OQ = cot — cot ; and therefore y = r cos (jot — o)£ ) = r cos (o>£ + e) if we write e for the constant, — cot . 56. In regard to the motion of P, we notice at once that it must cross the circle on the diameter AA f and return to A in the time that Q is describing the circum- ference ; so its motion is vibratory. It starts with zero velocity, and must be going with its greatest velocity when at the center; for its direction of motion is then parallel to that of Q. To get a more precise knowledge of the motion of P, let us take e = 0, so that y = r cos cot. By doing this the equation gains in simplicity and the motion remains the same, but the time is counted from the instant when Q is at A instead of Q f . We now have y = r cos cot, (a) -Jl = — rco sin cot. (b~) dt V J -JL = — rco 2 cos cot. (c) dt\ Equation (6) shows that the velocity of the point is greatest when cot = 90° ; that is, when P is at the cen- 66 CALCULUS. ter. Equation (V) shows that the acceleration is great- est when cot = 0° and 180° ; that is, at the start and at A! . Also, the acceleration is least when cot = 90°. 57. The variation in the ordinate OP may be best appreciated by noticing the identity of the equation y = r cos cot with the equation y = m eosnx given in exercise 9, Art. 40. Equation (a) accordingly repre- sents a cosine curve. Further, if the velocity equation (6) be graphically shown, its curve must be the sinus- oid. And finally, the acceleration equation ( 2 , which has replaced the coefficient r in equation (a). It is w^ell worth the student's while to construct care- fully the graphs for the three equations (a), (/>), (V), using the same unit of length for all three. The usual #-axis now becomes a time axis in each case, sii\ce the abscissas are times. The y-axis for (a) is a displace- ment axis ; for (J) it is a velocity axis ; and for (c) an acceleration axis. 58. We are now familiar with the geometrical mean- ing of —^ when y =/(#). If y =/(£), — ^ is analogous 7 (XX Ct'L to -^, and must have the same geometrical meaning. dx That is, viewed geometrically rather than kinematically, -^ is the tangent of the angle which the tangent to the Ctv curve y = f (t) makes with the £-axis. Accordingly, equation (6), Art. 56, might be called the curve of the tangent to (a) ; for any ordinate (with the abscissa t f ) SIMPLE HARMONIC MOTION. 67 in the graph of (5) represents the magnitude of the slope of (a) at the point whose abscissa is t'. Evidently the curve of (c) is related to (6) just as (&) is to (a). 59. The point P, vibrating back and forth across the circle (Fig. 6), is said to have simple harmonic motion. It is such motion as this that Jupiter's satellites seem to have as we look at his orbit " edge on." The range OA or OA! on one side or the other of the middle point is called the amplitude ; and the ordi- nate OP is the displacement. The period of a simple harmonic motion is the time which elapses from any instant until the point moves again in the same direc- tion through the same position ; that is, the time required by P to move from P ! to A! , thence back to J., and finally to the initial position P f , is the period. The phase is the fraction - — of the period of vibration. 2 7T The epoch is the angle e. "This expression y = r cos (wt + e) is to be found, perhaps more frequently than any other, in all branches of mathematical physics. It is in terms, or series of terms, of this form that every periodic phenomenon can be described mathematically. From the expres- sions for the longitude and radius vector of a planet or satellite to those of the most complex undulations, whether in water, in air, or in the luminiferous medium, all are alike dependent upon it." — Tait's Dynamics. Example. Find an expression for the up and down motion of the connecting-rod of a locomotive. 60. The downward fall of an unsupported bocly is due to the accelerating force exerted by the earth and known as gravity. At small distances above the earth's 68 CALCULUS. surface this force is practically constant ; the accel- eration caused by it is denoted by g. When g is de- termined at different places on the earth, it is found to vary within narrow limits. This variation is due to several causes, the chief one being the rotation of the earth on its axis, g has its least value at the equator and its greatest value at the poles. At Washington, D.C., g is 980.098 dynes ; * that is, the observed accel- eration due to gravity is, at that point on the earth's surface, 32.155 feet per second. Taking the origin at the point from which a body falls, with the positive end of the ^-axis downward, we now have a fdy\_ Jt\dtJ 9 ' therefore, after integrating, cl l = v = gt+ C. (Art. 15.) at If the body falls from rest, v = when t = ; there- fore (7=0, and the equation becomes !-* w Multiplying by dt and integrating again, y = ±Cjt*+C l . Since y = when t = 0, O f = ; therefore y = \ gt 2 . (5) * U. S. Coast and Geodetic Survey. FALLING BODIES. 69 Combining equations (a) and (b) so as to eliminate t, Equation (e) enables us to find the velocity with which a body is moving when it has fallen through a given space. For example, the monument at Washing- ton is 555 feet high ; if a ball is dropped from the top, what is its velocity upon reaching the ground ? We may take g = 32, a value sufficiently accurate in this example and similar ones. Then v = 8V555 = 188 feet per second, approximately. 61. If the body is projected directly upward, dt\dt) *' because the acceleration is now a retardation tending to diminish y. Integrating as before, If the body is projected with the velocity V, ■-$- = V when t == ; therefore C = FJ and the equation becomes Multiplying by dt and integrating again, and since y = when t = 0, (7' = ; and we have 70 CALCULUS. Combining equations (J) and (e) so as to eliminate t, 62. By means of the equations of the two preceding articles we can readily show that if a body is projected vertically upward, it takes the same time to come down that it does to go up ; also, upon reaching the point from which it was projected, it has the same velocity as that with which it was projected. From (d), when dt g time up ; from CO, when dt ' V 2g space up ; from (5), when -0 9 time down ; from (e), when V 2 2 9 In the derivation of formulas (a) to (/), no account has been taken of the resistance offered by the air to the fall or rise of a body. The formulas are strictly true only on the supposition that the acceleration is constant, and that the motion takes place in a vacuum. 63. We may now consider the case when the height is so great that the acceleration cannot be regarded as constant. What is the velocity of the body on reach- ing the earth ? A homogeneous sphere, or a sphere composed of con- centric layers with the density varying only from one layer to another, attracts an external body with an FALLING BODIES. 71 intensity varying inversely as the square of the dis- tance of the bocty from the center of the sphere. Let g be the acceleration due to the earth when the body is at the earth's surface, and/ the acceleration at the dis- tance y from the center. (Notice that the center thus becomes the origin.) Then, if R is the earth's radius, gB? 9' R2 +1 f = — k ; that is. y 1 ./ = and therefore we have d ( d y\ _ dt\dt) V 2, The minus sign is taken because y is diminishing as the body falls ; that is, dy is negative, and since -^ is increasing numerically, -q-\-j~) must also be negative. If we multiply by dt, as in the previous articles, and attempt to integrate, we have dt~J y* dt ' an indicated operation which cannot be performed un- less we know what function y is of t ; and this we do not know in advance. But multiplying by dy instead of dt, d fdy\ __ dy fdy\ __ gR 2 d n\tt) = iti d Kdtr-^ cly - The first member is immediately integrated by ob- serving that it is of the form xdx, and that /■ JU CI Jb c) 72- CALCULUS. We have, therefore, gjr If the body falls from the height h above the earth's (Jb II (1 RP* surface so that y = R + h when -^- =0, C = — -~ — r > at R + h and the equation becomes yw= qm (\ L_ 2\dtJ y \y R + h The same result is reached by writing a definite integral (Art. 16) whose limits are R + h and y. We then have 1W = p _/z^ : X 2\efc/ «^+a */ 2 = -<^ dy 1" = -^ 2 - = + ■ ^ 2 # -B + A. \y R + h Suppose that dy_ dt v l when y = R\ that is, v 1 is the velocity which the body has when it reaches the earth's surface. Then iw=#-„- R R + h = gR R + h ■ gh R R + h FALLING BODIES. 73 and therefore ^V^-i+I- r m If h_R, the series is diverging; the formula con- taining it cannot therefore be used, and we return to one of the other expressions. For example, suppose a body falls from an indefinitely great distance ; what will be its velocity on reaching the surface of the earth, all forces besides the earth's attraction being disregarded ? We have £ « that is, # = a cos V/x£. Comparing this result with equation (a), Art. 56, we conclude that a body subjected to an attractive force varying directly as the distance will move with simple harmonic motion. 65. Suppose the body is driven away from the origin by a force varying directly as the distance of the body. and proceeding as before, I — - J = /jlx 2 + C = fji (x 2 — a 2 ) ; (a) that is, Vfidt -Vx 2 — a 2 Integrating again, we have tVJi+ C r = log (x + Vx 2 - a 2 ) . Notice that the constant of integration m&y be written in either member of the equation as suits our conven- ience. Heretofore it has been written in the right-hand member. 76 CALCULUS. Now suppose that x = a when t = ; then . C f = log a, and tVfi + log a = log (x + V^ 2 — a 2 ), , /- i (x + Va 2 — a 2 \ ^V/x = log — ]. \ a From this expression we have „* J 2 Equations (5) and (c) show that as t increases, the body is driven farther and farther from the origin with ever increasing velocity. These equations involve the RECTILINEAR MOTION. 77 supposition that the initial velocity is zero. Let us now suppose that the initial velocity is — aVft. Resuming the equation , 7 xo since [_p\== — aVa when x = a. (7=0: \dt) and the equation becomes fdx^ 2 dx /— , or — = — V udt, x the minus sign being used because the motion is toward the origin. We now have log x = — Vfit + G\ and since x = a when t = 0, C f = log a ; therefore — V/jit = log --> and x = ae'^* 1 *. This equation shows that with the initial velocity — a V/x the body constantly approaches the origin, but never reaches it. 66. Suppose that a body instead of being projected vertically, is projected in a direction making the angle a with the horizontal plane, V being the velocity of pro- jection. The body thus has a vertical velocity and a horizontal velocity. The horizontal velocity is evi- dently unaccelerated, whilst the vertical velocity is being retarded by gravity. That is, taking the hori- 78 CALCULUS. zontal side of the angle a for the #-axis, and taking the y-axis vertical and positive upward with the point from which the body is projected as origin, d fdx\ _ n . d fdy\ _ dt\dtJ~ ; It\di)~ 9 ' These two statements are the " equations of motion " of the body. Examples of such equations have already occurred in preceding articles. Integrating the first (1 v one, — = J^cos a, the constant horizontal velocity. In- tegrating again, x = tv cos a, {a) the constant of integration being zero if t = when x= Q. Integrating the second equation of motion, When t = 0, the time of projection, -^ is the vertical 1 J dt component of the velocity for the same instant. This initial vertical velocity being F~sin a, we have — ^ = — at + F"sin «, dt J and integrating again, y = — \gt 2 + t V sin a. (J) Equations (a) and (b) give the coordinates of the body at any time t. Eliminating £, we have y = x tan a - f x\ (» 2 F 2 cos 2 a the equation to the path of the body. PARABOLIC MOTION. 