Book, ' CfcT Copighl X° COPYRIGHT DEPOSIT. ELEMENTS OF THE INFINITESIMAL CALCULUS BY G. H. CHANDLEK, M.A. Professor of Applied Mathematics, McGill University \ Montreal TRIED EDITION, REWRITTEN FIRST THOUSAND . <5 - NEW YORK JOHN WILEY & SONS London : CHAPMAN & HALL, Limited 1907 LIBRARY of CONGRESS Two Copies Received JAN 7 1907 Copyright Entry LASS J\ XXc„No. COPY B. ■ U MII I lull Copyright, 1907 BY G. H. CHANDLER ROBERT DRUMMOND, PRINTER, NEW YORK. PREFACE. The following pages are intended to serve as an introductory manual of Infinitesimal Calculus for beginners generally, but more especially for students of Engineering and other branches of Applied Science. As a logical foundation of the Infinitesimal Calculus the doctrine of Limits must be accepted as essential, but an attempt has been made at an early stage to accustom the reader to those principles and operations which are used in the practical applications of the subject. The order in which the subject is developed, though differ- ing from that of many text-books, is believed to be well calcu- lated to meet the difficulties and secure the interest of the student. In the present edition many changes and some additions have been made which it is hoped will bring the book into accord with present-day treatment and needs. To supplement the ordinary Mathematical Tables I have added short tables which are intended to facilitate curve tracing as well as the rapid calculation of integrals, etc. My thanks are due to Professor Murray Macneill for sug- gestions and for assistance in proof-reading and in the verifi- cation of examples. G. H. Chandler. Montreal, December, 19C6. iii CONTENTS. CHAPTER PAGE I. Limits. Infinitesimals 1 II. Functions. Derivatives. Differentials . . .11 III. Differential of a Power, a Product, and a Quotient . ■ 16 IV. Tangents and Normals 20 V. Differentials of Exponentials and Logarithms, 25 VI. Differentials of Direct Circular Functions. . 28 VII. Differentials of Inverse Circular Functions . 32 VIII. Differentials of Hyperbolic Functions ... 34 IX. Differentials as Infinitesimals 37 X. Functions of more than one Variable ... 41 XI. Small Differences . . . 47 XII. Multiple Points 51 XIII. Asymptotes 57 XIV. Tangent Planes. Tangents to Curves in Space 62 XV. Successive Differentiation 68 XVI. Rates 72 XVII. Maxima and Minima 75 XVIII. Curvature 82 XIX. Integration. Elementary Illustrations . . 93 XX. Fundamental Integrals I .98 XXI. Fundamental Integrals II 102 XXII. Fundamental Integrals III 105 XXIII. Fundamental Integrals IV 109 XXIV. Integration of Rational Fractions . . . .112 XXV. Integration by Substitution 114 XXVI. Integration by Parts 120 XXVII. Successive Reduction 124 XXVIII. Certain Definite Integrals . . , 5 . .129 v VI CONTENTS. CHAPTER PAGE XXIX. Areas and Lengths of Plane Curves. Surfaces and Volumes of Solids of Revolution . . .134 XXX. Simpson's Rule. Volumes from Parallel Sec- tions. The Prismoidal Formula. Length of a Curve in Space 142 XXXI. Polar Coordinates . .149 XXXII. Associated Curves . . . ... . . . .165 XXXIII. Centres of Gravity 176 XXXIV. Moments of Inertia 181 XXXV. Successive Integration 189 XXXVL Mean Values 202 XXXVII. Intrinsic Equation of a Curve. The Tractrix. The Catenary 205 XXXVIII. Infinite Series 211 XXXIX. Taylor's Theorem 223 XL. Fourier's Series 228 XLI. Approximate Integration. Elliptic Integrals 239 XLII. Singular Forms 244 XLIII. Successive Differentials of Functions of more than one Variable. Extension of Taylor's Theorem. Maxima and Minima from Taylor's Theorem 250 XLIV. Differential Equations of the First Order . 259 XLV. Differential Equations of the Second Order. 272 Appendix. Note A. Partial Fractions 287 Note B. Curve Tracing 290 Note C. Hyperbolic Functions 294 Note D. Mechanical Integration .... 299 Miscellaneous Examples 305 Tables. 1. Powers, Napierian Logarithms, etc. . . . 310 ' 2. Circular Functions I 312 3. Circular Functions II 313 4. Hyperbolic Functions 314 5. Lambda Function 315 6. Gamma Function 315 7. First Elliptic Integral 316 8. Second Elliptic Integral 316 Index 317 ELEMENTS OF THE INFINITESIMAL CALCULUS. CHAPTER I. LIMITS. INFINITESIMALS. i. Constant. Variable. When a quantity remains un- changed while another quantity changes, the former is called a constant, the latter a variable. 2. Limit of a variable. If the value of a variable v ap- proaches nearer and nearer to that of a constant a in such a way that their difference becomes and remains less in absolute * value than any given positive number, however small, v is said to approach a as a limit, and a is called the limit of v. If a; = 2,f the definition implies that the absolute value of x — 2 becomes and remains less than any positive number we choose to assign; e.g., it becomes <10 -3 ; it further * The absolute (or arithmetical or numerical) value is the value without regard to sign. Parallel vertical lines are used to indicate an absolute relation; thus |— 2| =2 expresses that the absolute value of — 2 is 2. So also x|<|a| or :r|<|a indicates that the absolute value of x is less than that of a . f The symbol = signifies an approach to a limit. Thus x = 2 may be read: x approaches 2 (as a limit). If the words "as a limit " are not expressed, they must be always understood. 2 INFINITESIMAL CALCULUS. [Ch. I. diminishes, becoming less than any smaller given positive number (say 10 -6 ), and so on. While x = 2. x—2 cannot be zero: the definition does not imply that x acquires the value 2. If it does become 2. it is no longer approaching 2 as a limit. Ex. 1. The value of (j : -4 x-2) is equal to that of x + 2 when any number except 2 is substituted for x. When j*=2 the fraction takes the form 0. an expression which is undefined and meaningless, but when x = 2 the limit of the value of the fraction is equal to that of J*-r2, i.e., 4, or in symbols * <~£?) - Similarly, if y=2x-x : . - = =2. 2. being the radian measure t of an acute angle, 6 lies be- tween sin and tan 0. Hence, dividing each of these into sin 0, sin 0/0 lies between 1 and cos 6. But cos = 1 when = 0. Hence /sin 6\ 3. The sum of the terms of the series 1— i + J— i-K . . has a limit as the number of terms increases without bound. Fig. 1. Let s n stand for the suni of the first n terms. Take 0P 1 =1, P,P_=-±. PJ* a =h etc. Then s x =0P ly s 2 =0P z , s,=0P, y etc. The points with odd subscripts continue to move to the left, * The symbcl £ is used for •"the limit of" (Echols. Differential and Integral Calculi. - The radian | =&7°'2958 . . .=206265") will be always understood to be the unit angle unless the contrary is manifest. 3. 4.] LIMITS. INFINITESIMALS. those with even subscripts to the right ; they tend toward meeting at some point P. since the fractions which are added = 0. Hei \OP — s n \ becomes and remains less than any given positive number, however small: therefore OP is the sum limit. The limit is that of the endless decimal '69314 . , . A series which has a limit is said to be convergent. 4. If the series \— f+f — £+. . . be similarly represented, the two sets of points cannot come within a distance 1 of each other, since the absolute value of the fractions added =1. Sup- pose the even P's to approach a point P (really the point P of Ex. 3) as a limit of position. Then OP— s n \ may become less than any given positive number, however small, but it will not remain so, for the addition of a single term changes OP — : by an amount approximately equal to 1. Consequently there is no limit, or the series is non-convergent. (Observe the signifi- cance of the words "and remains'' in the definition of § 2). 3. A variable quantity may or may not be capable of as- suming a value equal to that of the limit which it approach— Thus in Ex. 1. x — 2 = 4 when x = 2. and x — 2 = 4 when x = 2. Also (x 2 — 4) {x— 2) =4 when x = 2. but it cannot =4. for when x = 2 there is no fraction. Again, the series 1 — J-fJ— . . . can never equal its limit. In certain cases the limit and the value are entirely different. For example (Ch. XL), sin x—\ sin 2x — \ sin 3.r— . . .=\~ or — \- ac- cording as x=n by increasing or decreasing, but =0 when rm mm wU /. . 4, A variable may approach nearer and nearer to its limit in three ways: (1) by increasing only. (2) by decreasing only, (3) and by increasing and decreasing. Thus. Fig. 2 ; 4 INFINITESIMAL CALCULUS. [Ch. I. when M moves to the right v may be supposed to approach the limit a if P moves along A B only, (2) along CD only, or (3) from AB to CD, back to AB, etc., its value continually approaching a. 5. Limit of a constant. It is sometimes convenient to regard a constant as its own limit. Thus £ x=0 (ax + b) = b* and hence if a = 0, £ x ± b = b. So also £ Xz ± ax/x = a. 6. Infinitesimals. An infinitesimal is a variable whose limit is zero. It is therefore a quantity which approaches nearer and nearer to in such a way that its absolute value becomes and remains less than any given positive number, however small. Thus x is infinitesimal when # = 0; if x actually becomes it is no longer infinitesimal. Ex. When x is infinitesimal the following are also infinitesimals: x 2 , sin x, tan x, 1 —cos x, log (1 +x), 1 —2 X . 7. Quantities which are infinitesimal are such solely on account of their tending to a limit zero, not because of their having arrived at any particular degree of smallness; in other words, their chief characteristic is not being small but getting smaller. Nevertheless, it is often convenient and sometimes necessary to suppose them very small when they begin to = ; hence it is customary to regard them as very small in all cases. 8. From the definition' of § 2 it follows that the difference between a variable v and its limit a is an infinitesimal. Hence v — a = i, or v = a+i, where i is infinitesimal. Also if v — a=i, or v = a + i, where v is a variable, a a constant, and i an infinitesimal, a is the limit of v. Notice that the sign of v is the same as that of a as soon as the absolute value of i becomes less than that of a. 9. Infinites. A variable which is increasing (or decreasing) without bound, i.e., so as to exceed in absolute value any * As in algebra, letters near the beginning of the alphabet are used for constants unless the contrary is obvious. 