79 67. If we transform equation (e) by passing to a new pair of axes parallel to the first with V 2 sin a cos a V 2 sin 2 a 9 2 9 for the coordinates of the new origin, we have (Art. 100), Y^sin 2 a f , F 2 sin a cos a y H - = tan a[ x -\ 2 9 V g g_ f F" 2 sinacos^ 2 2 V 2 cos 2 a\ g After reduction this becomes « 2 F 2 cos 2 a x 2 = y, 9 which is seen to be a parabola convex upward with its vertex at the origin of coordinates. (Art. 129.) This curve is approximately shown in a stream of water issuing from a hose. It may also be traced by watching a tennis-ball or base-ball as the ball moves through the air. 68. To find the horizontal range, we put y = in equation (f— r 2 i\/sin 2 _:_»0 The expression here presented for integration looks quite simple, but it cannot be expressed in finite terms by means of the ordinary algebraic or trigonometric func- tions. If, however, we expand sin 2 - by Maelaurin's 2 theorem (Art. 34, ex. 14), and then take a so small that we may neglect powers of a (and 0) beyond the second, we shall have 4j"sin 2 |-sin 2 |) = « 2 -^ 2 . 84 CALCULUS. The above integral then becomes and this is integrated by formula ll r Chap. V, so that we obtain C Vf- a = cos" 1 cos -1 - = cos L - a a a a Solving for 0, = a cos y k When (9=0, -^2* = cos^O = -; hence f , the time from J. to (9, is —\— 2 V If T 7 be the time of an oscillation from J. to A 1 (on the other side of 0), This result is true only when a is small, as above shown. It is independent of «, and therefore the time of an oscillation is the same for all small arcs in the same circle. That is, if a and a! are two small but unequal arcs, the times of oscillation for the same pendulum are equal. 72. It will be noticed that the equation 6 = a cos\f- t AREAS. 85 is of the form of the equation expressing simple har- monic motion ; therefore the pendulum-bob. has simple harmonic motion in a circle which lies in a plane pass- ing through OX perpendicular to OF. The radius of this circle is «, and the displacement at any time t is 6. If a is given in degrees, it must be divided by the radian (57°. 295779 •••). For instance, if I = 50 inches and a = 1°, the radius of the circle across which the harmonic motion takes place is — - — i- = — inches, . , , L 57.°+ 57 approximately. 73. Areas. Let PS be a portion of the curve y =f(x) ; and let it be required to find the area bounded by this arc, the ordinates PM and SN, and the #-axis. Fig. 8. Let OM=a, ON=b, OT=z, and OV=z + 8p; then QT=y, and RV=y + 8y. If the area OLQT, any varying portion of the area OLSN, equals ^4, area OLR V=A + 8 A, and 8A = TQR V. Now, if the short 86 CALCULUS. arc QR were a straight line, the area TQRS would be a trapezoid, and we should have 8A = Sx±(QT + RV)= Sx(y + ±8i,) ; and — = y + \ Si/, ox In the limit QR becomes a straight line, and dA that is, dA = ydx = f(x)dx ; and this is a representative strip taken anywhere in the area OLSN. Suppose \f(x)dx = (x) + 0; then A = 4>(x)+C. Since we are measuring areas from the y-axis, when x = 0, A = ; when x = a, JL = area OLPM; when # = 6, -4 = area OLSN; therefore area OLSN= <£(J) + (7, area OLPM=(a) + O. Subtracting this last expression from the one pre- ceding it, area OLSN- area OLPM= mmMPSN f(x)dx. a ABE AS. 87 We have, then, for the area between the ordinates, whose distances from the ?/-axis are a and b respec- tively, the definite integral r f(x)dx. 74. For example, suppose we wish to find the area bounded by the parabola y 2 = \px, the a>axis, and any ordinate y r (the accompanying abscissa being x' ). A= Cydx = C x 2p i x*dx: 4:p 2 X 2 3 4 /V)2/v^2 We notice that the rectangle x'y f = 2p 2 x' 2 ' ; hence the area in question equals two-thirds the circumscribed rectangle. Examples. 1. Find the area of the upper right-hand quarter of the ellipse •— + &-■ = 1. In this case o ydv=J ~^Ja 2 -x l dx h \ x r* 2 , a2 • -l ^~ _ 7raS The area of the entire ellipse is therefore it ah. ira 2 , the area of the circle x 2 + y 2 = a 2 , may now be viewed as a special case of the ellipse in which b = a. 88 CALCULUS. 2. Find the area between the isotherm pv = c, the v-axis, and the two ordinates whose distances from the #-axis are a and b respectively. Aits, c log — a 3. Find the area bounded by the #-axis and the curve y = sin x, from x = 0° to x = 180°. Ans. 2.. 75. Mean values. The mean or average value of n quantities is the nt\\ part of their sum. If the quanti- ties to be averaged are successive values of a function of some variable, their magnitudes depend not only on the nature of the function, but also on the law of varia- tion of the fundamental. Thus, suppose we have the isotherm pv = c and wish to know the average pressure between the volumes i\ and v 2 . It is necessary to make some assumption in regard to the variation of v. (1) If its increments are supposed equal, we understand by the "mean value" of the pressure the average of the press- ures corresponding to the arithmetic series : v, v + civ, v + 2 civ, etc. (2) If the volume is assumed to depend on some other variable in such a manner that the abscissa increments are not equal, the mean value will now be the average of a new series of pressure ordinates corresponding to the new series of values of v arising under the second assumption. Evidently the two means will, in general, be unequal ; but one is just as properly the average as the other. An important illustration is afforded if we ask : what is the mean distance of a planet from the sun ? If a planet moved in its elliptic orbit in such a way that the radius vector described equal angles in equal times, that is, if — , its angular at MEAN VALUES. 89 velocity, were a constant, the mean length of its radius vector could be shown to be aVl — e 2 , a being the semi- major axis, and e the eccentricity of its orbit. But we know that the law of gravitation requires that the areal velocity shall be a constant ; that is, the radius vector describes equal areas, instead of equal angles, in equal times (Art. 88). In one case — is constant; in the other, A r 2 — is constant. A little consideration will 2 dt show that the mean value of r cannot be the same in the two cases. 76. If y=f(x) and all of the dx's are equal, the average length of y between x = a and x = b is at once found by dividing the area I f(x)dx by b — a; for returning to Fig. 8, if the area MPSN be divided by its base b — a, the quotient is the altitude of an equiva- lent rectangle of base b — a] and the altitude of the rectangle is the average altitude of the strips repre- sented by TQRV\ that is, of the ys. Examples. 1. Find the average length of the ordinates of a semicircle, supposing the series taken equidistant. We have x 2 + y 2 = r 2 ; or, y = Vr 2 — x 2 ; therefore M= — I Vr 2 — x 2 dx =+7rr. From this result it appears that the average ordinate equals the length of an arc of 45°. 90 CALCULUS. 2. Find the average length of the ordinates, sup- posing they are drawn through equidistant points on the circumference. In this case 2r 1 C n . = — I r si M=- rsinddO IT 3. Given pv = c ; show that the mean pressure between the volumes v x and v» is log- 2 , v chang- ing by equal increments. 211 4. A particle has simple harmonic motion. Find its mean velocity as it passes from the extremity of the radius to the center of the circle. 77. The above geometric conception of mean values may be adopted when a function is expressed in polar coordinates. If r =/(#), let x be written for 6, and y for r, so that we have y=f(x). This equation furnishes a curve which sustains peculiar relations to the original polar curve. The radii vectores lose their fan-shaped arrangement, and are placed parallel and equidistant (if 6 is an equicrescent variable) with their extremities 011 a common line, the :r-axis. The pole may be viewed as developing into this axis, — just as if a draw-string were let out, — while a circle of unit radius with the pole as center develops into a straight line parallel to the #-axis, the radii vectores keeping their position of perpendicularity with respect to the circumference of the circle. The mean value of the radius vector then 1 r b becomes I f(x) dx, as before. b — aJ* WOBK. 91 For example, to find r , the mean length of the radius a (\ _ e 2\ vector of the ellipse r = — - '-* 6 being an equi- F 1 + e cos 6 H crescent variable, we have, using one-half of the ellipse, 1 r«a(l - e 2 ) 7 r A o r = — I — ^ —ax = a VI — e*. irJol + e cos # The radii vectores, now in the role of ordinates, are distributed at equal intervals through an area A whose base is 7r. Example. Find the average length of the radius vector of the cardioid r = a(l — cos #). M=- \ a (1 — cos #) cfe = — 7T«yo 7T smo; = a. 78. Work. If a force _F acts on a body of mass m, giving it an acceleration — -, XT d 2 s F=m dT* Multiplying by ds, Fds = m — f — )ds. dt\dtj Integrating between the limits v and V, V being the velocity when s = 0, and v the velocity when s = s, If F= 0, fFds=±mv 2 . 92 CALCULUS. Fds is defined as the work done on the body m as it is moved through the space ds. I Fds is the work done in moving the body over the arc s. J mv 2 is defined as the kinetic energy which the body possesses because work has been expended upon it, the kinetic energy representing the work stored up in the body. In order to perform the operation indicated by I Fds we of course need to know what function F is of s in case F is a variable depending on s. Suppose that F=■)■ Similarly, the surface may be found by noticing that the arc 8s generates the frustum of a cone whose sur- face is known from elementary geometry to be or, in the limit, 2iryJs\ so that, if S is the area of the surface, / , 7 N0 dy\ 2 H>W 1+ @D* Examples. 0-2 n,2 1. The ellipse ^- + f- = l revolves about its major axis. What is the volume generated ? 1 V= ( iry 2 dx = ( it — (a 2 — x 2 ~) dx = — - I (a 2 — £ 2 ) dx Jo Jo a 1 a 1 Jo irb 2 [ 2 VOLUMES AND SURFACES. 95 The entire volume is therefore |- irab 2 . The volume of the sphere, ^ 7m 3 , is a special case, in which b = a. 2. Find the area of the surface generated as the ellipse revolves about its major axis. %s +ffi* 2irb — CV&-(a 2 -b 2 )x 2 fdx a 2 Jo = irb\ b + w ^J a 2 — b 2 therefore the whole surface is w • _iVa 2 — b 2 ' sin a 2irb = 2irb b + b + . _iVa 2 — b 2 _sm / a 2 _ ^2 a a 2 1 6" — — — cos - - Va 2 - b 2 a. 3. Find the area of the surface of a sphere whose radius is a. If we make b = a, we have, from the result in the preceding example, for the area of the surface of the sphere, 2 ira a + a 2 cos - - a ~Va 2 — a 2 _ We must now find the value of 2ira a + a 2 ) Va 2 — b 2 when b = a. 96 CALCULUS. Treating b as a variable and applying the principle of Art. 32, 1 b - COS - - a ,4( C0S ~ ,7 ■31 Va 2 - b*J b=a — Va 2 - db w> 2 ^-5 n i b Va 2 -6 2 Therefore, $ = 2 7ra ( a + a 2 - ) = 4 ira 2 . This result for the area of the surface of a sphere agrees, of course, with the one obtained by the method of elementary geometry. 82. The area integral I f(x) dx represents the sum %sa of strips whose height is y and breadth dx. We may reach the same result by starting with the elementary rectangle dxdy and using two integral signs, — one to indicate that we add such rectangles together to make a strip y in height, and a second to indicate that the strips are to be added together, making the area from a to h (Fig. 8). For example, the area of the ellipse may be found by adding together the areas dxdy from the major axis to the curve itself ; then adding together the strips from the minor axis to the end of the major axis. To indicate this double operation, we write - I I dxdy, using the right-hand integral sign with dy. DOUBLE INTEGRALS. 97 Performing the first operation, I dxdy = I ydi •/} *^0 The remaining part of the work is the same as in Art. 74, example 2. The above procedure in finding areas involves what is known as a double integral. Similarly, three succes- sive indicated integrations constitute a triple integral. Examples of double integrals will occur in subsequent articles. 