5-12.1 LIMITS. INFINITESIMALS. 5 given positive number, however large, is called an infinite, and is represented by oo (or — oo ). Such a quantity has no limit which accords with the definition of § 2. It should be noticed that an infinite, like an infinitesimal, is a variable. If z = 0, l/z=oo or — oo, and if z=oo* or — oo, l/z = 0. Ex. When x = 2, x— 2 = 0, and l/(x — 2) = oo or — oo accord- ing as x approaches its limit by decreasing or increasing. When x = J^, tan x = oo or — oo according as x is increasing or decreasing. io. Quantities which are neither infinitesimal nor infinite are said to be finite. A finite variable is therefore one whose value stops short at some number which can be assigned. ii. If i is an infinitesimal and n any constant, ni is an infinitesimal. For |ra|< any assigned positive number a if i\<\a/n; but i does become | < \a/n f since a/n is also an assignable number. In special cases n may be 0, then ni remains =0. If n is infinite, ni may be infinite, or it may have a limit, which may or may not be zero. 12, The sum of any finite number n of infinitesimals is an infinitesimal. Let i be a positive infinitesimal which is and remains greater than the absolute value of any of the given infini- tesimals. Then the sum of the given infinitesimals —ni, and is therefore infinitesimal (§ 11). If n is infinite, the sum may have a finite limit. The deter- mination of such a limit is the fundamental problem of that part of the subject which is known as the Integral Calculus. Ex. 1+1+.. . +^ = I^±l ) =^(i + ^). The limit of the n 2 n 2 n 2 2 n 2 sum for n infinite is therefore J. *z = co should be read "x is a positive infinite," or u x increases without bound"; x— —go, u x is a negative infinite," or "x decreases without bound," 6 INFINITESIMAL CALCULUS Ch. L 13. Propositions relating to limits. Let v 1; v 2 be two variables which have limits a x , a 2 . Then V\-=a,i+ii and V2 z =a2 + i2i where i\ and i 2 are infinitesimals. (A) The limit of the sum (or difference) of the variables is equal to the sum (or difference) of their limits. For, (vi +v 2 ) - (ai +a 2 ) = i\ +%& which is infinitesimal. Hence £(^1 + ^2) = 7 -i +&2- The proposition is evidently true for the sum of any finite number of variables. (B) The limit of the product of the variables is equal to the product of their limits. For, v\V 2 — a x a 2 = {a\ + i\ ) (a 2 + i 2 ) — a x a 2 = a 2 i\+a\i 2 +i\i 2 > which is infinitesimal. Hence £v\V 2 = aia 2 . This also is true for any finite number of variables. Thus if x = a, £ 2 = a 2 , x 3 = a 3 , etc. (C) The limit of the quotient of the variables is equal to the quotient of their limits, provided that the limit of the divisor is not zero. For, ifa 2 ^0, ^-^ ai+k Ul a ^~ a ^ v 2 a 2 a 2 +i 2 a 2 a 2 2 +a 2 i 2 '' which is infinitesimal, since the numerator = 0, while the denominator = a 2 2 . Hence £— = — . ^2 &2 If a 2 = and ai^O, V\/v 2 is infinite and therefore has no limit. If a 2 = and ai = 0, V\/v 2 is the quotient of two infinitesimals and may have a limit, as in Exs. 1 and 2 of § 2. The determination of such a limit is the fundamental problem of that part of the subject which is known as the Differential Calculus, 13, 14.] LIMITS. INFINITESIMALS. sin ■m i «. . „ sin tan Ex. 1. Since tan = cos 0' cos Hence (§2, Ex. 2), £ d ^(^^jJ- = l. /sin 0\ 2 2. Since 1 -cos = — — , . . — =-— ; 1 + cos 2 1 + cos • r A-cosfl \ 1 _1_ ... sin 3 . . /tan0-sin0\ 1 3. tan -sin 0= — — rr, . - £ e =o\' Zi ) = o* COS 0(1 + COS 0) o v\ Q3 / 2 n-2~\ 2' " ^°°\n-2/ * a + b cosi(A— B) 1 n 5. In any plane triangle, fl cosiU+jB) - If a and & are tangents at two points near * one another on a curve, and the points ap- proach coincidence, A and B = and there- fore the right-hand member of the above ^= 1 . Hence £(a + b)/c = l when c = 0. We may assume that the arc > c and provided that a is x=a 4x 3 + x — 2 4a 3 + a-2 not a root of the equation 4# 3 + £ — 2 =0. 14. Orders of infinitesimals. If the limit of /?/«, the quotient of two infinitesimals, is zero, /? is said to be of a higher order than a. If /?/a has a limit which is not zero, 3 is said to be of the same order as a. If /?/a n (where n is a instant) has a limit which is not zero, /? is said to be of le nth order, a being assumed to be of the first order. * It is assumed that the points may be taken so near each other that the arc is everywhere concave to the chord. 8 INFINITESIMAL CALCULUS. [Ch. I. Ex. 1. a and /? being of the first order, a/? is of the second order, a 2 /?, a/? 2 of the third, a 3 /?, a 2 /? 2 , a/? 3 of the fourth. 2. If is infinitesimal and of the first order, sin 6 and tan d are of the first order, 1 —cos d of the second, tan 6— sin 6 of the third. 15. Definition. When the limit of the quotient of two infinitesimals is 1 the infinitesimals are said to be equiva- lent.* Thus if 6 is infinitesimal, #, sin d, and tan 6 are equivalent; also an infinitesimal arc and its chord. If the limit of ft /a is h (h being any constant, not zero), the limit of ft /ha is 1, hence ft is equivalent to ha. Thus 1 — cos 6 is equivalent to \6 2 , tan 6 — sin 6 to |# 3 . 16. If the difference of two infinitesimals of the same order is of a higher order, they are equivalent. For, if p=a+i, ft/a = l+i/a, .'. £{ft/a) = l if £(i/a) = 0. Conversely, the difference of two equivalent infinitesimals is of a higher order. For, if ft/a = l+i, ft=a+ia, and ia is of a higher order than a. The letter / will be used as a symbol for higher infini- tesimals. Thus if 6 is infinitesmal, sin 6 = 6+1, tan 6=6 +I\) also, since 1 — cos 6 is of the second order, cos 6=1+I 2 - 17. The limit of the quotient of two infinitesimals is not changed when either is replaced by an equivalent infini- tesimal. a ft' ' a' ' a' .'• £~=£^r if ^|=1 and £-,= !. a a ft a Hence if a and ft consist of infinitesimals of different orders, the limit of ft/a depends only on the infinitesimals of the lowest order in each. * Not equal, but equivalent in the sense of being interchangeable in the determination of the limit of a quotient or of a sum (§§ 17, 91). For 15-17.] LIMITS. INFINITESIMALS. Examples. 1. Let A B be a, circular arc of radius a subtending an infinitesi- mal angle 6 at the centre, BC perpendicular to OA, AD and BE tangents. Let be regarded as of the first order. Then (1) The arc AB=ad, and is therefore of the first order. . v ^chord AB _ J2a sin \Q e o =£ 2 -^ (§§15, 17) -a. Fig. 4. B^,D c A The chord is therefore of the first order and equivalent to ad, the arc. ,«v r CA /• <*(! -cos 0) i : : Hence CA is of the second order and equivalent to %ad 2 . AD-CB q(tan 0-sin 0) — & . 2« (4) £ ^ 03 Hence AD—CB is of the third order and equivalent to Ja0 3 . ,*. r £D-CA r (CA/cos0)-CA r a(l-cosd) 2 ( 5 ) £ 71 — =£" 71 =£ 6- =£ add 2 )' 6- = \a. 4 cos Hence BD —CA is of the fourth order and equivalent to \a0*. A T\ fl D 2. Show that the limit of ^r— — — =2. 3. Show that (AE + EB) —chord AB is equivalent to |a0 3 , and hence that the difference of arc and chord is an infinitesimal of at least the third order. 4. Find £ X-2 x 3 — x 2 — 2x whena;=l and x = 2. Arts. (1) J, (2) £. 10 INFINITESIMAL CALCULUS. [Ch. I. Show that there is no limit when x = or —1. j. cosecz— cot x £ 1— cos 3 0_ 3 u sinx sin 2 (n -l)(w- 3) n(n— 2) 7. £ n =oo ; 7T\ "~1« 8. £j;«— 00 (2-|-3 a; )— 2. v'l+a; —1 9 - £c=o — = i- [Rationalize the numerator.] r sin a:— sin a 10. £3. . n =cos a. x=a x-a CHAPTER II. FUNCTIONS. DERIVATIVES. DIFFERENTIALS. 18. Function. When a variable quantity depends for its values upon those of another variable quantity, the first is said to be a function of the second, and the second is called the variable or argument of the first; e.g., x 2 — 2:r + l, x x , log (a+x), sin ax, are functions of the variable x. The expression f(x) is used as a symbol for a function of x, f(a) being the value of the function when x = a; e.g., if f(x) = l-x 2 , /(0) = 1, /(1) = 0, /(2)= -3, /(a)=l-a». For a similar purpose F(x), f(x), etc., may be used, and f(x, y), F(x, y), etc., for functions of x and y. The variable of a function may itself be a function of another variable, or it may be an independent variable — one to which arbitrary values may be assigned. 19. Implicit functions. In any equation containing two variables x and y, e.g., y 2 =4ax, log (x + y) = 2, either of the variables is virtually or implicitly a function of the other, or an implicit function of the other, since the value of either is determined when that of the other is assigned. If we solve for y in terms of x, y becomes explicitly a function of x, or an explicit function of x. 20. Graphs. The curve whose equation is y = f(x) is the graph or geometrical representation of the function /(#). The ordinate corresponding to any abscissa x is the value of the function when the value of the variable is x. When for a value of the variable there is only one corre- sponding value of the function, the function is said to be 11 12 INFINITESIMAL CALCULUS. [Ch. II single- valued. Thus e x , Fig. 5, and log x, Fig. 6, are single- valued functions. The function sin -1 x, Fig. 7, is multiple- valued. It will in general be assumed that a function is single- valued ; when such is not the case the function may be treated as single-valued by considering a limited range of its values. Thus sin -1 x is single-valued if n values > -j: and < it — are excluded. The ordinate of the curve y 2 = 4x is a double- valued function of x, but may be represented by two single-valued functions 2\/^and -2\/x. Fig. 6. Fig. 7. 21. Continuity. In general a gradual change in the value of a variable produces a gradual change in the value of the function, but it is possible that a slight change in the variable may produce an abrupt finite or infinite change in the func- tion. In more precise language, f(x) is continuous for the value a of the variable when, as /i = (h being a small change in x), £f(a + h) = £f(a-h) = f(a), and discontinuous if this relation is not true. LetOA = a,BA = AC = h. In Fig. 8, AP = /(a), CR = f(a + h), BQ = f(a-h). Also as ft=0, £CR=AP, and £BQ = AP, hence the ordinate is continuous at A. But in Fig. 9, £CR = AP, £BQ = AP'; these are not equal, hence y is dis- continuous at A. In Fig. 10, BQ becomes infinite when h = 0, and y is discontinuous. When the function changes abruptly from one finite value to another finite value it is said to have finite discontinuity, 21-24.] CONTINUITY. DERIVATIVE. 13 when the function becomes infinite it is said to have infinite discontinuity. Ex. 1. /Or) has finite discontinuity at x =0. For, when 2*+l h = 0, £}(h)=0 and £f(—h) = \. Thus when x increases through the value the function drops suddenly in value from slightly less than 1 to slightly more than 0, without passing through the intermediate values. It cannot be said to have any value when z=0. 2. The following have infinite discontinuity for x = l: (x+l)/(x-l), (x-l)~i, 3^~^~\ tani^. 3. Examine the function 2 X for x=0. 22. Interval. Values of the variable between two assigned values a and b are said to lie in the interval from a to b. The interval may be conveniently indicated by [a, b]. Re- versal of a bracket indicates the exclusion of the adjacent end value; e.g., [a, b[ indicates values from a to b, including a but excluding b. Thus (1 — x 2 ) 1 * is real for the interval [— 1, 1], (1 — z 2 ) - * is continuous for the interval ] — 1, 1[. 23. Increment. Any change in the value of a quantity is called an increment or difference of that quantity. An increment is positive or negative according as the quantity is increased or decreased. The symbols Ax, dx, are used for increments or differences of x. 24. Derivative. Let there be a variable and a function of that variable. A particular value of the variable being x 14 INFINITESIMAL CALCULUS. [Ch. II let the corresponding value of the function be y. Let x receive an increment Ax, and let Ay be the corresponding increment of y. The limit of Ay/ Ax when Ax = is called the derivative of y. Thus the derivative is the limit of the quotient of the increment of the value of the function by the increment of the value of the variable when the latter in- crement is infinitesimal. The primary object of the Infini- tesimal Calculus is to determine this limit for various kinds of functions. Ex. 1. liy=x 2 , Jy = (x+Jx) 2 -x 2 =2xJx+(Jx) 2 . .*. Jy/Jx=2x + Jx. .'. £{Ay/Ax)=2x when £Jx=0. Similarly, if y=ax 2 , £(Jy/Jx) =a . 2x, a being any constant. 2. If y =ax 3 , show that £(Jy/Jx) =a. 3x 2 . 3. If y=x, Ay = Ax, .'. Ay/Ax = \, or £(Ay/Jx)=l (§5). Similarly if y=ax, £{Ay/Ax) =a. 4. y=4x 5 -3x 2 + 2x-l. The method of obtaining Ay /Ax shows that the result will be the same as if each term were treated separately and the results added, also that a constant term disappears in subtracting. /. £(Jy/Jx) =4: .3x 2 -3 .2x + 2 = 12x 2 -6x + 2. 25. The general method illustrated in the above examples may be described as follows: Let y=f(x). The value of the function for the value x-\-Ax of the variable is }(x + Ax); hence Ay, the increment of function, is f(x + Ax) — f(x), and Ay __f(x + Ax) — f(x) Ax Ax This expression is simplified and its limit taken when Ax = 0. This limit is the derivative, and is, for various values of x, a new function of x. It is called the derived function, or derivative function, or simply the derivative, of the given function. Let it be written f'(x). Thus if f(x) = x 2 , f'(x) = 2x; if./(x) = ax 3 , f'(x) = 3ax 2 ; if f(x) = x 2 + 2ax, f'(x) = 2(x + a). It should be noticed that there cannot be a limit (a deriv- ative) unless Ay = as well as Ax; i.e., unless £f{x J rAx) = j(x) when Ax = 0, or (§ 21) unless ](x) is continuous for the value of x in question. 25-27.] DIFFERENTIAL. 15 Y a p^£ Ay Ax ^$ y y o a? Aa? Fig. 11. 26. Geometrical illustration. Let y = f(x) be the equation * of a curve of which P(x, y) is a point. Then Ay/ Ax is the slope or gradient of the secant PQ. When Ax^O, Q approaches P, and the limit of pc sit ion of the secant is (by defi- nition) the tangent at P. Hence £(Ay/Ax) or f(x) is .tan 0, the slope of the tangent at (x, y). Thus for the curve y = x 2 , tan<£ = 2:z; for y=x 3 , tan<^> = 3x 2 . 27. Differential. Def. The differential of the variable of a function is any increment of that variable; the differential of the function is the derivative of the function multiplied by the differential of the variable. The letter d is used as an abbreviation for "the differential of." If then y or f(x) is a function of x, the definition states that dx is any increment of x, and that dy or df(x) is fix) dx.~\ Hence g=f(s) = 4>en^0. Thus dx and dy are defined in such a way that dy/dx is equal to the limit of Ay/ Ax, or the differential quotient is the limit of the difference or increment-quo- tient. Geometrically, dy/dx is the slope X of the tangent at (x, y), Fig. 12, and dy is the increment of the ordinate of the tangent corresponding to the increment dx of the abscissa. It should be noticed that, although dx may have any value, the value of dy/dx is independent of dx. * The angle between the axes is assumed to be a right angle in all cases unless the contrary is mentioned. t From its position as a multiplier of dx the derivative f\x) is also called t e differential coefficient of fix). % If the angle between the axes is co, dy/dx = sin <£/sin (). CHAPTER III. DIFFERENTIAL OF A POWER, A PRODUCT, AND A QUOTIENT. 28. The operation of obtaining derivatives or differentials is called differentiation. We now consider a few general formulae which will assist in differentiating, first showing that the differential of the algebraical sum of any finite number of terms is equal to the algebraical sum of the differentials of the terms; also that a constant factor in a term appears as a factor in the differential of that term, and that a constant term dis- appears in differentiating, or has for its differential. Let y=au + v — w + c, where u, v, w are continuous func- tions of x, and a and c are constants. Let the increment Ax in x cause increments Au, Av, Aw in u, v, w, Ay being the resultant increment of y. Then Ay = [a(u + Au) + (v + Av) — (w + Aw) + c] — (au +v — w+c) = a Au+Av — Aw. Ay _ Au Av Aw Ax Ax Ax Ax' Hence, taking the limits when Ax±=0, dv du dv dw , 77, -r £ = a- r -+- —, or dy=a du-\-dv — dw. dx dx dx dx .'. d(au + v — w + c) = a du+dv — dw. 29. Differential of a power. Let v be a function of x, to find d(v n ), n being any constant. Let y = v n . Then Av\ n / dv\ n Ay=(v+Av) n -v n = v n (l-\ — ) -1 16 28, 29.] POWER. PRODUCT. QUOTIENT. 17 Taking Av\<\v, expanding by the binomial theorem, and dividing by Ax, Ay - Av ^ nv n- I ' n—\ Av \ Ax Ax Taking the limits when Ax^O, and .'. Av also = 0, dy , dv dx dx J or d (y n ) = nv n ~ l dv. (A) This result is true for all values of n. The case in which n = \ deserves special mention; (A) then becomes d(V?)« * (B) 2Vv When n= — 1, (A) becomes Ex, 1. d(x 5 )=5x*dx. 2, d(3x 5 + 2)=3d(x 5 )=l5x*dx. 3 d(3x^_-2x 2 + 6)=3 Ax 3 dx-2.2xdx=4x (3x 2 -l)dx. 4. dV x 2 =d(x*) =%x~$ dx. 5. d(-) =d(x~ 2 )=-2x- z dx= -. \x 2 / X 3 6* d(a 2 + x 2 y=3(a 2 + x 2 ) 2 d(a 2 + x 2 ) =S(a 2 + x 2 ) 2 2xdx=6x(a 2 + x 2 ) 2 dx. In this example v=a 2 + x 2 , and n=3, a being constant. „ j/a—xX 1 7, x dx 7. a I = -d(a—x)= -. \a—b/ a—b a — b 8 d . =d(ax + bx 2 )~* v(ax + bx 2 ) 3 = -%(ax + bx y-r* d(ax + bx 2 )= -Uax + bx 2 )-Ha + 2bx) dx. n W"l 2 d{a 2 -x 2 ) /T>N -2xdx x dx 9. dv a *-x 2 = — - \ by (B), = — = 2V a 2 -x 2 2Va 2 -x 2 v 'a 2 - ( 1 \ d(a 2 -x 2 ) 2sds 18 INFINITESIMAL CALCULUS. [Cn. III. 30. Differential of a product. Let y=uv, where u and v are functions of x. Then Ay = (u+Au)(v + Av)—uv = v Au+u Av+Au Av. Ay Au Av . Av .'. -r = v—+u-7-+Au—. Ax Ax Ax Ax The limit of the last term is 0. dy du dv • • 7 v^ 1 t^ ^ . ax ax ax or d(uv) = v du + u dv. (C) Similarly, d(uvw) = vw du+wu dv +uv dw. (C^) Ex. d(4x + 3)(x 2 -l) = (x 2 -l)d(4:X + 3) + (4x + 3)d(x 2 -l) = (x 2 -l)4:dx + (4:x + 3)2xdx=2(6x 2 + 3x-2)dx. 31. Differential of a quotient or fraction. Let the frac- tion be u/v, u and v both being variable. Then d ^) =d \ U v) = v du+U \ : ^)' by (C) and (Bl) * TT Ju\ vdu — udv /T ^ X Hence d[ — )= 5 . (D) _ 7 /V-l\ (x 2 + l)d(^ 2 -l)-(x 2 -l)d(o: 2 + l) 4zdr .Lx. d x 2 +V (x 2 + l) 2 (z 2 + l) 2 ' Examples. 1. d(a 2 — x 2 ) 3 = — 6x(a 2 — x 2 ) 2 dx. 2. d^\Vx 2 =xdx/^\+x 2 . 3. If j(x)=ax 2 + 2bx + c, f'(x) =2(ax + b)* 4. y = Vx' d —a 3 , dy/dx=3x 2 /2\ // x 3 —a 3 . 5. d [ax(x 2 - l)(x + 1)] = a(x + l)(4o: 2 -3 - l)dc. /^-1\ 8a: 3 da; 7. d(l+x)\ / l-a; = (l-3a;)da;/2\ / l-a;. *A*)-^/(*)/*i|27. 30,31.] POWER. PRODUCT. QUOTIENT. 19 8. y = 3(a + bx 2 )*, dy^lObx(a + bx 2 )sdx. 9. y = vV + {b -x) 2 , dy=(x- b)dx/V a 2 + (b -x)\ 0. y={a 3 -x 3 )~\ dy/dx = 3x 2 /(a 3 -x 3 ) 2 . 1. i/ = o; 3 /(«+^) 2 , dy/dx= (3a+x)x 2 /(a+x) 3 . 2. y=\/a 2 —x 2 ,dy=—xdx/^a 2 —x 2 . 3. y=^2ax—x 2 , dy=(a—x)dx/v / 2ax—x 2 . 4. y = x*/(a 2 -x 2 ), dy/dx= (3a 2 -x 2 )x 2 /(a 2 -x 2 )\ 5. d[z n /(l +x) ri ] = nx^- 1 dz/(l +x) ri + l . 6. d(a 2 -z 2 )- 1 = 2xda;/(a 2 -a; 2 ) 2 . 7. i/=a; 2 /v / l+x 4 , dy/dx = 2x/(l+x 4 )*. 8. 2/ = (x-a)/v / x, dy/dx = (a + x)/2Vx*. 9. y=ax/\ // 2ax—x 2 , dy/dx=a 2 x/(2ax—x 2 )*. on \ aJrX j adx a ~* (a-a;)\/a 2 -x 2 21. 2/=2x/Va 2 + z 2 , d?/=2a 2 ^/(a 2 + x 2 )i. 22. .y=z(a 3 + z 2 )>/a 2 -x 2 , cfy/dx = (a 4 + a 2 x 2 -±x 4 )/^a 2 -x 2 . 23. f(x)=V x + Vl+x 2 , J'(x)=Vx + Vl+x 2 /2Vl+x 2 . x 2 y 7, 24. — + --=1. Differentiating each term, 2x dx 2y dy , dy b 2 x a 2 b 2 ' ' ' dx a 2 y 25. y 2 =£ax, dy/dx = (a/x)*=2a/y. 26. x 2 y + b 2 x-a 2 y=0 y dy/dx = (b 2 + 2xy)/(a 2 -x 2 ). 27. x 3 + y 3 =3axy, dy/dx= — (x 2 —ay)/(y 2 —ax). 28. x 2 y—xy 2 =a 3 , dy/dx = (y 2 —2xy)/(x 2 —2xy). 29. If y-i show that ^_ + — ^==0. * Vl+z 4 Vl + ^ CHAPTER IV. TANGENTS AND NORMALS. 