83.* Suppose a point to travel once round the closed oval area J., an indicator diagram, for instance, so as always to have the interior of the curve on the left Fig. 9 hand. Let B be the minimum point, and C the maxi- mum point with respect to the #-axis ; D the minimum point, and E the maximum point with respect to the ^/-axis. * Greenhill's Differential and Integral Calculus. 98 CALCULUS. Then A = ( J dxdy = I xdy, taken round the perimeter of the curve. From B to O along BPC, dy is positive, and fxdy = area MBP ON. From (7 to B along C(?Z?, tf ?/ is negative, and j*xdy = - area 3IBQCJST; so that, taken round the curve, fxdy = area MBPCN- area 3IBQCN= A, the area of the closed curve. But 1 1 dxdy = ( ycfo ; and from E to D along EPD, dx is negative, so that J^/.r = - area LEQJDK; and from i) to JE along DBE, dx is positive, so that fydx = area LBBBK; and therefore, taken round the curve, I ydx = — A. Therefore taken round the curve, j (ydx + xdy) = ; and ydx + xdy = d (xy) is called a perfect differential. Its integral between two limits is independent of the intermediate values of x and y and of the path described MOMENT OF INERTIA. 99 between the limits; so that, taken round any closed path, the integral is zero. When Fig. 9 represents an indicator diagram, and KL the reduced stroke of the piston, while the ordinate y represents the pressure of the steam, the pencil will describe the contour with the area to the left, when the steam pressure is urging the piston from L to K. The diagram taken on the return stroke from the other end of the cylinder will be described in the opposite sense, with the area on the right hand of the describing pencil. 84. Moment of inertia. When a rigid body rotates about an axis, the linear velocity of any particle of the boc ^ is ds rae v = — = = r&), dt dt a) being the angular velocity of the particle, and r its distance from the axis. Its kinetic energy of rotation is therefore ^ mv 2 = ^ mr 2 co 2 , m being the mass of the par- ticle ; and the kinetic energy of rotation of the whole body is 1 mv i _|_ i m V 2 + | m n v ,!2 + ••• = 2 ^ mv 2 = 2 ^ mco 2 r 2 = ^ ft) 2 Smr 2 ; that is, one-half the product of the square of the angular velocity and 2mr 2 . The symbol 2 is used to indicate a pol} r nomial in which the terms are similarly constituted, as in the case before us. Since such an expression as 2 J mv 2 is in reality a polynomial, only common factors can be re- moved and placed before 2, the symbol of summation. Thus in 2 \ mco 2 r 2 , \ is, of course, a common factor ; co 2 is a common factor because the rotating body is supposed to LofC. 100 CALCULUS. be rigid, and consequently all of its parts have the same angular velocity ; but m is not a common factor because it is not supposed that all of the particles have equal masses ; neither is r a common factor, for the particles are at different distances from the axis of rotation. The quantity 2 mr 2 is called the moment of inertia of the body with respect to the axis, and is seen to be the sum of the products obtained by multiplying the mass of each particle by the square of its distance from the axis. If a body rotates with a given angular velocity about different axes, the kinetic energy of rotation with respect to any axis must be proportional to 2 mr 2 ; consequently, the moment of inertia measures the capacity of a body to store up kinetic energy during rotation about the axis with respect to which the moment of inertia is taken. Examples. 85. l. A sheet of metal, rectangular in shape and of uniform density, is made to rotate about an axis coinciding with one end. What is its moment of inertia ? Take the axis of rota- tion for the #-axis with the origin at the left-hand corner of the rectangle. Let b be the breadth and -2 % d the height of the rec- tangle. If p is the density of the metal, pdydx is the mass of the indefi- nitely small rectangle dy dx cut anywhere from the i Fig. 10. MOMENT OF INERTIA. 101 sheet ; and (pdydx^y 2 is the moment of inertia of this small piece. Hence the moment of inertia of the entire sheet becomes i Jo ^ f ^ = al yUydx = phj*fcly = f*f. From this example it is plain that in all cases in which the density is constant throughout the body, the density factor may as well be set aside until the integration is completed. If, however, the density varies from point to point, so that p is some specified function of x and y, it must be kept under the sign of integration and be taken account of in the process of integrating. 2. A straight slender rod of length Z, whose density varies directly as the distance from one end, rotates about an axis perpendicular to it and passing through the end having the least density. What is the moment of inertia with reference to this axis ? Take the given axis as the #-axis, with the origin at the end of the rod. pccy ; therefore p = ky if Ar is the density at a unit's distance from the end. Then the moment of inertia is £py 2 dy = JkyHy = — • 3. Find the moment of inertia of a circle with refer- ence to an axis through its center and perpendicular to it, p being a constant. 102 CALCULUS. Let R be the radius of the circle, r the distance of any particle from the axis, and 6 the variable angle measured from some chosen radius. Consider an ele- mentary portion bounded by the circles whose radii are r and r + dr, and by the radii forming the angle d0. In the limit this bit of area becomes the rectangle (rd0)dr; hence, the integral is C ' C \\rdrd6) = 2tt Crhlr=-W; and therefore the moment of inertia is ™ 2 86. Kepler's laws. It is shown in works on the determination of orbits* that the equations for the un- disturbed motion of a planet or comet relative to the sun are : g + F(l + ™)£ = 0, (1) g + F(l + m)5 = 0, (2) g + *. ( l + Bl )* = 0, (3) in which x, y, z are the coordinates of the heavenly body referred to the sun as origin, — -, — f, — - are dt l dt l dt z the accelerations parallel to the three axes of reference, r is the distance of the body from the sun, k 2 is the mass of the sun, and m the ratio of the mass of the body to the mass of the sun. Having these three equations, we * Watson, Theoretical Astronomy; Dziobek, Planeten-Bewe- gungen ; Tisserand, Determination des Orbites. KEPLER'S LAWS. 103 can at once establish Kepler's laws. Arts. 87, 88, 89, 90, 93 are taken, with slight changes, from Watson's Theoretical Astronomy. 87. If we multiply equation (1) of the preceding article by y, and equation (2) by x, and subtract the last product from the first, we shall have, after inte- grating the result, xdy — ydx _ n Jt =C ' C being the constant of integration. In a similar manner we obtain xdz — zdx _ pi ydz — zdy _ „,, dt dt If we multiply these three equations respectively by 2, — y, and #, and add the products, Cz-C f y + C"x=0. This is the equation to a plane passing through the origin of coordinates (Art. 142). Since x, y, z are the coordinates of the heavenly body, it must remain in this plane. The path of the heavenly body relative to the sun is therefore a plane curve, and the plane of the orbit passes through the center of the sun. 88. If we multiply equations (1), (2), and (3) re- spectively by 2 dx, 2 dy, and 2 f' S t . (16) A 2 (1 + m f ) V y Comparing equations (15) and (16), we see that (1 + 111) T 2 = a 3 ,.rj, (l + m')r' 2 a' 8 " ^ ' If the masses of the two planets are very nearly the same, we may take 1 + m = 1 + ?n' ; and hence, in this case, it follows that the squares of the periodic times of tivo planets are to each other as the cubes of the semi- major axes. (Kepler's third law.) CHAPTER IV. ANALYTIC GEOMETRY. 94. In this chapter it is proposed to present the elementary principles of analytic (coordinate) geometry, with especial reference to conic sections. The Cartesian system* of coordinates has already been explained in Art. 36. Here, as there, we shall speak of the curve F(x, y) = 0, the curve y =f(x'), the line ax + by + c = 0, etc., instead of saying " the curve which the equation F(x, y) = represents," etc. 95. If the equations y = 0O), (6) are treated as simultaneous, the x and y of equation (a) must mean the same as the x and y of equation (6); consequently, as coordinates they are restricted to the point or points common to the two curves (a) and (5). If equations (a) and (6) have been so combined as to eliminate one of the coordinates, say y, the x of the resulting equation is the abscissa of the point of inter- section of the two curves, and the curves intersect in as many real points as there are real roots of this new equation. * Called the "Cartesian system," after Rene' Descartes (1596- 1650), the inventor of coordinate geometry. Ill 112 CALCULUS. For example, if y be eliminated between the two equations x 2 + y 2 — 4 = and x + y — 1 = 0, we have x 2 — x — f = 0. \ ±\ V7 are therefore the abscissas of the points where x 2 + y 2 — 4 = and x + y — 1 = intersect. 96. If, however, equations (a) and (5) have been so combined that neither x nor y is eliminated, the x and ^/ now refer primarily to the points common to the curves of (a) and (?>) ; but we may treat them as a new x and y, — the current coordinates of a point describing a new curve, which passes through the intersections of the curves (a) and (J). For example, suppose we have the equations y=2x — 2 and 2y = x + 2. Adding them, y = x, a straight line distinct from the given lines, but passing through their point of intersection. 97. If the coordinates of a given point satisfy a given equation, the point evidently lies on the curve which the equation represents. Conversely, if a point is on a curve, its coordinates will satisfy the equation to the curve. If an equation F(x, y) = can be written f(x, y)<£(>, y)=o, the curve of the given equation is made up of the com- bined curves of f(x, y} = and $(x, y)= ; for any point whose coordinates cause m f(x, y) to vanish, thus satisfying the equation f(x,y')=0, will also cause /(#, y}<\>(x, y} to vanish. Hence, all points 011/(2;, y^} = are also points on f(x, y^)$(x, y) = 0. Similarly, all points on $(2;, #)= are points on f(x, y^)$(x, ^/)= 0. ANALYTIC GEOMETRY. 113 Further, there are no other points on f(x, y^)c\>(x, y) = 0, because f(x, ^)(#, y) cannot vanish except by the vanishing of either f(x, y) or <£(#, y). For example, x 2 + 2 y 2 + 3 xy — x — y = (x + y) (x + 2 y — 1) ; hence, the curve which x 2 + 2 y 2 + 3 xy — x — y = represents is made up of the straight lines x + y = and x + 2y — 1 = 0. (The term " curve " is here used as inclusive of straight lines.) 98. If we have the equations y =f(x) and y = <£(#), the equation y =f(x)(x) represents a curve whose ordinate for any abscissa x ! is the product of the orcli- nates corresponding to x' in the two primary curves. For example, if y = x and y = log x, y = x log x is a third curve whose ordinate at any point equals the product of the corresponding ordinates. In drawing such a set of curves to the same axes of reference it is well to use colored pencils or crayons. For instance, if the straight line y = x is drawn in red, the logarithmic curve in yellow, and the curve y = x log x in blue, the resulting diagram appeals to the eye much more forcibly than if all were done in black or white. 99. If y =f(x) and y = <\>(x), the equation represents a curve whose ordinate at any point is the sum of the corresponding ordinates of the given curves. For example, the so-called " equation of time" is made up of two parts : one due to the eccentricity of 114 CALCULUS. the earth's orbit, the other to the obliquity of the ecliptic. If E x and E 2 represent these two parts re- spectively, E, the whole equation of time, equals E x + E v With a scale of dates one year long for the x-axis, and a scale marked to minutes for the ?