32. Let P and Q be two points near one another on a curve of which the equation is given. Let the coordinates of P be (x, y), then x = OA, y = AP. When x has the in- Y F /4i / c /<& Fig. 13. Fig. 14. crement Ax or AB, the new value of y is BQ, hence CQ is Ay. Let the tangent at P make an angle with the x-axis. Then as in § 26, when Q approaches P as a limit of position, Ax = 0, and tan (j) = £ tan CPQ = £(Ay/Ax) = dy/dx. Let the length of the curve measured from some point up to P be s, and let the length of the arc PQ be As, and the length of the chord PQ be q. Then cos = £ cos CPQ = £(Ax/q) = £(Ax/ As) (§17) =dx/ds. Similarly, sin cf) = dy/ds. Thus, cos^ = g(l), sin<£ = |j (2), tan«£=g (3). 20 32, 33.] TANGENTS AND NORMALS. 21 or Squaring (1) and (2) and adding, * ds 2 = dx 2 + dy 2 . o N A Fig. 15. Fig. 16. These relations show that if dx is PD (Figs. 15, 16), dy is DE, and ds is PE. dx ds 33. The subtangent TA = y—, the tangent^ TP=y—, the subnormal AN = y -p-, the normal NP = y -r-. The intercepts of the tangent on the axes are OT = x-y^, OS= -OT tan = y-x& y - y *= (fX {x ~ xi) Also, is the equation of the tangent, and * Powers of a differential dx are written dx 2 , dx 3 , etc. They must be distinguished from d(x 2 ), d(x 3 ), etc., which are differentials of powers of x. t The line-segments known as the tangent and normal are the portions of the tangent and normal which join the x-axis to the point of contact. 22 INFINITESIMAL CALCULUS. [Ch. IV. the equation of the normal, at a point whose coordinates are (x\, 2/1), the parentheses ( ) x indicating the particular value which the enclosed quantity has when x\ and y\ are substituted for x and y. It will be convenient to take dx as +, i.e., measured in the + direction of the a>axis, and to suppose (p to be a positive or a negative acute angle; hence cos and ds are always + , and sin have the same sign as dy. It should be noticed that the curve rises or falls (y increases or decreases) according as dy is + or — . Examples. 1. The curve a 2 y=x(x 2 —a 2 ), Fig. 17. Differentiating we have tan =dy/dx = (3x 2 —a 2 )/a 2 . At the origin x=0, .'. tan = — 1 and .'. (f>= —45°. At A or 5, x= ±a, .'. tan <£ =2 and =63° 26'. When x= ±a/Vs, tan <£=0, /. <£=0. Y -— ^ y / / / /S \ \ / / y V \ 1 / l"^s / 1 X \ / \ / \ / X Fig. 17. Fig. 13. The equation of the tangent at any point (x ly y x ) is y-y* 2. The common parabola y 2 =4ax. Differentiating each term, 2y dy =4:0, dx, .'. dy/dx=2a/y. .'. y —y l = {2a/y l ){x—x l ) is the equation of the tangent at (x lf y x ), and reduces to y 1 y=2a(x + x 1 ). The subnormal =y dy /dx = 2a, a constant. 3. The equation x% + y*=a$ represents an astroid, or four-cusped hypocycloid (Fig. 18), i.e., the locus of a point in the circum- 33.] TANGENTS AND NORMALS. 23 ference of a circle which rolls inside the circumference of a fixed circle, the diameter of the latter being four times that of the former. Differentiating the equation, we get %x~$ dx + %y~% dy=0, whence dx/dy = — (x/y)$. The intercepts of the tangent on the axes will be found to be cfi x% and a$ y$. Squaring, adding, and taking the square root we find that the part of the tangent intercepted between the axes is of constant length, viz., a. Hence, if a straight line of length a slide with its extremities on two given lines at right angles to one another, it will constantly touch this curve. 4. To find tan <$> at any point of the curve x 2 y—xy 2 =2. Differentiating each term by (C), x 2 dy + 2xy dx—2xy dy—y 2 dx=0, .*. dy/dx = (y 2 —2xy)/(x 2 —2xy). 5. Find the equations of the tangents at the points (—1, 1), (2, 1) on this curve. Arts. x—y + 2=0, x=2. 6. Of the rectangular hyperbola xy=k 2 show that (1) the equation of the tangent at (x 1} y x ) is x/x 1 -{-y/y l =2 f (2) the equation of the normal at (k, k) is y=x } (3) the subtangent always = —the abscissa, (4) the tangent makes with the axes a triangle of constant area, viz., 2k 2 . 7. Show that the tangent to the curve (x + a) 2 y=a 2 x is parallel to the axis of x when x=a f perpendicular to it when x= — a, and that the tangent at the origin bisects the angle between the axes. 8. Find the equations of the tangent and normal at the point (a, a) on the curve ay 2 =x 3 . Arts. 3x—2y=a, 2x + 3y=5a. Also show that the subtangent = fa, the subnormal = fa, the tangent =^aVl 3, the normal =JaVl3. 9. On the curve x 2 y + b 2 x=a 2 y, show that tan 4>=b 2 /a 2 when x=0, 20b 2 /9a 2 when x=^a y and 5b 2 /9a 2 when x=2a. 10. Show that the curves y(4: + x 2 )=8, 4y=x 2 , intersect at an angle tan -1 3. 11. Find the equations of the tangents of the following curves at the given points : (1) xy = l+x*eLt (1,2). Arts. x-y + l=0. (2) x 2 + y 2 =x s at (2, 2). 2x-y=2. (3) xn + yn=xfi+ l &t (2, 2). (n + 2)x-ny =4. 24 INFINITESIMAL CALCULUS. [Ch. IV. (4) a 2 y=x* at (x u y x ). (5) y 2 =3x+l at {x l} y x ). a 2 o 2 Sa^lc — a 2 !/=2:r, 8 . 3^-22/^ + 3^4-2=0. a 2 6* the 12. 4:xy=4: + x 3 ; show that at (2, f) the subnormal = f, subtangent =2, the normal = V, the tangent = f. 13. y 2 =3x + l; show that when y=—£ the subnormal = f , the subtangent =-3-, the normal = — JV73, the tangent =1^73. 14. x$ + y$=a$, Fig. 18; show that ds = (a/x)$dx. 15. Find an expression for the length of the perpendicular from the origin on the tangent at any point (x, y) of any curve. Ans. (x dy—y dx)/ds. 16. In the case of the parabola y 2 =4:ax, show that the length of this perpendicular = x Va/ (a + x). CHAPTER V. DIFFERENTIALS OF EXPONENTIALS AND LOGARITHMS. 34. Differentials of the exponentials a* and e v . Let a be any constant, v any function of x. Then A(a v ) = a v + jv -a v = a v (a* v -l) = a*[A Av + J A 2 (Av) 2 + ...], A = loge a, by the exponential theorem, the series being convergent for all values of Av. Ax Ax and taking the limits, d(a v ) _ a v dv dx dx 1 or d (a v ) = Aa v dv. (E) When a = e = 2.71828 . . . , A is 1. Hence d«) = ^cfa. (F) Ex. 1. d(e 3x ) = e 3x d(3x) = 3e 3x dx. 2. d{e~ x ) = e~ x d( —x) = —e~ x dx. 3. d(2~ x ) = (loge 2)2-* d( -*) = - (loge 2)2"* dx. 35. Differentials of logarithms. First suppose the loga- rithms to be Napierian (or hyperbolic or natural) logarithms, the base being 6 = 271828 . . . Let y = \og e v, 25 26 INFINITESIMAL CALCULUS. [Ch. V. then v = e v , .'. dv=e v dy=v dy, dv .*. dy, or d(\og e v) = — . (G) Secondly, let the base be any number a. Then V log a v = M loge v, where M= 1/loge a, or = loga e, .-. d(\og a v) = M— . (Gi) V M is the modulus of the system of logarithms with base a. Unless the contrary is indicated, the logarithms are always assumed to be Napierian. Ex. 1. dlog (ax 3 )=d(ax*)/ax z =3 ax 2 dx/ax 3 =S dx/x. 2. d(x log x) =log x dx-Vx d(log x) = (log x + l)dx. 36. To differentiate u v , where both u and v are variable quantities. Let y = u v , then log y= (log u)v; hence, differen- tiating, — = v \- (log u)dv, .'. dy = y\v h (log u) dv \ y it i— tc -J .*. d(u v ) = v u v ~ l du+ (log u)u v dv, (G 2 ) i.e., the differential is obtained by supposing u and v in turn to vary while the other remains constant, and adding the results. 37. When an expression is made up of factors it is often simpler to take logarithms before differentiating. Ex. 2/ = (z + l)*(x + 3)V(z + 4) 4 , logy=ilog (x + l)+f log (:r + 3)-41og (x+4), dy 1 dx 9 dx t dx y 2 x + \ 2x + 3 x + 4' whence dy. 36,37.] EXPONENTIALS AND LOGARITHMS. 27 Examples. 1. y=- log— — = — [log (a; -a) -log (x + a)], dy = 2a x + a 2a x 2 —a \—i 2. y=\og (x/Vl + x 2 ) =log x— \ log (1+x 2 ), dy=dx/(x + x 3 )> 3. }(x) = Vx- \og(l + Vx), f , (x)= j(l + y/x)- 4. dlog (z + V^ia^-dx/V^ia 2 . 5. y=\og[e x /(l+e x )], dy=dx/(l+e x ). 6. y=e xn , dy=ne xn x n ~ 1 dx. 7. dlog (Vx-aH- V / x-6)=idx/v / (a:-a)(a;-6)» 8. y= a loex = (e A ) log * = (e log *) A =x A , y=uvw, and y=u/v, after taking logarithms, and compare the results with formulae (C), (CJ, and (D). 13. Differentiate y=v n after taking logarithms, thus showing that d(v n ) =nv n ~ l dv. 14. Show that the subtangent of the exponential curve y=a x is constant and =log a e. 15. Find the subnormal of the curve y 2 =a 2 log x. Arts. a 2 /2x. Ad -fc Ad d sin v dd = cos V dv dd' d sin v = = cos V dv. d cos v = ■■ — sin v dv. CHAPTER VI. DIFFERENTIALS OF DIRECT CIRCULAR (TRIGONOMET- RICAL) FUNCTIONS. 38. To differentiate sin v. Suppose v to be a function of 6. Then A sin v = sin (v + Av) — sin v = 2 cos (v + \Av) sin £ Jv,* . r i sin v r 2 cos v . \Av /ff v _ •-£—nr- = £ 7^ > (§ 17 ) or (1) Similarly, d cos v = — sin v rfy. (2) 39. The differentials of the remaining functions may be found by first expressing them in terms of sine and cosine The results with (1) and (2) above are: t d sin v = cos v dv, (H) d cos v = — sin v dv, (I) d tan v = sec 2 v dv, (J) d cot v = — cosec 2 ?; <#y, (K) d sec v = sec v tan 7; cfa, (L) d cosec v = — cosec v cot v dv. (M) * Since sin A— sin # = 2 cos i(A + i?) sin h(A— B). t To these results may be added: d vers v = d(l —cos r)=sin v dv, d covers v = d(l— sin v)= —cos v dv. 28 B3,39.] DIRECT CIRCULAR FUNCTIONS. 29 Examples. 1. d sin nO = c(^s nO d(nO) =n cos nO dO. 2. d sin (tan 0) = cos (tan 0) d(tan 0) =cos (tan 0) sec 2 dO. 3. y=tan0-0, dy = tsa\ 2 ddd. 4. d(£0-isin20)=sin*0d0. 5. d(i0 + isin20)=cos 2 0d0. 6. d(sec + tan 0) = (l+sin 0) d0/cos 2 0. 7. y=J tan 3 £ — tanx+xj dy=t2Ln 4 xdx. 8. /(a;) = sin a: —J sin 3 a;, /'(a;) = cos 3 x. 9. d(sin 2 .r cos 2 x) = £ sin 4x cfo. 10. ?/ =log tan J0, dy= cosec dO. 11. 2/= log tan (i?r + £0), dy=secdd0. 12. d(sec + log tan £0) = sec 2 cosec d0. „_ ./ sin x \ (cos 3 x—sm^x)dx 13. d[~ I =-—. — . U + tanov (sin a: + cos xy 14. /(x) =sin (log x), f'(x) =x~ l cos (log x). 15. de* cos x =e x (cos x —sin x) dx. 16. d log sin = cot d0. 17. d log cos = —tan d0. 18. d log tan =sec cosec d0. 19. d log sec 0=tan dO. 20. d log (sec + tan 0) =sec d0. 21. y =log Vsin x + log Vcos x, dy/dx = cot 2x. 22. y=2/(l+tanix), dy/dx = -1/(1+ sin z). 23. rilogV(l-cos 0)/(l + cos 0) = cosec Odd. 24. d sin nd sin n = n sin n_1 sin (n + 1) d0. 25. By differentiating sin 20=2 sin cos 0, show that cos 20=cos 2 0-sin 2 0. 26. The cycloid. This is the curve traced by a point in the circumference of a circle which rolls along a straight line, Fig. 19. Let 0=the angle through which the circle (of radius a) rolls while the tracing-point moves from to P. Then x = OM = OB-MB=2lyc PB-PD = ad-asmd, y = MP = BC-DC = a -a cos 0. From these two results may be eliminated; but as the result- 30 INFINITESIMAL CALCULUS. [Ch. VI, ing equation is not algebraical we shall suppose the locus deter mined by the simultaneous equations x=a(0 — sin 0), y =a(l — cos 6). o M Fig. 19. For a single arch of the curve varies from to 2n\ for greater or smaller values of the curve is repeated indefinitely in both directions. Produce BC to meet the circle in E 1 then PE is the tangent and PB the normal at P. For dx=a(l -cos 6) dd =BD dd, dy=a sin 6dd=DP dd, /.if the tangent makes an angle (j> with the axis of x, tan =dy/dx=DP:BD=tfmDBP = ttmDPE } BPE m an angle in a semicircle, being a right angle. Therefore PE is the tangent. Hence at each instant the circle may be supposed to be turning about its lowest point as an instantaneous centre of rotation." Since CE = CP,CEP=W J ,\ the normal PB = 2a sin £0. 30.] DIRECT CIRCULAR FUNCTIONS. 31 27. If the axes of the cycloid be taken as in Fig. 20, its equa- tions are x=a(l — cos 0), y =a(#-fsin 0), being the angle through which the circle has rolled from R. The locus of Q (of D in Fig. 19) is called the " companion to the cycloid." Its equations are x=a(l — cos 0), y=aO. Show that in this curve tan <£=cosec 0, and hence that <£ is least when x=a. 28. At any point of the cycloid, show that ax y y y dy \2a „ V2ax— x 2 • 29. In the cycloid, Fig. 20, show that ds/dx = \/(2a)/x. CHAPTER VII. DIFFERENTIALS OF INVERSE CIRCULAR (TRIGONOMET- RICAL) FUNCTIONS. v 40. To differentiate sin -1 — the radian measure of the a angle whose sine is v/a, a being a constant. v Let y=sin _1 — , then v = a sin y. I v 2 / c°. dv = a cos y dy=a^l — ^ dy=V a 2 — v 2 dy. .'. dy or v dv ^ dv d sin" 1 - = , (N), or dsin~ 1 v = . , if a=l. a vV-v 2 Vl-v 2 Similarly, , _J dv , dt> .. a cos x -= (Ni), or d cos 1 v= . . if a=l: a Va 2 -v 2 Vl-v 2 dtan -1 - = -^- — ^ (P), or d tan -1 i> = T — —5, if a=l; a cr + ir 1+v 2 , . J a dv - dv a cot l ~ = — 5-7 — 9 (Pi), or d cot 1 v= — — — 5, if a=l; a a 2 + v 2 1+v 2 v adv ^ , dv dsec x -= — , (Q), or d sec x v = — ,. , if a=l; a v V ^2 _ a 2 ^\/ ^2 _ I 7 , v a dv ,~ N 7 1 dv d cosec -1 - = (Qi)> or d cosec _1 v= . , a v \/ v 2_ a 2 -yVv 2 — 1 if a=l. * This formula should be preceded by a minus sign if cos?/ is — , i.e., if the angle is a second or third quadrant angle (see Fig. 7). The f rmulse as given may be supposed to apply only to first-quadrant angles 32 40.] INVERSE CIRCULAR FUNCTIONS. 33 1. dsin- 1 (2x 2 ) = Examples. d(2x 2 ) 4x dx Vl-(2x 2 ) 2 Vl-4x 4 dtan-^ dx 2. d(log tan _1 x) = tan _1 x (l+x 2 ) tan _1 x" 6x da; ,1 j , a a ^ 3. a sin -1 3x 2 = — , 4. d cos -1 — = V1-9X 4 a; x vV-a : _ , . x— a a dx _ , , a— a; dx 5. asm -1 = — . 6. a cos -1 ^ xV / 2ax— a 2 a V2ax— x 2 m , _, x 2 2x dx dx 7. a sin l — = — - . 8. d tan -1 ^ = a 2 vV _£4 e* + e~* 9. d sec- 1 Vl+x 2 = dx/(l +x 2 ). 10. d tan- 1 (Vl+x 2 - x ) = -\dx/{\ +x 2 ). 11. dsin- 1 (x/\ / T+x 2 )==dx/(l+x 2 ). 12. d sin- 1 [(1 -x 2 )/(l H-x 2 )] = -2dx/(l+x*). 13. d tan- 1 [2z/(l -o; 2 )l = 2d:c/(l + 2 2 ). 14. d sin-^sin a;=£Vl +cosec a; dx. 15. d vers -1 — =d cos -1 (1 ) = — — a \ a/ \Z2ax-x 2 U-a da; 16. a sin -1 &-a 2V(x-a)(6-x)' 17. y=asm- l (x/a) + \ / a 2 —x 2 , dy=dx\/(a—x)/(a + x). 18. i/ = \/x 2 — a 2 — a sec -1 (x/a), dy=dxVx 2 —a 2 /x. 19. i/= a; tan _1 a; — log Vl+x 2 , dy=ta,n~ l x dx. on * i .1 * ^ (l+x)dx 20 o y=tan- 1 x + log^=, ^-^jj+^y. x — a x dy 2ax 2 21 2/=log v — — + tan~ 1 -, x + a a dx x 4 —a* CHAPTER VIII. DIFFERENTIALS OF HYPERBOLIC FUNCTIONS. 41. Def. The quantities \{e x — e~ x ), \(e x + e~ x ) are called, Fig. 21. respectively, the hyperbolic sine (sinh x *) and hyperbolic cosine (cosh x) of x. The hyperbolic tangent cf x is defined * This may be read "sine h of 2." 34 41 J HYPERBOLIC FUNCTIONS. 35 i to be sinh .r/cosh x, and the hyperbolic secant, cosecant, and cotangent to be the reciprocals of the cosine, sine, and tangent, respectively. The graphs of the functions are represented in Fig. 21. Observe that sinh x may have any valua, cosh z^ 1, tanhx> — 1 and <1, coth:r>l or < — 1, etc., sinh = 0, cosh0= 1, etc. The fundamental relations cosh 2 x — sinh 2 £ = 1 , sech 2 a; = 1 — tanh 2 z, cosech 2 a; == coth 2 £ — 1 , are easily verified. The differentials of the hyperbolic functions are similar to those of the circular functions. Only the most important are given here. (For the others see Appendix, Note C.) Differentiating sinh v = \(e v — e~ v ) y co^\iv = ^{e v + e~ v ) } we have d sinh v = cosh v dv, d cosh v = sinh v dv, whence may be deduced j • i i v dv a sinh x - = a Vv 2 +a 2 ' j i i v dv a cosh 1 - = <*> \ v 2 -a 2 v a dv a a 2 — v A dtanh i-==— — v\<]a, 7 ,i _i v adv . - dcoth 1 -^-^ -, v\>\a. a a 2 — v 2 l ' Examples. 1. y =log cosh x, dy/dx =ta,nh x -* x < u x x j dx Ix + a 2. y = sec J — + cosh~ 1 — , dy=-.\ . * a a xyx-a 36 INFINITESIMAL CALCULUS. [Ch. VIII. 3. y=xVa 2 + z 2 + a 2 sinh- 1 (x/a), dy/dx=2Va 2 + x 2 . 4. Show that sinh -1 — = log( ), a \ a / tanh~ 1 -=-log ( ). a 2 \a—xl [Let smb.' 1 x/a=z and e z =u. Then #/a=sinh z=%(u—ur l ). Solve for u in terms of x.] 5. Gudermannian. If x = log tan (in + id), or log (sec + tan0), 6 is called the gudermannian of x (gd x) and x is gd _1 0.* Prove that d gd x=sech x dx } d gd _1 x=sec x dx. [Differentiate, gd # =2 tan -1 ^ — in, and gd -1 £ =log (sec x + tsm x).] 6. If z=log (sec + tan d), prove that cosh x =sec 0, sinh x =tan #, tanh a: = sin 0. * The inverse gudermannian gd —1 # is also written X{6), i.e., A(#)=l g tan (i^+ J<9)=log (sec 0+tan 6). CHAPTER IX. DIFFERENTIALS AS INFINITESIMALS. 42. Let y be a function of x, dx an increment of x, and suppose y and its derivative to be continuous from x to x + dx. Let Ay, dy, be the increment and differential of y corresponding to dx. Let dx become smaller and smaller and « 0, then Ay and (in general) dy are also infinitesimals. Since Ay _dy ^dx~dx' Ay dx dy dx +h where i is infinitesimal. Hence Ay = dy + I, (1) where / is an infinitesimal of an order higher than that of dx and dy. If dy^O, l/dy = 0, and / becomes a very small part of dy. Hence dy, when very small, is a close approximation to Ay. In reality (1) implies that dy is what remains of Ay when the higher infinitesimals are omitted; in other words, if higher infinitesimals are left out of account dy may be used as if it were the increment of y corresponding to the incre- ment dx of x. If 2/ = /Or), (1) may be written f(x + dx) - f(x) = f {x)dx + 1, (2) where l/dx = 0, and hence / is a very small part of dx when dx is very small. 37 38 INFINITESIMAL CALCULUS. [Ch. IX 43. Differentiation by the omission of the higher infini- tesimals is much used in the practical applications of the subject, and may be illustrated by the following examples. It must be remembered that dx is now regarded as infini- tesimal, and that the higher infinitesimals are not omitted because they are of trifling numerical value, but because they do not affect the final limit expressed by dy/dx. (See § 17.) The result is in no sense an approximation. Ex. 1. If y=x n , Ay = (x + dx) n —x n = x n + nx n ~ 1 dx + . . . — x n =nx n ~ l dx + . . . , the terms indicated by . . . being higher infinitesimals. When these terms are omitted A changes into d. .'. d(x n ) =nx n ~ 1 dx. 2. y=e x , Ay =e x + dx -e x = e x (e dx -1) =e x (l+dx + . . .-1). .'. dy=e x dx. 3. y=sinx, Ay=sin (x + dx) — sin x = sin x cos dx + cos x sin dx— sin x. But cos dx = l+/i, sin dx=dx+I 2) (§16). .' . dy = cos x dx. 4. To find the differential of the area A, Fig. 23. ■ Y o x Fig. 23. Fig. 24. For the increment dx of x the increment dA of the area = PMNQ = PN + PRQ. PN=y dx, and PRQ is a part of RS, which =dx Ay and is /.a higher infinitesimal. .'. dA=y dx. 43.] DIFFERENTIALS \s INFINITESIMALS 39 For example, for the curve y x 8 , Fig. 24, dA x*dx } hence the relation connecting .1 and x is in this case .1 -{.r 1 . 5. Barometric measurement of heights. Lei //,, be the height of a cubic inch o{ air at pressure p . Then the weight of a cubic inch at pressure p is (Boyle's law) u? p Po, ' l the temperature is tho same as before. Of a column of air of uniform temperature and one square inch in horizontal section consider the portion between sections at distances X, x + dx from the top, and let p, p+Jp be the pressures at top and bottom of this portion. Then Jp is the weight of this portion, dx its volume, its average pressure >p and

(3) which brings out clearly the fact that the coefficients of dx and dy are the partial derivatives of u, which are equal to the partial differential quotients. The subscripts are usually omitted, (3) becoming 7 du 1 du _ au = — ax-\-— dy. (4) The symbol d is frequently employed to express partial differential quotients or derivatives, (4) being written * , du , da , du = — <±c + — dy. (5) * du/dx may be read "partial du by dx." 46.] FUNCTIONS OF MORE THAN ONE VARIABLE. 43 Ex. 1. w=sin {x 2 + xy). Differentiating, first regarding x only as variable, and afterwards regarding y as the variable, du/dx =cos {x 2 + xy) . (2x + y), du/dy = cos (x 2 + xy) x. 2. u = x 3 + y 3 + z 3 — 3xyz, du/dx =3(x 2 — yz), du/dy =3(y 2 —zx), du/dz=3(z 2 — xy). 46. If u = f(x, y), dx and dy, being differentials of the variables, are increments of x and y. If dx and dy are taken as infinitesimal increments, du is not the same as Ju, the infinitesimal increment of the function. Since it may be obtained by the ordinary rules of differentiating, du is Ju when the higher infinitesimals are omitted (§ 42), or Ju=du+I, Hence when dx and dy are very small du is a close approxi- mation to Ju. Examples. 1. w=sin (x 2 — y 2 ), dxu =2x cos (x 2 — y 2 ) dx, d y u = —2y cos (x 2 —y 2 ) dy. 2. u = (x—y)/(x + y), du = 2(y dx— x dy)/(x + y) 2 . 3. u = (ax 2 + by 2 + cz 2 ) n , n-l du=2nu n (ax dx + by dy +cz dz). 4. If tan 6 =y /x, (x 2 + y 2 ) dd =x dy —y dx. 5. u=x y , d x u=y x v ~ x dx, d v u = (log x) x v dy. : . du=y x v ~ l dx+ (log x) xv dy, as in (G 2 ). 