/-axis, we may con- struct the curve of E 1 and also the curve of E v A third .curve, whose ordinate for any date is the sum of the ordinates of the first two curves for that date, then represents E. See Young's General Astronomy, Fig. 64, edition of 1898. The principles of this section and the preceding one f(x) can evidently be extended to such forms as y = ^ , y =/(*)- et€. * w 100. If we move the #-axis parallel to itself through the distance y\ every ordinate is changed by the amount y' . Similarly, if the y-axis is moved parallel to itself through the distance x', every abscissa is changed by the amount x 1 '. So, if X and Y are the new current coordinates, x = X + x' and y = Y + y\ in which x r and y' are the coordinates of the new origin referred to the old axes. Hence, if in any equation F(x, y) = 0, we write x + x' for x, and y + y' for y, so that the equation becomes F(x + x', y + y r ) = 0, the geometric result se- cured is a change of origin to a new point (x } ', y f ), with new axes parallel to the old ones. The new current coordinates may be written x, y, instead of X, I 7 , since they do not occur in connection with the old coordinates and therefore cannot be con- fused with them. For example, x 2 + y 2 = r 2 being the equation to a circle with its center at the origin, (x — a) 2 + (y — b) 2 = r 2 is ANALYTIC GEOMETRY, 115 the same circle with its center at (a, 5). The coordi- nates of the new origin referred to the old axes are — a, — h. 101. Suppose the axes to rotate around the origin through the angle «. The new coordinates of any point P are X= 0B f ; Y=PB f . Fig. 11 Now OB 1 cosa = 0C=0B + BC=x + BC=z + PB f sin «. Therefore x = OB' cos a — PB f sin « that is, x — X cos a — Y sin a. Similarly, ^ = Xsin a -f- 1" cos a, 116 CALCULUS. Hence, if in any equation F(x, y)=0 we write x cos a — y sin a for x and # sin a + y cos a for y, the geometric result is the rotation of the axes through the angle a, in which a may have any value and be positive or negative. 102. In the formulas just derived, the old coordi- nates x and y are explicit functions of the new coordi- nates X and Y. If we multiply the first formula by cos a and the second by sin a, and add the products, we have X = x cos a + y sin a. Similarly, Y — — x sin a -f- y cos a. The new coordinates are now explicit functions of the old ones. 103. The formulas derived in the three preceding articles are indispensable in astronomy. As an example of the use of the two in Art. 102, suppose a planet is referred to the line in which the plane of its orbit cuts the ecliptic as the a>axis, with a line at right angles to it in the ecliptic as the ?/-axis. The planet may be referred to a new #-axis having its positive end directed toward the vernal equinox, with a correspond- ing new ?/-axis, if we use the relations X = x cos Q> — y sin Q> , Y — x sin Q> + y cos Q> . The axes are here moved backward, that is, in the negative direction, through the angle &> (the longitude ANALYTIC GEOMETRY. 117 of the ascending node), Q> being the angle between the vernal end of the equinoctial line and the line passing through the sun and the point through which the planet moves in going from the south to the north side of the ecliptic. 104. We have seen (Art. 39) that ax + by + c = represents a straight line because -£ = , a constant. dx b It follows that if in any two equations y = mx + ?i, y = m ! x + n f , m ! = m, the lines are parallel, for they have the same slope. Also, if m 1 = , the lines are at right angles to m each other ; for m and m' are now the tangents of a and 90° + a ; since tan (90° + «) = — cot a. 105. If x is put equal to zero in the equation F(x, y) = 0, the resulting value of y must be the ordinate of the point where the curve crosses the y-axis. Now if x = in ax + by + c = 0, y = ; but — - is the constant b b term when the equation is written in the form a g y = — - x — -. It follows that if any linear equation b b be written in the form y = mx + n, the constant term is the distance from the origin to the point where the line crosses the y-axis. 118 CALCULUS. 106. Suppose that P 1 '(V, ?/), P" (x" , y ,f } are any two points on a straight line, while P(x, y) is the moving point. From similar triangles (Fig. 12), PQ r _P"D. F'Q' PD' that is, y - >/' _y"-y' x — x' x" — x' This equation we describe as the equa- tion to the straight line in terms of the coordinates of two points through which it passes. For example, the points (—2, 1), (3, — 4) determine a line whose equation is y 1 4-1 z-(-2) 3-(-2y which becomes, after reducing, x + y + 1 = 0. If P ! and P n are indefinitely near each other, y n — y f becomes dy, and x n — x' becomes dx. Hence y - y f = ^f^ - ^)' which is the equation to the straight line in terms of its gradient (or slope) and one point through which it passes. ANALYTIC GEOMETRY. 119 107, Referring to Fig. 12, Ave see that if D is the distance between any two points, j) = v that is, by — , we shall have the perpendicular Vl + m 2 distance between the two lines, and hence the distance from (V ', y f ) to y = w?^ + w. The formula for the dis- tance from any point (V, y f ) to any line y = mx + n is therefore y — W22/ — w Vl + m 2 Or, if the equation to the line be in the form ax + by + c = 0, 120 the formula is CALCULUS. r , a f , c which equals ax f + by ! + c Va 2 + b 2 It appears, then, that we have simply to evaluate the function ax + by + c for the coordinates of the given point and divide by the square root of the sum of the squares of the coefficients of x and y. Exercises. 1. Given the equation ax + by + c = 0, show that if it be written in the form c c the quantities standing beneath x and y are the intercepts on the axes of x and y respectively. 2. Write the formula for the distance from the origin to the line ax + by + c = 0. 3. Find the equation to a straight line which passes through a given point (_p, g) and makes equal angles with the axes. 4. Find the length of the perpendicular from the origin on the line a (x — a) + b (y — b) = 0. Also, find the portion of this line intercepted by the axes. ANALYTIC GEOMETRY. 121 5. Write the equation to a line passing through the origin and making an angle of 120° with the a>axis. 6. The coordinates of the vertices of a triangle are (1, 2), (—3, i) 3 (4, a/2); write the equations to its sides. 7. Find the equation to a line which passes through the intersection of the lines x = a, x +y + a = 0, and through the origin. 8. Show that the lines y = 2 x -{■ 3, y = 3 x + 4, y = 4 # + 5 all pass through one point. 9. Find the slope of the line y = mx + 3 in order that it may pass through the intersection of the lines y = x + 1 and y = 2 x + 2. 10. Find the equation to the straight line which is equidistant from the two lines y = mx + n ± ?i f . 11. If a is the angle between the lines y = mx+n and y = m'x + n ! , show that tan a = — — 1 + wim' 109. The ellipse. Suppose that the circle x 2 + y 2 = r 2 is in a plane JfeT which makes an angle 7 with another plane If. Let the #-axis to which the circle is referred be parallel to plane If. If perpendiculars to jV are dropped from the extremities of the ordinates of the circle, the feet of these perpendiculars will form in If a new curve which is the projection of the circle on the plane If; and any ordinate y f of the circle be- comes the ordinate y f cosy in the new curve. Hence, if we write the circle-equation in the form y = ±Va 2 — z 2 , 122 CALCULUS. the new curve, called the ellipse, has for its equation y = ± cos 7 Va 2 — x 2 . a cos 7 is a constant, and is evidently that line of the ellipse which replaces that radius of the circle which is at right angles to the #-axis. Put acosy = b; then cos 7 = -, and the ellipse-equation becomes y — ±- Va 2 ~2. a .2 2 that is, — + ^. = 1. a 1 b z a is called the semi-major axis, and b the semi-minor axis of the ellipse. The student should distinguish carefully between the axis of reference (a?-axis) and the major axis. An axis of reference is a mere convenience, and we might study the ellipse with such an axis occupying some other posi- tion in relation to the curve, or even without any such axis ; but the major axis is an essential line of the ellipse, occupying a special position within it. The same is true of the minor axis. 110. Now let the ellipse and the circle be drawn in the same plane, — the ellipse inside the circle with the major axis coinciding with that diameter of the circle which lies in the #-axis. The ellipse has the appearance of a circle flattened in the direction YY f . AA! = 2 a is the major axis, and BB ! = 26 is the minor axis. A and A 1 \ the extremi- ties of the major axis, are called the vertices. ANALYTIC GEOMETRY 123 With jB, one extremity of the minor axis, as a center and with a radius equal to a, strike an arc. This arc will cut AA! at points F and F' equally distant from 0, the common center of the circle and the ellipse. Evi- dently the flatter the ellipse is, the farther these points will be from ; hence, if we know the ratio of OF to OA, we know how flat the ellipse is, compared with the circumscribed circle. BF=a> £0 = b; therefore OF = Va 2 - 6 2 , and OF OA Va 2 — b 2 Fig. 13. This important ratio is called the eccentricity of the ellipse, and is denoted by e. The somewhat similar (£ Q ratio, , is called the ellipticity of the ellipse. ct The eccentricity is evidently a proper fraction. The points F and F' are called foci ; and the double ordi- nate through either focus is known as the latus rectum. !4 Since OJ 7 C4Z,CDX?7,S. 111. = V« 2 - = a(l - 6 s , Va 2 — F FA = a - Va 2 Similarly, -6 2 = a(l-e). From the way in which the points F and F f Avere found we see that the sum of the distances BF and BF ! is 2 a. It may now be shown that the sum of the dis- tances from any point P(x\ ?/) on the ellipse to the foci is 2 a. The coordinates of F, the right-hand focus, are ae, ; hence, by Art. 107, (FF) 2 = O' - aey + ij 2 — x 12 — 2 aex 1 + a 2 e 2 + y' 2 . Since P is on the ellipse, its coordinates must satisfy the equation to the ellipse, and we have b 2 y' 2 = — (a 2 — x 12 } = (l-e 2 )(a 2 -x f2 ). Substituting this value of y 12 in the expression for (Fpy, (FP) 2 = x ,2 -2 aex f + a 2 e 2 + (1 - e 2 )(a 2 - x /2 ) = a 2 — 2 aex f + e 2 x' 2 ; therefore FP = a — ex 1 . Notice that the other root, — (a— ea/), is rejected because a > ex 1 and FP is positive. Repeating the ANALYTIC GEOMETRY. 125 argument for the distance F'P, the coordinates of F ! being — ae, 0, F'P = a + ex 1 ; hence FP + F f P = 2 a. 112. The proposition just established affords a way of mechanically constructing an ellipse. Fasten one end of a string at a point F on the blackboard or paper, and the other at a point F', taking the distance FF f • somewhat less than the length of the string. Pass the string around a pencil and move the point of the pencil over the paper, keeping the string taut. An ellipse will be described. It is clear now that we might define an ellipse as the path of a point which moves so that the sum of its dis- tances from two fixed points is a constant. 113. If b = a in — + *f- = 1, the equation returns to the circle-equation x 2 + y 2 = a 2 ; also, e = = 0. It thus appears that a circle is merely an ellipse with equal axes and eccentricity equal to zero. 114. Let a line DP f (Fig. 14), be drawn parallel to the minor axis. If the equation to the ellipse is — + &- = 1, the minor axis lies in the ?/-axis, and a 2 o 2 hence the line DD f is parallel to the ?