6. u=log(e? + eV), du/dx + du/dy = l. 7. u=t&n- 1 (x/y), du/dx = y/(x 2 + y 2 ), d u /dy = — x/(x 2 + y 2 ). 8. u =102: (tan x + tan y). sin 2a:— + sin 2 v— =2. & * ' dx *dy 9. u=\og y X, UX du/dx + y du/dy =0. [Note, logy x = log, x /logo y.] 10. Given x =r cos 0, y =r sin 0, show that dx 2 + dy 2 = dr 2 + r 2 d0 2 , x dy —y dx =r 2 dQ. 44 INFINITESIMAL CALCULUS. [Ch. X. 11. (1) If a function u consists of terms such as axPyQ, and p + q is the same number n in each of the terms, u is said to be homogeneous and of the degree n. Show that for such a function * du du x — \-y — =nu. dx oy (2) If u=f(v) and v is homogeneous as defined, show that du du „.- x— + y—=nvf (y). dx dy These propositions may obviously be extended to functions of three or more variables. Verify in the case of Exs. 1 and 3. 47. Tangent and normal. Let f(x, y) = c (c constant) be the equation of a curve. The. first member of the equation is a function of x and y; calling it u and differentiating the equation we have, by § 45 (5), ¥/ X + ¥y dy=0 > - (1) whence dy/dx, the slope of the tangent at (x, y) } is — — / — • Let (xi, y\) be the point of contact of a tangent to the curve, (x, y) any other point on the tangent. Then x — x\ and y — yi are proportional to dx and dy. Hence, from (1), (!),<— > + (l), ( s<-^ (2 > is the equation of the tangent, and x — x^ y — yi '?)u\ /du\ fix) 1 \dy)i (3) the equation of the normal, at (x\, y\). These equations are often more convenient than those of § 33. * A particular case of Euler's theorem on homogeneous functions (Ex. 1, §234). •17. is.] FUNCTIONS OP MORE THAN ONE VARIABLE. 45 Ex. Find the equations of the tangent and normal at the point (a, a) on the curve x 3 -f y 3 — 2a xy=0. du /dx = 3x 2 -2ay =3a 3 -2a 2 - a 2 for (a, a). du /dy =3y 2 -2ax =3a 2 -2a 2 =a 2 for (a, a). :. the tangent is a 2 (x — a)-\-a 2 (y — a) = 0, or x + y=2a, . . . . x —a y —a and the normal is — -- = — — , or x=y. a 2 a 2 48. Centre of a conic. Let the general equation of a conic be ax 2 + 2hxy + by 2 + 2gx + 2fy + c = 0, or a = 0. When referred to parallel axes through the point (xi, 2/1) the terms of the first degree are 2(axi +hy 1 +g)x + 2(hxi+byi+f)y, /du\ /du\ The new origin is therefore the centre if ( — ) =0 and \ ox) 1 (^— ) =0. Hence the centre of the conic is the intersection of 'du/dx = and du/dy = 0. If the coordinates of the point thus found satisfy the given equation, the centre is on the conic, which therefore consists of a pair of straight lines. Examples. 1. Find the equation of the tangent at any point of the curve (x/a) m + (y/b) m =2, and show that x/a + y/b=2 is the tangent at the point (a, 6). 2. Show that the length of the perpendicular from the origin on the tangent at (x, y) to the curve u=c is du du\ I ,du\ 2 /dio + / du du\ I y/ 1 N \°xl \dy 46 INFINITESIMAL CALCULUS. [Ch. X. 3. In the case of the curve x% + y5=a%, Fig. 18, show that this perpendicular =^/axy. 4. In the case of the parabola (x/a)% + (y/b)*=l, show that this perpendicular =[abxy/(ax + by]*. 5. Find the centres of the conies (1) x 2 -4xy-2y 2 + 6y=2, Arts. (1, £). (2) 18x 2 -8xy + 3y 2 + 8x-6y-5=0. (0,1). 6. Show that 3x 2 + 5xy—2y 2 —x + 5y=2 represents a pair of straight lines. CHAPTER XI. SMALL DIFFERENCES 49. When the differentials of the variables of a function are small increments, the differential of the function is a close approximation to the increment of the function (§§ 42, 46). Examples. 1. Given sin 30° =i cos 30° =£^3", find sin 30° 1'. Here the angle increases by a small amount and it is required to find the small increment in the sine. We have d sin d = cos dO; cos 0=£\/3, dO =60/206265 rdn., .'. dsin ='0002519, .". sin 30° V ='5002519, which is correct to the last decimal place. 2. How much must be added to log l0 sin 30° to get log 10 sin 30° 1'? We have d log sin =cos dd/s'm 6 ='0005038, which is the in- crease of the Napierian log. ; the increase of the common log. is obtained by multiplying by the modulus. .'. '0005038 X* 4342945 ='0002188 is the required increment. The difference columns in the mathematical tables are found or verified in this way. 3. The radius of a right circular cone is 3 inches and the height is 4 inches; if the radius were '006 in. more, and the height '003 in. less, what would be the change in the volume? The volume v=%7zr 2 h, .'. dv=\-(2rh dr + r 2 dh) = Jtt(2 X3 X4 X'006 -3 2 X'003) =' 1225 cub . in. 4. Assuming that the radius of an iron ball increases by '000011 of its original length for each degree of temperature, what will 47 48 INFINITESIMAL CALCULUS. [Ch. XT. be the increase in volume of an iron ball of 8 in. radius when the temperature is raised 25 degrees? The volume v =f 7rr 3 , .* . dv =47rr 2 dr =4ttX8 2 x25x*000011x8 = 1.77 cub. in. 5. In a certain triangle, b =445, c =606, A =62° 51' 33", whence a is calculated and found to be 565; it is then noticed that A should have been 62° 53' 31"; what is the correction to a? The change in A is V 58" = 118" = 118/206265 rdn. Also, a 2 = b 2 + c 2 —26c cos A ; differentiating this, supposing b and c constant, we have 2a da=2bc sin A dA, .'. da=bc sin A dA/a = 445 X 606 X sin 62° 51' 33" X 118/206265x565 =243. An approximate value of sin A is sufficient in this place. 6. Given loge 900=6*8024, find log e 901. Increase of log x =dx/x = l/900. Arts, 6'8035. 7. Given log 10 1000 =3, find log 10 1002. Arts. 3*00087. 8. Find tan 45° 1'. Arts. 1*00058. 9. On account of the rotation of the earth the correction to the weight w of a body is — w cos 2 ^/289, where ^ is the latitude. What is the change in this correction for one mile north of lati- tude 45° N.? the radius of the earth being assumed to be 4000 miles. Ans. w/(289x4000). 10. Find the relation connecting small differences of t and d in the equation sin h=sin sin d + cos cos d cos t, tan d\ dt=[-7~-, )dd. \ sin t tan t' This is the ''Equation of Equal Altitudes' ' in astronomy. 11. Find the relation connecting small differences of d and A in the equation sin d =sin <£ sin h — cos cos h cos A f and h being constant. Ans, dA =cos d d is given by the equation a\ cos + b\ sin cf> = 0. (1) If (x, y) is a point on the tangent at a distance t from the origin, cos cf) = x/t and sin $ = y/t. Substituting in (1) we have a\x + biy=Q for the equation of the tangent at the origin, i.e., the terms of the first degree in the given equa- tion, equated to zero, represent the tangent at the origin. If there are no terms of the first degree, it may be shown in the same way that a 2 x 2 + b 2 xy + c 2 y 2 = is the equation of a pair of tangents at the origin; and generally, yjhen the origin is a point on the curve, the terms of the lowest degree, equated to zero, represent the tangents at the origin. 53. Multiple points. A point at which there are two or more tangents (i.e., where two or more branches of a curve intersect) is called a multiple point; it is called a double point, a triple point, etc., according as two, three, etc., branches intersect at the point. When the equation a 2 x 2j rb 2 xy + c 2 y 2 = represents a pair of distinct lines the point is called a node (Figs. 27, 28). When the lines are coincident the two branches of the curve touch one another and the tangent may be considered as a double tangent. Such a point is called a cusp, w 7 hich is said to be of the first or second species according as the two branches of the curve lie on opposite sides (Figs. 29, 30) or on the same side (Fig. 31) of their common tangent; and to be double or single according as the branches lie on both sides (Fig. 34) or on one side only (Fig. 29) of their common normal. A cusp is also called a stationary point; for, con- sidering the curve as the path of a moving point, at a cusp the point must come to rest and reverse its motion. When the lines are imaginary the point is called a con- jugate point. The coordinates of such a point satisfy th§ 53.] MULTIPLE POINTS. 53 equation of the curve, but the point is isolated from the rest of the locus which the equation represents. Examples. 1. The lemniscate* a 2 (y 2 -x 2 ) + (y 2 + x 2 ) 2 =0, Fig. 27. The origin is a node at which the tangents are y 2 0, i.e., y = x and y = — x. .A Fig. 27. Fig. 28. 2. The folium t x 3 + y 3 =3axy, Fig. 28. The origin is a node, the tangents being given by xy=0, i.e., 2=0, y=0, the axes. 3. The semi-cubical parabola ay 2 =x 3 , Fig. 29. The origin is a cusp, the tangents being given by y 2 =0, i.e., two lines coinciding with the axis of x. Moreover, the curve is symmetrical with regard to the axis of x } and y is impossible if x is negative; hence the cusp is single and of the first species. 4. In the curve (y—x) 2 =x 3 , Fig. 30, the origin is a cusp at which the tangent is y=x; also, since y=x±x%, y> x on one branch and am/(\-{-m 3 ) and y = mx or Sam 2 /(\ -f m 3 ). Thus x and y are expressed in terms of a third variable, and by giving arbitrary values to m the coordinates of any number of points on the curve may be calculated. The same substitution may be employed in other cases (e.g., Figs. 36, 37, 38) in which the equation contains terms of two degrees only. It should be noticed that /// is the slope of the line drawn from the origin to the point (x, y) on the curve. 54 INFINITESIMAL CALCULUS. [Ch. XII. 5. In the curve (y—x 2 ) 2 =x*, or y=x 2 (l±Vx), Fig. 31, the origin is a cusp, the tangent at which is y=0; also, y is + on both branches until x = 1, and / . the cusp is of the second species. Fig. 29. Fig. 30. Fig. 31. 6. In the curve y 2 =x 2 (2x + l), Fig. 32, the origin is a node at which the tangents are y=±x. But in the curve y 2 =x 2 (2x — l), Fig. 33, the tangents are y 2 =—x 2 , and are .*. imaginary, and hence the origin is a conjugate point. Fig. 32. Fig. 33. Fig. 34. Fig. 35. 7 There are certain cases in which the origin is a conjugate point even when the terms of the second degree are a perfect square. Thus, in the curves y 2 = x 4 (2x + l), Fig. 34, and y 2 =x 4 (2x — 1), Fig. 35, the origin is a double point and the tan- gents are given by y 2 =0; in the first curve the origin is a double cusp, in the second a conjugate point, since y is imaginary for any value of x less than \. 