/-axis. If its dis- tance from the axis is -, its equation is x = — , the e e equation affirming that whatever may be the ordinate of the point tracing the line, the abscissa is constantly -. Now the distance from any point P(x } ', y 1 ^) on the e 126 CALCULUS. curve to this line is - minus the distance of the point from the y-axis ; that is, - — x r , or a — ex We have already seen (Art. Ill) that the distance from P to the focus is a — ex 9 . Hence the distance from any point on the ellipse to the focus and the distance from the point to the line x = - are in the ratio e a — ex f a — ex' ' that is, e. Accordingly the ellipse may be defined as the path of a point which moves so that its distance from a given jived point and its distance from a given fixed line have a constant ratio less than unity. The line x = - is called the directrix. e 115. In Fig. 14 let FP = r and angle EFP = 6. Y D P C ( ° ^ E X, L F J D' Fig. 14. ANALYTIC GEOMETRY. 127 We have FF=--ae; e jPi = rcos(18O°-0) = — r cos 6, and also FP PC""' then FP = :ePC=e(FF+FL); that is, fa r = el ae — r cos \e = a — ae 2 — er cos i Solving for r, 1 + e cos This is the equation to the ellipse in polar coordi- nates with the pole at the right-hand focus. From the point of view of astronomy it is the most important of all forms of ellipse-equations. See Art. 90. 116. It is of interest, logically, to note that the theory of the ellipse may be developed from the defini- tion given in Art. 112, or the one in Art. Ill, or, indeed, from any fundamental property. In the present instance we have chosen to begin by viewing the ellipse as the projection of a circle on a plane making a given angle with the plane of the circle. -A ir 117. The hyperbola. About the ellipse — + *- = 1 circumscribe a rectangle with its sides parallel to the axes of the ellipse. Draw the diagonals of the rect- 128 CALCULUS. angle. Half of one of the diagonals is Va 2 + b 2 . With a radius of this length and with the center of the ellipse for center, strike an arc cutting the major axis produced in the points F and F f . Now take -vi b 2 and draw a line DD f (Fig. 15) whose Fig. 15. equation is x = -. This line will cut the ellipse, because a , < a. e Following the analogy of the ellipse, we proceed to find the path of a point whose distance from the fixed point F is to its distance from the fixed line DD r in ANALYTIC GEOMETRY. 129 a constant ratio, — this ratio e being here defined as Va 2 + b 2 , and hence greater than unity. a Let P be the moving point whose coordinates are x = PN, y = PL. PN cuts DD f at (7, and DD f cuts the #-axis at P. ^ = e; that is, PF=ePC=e(x-^ PC \ e Also, (PFy = (Pi) 2 + (PL) 2 Equating the two expressions for (PP) 2 , e 2 ix j = y 2 + (x — ae) 2 . Expanding and reducing, &(# _ 1) _ y 2 = a 2Q e 2 _ ly 9 a 2 a 2 (e 2 - 1) that is, — -|- = 1. 118. The curve whose equation we have now found is the hyperbola. Although closely related to the ellipse, it differs from that curve in various important respects : 1. a and b being the same in magnitude and position for the two curves, no portion of the hyperbola lies within the area occupied by the ellipse ; for as soon as x < a, y is imaginary. 130 CALCULUS. 2. The hyperbola has two parts or branches sym- metrically placed with respect to the axial line in which BB' lies ; for if we assign values — x\ — x n , — x ,n , etc., to #, we obtain the same values for y that are obtained when + x' \ + x n ', + x'", etc., are the values assigned. 3. Values indefinitely large may be assigned to x without making y imaginary. The curve, therefore, extends to infinity. 119. The equation to the hyperbola may be written b y =± a Vx 2 — a 2 ; or, expanding (x 2 — a 2 ) 1 by the binomial theorem, bf a 2 a* \ , . The equations to the two diagonals (produced) of the rectangle (Fig. 15) are seen to be y = ±-x. (5) ct Comparing equations (a) and (6), we observe that any ordinate of the hyperbola is less than the corre- sponding ordinate of the lines ; but we also notice that as x becomes larger and larger, the ordinates, according to equations (a) and (J), approach equality. Whenever such a relation exists between a line and a curve, — the distance between them becoming indefi- nitely small as the points describing them recede to infinity, — the straight line is called an asymptote. The hyperbola — — &— = 1 has therefore the asymp- 7 ct o totes y = ± - x. ANALYTIC GEOMETRY. 131 120. The line AA! (Fig. 15), 2 a in length, is called the transverse axis of the hyperbola ; and BB f , 2 b in length, is the conjugate axis. If we consider the equation b 2 a 2 we find it situated with respect to the y-axis just as the first hyperbola is with respect to the #-axis. This curve, 2- 5=1, is known as the conjugate hyperbola b 2 a 2 in distinction from the primary or transverse hyperbola. Following the method of Art. 119, we find that the lines y — ±- x are asymptotes of the conjugate hyperbola also. 121. The points A and A! where the transverse axis meets the curve are its vertices. Similarly, B and B r are the vertices of the secondary or conjugate hyperbola. The transverse and conjugate axes, the asymptotes, and the directrix are all essential lines of the hyperbola, and sustain a fixed geometric relation to it like a rigid framework, so that if the position of the hyperbola is changed with respect to the x- and ^-axes, these lines go with it. 122. If b = a, the rectangle (Fig. 15) becomes a square ; the asymptotes become y = ± x, the two lines now crossing each other at right angles (Art. 104) ; the equation to the hyperbola itself becomes x 2 — y 2 = a 2 , and is known as the equilateral hyperbola. It is evi- 132 CALCULUS. dently the hyperbola that would appear in plane M, Art. 109, in connection with the circle x 2 + y 2 = a 2 and the circumscribed square. 123. If we rotate the axes in the negative direction through the angle — 45°, we shall have these axes coin- ciding with the asymptotes of the hyperbola x 2 — y 2 = a 2 . To do this we use the formulas x = x cos a — y sin tf, y = x sin a -f y cos a, (Art. 101) which become, for a = — 45°, x = x %V2 + y 1V2 = |V2(> + y), y = -xW2 + yW2 = W 7 2(y -. x). Substituting these values for x and y in the equation x 2 — y 2 = a 2 , we have that is, i ( .r + y) 2 — \ (y — x) 2 — a 2 , which becomes, after reduction, a 2 This is the equation to an equilateral hyperbola referred to its asymptotes. The isotherm pv — c (Art. 36) is an important illus- tration. (See Maxwell's Theory of Heat, Chap. VI.) 124. Following the method of Art. Ill, and using Fig. 15, it is found that (FP) 2 = e 2 z' 2 - 2 aex' + a 2 . ANALYTIC GEOMETRY. 133 Taking FP positive, and noticing that ex' > a, we have FP = ex f - a. Similarly, FP = ex 1 + a ; therefore FP -FP = 2a, and the hyperbola may be defined as the path of a point moving so that the difference of its distances from two fixed points is a constant. 125. The polar equation to the hyperbola may be readily obtained from Fig. 15. Let FP = r, FP = r ! , and the angle LFP = 6. We have also r f — r = 2 a and F f F= 2 ae. Then r 12 = r 2 + 4 ar + 4 a 2 , and since r f is one side of the triangle F'PF, r !2 = r 2 + 4 ^2 _ 4 ^ r CQS ( 180 o _ ffy^ Equating these two values of r /2 , r 2 +■ 4 ar + 4 a 2 = r 2 + 4 a 2 6 2 + 4 a#r cos # ; a(e 2 -l) hence r = —^ 4- 1 — e cos # This equation may be described as the right-hand focal polar equation to the hyperbola. When 0=0, r = — a(l + e), and the feather-end of the arrow (Art. 45) gives the vertex of the left-hand branch of the curve. As the radius vector continues to revolve in the positive direction, we continue to get negative values of r and points on the left-hand branch 134 CALCULUS. until 6 = cos l - ; r is then infinite, and the radius e j vector is parallel to the asymptote y=-x, because -il * -\b cos i - = tan L - . 6 a 1 1 From = cos *- to 6 = 360° — cos -1 -, r is positive, and the right-hand branch is being traced. Finally, when 6 changes from 300° — cos -1 - to 360°, the re- maining part of the left-hand branch is traced. 126. The parabola. It remains to inquire what kind of a curve we have when a point moves so that its dis- tance from a fixed point is to its distance from a fixed line in the constant ratio unity. D N Y c p^- B X i D \F J F] r [G. 16. Let DD\ Fig. 16, be the fixed line, and F the fixed point. Let 2p be the length of FB, the distance from F to DD f . Take a line through F perpendicular to ANALYTIC GEOMETRY. 13 oo DD f for the x-axis, and a line parallel to DD\ bisecting FB, for the ^-axis. Let P be the moving point with the coordinates OL and PL. Then, FP = PN=x+p; also, (FPy = (PL) 2 + (PX) 2 = y 2 + (x — p) 2 . Hence, (x + ff = y 2 + (x — p) 2 ; that is, y 2 = 4 px. This curve is called the parabola. Since it is the path of a point whose distance from the fixed point is to its distance from the fixed line in the ratio unity, it must be regarded as the transition curve between the ellipse and the hyperbola. Its eccentricity 0, being the ratio of the two distances, is of course unity. 127. Considering the equation y 2 = 4px, we see that the parabola has a line of symmetry which has been used as the a>axis ; for if any value be assigned to #, y has two values numerically equal and with opposite signs ; so that if the area above the #-axis were folded over, the part of the curve in the upper area would exactly fit the part in the lower. This line of sym- metry is called the axis of the curve, and the point where it meets the curve is the vertex. The point F is the focus, and the double ordinate through the focus is the latus rectum, as in the case of the, ellipse. When x—p, y = 2p; therefore the semi-latus rectum is twice the distance of the focus from the vertex. 136 CALCULUS. 128. If p is positive in the equation y 2 = \px, posi- tive values may be assigned to x without making y imaginary. Hence, the parabola like the hyperbola extends to infinity. It differs from the hyperbola, however, in this important respect : it has only one real branch, because negative values of x make y imaginary. 129. If p is negative in the equation y 2 = \px, we have the same law as before, governing the motion of the point P ; but the path is now wholly on the nega- tive side of the ^/-axis, for only negative values can now be assigned to x. Similarly, x 2 = 2 py is a parabola above the z-axis, with the ?/-axis for its line of symmetry, if p is positive ; while x 2 = 2py is the same curve below the #-axis if p is negative. 130. The polar equation to the parabola, the focus being pole, is obtained from Fig. 16. FP is r, and the angle XFP is 6. Then FL = r cos #, but r = PJV=FL + 2p; therefore, r = r cos 9 + 2 p ; that is, r = 2— cos 6 131. The ellipse, parabola, and hyperbola are known as conic sections ; for it can be shown that if a cone of revolution is cut by a plane in any manner whatever, the cross-section is one of these three curves. See ANALYTIC GEOMETRY. 137 Puckle's Conic Sections, Arts. 323-325, together with Chap. VIII of that treatise. 132. A straight line becomes a tangent to a curve at any point (x', y'} if (1) it passes through that point, and (2) if it has the same slope as the curve at that point. We have already seen (Art. 38) that if y=zf(£) is the equation to a curve, -^ gives its slope dx or gradient at each point. It has also been shown (Art. 107) that y - y ! = ^ O - *0 is the equation to a straight line passing through the point (z f , y'*) with the slope -^. It follows that if (x } ', y) is a point P ! on the curve y =f(x) and if -r~ is specialized for that point, becoming — ^ or -JL-, y - y' = ^7 O - *') is the equation to the tangent at P r . For example, let us find the equation to the tangent at the upper extremity of the latus rectum of the par- (a 1J 2 77 abola y 2 = 4 px. Differentiating, we have -f- = ■— *-. ax y The coordinates of the upper extremity of the latus rectum are #, 2 p. Specializing — ^ for this point. dx df] = %£ = i . 138 CALCULUS. and the general equation gives y - 2p = x - p; that is, y = x + p, which is the equation to the tangent in question. We notice that this particular tangent makes an angle of 45° with the axis of the curve, which is here the #-axis, and cuts the axis produced where the directrix DD f cuts it. (See Fig. 10.) Example. Find the general equation to the tangent to the ellipse ^-+'-^ = 1. Differentiating and solving for -^, we have (XX 7 70 ay _ _ b l x dx a 2 y and the general equation to the tangent becomes, for the ellipse, 72 / y - y =-- 2 -( x -v)- a A y' For instance, the coordinates of the upper extremity of ty the left-hand latus rectum are — ae, bVl — e 2 ; ~ir therefore becomes b 2, ae ■ ; that is, e ; a 2 bVl- e* and the tangent at the point named is y — b Vl — e 2 = e(x -f ae). It will be noticed that this particular tangent makes with the major axis an angle whose tangent is the eccentricity. ANALYTIC GEOMETRY. 139 The line perpendicular to the tangent at the point of tangency is called the normal. Its equation is evidently y - y = -j-j( x - x ^ 133. The general equation to the tangent to the parabola y 2 = \px, in terms of the coordinates of the point of tangency, is seen to be y - y = -6- O - * ) ; that is, yy 1 = 2px + y' 2 — 2px ! ; or, since y f2 = i px f , (x\ y !> ) being on the parabola, W 1 = 2jp(a? + aO, (1) and y== ^(aj + ^). (2) Writing the equation to the line passing through the focus and P(x ] ', y), we have, after reducing, If -A is the point where the tangent cuts the axis pro- duced, FPA is an isosceles triangle. To prove this, 2 p let tan a= — 4-, the coefficient of # in (2). Then y tan 2 a = ~^ = 4j?y ' = 4 ^ = _J^_ . 1 4p 2 y f2 — 4: p 2 ipx'—ip 2 x f — p 1 ~y¥ 140 CALCULUS. But this is the coefficient of x in (3). Hence, angle PAF = angle FPA. It immediately follows that if a line Pi is drawn on the concave side of the parabola, parallel to its axis, FP and PL make equal angles with the tangent.* Advantage is taken of this property of the parabola in the construction of reflectors. Since the angle of reflection of a ray of heat or light equals the angle of incidence, if a light be placed at the focus of a para- bolic reflector, the light is reflected in a system of (approximately) parallel rays. Illumination of a rail- road track for a long distance in front of the locomotive is secured by means of such a reflector. Conversely, if rays of heat or light, parallel to the axis of a para- bolic reflector, fall upon its concave surface, they will converge at the focus. 134. It is required to find the locus (path) of the middle point of any ellipse-chord moving parallel to itself. Let C(x n , y rf ) and G\x\ y f ) be the points where 9 9 the chord meets the ellipse — + ~ = 1, and let M(x, y) bisect the chord CC f . * " Hertz, in the first of his celebrated experiments on the propa- gation of electric rays, made use of this property of parabolic surfaces. He employed large reflectors of sheet zinc bent into the form of para- bolic cylinders, in whose focal line the transmitter and the receiver of the electric waves were placed. The electric rays passed from the transmitter to the first parabolic reflector, were there reflected so as to become parallel, and were then reflected from the second reflector to the receiver placed at its focus." — Young and Linebarger's Calculus. ANALYTIC GEOMETRY. Ill Then x = J(V + x fr ), and y = ^(y f + y ,f }; that is, x' = 2 x — z rf , and y f = 2y — y lf . Since (V, y') and (V', 2/") are each on the ellipse, ^ + ^ = !> (1) £ T + 'V= L ( 2 ) Substituting in (1) the values just noticed for x r and y f , we have (2x-x")* C^y-y^y _ a? + b* 4x 2 -±xx" x" 2 4y 2 -4yy n y n * H or I -\ — \- - = 1 a 2 + a? ^ b* + b* Introducing relation (2), x(x - x") y(y - y") _ a 2 + 62 - u ' and therefore ^ „ 6 2 # a; — x n a 2 y Now let y — y n = m(x — x ff ) be the equation to the chord OC f ; then - — Ki = m, and this is true, of course, when (#, y), the point tracing the line (7(7', is restricted to the point M. 142 CALCULUS. Equating the two values of l ^- rr , x — x n b 2 x a 2 y b 2 that is, y = — - x. arm The path of the middle point of any chord of slope m, moving parallel to itself across the ellipse — + ^- = 1 a 2 b 2 is therefore a straight line passing through the center of the ellipse. If m = 0, the equation becomes x = 0, the equation to the minor axis ; and if m = oc, y = 0, the equation to the major axis. Writing — b 2 for ?> 2 , we have b 2 the corresponding equation in relation to the hyperbola. 135. By means of the result in the preceding article, we are now able to find the center and construct the axes of any ellipse. Draw any two par?dlel chords and bisect them. The chord passing through the points of bisection must pass through the center of ellipse ; and the point of bisection of this third chord is the center. With the center now found and a radius of any convenient length, strike a circle cutting the ellipse. Draw one chord common to both ellipse and circle, and finally draw an ellipse-chord perpendicular to the preceding chord ANALYTIC GEOMETRY. 143 at its middle point. It will be the major (or minor) axis of the ellipse. Having given the hyperbola with its accompanying conjugate hyperbola, the construction of the axes is the same as for the ellipse. 136. The tangent-equation, Art. 132, involves the coordinates of the point of tangency. It is desirable to obtain a form in which these coordinates do not appear. What conditions must be imposed on the line y = mx + n so that it shall keep the slope m and yet be a tangent to the ellipse —, + — = !? Eliminating y between the equations y = mx + n, (1) x 2 y 2 a* + P = 1 ' < 2 > the resulting equation, x 2 {mx + w) 2 _ i a 2 b 2 has for its roots the abscissas of the points of intersec- tion of (1) and (2). These roots are mn / m 2 n 2 n 2 — b 2 b 2 b* b 2 1 m 2 I / 1 m 2 \ 2 1 m 2 a? ¥ \\a 2 ¥J ~a 2 T 2 Thus far the straight line is merely a secant (real or imaginary) of the ellipse. If it is to become a 144 CALCULUS. tangent, the two points of its intersection with the ellipse must be indefinitely near to each other; that is, the two abscissas must be equal. Hence, the radi- cal, which now makes them unequal, must vanish, and we have m 2 n 2 n 2 — b 2 f \ m* Y 1 m 2 Co 2 W ~a? ¥ From this equation of condition we obtain n = ± Va 2 m 2 + i 2 , which is therefore the relation which must hold between n and a, m and b in order that (1) shall be tangent to (2), and we have y = mx ± ^fa 2 m 2 + b 2 . (3) The double sign in (3) plainly means two tangents parallel to each other, one cutting the ^/-axis at the distance VoW + P above the origin, and the other at the same distance below it. The corresponding equation for the tangent to the hyperbola may be obtained at once by writing — b 2 for b 2 in (3), and we have y = mx ± Va 2 m 2 — b 2 . (4) If b = a, so that the ellipse becomes a circle, (3) becomes y = mx ± aVm 2 + 1. (5) Similarly, if the hyperbola is equilateral, (4) becomes y = mx ± a^Jm 2 — 1. (6) ANALYTIC GEOMETRY . 145 Exercises. 137. 1. Construct the ellipse £ + £ = L 4 9 What is its eccentricity ? How must the formula for e be written in this case ? 2. Find the points of intersection of the ellipse and hyperbola whose equations are aj" + 2^ = l, 3x 2 -6f = l, and show that at each of these points the tangent to the ellipse is the normal to the hyperbola. (Puckle's Conic Sections.) 3. Find the equations to the asymptotes of the hyperbola 3 x 2 — 6 y 2 = 1. 4. Find the distance between the right-hand focus of x 2 + 2 y 2 = 1 and the right-hand focus of 3xr — 6y 2 = l. 5. Write the equation to a circle which shall have its center coincident with the focus of the parabola y 2 = Ap>x, and shall be tangent to the parabola. 6. Find an expression for the perpendicular distance 9 9 XT 1/~ from the right-hand focus of the ellipse —^ 4-^=1 to the tangent y = mx + Va 2 m 2 + b 2 . 7. Find an expression for the perpendicular distance from the focus of the parabola y 2 = £px to any normal. 8. Show that the line y = mx + n becomes a tangent to the parabola y 2 = 4^px if n = — 9. Show that the path of the middle point of any parab- ola-chord moving parallel to itself is a line parallel to the axis of the parabola. 146 CALCULUS. 10. Given a parabola, find its axis and focus. 11. The locus of the foot of the perpendicular from the center of the equilateral hyperbola x 2 — y 2 = a 2 is the lem- niscate (x 2 + y 2 ) 2 = a 2 (as 2 — y 2 ). Use y = mx + a Vm 2 — 1. The line perpendicular to it passing through the center is y = x. It is required to find the path of the inter- section of these two lines as m passes through all values. Eliminate m. 12. Show that the locus of the foot of the perpendicular dropped from the focus of the parabola on its tangent is the tangent at the vertex. 13. If a source of light or heat is placed in one focus of an ellipse, the rays will be reflected so as to meet in the other focus. 14. A. planet at P is moving in the direction PQ. Its distance PS from the sun at S (one focus of its elliptic orbit) is J its major axis. Construct the orbit. 15. Given one focus and any point P and the length of the major axis of an ellipse; show that the eccentricity depends on the direction of the tangent at P. Construct the major and minor axes of ellipses corresponding to various tangents through P. 16. The tangent at any point of a hyperbola is produced to meet the asymptotes ; show that the triangle cut off is of constant area 17. Find the equation to the path of the center of a circle which is tangent to two given circles. ANALYTIC GEOMETRY. 147 138. Just as a point in a plane may be determined by referring it to two lines at right angles to each other, so a point in space may be determined by referring it to three planes, each plane intersecting the other two at right angles. The point common to the three planes z • M / 7/^ P X 2 f Fig. 17. is called the origin, and the lines of intersection of the planes are known as the axes of x, y, and z. The posi- tive directions of the axes are usually taken to be repre- sented in the figure by OX, OY, and 0Z\ the negative directions are then OX', OY', 0Z r . If P (Fig. 17) is any point in space, then PL, PM, PN, its perpendicular distances from the planes YOZ, ZOX, and XOY respectively are the coordinates x, y, and z respectively. 139. The three planes evidently divide the space around the origin into eight equal triedral angles. If a point is in the upper front right-hand angle, its 148 CALCULUS. coordinates are all positive, because each one is meas- ured parallel to its own axis and in the positive direc- tion. Again, if a point is in the upper front left-hand angle, the y and z coordinates are positive, but the x coordinate is negative because measured in the nega- tive direction parallel to OX' . In like manner we are able to state the character of each coordinate for points situated in each one of the other six angles. 140. OP, the distance of P from the origin, is the diagonal of the rectangular parallelopiped, three of whose edges are PZ, PM, PN. Therefore, (OPf = (Pi) 2 + (PJ/) 2 + (PX) 2 - a? + } f + z i. 141. Suppose we have any two points P ! (x', y' , z') and P" (x" , y" , z"). Let planes be passed through P' and P" parallel to the three planes of reference. There is thus formed a rectangular parallelopiped whose diagonal is the line P f P", and three of whose edges are x" — x\ y"—y f , z" — z' . Therefore, (jp>p"y = o" - x r ) 2 + o" - y'y + (z" - z'y. This formula for the distance between two points in space should be compared with the formula in Art. 107. 142. We have seen (Art. 39) that the general equa- tion of the first degree in two variables represents a straight line. It may now be asked, What is repre- sented by Ax + By + Cz + D = 0, (1) the general equation of the first degree in three variables ? AXALYTIC GEOMETRY. 