8. The curve ay 3 —3ax 2 y =x 4 , Fig. 36. The origin is a triple point at which the tangents are ay 3 — 3ax 2 y =0; i.e., y =0, y = ±x\ / S. 9. In the curve ay 4 — ax 2 y 2 = x : \ Fig. 37, the origin is a quad- ruple point, at which the tangents are y=0 y y=0,y = ±x. 10. (x— y) 2 = (x — 1)\ The point (1, 1) is a cusp, for the equa- tion referred to parallel axes through (1, 1) is (x— y) 2 =x*. 54.] MULTIPLE POINTS. 55 11. Find the tangents to the following curves at the origin: (1) (a 2 -\-x 2 )y 2 = {a 2 -x 2 )x 2 . Ans. y=±x. (2) a 2 y(x + y)=x 4 . y =0, x + y = 0. (3) x{y-x) 2 =y\ x =0, y=x, y=x. (4) a(y-x)(y 2 + x 2 ) + x 4 =0. y=x. (5) y 3 (y-x)=a(y 3 + x' d ). x + y=0. (6) (x-a)y=x(x-2a). y=2x. (7) y 2 =(x-l)x 2 . Imaginary. Fig. 36. Fig. 37. 12. Show that the origin is a single cusp of the first species on the cissoid y 2 (a—x)=x 3 , Fig. 41. 13. Show that there is a node at the point (1, 2), on the curve (y-2) 2 = (x-l) 2 x. 14. Show that the point (2a, 0) is a node on the curve ay 2 = (z-a)(z-2a) 2 . 15. Show that the point (—a, 0) is a conjugate point on the curve ay 2 =x{a + x) 2 . 54. Let the equation of a curve, freed (if necessary) from fractions and radicals affecting the coordinates, be f(x, y)=c or u = c. The tangent, § 47 (2), when referred to parallel axes through the point of contact {x\, y{) is ( cm\ (du\ n 56 INFINITESIMAL CALCULUS. [Ch. XII. Hence if (— ) =0 and (^-) = the equation of the Xdx/i \dy/i curve referred to the new axes will have no terms of the first degree. Conversely, points whose coordinates satisfy du/dx = and du/dy = as well as the given equation u = c are multiple points of the curve. Ex. 1. To examine the curve (x — l) 5 — (2x — y) 2 =0 for multiple points. du/dx =5(x-l) 4 -4(2a: -y) =0, du/dy =2(2x -y) =0, whence x = l } y=2. These coordinates satisfy the given equa- tion, hence (1, 2) is a multiple point. Transforming to parallel axes through (1, 2) the equation becomes x 5 — (2x— y) 2 =0, hence the point is a cusp at which the tangent is 2x ==y. 2. Examine the curve x 3 — 2x 2 + y 2 — 4x-f 2y-\-§=0 for multiple points. Arts. A conjugate point at (2, —1). CHAPTER XIII. ASYMPTOTES. 55. Definition. An asymptote of a curve is the limit of position of a secant when two of its points of intersection with the curve move away to an infinite distance, and hence also the limit of position of a tangent when the point of contact moves to an infinite distance. 56. Asymptotes by substitution. Ex. 1. Of the curve x 3 + y 3 =3axy, Fig. 28, the line y=mx + b is an asymptote if m and b are determined so that the line may meet the curve in two points infinitely distant. Substituting mx + b for y in the equation of the curve we have (l+m 3 )# 3 + 3(ra 2 6— am)x 2 + . . . =0, (1) the roots of which are the abscissas of the points of intersection of the line and the curve. Two of the roots become infinite * when m and b change so as to cause the coefficients of the two highest powers in (1) to =0. Hence the required values of m and b are obtained by solving the equations l+m 3 =0, m 2 b—am=0. .'.m=—l and b=—a. Hence the asymptote is y=—x—a, or x + y + a=0. The result might have been obtained equally well by the substitution x=my + b. * The equation a x n + a x x n - l -\- . . . J ra n - l x-\-a n = (2) is obtained from a n x n -\-a n - l x n - l -\- . . .-\-a i x-\-a = (3) by changing x into 1/x. The roots of (2) are the reciprocals of those of (3). Hence if a and a x change and = 0, two roots of (3) = and • *. two roots of (2) become infinite. 57 58 INFINITESIMAL CALCULUS. [Ch. XIII 2. Find the asymptotes of the hyperbola ——^- = 1. a h Ans. y= ±—x. a 57. Asymptotes by expansion. The following definition of an asymptote gives a better idea of the relation of the line to the curve: Def. When the distance (measured parallel to an axis) between a line and a curve is infinitesimal as both recede to an infinite distance, the line is said to be an asymptote to the curve. Such lines may be rectilinear or curvilinear. If the equation of a curve, when y is expressed as a series of descending powers of x, take the form c d y = ax + b-\ \--z + . . . , (1) x x z the line y=ax + b will be a rectilinear asymptote. For the difference between the y of the curve and the y of the line is c/x + d/x 2 + . . . , which is infinitesimal when x is infinite. The line y = ax + b is also the limit of a tangent of the curve (1). For the slope of the tangent = dy/dx = a — cx~ 2 — . . . =a for x infinite, and the ^/-intercept of the tangent = y—x dy/dx = b + 2cx~ 1 + . . . =b. The sign of the term c/x in (1) will determine whether the curve lies above or below the asymptote when x is very large. If the equation take the form y=ax 2 + bx + c-\ h-« + . . . , xx 1 there will be a curvilinear asymptote, viz., the parabola y = ax 2 + bx-\-c. Ex. 1. i/ 3 =x 3 + 3ax 2 , Fig. 38. / 3a\ / 3a\ * . . . . We have y 3 =x 3 il-\ ), or y=xil-\ — I , which by the 57.] ASYMPTOTES. 59 binomial theorem (when a;|>|3a) a* or y=x + a h...; x .*. y=x + a is an asymptote. The curve lies below the asymptote when a: is a large positive number, and above it when a; is a large negative number. x 2 ii 2 bx / x 2 \ * 2. The hyperbola — -f- = 1 . Here y=±—(l — -) JC a 2 b 2 a \ a 2 / bx I a 2 \ bx ab bx .*. the asymptotes are y=±—' Fig. 38. 3. fy(x-l)=x 3 , Fig. 39. x 3 By division, 4y = x 2 + x + l + Fig. 39. 1 x — 1 ' x — 1" When x is very large the last term is very small and =0, and the ordinate of the curve = that of the parabola 4y=x 2 + x + l, which is called a parabolic asymptote. (The line AB is the axis of the parabola.) It will be noticed that this curve is asymptotic to the given curve both when x is + and when x is — . The line x = l is a rectilinear asymptote, as will be seen from § 58. 60 INFINITESIMAL CALCULUS, [Ch. XIIL 58. Asymptotes parallel to the axes. Let the algebraical equation of a curve, freed (if necessary) from fractions and radicals affecting the coordinates, and arranged in descending powers of x, be fi(y)x™ + J 2 (y)x m - 1 + f 3 (y)x m - 2 + . . . = 0, whence /i(2/)+/ 2 (^+/ 3 (2/)4+...=0. (1) If there is an asymptote parallel to the #-axis, y remains finite when x is infinite. Hence all the terms of (1) after the first become infinitesimal, and the y of the curve ap- proaches a limit which satisfies /i(j/) = 0, i.e., 3/ = the y of a line y—a = Q, y~a being a factor of fi(y). Hence, when the equation of a curve is arranged according to powers of x, the coefficient of the highest power, equated to zero, represents the asymptotes which are parallel to the z-axis. The asymptotes parallel to the y-axis may be found in the same way. Y i — ~~~— -^ / ^_^-^-^c \ X . \ 1 1 Fig. 40. Fig. 41. Ex. 1. x 2 y 2 -3xy 2 -x 2 + 2y 2 =0, Fig. 40. Arranged according to powers of x the equation is (y 2 -l)x 2 -3y 2 x + 2y 2 =0, 58, 59.] ASYMPTOTES. 61 and according to powers of y } (x-l)(x-2)y 2 -x 2 =0. Hence y = ±1, and x = l, x=2 y are asymptotes parallel to the axes. 2. The cissoid y 2 (a—x)=x 3 , Fig. 41. The line a —x = 0, or x = a is an asymptote parallel to the ?/-axis. 59. In any equation the terms of the highest degree, equated to zero, represent lines drawn through the origin. The equation which gives the slopes of these lines is found by substituting mx for y, or m for y/x. This is the same equa- tion as that which determines the slopes of the asymptotes (§ 56). Hence the terms of the highest degree, equated to zero, represent lines drawn through the origin in the direc- tion of the infinite branches of the curve. Examples. 1. x 3 —y 3 =3axy. Arts. y=x—a. 2. y 3 =x 2 y + 2x 2 . y= ±'x + l, y + 2 = 0. 3. x 4 =xy 3 + 3y 3 . x-y = l, z + 3=0. 4. x 2 y = x 3 -\- x -{- y . y=x,x=±l. 5. x 4 — y 4 + x 2 =4txy 2 . x±y = \. 6. x'=x 2 y 3 -(l-x)y 3 . 3(x-y) = l, 2x~ ±VK-1. 7. axy=x 3 —a 3 . x=0 y ay=x 2 . 8. x 3 + y*=a 3 . x + y=0. 9. x 3 -27y*=2x 2 . 3x-9y-2=0. 10. y + xy=x 3 . x + l = 0, y =x 2 — x + 1. 11. y =tan x. The y of the curve =00 when the x = \n y : . the line x=\tz is an asymptote (§ 57). Similarly x = (n + i)n, n any integer, is an asymptote. The same lines are asymptotes to y=sec x. 12. Show that y = l and y= — 1 are asymptotes of ?/=tanh x. 13. Show that the curve 2y=e x is asymptotic to 2/=sinh x, y = cosh x, and y=Q, CHAPTER XIV. TANGENT PLANES. TANGENTS TO CURVES IN SPACE. 60. Geometrical illustration of partial and total differen- tials. Let z = f(x, y). Values of x and y determine z, and hence a point (x, y, z) in space referred to axes which we shall assume to be rectangular. Points thus obtained lie on a surface which is the locus of the equation z=f(x, y). This surface is a geometrical representation of the function. z F i\^^^Z-^ 0/ K J 1 H G E R / F 1 /" H* M r N C » Fig. 42. Let OA = x, AM=y, and MP = z. Then P is a point on the surface. Let OB=x + dx, BC=y+dy, and let the new value of z be CQ. Then Q is another point on the surface. The plane PF parallel to XOY cuts off CF = MP, hence FQ is Az y the increment of z. Planes through P and Q parallel to XOZ and ZOY cut the surface in PH, HQ, QK, KP. 62 60-62.] . TANGENT PLANES. 63 Draw PJ and PG tangents to PK, PH, and let PGRJ be the plane through PJ, PG. If we suppose y to be constant, we are confined to the plane PN (produced if necessary); Pl = dx and PJ touches PK, hence I J is d x z. Similarly EG is d u z. Let a line through the middle point of MC parallel to OZ meet the plane PGRJ. This lme=$(MP+CR) in the trapezium PMCR, and also =%(DG + NJ) in the trape- zium GDNJ. .'. FR = IJ + EG = d x z + d y z. But dz = d x z + d y z (§ 45). Hence FR is dz. The tangents of the angles IPJ, EPG are the partial derivatives of z with respect to x and y, i.e., they are dz/dx and dz/dy. 6i. Tangent plane. When Jz and dz are infinitesimal the latter is the part of the former, which contains the infini- tesimals of the lowest order (§ 46). Thus FQ and FR correspond in the plane PMCQ to Ay and dy of § 42. Hence the straight line PR touches the section of the surface made by the plane PMCQ, and therefore the plane PGRJ is the locus of all such tangent lines at P, for dx and dy are any increments. Such a plane is defined to be the tangent plane at P. Notice that if {x, y, z) is the point of contact, and dx, dy are any increments, {x + dx, y + dy, z + dz) is any other point in the tangent plane. 