149 1. It represents a surface and not a solid. For let (a, 6) be a point in the plane YOX, and suppose that a straight line be drawn through this point parallel to the z-axis to meet the locus of equation (1), whatever kind of locus it may be. We now have -Aa-Bh-D z = a ; therefore the straight line meets the locus in one definite point at the distance — — from the plane YOX, and consequently the locus cannot be made up of layers either adjacent to one another or occurring at intervals. 2. The surface is a plane. For suppose that a point moving in the surface be so restricted that it must remain at a constant distance from the plane YOX ; that is, let z have a constant value, say c. Equation (1) is now reduced to Ax + By + Cc + D = 0. Therefore the point moving in the surface and at a constant distance from the plane YOX is moving in a straight line. In other words, any section of the sur- face made by a plane parallel to the plane YOX is a straight line. Hence, if Ax + By 4- Oz + D = is not a plane surface, it must be a wavy surface, something like a corrugated tin roof with the corrugation lines parallel to the plane YOX. But repeating the argu- ment, making y a constant, we find that all sections made by planes parallel to the plane ZOX are straight lines. The surface in question must therefore be a plane. In case the constant term D is zero, the coordinates 150 CALCULUS. of the origin (0, 0, 0) satisfy the equation, and the plane passes through the origin. Thus the equation, Cz -C'i/ + C"x = (p. 103), represents a plane passing through the origin ; and any moving point whose coordinates satisfy this equa- tion at each instant, must be moving in the plane and hence in a plane curve. Arts. 138-1-12 have been introduced for the sake of Arts. 86-88. dx dx doc CHAPTER V. FORMULAS. li- ( ( ( ^dx + ^dx) = u + v. J \dx doc J 2. J^uv = v^+u dv . a ,. M*—*» + "— *»)=*"' doc doc doc i J \ doc doc J 3. jL u n = nu n - 1 ^ 3,. (u tn —dx= u ™ +1 . dx dx J dx m + 1 4. -4^- sin m = cos w — • 4,. I cos ^ — dx = sin if. dx dx J dx 5. — 'cos w = - sin «« — • 5,. \ sin w — dx - - cos u . dx dx J dx 6. -^- tan ^ = sec 2 u ~- 6,. f sec 2 if ~ *» = tan u . dx dx J dx 7 . — cot u = - cosec 2 m — • 7, . f cosec 2 u — dac=-cotu. dx dx J dx 8. -!^-secw=tanwsecw— • 8,. {t2LMi&eeu^dx=secu. dx dx J dx 9. -7- cosec u = — cot u cosec if — ^- ax dx , d?f 9, . 1 cot u cosec tf ^^ dx = — cosec ti. dtf du d . ! dx _, ^ f dx , . 1 10. — -Sin-l^rr: — -. 10^ J — - dX = Sin" 1 if. ax Vl _ ^2 vi _ ^ dtf dw 11. -^-cos 1 ^ — — Hi. f -^=dx = cos" 1 w. dx Vi_^2 J Vf^ 2 151 152 CALCULUS. du du 12. -=— tan -1 u - — • 12 L . J - — L — c dx = tan -1 u. dx 1 + w a J \ + w 2 <7?^ dw d d ^y* i d nr 13. — -cot _1 t^=-- 3« 13x. J — - — — ^ da? = cot 1 if. da? 1 + u 2 ^ 1 + if 2 d?f du ia d 1 c&e -. , f das , t 14 - -j— sec _1 i^ = — , 14 x . J doc = sec * n. d 1 f/.r 15. cosec~ * m = rca? Mvie- — 1 d?£ 15 J , ' 1 dx = cosec -1 u. d ^a? _ ^ac 16. ^e x = e x . 16 da? 17. ^-a r = a x log e «• 17,. f <*> x log" e ada? = a x . dx J 18. -*Le u = e u ^* 18,. fe*^da; = e M . da? da? J da? 19 . Va V*W - Va2 -* 2 -sm-^. J x 2 x a /X X i ' — dx = a vers -1 V2 ax — x 2 . ■\/9, r/cr. — cr? a 34. / V2 ax — x 2 dx V2 ax — x 2 xa/2 ax — x 2 ax V 2 ax — x 2 dx = — vers -1 - -\ V 2 ax — x 2 ' 2 a 2 36. I — — dec = a vers -1 - + V2 ax — x 2 . J x a 154 CALCULUS. 37 . I tan x dx = — log cos x. 38. j cot x dx = log sin x. 39. f-^ = log tan-- 40. f-^-=logtanf- + J sin. i- 2 J cos a; \2 4 41. I - = — cot x. 42. I — — = tan x. J suras J cos L # C ■ 7 i o sin 2 a? 43. I sin a; cos a? ao? = — J cos 2 a? = — - — 44. I — = log tan x. J sin x cos x 45. I x sin .r cfo = sin x — x cos x. 46. I ar sin x dx = — as 2 cos a; + 2 x sin a; + 2 cos x. 47 . J sin 2 x dx = — \ sin 2 a: + \ x. 48. J cos 2 x dx = \ sin 2x + % x. 49 . I log x dx = a; log a; — a:. 50. I sin -1 x dx = x sin -1 x + Vl — a?. 5 1 . f tan" 1 a; c?aj = a; tan" 1 x — \ log (1 + x 2 ) . . J sec -1 x dx = x sec -1 a? — log (x + Va? + 1) . . i fa>6 , f f = -| = tan-^J^tan| J Ja+6cosx Va 2 -6 2 V Xa + & 2 52 For other integrals see Peirce's Short Table of Integrals. FORMULAS. 155 54. f(z + x)=f{z)+f(z)x+f^ + £^^+.... 55. f(x) =/(Q) +/' (O) X + £^j^- + / "^°) * + .... 56. (a + rt)" = a" + ma-'x + !il!izll^ m (m — 1) (m — 2) a" 1 " 3 a^ 57. e ^l + , + | + | + .... 58. a- = l + g ; log e a + x2(log ' a)2 + g:3(1 ^ a)3 + -.. [2 [3 /y»2 /ytO /y»4 59. log(l + a; ) = x-| + |-|+.... 60. sm^ = ^-, — h, ■••- 61 • cos# = l — ■ — K ■•• [3 16 |2 li 62. sin ( - + « ] = cos a. 63. cos [ - + a ) = — sin a. 64. tan (- + «]=— cot a. 65. cot (5 + a] = — tan a. 66. sin (— a) == — sin a ; cos (— a) = cos a. 67. tan (— a) = — tan cc ; cot (— «) = — cot a. • o o -i ™ *. sin a sin a 68. sm- « + cos 2 a = 1. 69. tan a = cos a Vl - sin 2 a 70. sin (a + /?) = sin a cos /2 + cos a sin /3. 71. cos (a + /?) = cos a cos /J — sin a sin /?. 72. sin (a — /?) = sin a cos /? — cos « sin /?. 73. cos (a — j8) = cos a cos /? + sin a sin (3. 74. sin 2 a = 2 sin a cos a. 156 CALCULUS. 75. cos 2« = cos 2 a — sin 2 a = 2 cos 2 a — 1 = 1 — 2 sin 2 a. „„ , / , m tan ft + tan /? 76. tan (a + /?) = — • 1 — tan « tan /? „ . / n \ tan « — tan B 77. tan (ft — S) = ^-« 1 + tan ft tan /3 78. tan2«= 2tan " ■ 79. cot 2 a = COt2 <* ~ 1 1 — tan- ft 2 cot < ft /l — COS ft - ft /l 80. Sm =yj- 81. COS 2 = > /- ™ , ft /l — COS ft on , ft /1.+ COS ft 82. tan-=\- 83. cot-=\/— ?- 2 \ 1 + cos « 2 \ 1 — cos ft 84. sin « + sin /3 = 2 sin i (ft -f- /5) cos i (ft — /}). 85. sin « — sin f3 = 2 cos -J (ft + /?) sin J (ft — /3). 86. cos « + cos /} = 2 cos £ (ft + /J) cos | (ft — /J). 87. cos ft — cos (3 = — 2 sin J (ft + /J) sin \ (a — /?). 88. log afr = log a + log 6. 89. log - — log a — log b. b 1 i 90. log a n = n log a. 91. loga n = -loga. n 92. If asc 2 + bx + c = 0, & F& 2 c & 1 m — r 2 a \4 a 2 a 2a 2a 93. [* = l-2?3'-4 — n. 94. log 1=0. 95. Iog0=— oo. 96. log a a=l. 97. e = 2.7182818284 .... 98. log 10 e = 0.43429448 •• 99. 7T = 3.14159265-... 100. log 10 7r = 0.49714987 •■ FORMULAS. 157 101. #° = — = 57°.2957795 102. R" = 180 ' 60 ' 60 " = 206264".8 •• 103. log R° = 1.75812263 • •-. 104. log R" = 5.31442513 •••. 105. 30° 45° 60° sin, | iV2 |V3 cos, iV3 |V2 i 106. Base of right triangle = /^ cos y ; alt. = h sin y. Qi = hypotlienuse ; y = angle at base.) 107. Area of sector of circle = \ r(rO) = \ r*9. 108. Area of ellipse =wab. INDEX. The references are to pages. Abscissa, 42. Acceleration, 62. Amplitude, 67. Analytic geometry, 111. Anomaly, true, 107. Aphelion, 106. Apsides, 57. Archimedes, 2. Areal velocity, 105. Areas, 85. of surfaces of revolution, 94. Asymptote, 130. Asymptotes of hyperbola, 130. Attraction of homogeneous sphere, 70. Axes of coordinates, 41. Axes, change of, 115. major and minor, 122. transverse and conjugate, 131. Axis of parabola, 135. of symmetry, 135. Binomial theorem, 30. Bodies, falling, 67. Boyle's law, 2. Cartesian coordinates, 41, 111. Catenary, equation to, 93. Circle, equation to, 46. Circular functions, 20. Comets, orbits of, 108. Concavity of curves, 48. Conic sections, 136. exercises on, 145. Conjugate hyperbola, 131. Constant, derivative of, 8. of integration, 13. Convergence of series, 30. Coordinates, 42. current, 43. polar, 56. transformation of, 56, 115. 114, Definite integrals, 14. Derivative, 5. of constant, 8. of product, 9. of quotient, 15. of sum, 8. of cos x and sin x, 18. of x n , 10. Derivatives, second and higher, 24. partial, 24. Descartes, 111. Differential, 5. perfect, 98. total, 25. 159 160 INDEX. Differentiation, 8. Directrix of ellipse, 126. of hyperbola, 129. of parabola, 138. Displacement, 67. Distance between two points, 119. from point to line, 119. Double integrals, 97. Eccentricity of ellipse, 123. of hyperbola, 129. of parabola, 135. Ellipse, 121. area of, 87. axes of, 122. construction of, 125. determination of center and axes of, 142. directrix of, 126. eccentricity of, 123. ellipticity of, 123. equation to, 122. path of middle point of chord ' of, 140. polar equation to, 127. foci of, 123. vertices of, 122. Energy, kinetic, 92. Epoch. 67. Equation to a curve, 111. ax + by + c = 0, 45. Ax + By + Cz + D = 0, 148. Equations of motion, 78. Equilateral hyperbola, 131. Falling bodies, 67. Foci of ellipse, 123. of hyperbola, 128. Focus of parabola, 135. Formulas, collection of, 155. Function, algebraic, 4. explicit and implicit, 16. periodic, 47. Function, transcendental, 4. of several variables, 24. Gradient, 44. Graph, 41. Gravitation, law of, 2. Gravity, 67. Harmonic motion, 67. Heat, minimum intensity of, 54. Hertz, 140. Horizontal range, 79. Hyperbola, 127. asymptotes of, 130. axes of, 131. directrix of, 129. eccentricity of, 129. equation to, 129. polar equation to, 133. foci of, 128. Hyperbola, conjugate, 131. equilateral, 131. transverse, 131. Indeterminate forms, 31-33. Indicator diagram, 97. Inertia, moment of, 100. Inflexion, point of, 49. Integrals, double and triple, 97. definite and indefinite, 13. table of, 151. Integration, 12. constant of, 13. by parts, 37. Intercept, 119. Isotherm, 132. INDEX. 161 Kepler's laws, 102. Kinetic energy, 92. of rotation, 99. Latus rectum, 123. Law, Kepler's first, 107. Kepler's second, 105. Kepler's third, 110. Lengths of curves, 92. Limits of definite integrals, 14. Locus, 41. Logarithmic differentiation, 21. Logarithms, common and Napier- ian, 39. Maclaurin's theorem, 29. Maxima and minima, 48. exercises in, 52-55. Mean values, 88. Moment of inertia, 100. Motion, equations of, 78. of falling body, 68. of rising body, 69. in a parabola, 77. pendulum, 82. rectilinear, 74. simple harmonic, 67. in a vertical curve, 81. Newton, 2. Node, longitude of ascending, 116. Normal, 139. Ogee, 49. Operation, indicated, 12. Orbits, eccentricity of, 108. Ordinate, 42. Origin of coordinates, 41. change of, 114. Parabola, 134. axis of, 135. directrix of, 138. eccentricity of, 135. focus of, 135. latus rectum of, 135. equation to, 135. polar equation to, 136. vertex of, 135. Parabolic reflector, 140. Parallelism, condition of, 117. Partial differential coefficients, 24. Pendulum, 82. time of oscillation of, 84. Perfect differential, 98. Perihelion, 106. Period, 67. Periodic function, 47. time, 110. Perpendicularity, condition of, 117. Phase, 67. Point of inflexion, 49. Polar coordinates, 56. Pole, m. Projectile, path of, 78. Radian, 47, 85. Radius vector, 57. Range, horizontal, 79. on an incline, 80. Revolution, areas of surfaces of, 94. volumes of, 94. Sinusoid, 47. Slope, 44. Straight line, 117. exercises in, 120. Symmetry, line of, 135. 162 INDEX. Table of formulas, 155. of integrals, 151. Tangent in terms of slope and intercept, 143. to curve, 137. Taylor's theorem, 28. Transformation of coordinates, 56, 114, 115. Transverse hyperbola, 131. Triple integrals, 97. True anomaly, 107. Variable, 3. Velocity, areal, 105. angular, 63. . component, 60. linear, 58, 64. Vertices of ellipse, 122. Volumes of revolution, 94. Work, 91. SCIENCE. 51 Physics for University Students. By Professor HENRY S. CARHART, University of Michigan. Parti. Mechanics, Sound, and Light. With 154 Illustrations. i2mo, cloth, 330 pages. Price, $1.50. Part II. Heat, Electricity, and Magnetism. With 224 Illustrations. i2mo, cloth, 446 pages. Price, $1.50. THESE volumes, the outgrowth of long experience in teach- ing, offer a full course in University Physics. In preparing the work, the author has kept constantly in view the actual needs of the class-room. The result is a fresh, practical text-book, and not a cyclopaedia of physics. Particular attention has been given to the arrangement of topics, so as to secure a natural and logical sequence. In many demonstrations the method of the Calculus is used without its formal symbols ; and, in general, mathematics is called into ser- vice, not for its own sake, but wholly for the purpose of establish- ing the relations of physical quantities. At the same time the course in Physics represented by this book is supposed to pre- cede the study of calculus, and its methods will in a general way prepare the student for the study of higher mathematics. Professor W. LeConte Stevens, Rensselaer Polytech?tic Institute, Troy, N. Y. : After an examination of Carhart's University Physics, I have unhesitat- ingly decided to use it with my next class. The book is admirably arranged, clearly expressed, and bears the unmistakable mark of the work of a successful teacher. Professor Florian Cajori, Colorado College : The strong features of his Uni- versity Physics appear to me to be conciseness and accuracy of statement, the emphasis laid on the more important topics by the exclusion of minor details, the embodiment of recent researches whenever possible. Professor A. A. Atkinson, Ohio University, Athens, O. : I am very much pleased with the book. The important principles of physics and the essentials of energy are so well set forth for the student for which the book is designed, that it at once commends itself to the teacher. Professor A. E. Frost, Western University, Allegheny, Pa. : I think that it comes nearer meeting my special needs than any book I have examined, being far enough above the High School book to justify its name, and yet not so far above it as to be a discouragement to the average student. 52 SCIENCE. Primary Batteries. By Professor HENRY S. CARHART, University of Michigan. Sixty- seven Illustrations. i2mo, cloth, 202 pages. Price, $1.50. THIS is the only book on this subject in English, except a translation. It is a thoroughly scientific and systematic account of the construction, operation, and theory of all the best batteries. An entire chapter is devoted to a description of stand- ards of electromotive force for electrical measurements. An ac- count of battery tests, with results expressed graphically, occujDies forty pages of this book. This chapter forms an excellent outline guide for laboratory purposes. The tests have all been performed either by the author himself or under his immediate supervision. They are free from bias and exhibit some facts not heretofore ac- cessible to the public. The battery as a device for the transformation of energy is; kept constantly in view from first to last ; and the final chapter- on Thermal Relations concludes with the method of calculating electromotive force from thermal data. Professor John Trowbridge, Harvard University: I have found it of the greatest use, and it seems to me to supply a much needed want in the literature of the subject. Professor Eli W. Blake, Drown University : The book is very opportune, as putting on record, in clear and concise form, what is well worth know- ing, but not always easily gotten. Professor George F. Barker, University of Pennsylvania : I have read it with a great deal of interest, and congratulate you upon the admirable way in which you have put the facts concerning this subject. The latter portion of the book will be especially valuable for students, and I shall be glad to avail myself of it for that purpose. Professor John E. Davies, University of Wisconsin : I am so much pleased with it that I have asked all the electrical students to provide themselves with a copy of it. . . . 1 have assured them that if it is small in size, it is, nevertheless, very solid, and they will do well to study and work over it very carefully. Professor Alex. Macfarlane, University of Texas : Allow me to congratu- late you on producing a work which contains a great deal of information which cannot be obtained readily and compactly elsewhere. SCIENCE. 53 Electrical Measurements. By Professor Henry S. Carhart and Asst. Professor G. W. PATTER- SON, University of Michigan. i2mo, cloth, 344 pages. Price, $2.00. IN this book are presented a graded series of experiments for the use of classes in electrical measurements. Quantitative experiments only have been introduced, and these have been selected with the object of illustrating general methods rather than applications to specific departments of technical work. The several chapters have been introduced in what the authors believe to be the order of their difficulty involved. Explana- tions or demonstrations of the principles involved have been given, as well as descriptions of the methods employed. The Electrical Engineer, New York : We can recommend this book very highly to all teachers in elementary laboratory work. The Electrical Journal, Chicago: This is a very well-arranged text-book and an excellent laboratory guide. Exercises in Physical Measurement By Louis W. Austin, Ph.D., and Charles B. Thwing, Ph.D., University of Wisconsin. i2mo, cloth, 198 pages. Price, $1.50. THIS book puts in compact and convenient form such direc- tions for work and such data as are required by a student in his first year in the physical laboratory. The exercises in Part I. are essentially those included in the Praciicwn of the best German universities. They are exclu- sively quantitative, and the apparatus required is inexpensive. Part II. contains such suggestions regarding computations and important physical manipulations as will make unnecessary the purchase of a second laboratory manual. Part III. contains in tabular form such data as will be needed by the student in making computations and verifying results. Professor Sara.li F. Whiting, Wellesley College : It comprises very nearly the list of exercises which I have found practical in a first-year college course in Physics. I note that while the directions are brief, skill is shown in seizing the very points which need to be emphasized. The Introduction with Part II. gives a very clear presentation of the essential things in Measurements, and of the treatment of errors. 58 MATHEMATICS. Elements of Algebra. By Professor James M. Taylor, Colgate University, Hamilton, N.Y. At Press. IN this book Professor Taylor aims primarily at simplicity in method and statement, and at a natural and logical sequence in the series of steps which lead the pupil from his arithmetic through his algebra. An introductory chapter explains the meaning and object of literal notation, and illustrates the use of the equation in solving arithmetical problems. This is followed by a drill on particular numbers before the pupil is introduced to the use of letters to represent general algebraic numbers. General principles are brought out by induction from particular cases, and proofs are given in their natural places where the pupil will be unlikely to memorize without comprehending them. Nomenclature has been looked to carefully. Many of the misleading terms of the older text-books have been discarded and others more useful and help- ful have been applied. The methods of working examples have been chosen for their simplicity and the scope of their application. Suggestions as to method of attack are given, but formal rules are stated but rarely. Positive and negative numbers are so explained and defined as to give clear and true concepts, such as lead naturally to still broader views of numbers. Factoring is made a fundamental principle in the solution of quadratic and higher equations. Particular atten- tion is given to the theory of equivalent equations. In illustrating the meaning of numbers, equations, and systems of equations, the graphic method is used. In general the aim is to render as clear as possible to the pupil all fundamental processes and to simplify the statement of rules. The book is particularly adapted to beginners, and is intended at the same time to prepare for any college or scientific school, as each subject is so treated that the pupil will have nothing to unlearn as he advances in mathematics. MA THE MA TICS. 59 An Academic Algebra. By Professor J. M. Taylor, Colgate University, Hamilton, N.Y. i6mo, cloth, 348 pages. Price, $1.00. THIS book is adapted to beginners of any age and covers sufficient ground for admission to any American college or university. In it the fundamental laws of number, the literal notation, and the method of solving and using the simpler forms of equations, are made familiar before the idea of alge- braic number is introduced. The theory of equivalent equa- tions and systems of equations is fully and clearly presented. Factoring is made fundamental in the study and solution of equations. Fractions, ratios, and exponents are concisely and scientifically treated, and the theory of limits is briefly and clearly presented. Professor C. H. Judson, Fur?7ian University, Greenville, S.C. : I regard this and his college treatise as among the very best books on the subject, and shall take pleasure in commending the Academic Algebra to the schools of this State. Professor E. P. Thompson, Miami University, Oxford, O. : The book is compact, well printed, presenting just the subjects needed in preparation for college, and in just about the right proportion, and simpiy presented. I like the treatment of the theory of limits, and think the student should be introduced early to it. I am more pleased with the book the more I examine it. Logarithmic and Other Mathematical Tables. By William J. Hussey, Professor of Astronomy in the Leland Stan- ford Junior University, California. 8vo, cloth, 148 pages. Price, $1.00. IN compiling this book the needs of computers and of students have been kept in view. Auxiliary tables of proportional parts accompany the logarithmic portions of the book, and all needed helps are given for facilitating interpolation. Various mechanical devices make this work specially easy to consult ; and the large, clear, open page enables one readily to find the numbers sought. It commends itself at once to the eye as a piece of careful and successful book making. 60 MA THE MA TICS. A College Algebra. By Professor J. M. TAYLOR, Colgate University, Hamilton, N.Y. i6mo, cloth, 326 pages. Price, $1.50. A VI GO ROUS and scientific method characterizes this book. In it equations and systems of equations are treated as such, and not as equalities simply. A strong feature is the clearness and conciseness in the state- ment and proof of general principles, which are always followed by illustrative examples. Only a few examples are contained in the First Part, which is designed for reference or review. The Second Part contains numerous and well selected examples. Differentiation, and the subjects usually treated in university algebras, are brought within such limits that they can be success- fully pursued in the time allowed in classical courses. Each chapter is as nearly as possible complete in itself so that the order of their succession can be varied at the discretion of the teachers. Professor W. P. Durfee, Hobart College, Geneva, N, Y. : It seems to me a logical and modern treatment of the subject. I have no hesitation in pro- nouncing it, in my judgment, the best text-book on algebra published in this country. Professor George C. Edwards, University of California: It certainly is a most excellent book, and is to be commended for its consistent conciseness and clearness, together with the excellent quality of the mechanical work and material used. Professor Thomas E. Bo3/ce, Middlebury College, Vt. : I have examined with considerable care and interest Taylor's College Algebra, and can say that I am much pleased with it. I like the author's concise presentation of the subject, and the compact form of the work. Professor H. M. Perkins, Ohio Wesleyan University : I think it is an excellent work, both as to the selection of subjects, and the clear and concise method of treatment. S.J. Brown, Formerly of University of ' Wis cons i?i ; I am free to say that it is an ideal work for elementary college classes. I like particularly the introduction into pure algebra, elementary problems in Calculus, and ana- lytical growth. Of course, no book can replace the clear-sighted teacher ; for him, however, it is full of suggestion. 10V 3" 1900