62. Equation of the tangent plane. Let the equation of the surface be f(x, y, z) = c or u = c. Differentiating, ?)u , , du. , du . — d x + 7T- d y + — dz = . ox oy oz Let (x\, y\, z{) be the coordinates of the point of contact P, Fig. 42, (x, y, z) those of any other point in PR and there- fore of any other point in the tangent plane. Then x — x\, y — yi, z — Zi are proportional to dx, dy, dz. is the equation of the tangent plane at (x\, y%, z\). ~ 64 INFINITESIMAL CALCULUS. [Ch. XIV. Ex. To find the tangent plane at the point ( — 1, 1, 2) on the surface x 3 -x 2 y + y 2 + z = l. du/dx =3x 2 —2xy =5 for the point (— 1, 1, 2), du/dy= —x 2 + 2y = l for the point ( — 1, 1, 2), du/dz = l. Hence the tangent plane is 5(x + l) + (2/-l) + (3-2)=0, or 5x + y+z + 2=0, 63. Equations of the normal. The normal passes through (x\, 3/1, z\) and is perpendicular to the tangent plane; hence its equations are x-xi y-yi z-z x (du\ fdu\ /du\ ' w \dx/ 1 \dy)i Vdz/i 64. Tangent plane at the origin. Conical points. Let the equation of the surface be freed (if necessary) from fractions and radicals affecting the coordinates. If the origin is on the surface the equation will contain, no constant terms, and by substituting r cos a, r cos /?, r cos y for x. y, z, it may be shown exactly as in § 52 that the terms of the first degree, equated to zero, represent the tangent plane at the origin. Similarly, if there are no terms of the first degree, those of the lowest degree present will represent a surface touching the given surface at the origin. This tangent surface is generally a cone, in which case the origin is called a conical point; but it may be two or more planes.* As in § 54, it may be shown that the coordinates of a conical point or a point where there are two or more tan- gent planes will satisfy du/dx = 0, u/dy = 0, du/dz = 0, as well as the given equation u = c. * A homogeneous equation with no constant term represents a locus of straight lines passing through the origin. For, if satisfied by x, y, z, it is satisfied by ex, cy, cz, the coordinates of any other point on the line joining the origin to (x, y, z). 63-60.] TANGENT PLANES. 65 Ex. 1. Of the surface x 2 — y 2 — z 2 + x 3 =0, x 2 —y 2 —z 2 =0 is a tan- gent cone at the origin,, 2. Of the surface x 2 — y 2 — z 2 — 2yz =x 3 , x + y + z=0 and x—y—z = are tangent planes at the origin. 3. Find a conical point on the surface x 3 + y 2 + 2yz — 3x— 42=2. Arts. (1, 2, -2). 65. Centre of a quadric. Exactly as in § 48 it may be shown that the centre of any surface whose equation u = c is of the second degree is obtained by solving the simulta- neous equations ?>u/dx = 0, du/dy = 0, du/2)Z = 0. Ex. Find the centre of x 2 3y 2 -z 2 + 4yz-4:X + 8y-6z=0. Ans. (2, 2, 1). 66. Curve in space. Let P and Q be two points near one another on a curve, P being (x, y, z) and Q (x + dx, y + dy, z + dz). Then dx = AB=CE = PG, dy = ED = GF, and dz = FQ. Let the arc PQ = ds and the chord PQ = q. The tangent- PT is the limit of position of the secant PQ when Q approaches coincidence with P. Let a,/?, y be the direction angles of PT. Then a = HPT, and cos a = £ cos GPQ = £(dx/q) = £(dx/ds) (§ 17) =dx/d's. Similarly cos /? = dy/ds, cos y = dz/ds. Squaring and adding, Fig. 43. 'dx\ 2 (dy\ ds) 2 + (d 2 s) 2 . 17. ?/ 2 =4ax, 2y dy =4a dx, or y dy =2a dx; differentiating again, y d 2 y + dy 2 =0 (dx being constant), .*. y d 2 y+ (2adx/y) 2 =0, or d 2 y/dx 2 =—4:a 2 /y 3 . x 2 y 2 18. Given the ellipse — + - =1, show that a 2 ' b 2 dy dx b 2 x d 2 y a 2 y dx 2 V a 2 y d ' 70.] SUCCESSIVE DIFFERENTIATION. VI d 2 y/dx 2 may be found as in Ex. 17, or from dy/dx. Thus dy d 2 y b* y X dx dx 2 a 2 y Substitute for dy/dx and reduce. 19. If y =f(x) find /"(#)> given x=a cos 0, y=b sin 0, Ans. f"(x) = —b 4 /a 2 y\ 20. a 2 + y 2 =2xy, d 2 y/dx 2 =a 2 /(x—y) 3 . 21. x 3 + y 3 =3axy, d 2 y/dx 2 =2a 3 xy/(ax — y 2 ) 3 . j CHAPTER XVI. RATES. 71. Let y be a function of x of which Fig. 44 is the graph. When x increases by the amount Ax the change in y is dy, and Ay I Ax is called the average rate of change of y per unit of x (or briefly, the average z-rate of y) fof the change Ax in x. When Ax is taken smaller and smaller and = the average rate Ay/ Ax is taken for a gradually dimin- ishing change in x, and the limit of Ay I 'Ax, namely dy/dx, is defined to be the x-rate of y for the value x of the vari- able. Thus as x increases and reaches the value OA , the z-rate of the function y is dy/dx or tan <£>. This is an in- stantaneous and variable rate, and is the same as the con- stant rate which y would have if P should henceforward move along the tangent PD. If y=f(x), dy/dx = f ; (x); hence the £-rate of f(x) is /'(#), and for a similar reason the x-rate of any derivative is the succeeding derivative. 72. If y is a function of x, as x changes the function will increase or decrease according as its graph rises or falls, that is, according as dy is + or — . Also dx is + if x in- creases. Hence as x increases, y increases or decreases according as dy/dx is + or — . Thus a + value of the rate 72 71-74.1 RATES. 73 implies that a function is increasing as its variable increases, and a — value implies that the function is decreasing as the variable increases. 73. If x and y are functions of a third variable t, dy/dx= (dy/dt)/ (dx/dt). Hence dy/dx is the quotient of the simultaneous rates of change of y and x, or dy and dx are proportional to the rates of y and x.* If y = f{x) y and x is a function of t, dy/dt = f (x) . dx/dt, which gives the rate of y in terms of that of x. ' If u = f(x, y), and x and y are functions of t, then, § 45 (5), du _dudx du dy dt dx dt dy dt ' which gives the rate of u in terms of the rates of x and y. 74. If a point moving in a straight line is at a distance x from a fixed point in the line at the end of an interval of time whose measure is t, its velocity v is the £-rate of x, and is therefore dx/dt; and its acceleration a is the £-rate of v, and is therefore dv/dt. But 1 (dx\ dv_ \dt I _d 2 x ., dv__dvdx_dv dt dt dt 2 ' ' dt dx dt dx ■ dx , dv d 2 x dv Hence v=-r: and a = — =—r^ = v -r~- at dt dt z dx Similarly the angular velocity and angular acceleration of a revolving body are -7- and -7-^ respectively. Time rates are sometimes indicated by dots, x being the same as dx/dt, and x the same as d 2 x/dt 2 . * In some treatises dy and dx are defined to be rates. 74 INFINITESIMAL CALCULUS, [Ch. XVI. Examples. 1. The ordinate of the curve y = V25—x 2 is moving parallel to the ?/-axis at the rate 2 in. per sec. At what rate is its length changing when x = 3 ? dy x dx 3 _ . _ . n ^ T -7- = .. = — — when x =3 and dx /at = 2. Hence y is decreasing at the rate of \\ in. per sec. 2. At what points on the curve y= log sec # do £ and y change at the same rate? Arts. x = (n + \)7i, n an integer. 3. Find the acceleration if (1) v=u + bt, (2) x=ut + bt 2 , (3) v 2 =u 2 + bx, u and b being constants. Arts. (1)6, (2)26, (3)^6. 4. If z=a cos (&£ + c), show that the acceleration = — b 2 x. 5. If x=a sinh (bt + c), show that the acceleration = b 2 x. 6. Show that tan x always increases with x. 7. Three adjacent sides of a rectangular parallelepiped are 3, 4, 5 inches in length, and are each increasing at the rate of '02 in. per in. per min. At what rate is the volume increasing? Ans. 3*60 cu. in. per min. 8. One end of a ladder moves down a vertical wall with velocity v 1} while the other end moves along a horizontal plane with veloc- ity v 2 . Show that v 1 /v 2 =tan 0, where 6 is the angle which the ladder makes with the vertical. 9. Two straight lines of railway intersect at an angle 60°. On one a train is 8 miles from the junction and moving towards it at the rate of 40 miles per hour, on the other a train is 12 miles from the junction and moving from it at the rate of 10 miles per hour. Is the distance of the trains from each other increasing or decreasing ? CHAPTER XVII. MAXIMA AND MINIMA. 75* Suppose y to be a function of x and that x continually increases. Then (§ 72) y will increase or decrease accord- ing as dy/dx is + or — . When dy/dx changes from + to — , y ceases to increase and begins to decrease, and is then said to be a maximum; when dy/dx changes from — to +, y ceases to decrease and begins to increase, and is then said to be a minimum. Now in order that a quantity may change sign it must become or 00 or — go ; * hence as y becomes a max. or a min., dy/dx becomes or 00 or —00 and changes sign from + to — for a max. and from — to + for a min. Fig. 45. 76. Suppose, for example, that the curve of Fig. 45 repre- sents the graph of a function and that it is traced by a point moving from left to right so that dx is + . Then y decreases from A to B and dy/dx is — , between B and C y continually * A quantity may change sign on account of finite discontinuity without passing through the value 0, but this occurs so rarely that we need not consider it further. 75 76 INFINITESIMAL CALCULUS. [Ch, XVII. increases and dy/dx * is + ; at B y ceases to decrease and begins to increase, dy/dx changes from — to + through the value 0, and y is a min. Similarly at C y is a max., and again a min. at D. At E dy/dx becomes oo and changes from + to — , hence y is a max., and similarly y is a min. at F. Points such as A, B, etc., are called turning points) and the max. and min. values of y are called turning values. It will be noticed that a max. is not necessarily the greatest of all the values of y; it is greater than the values which im- mediately precede or follow it; and similarly a min. is not necessarily the least value of y. 77. To obtain the values of x which make a function y a max. or min. we must obtain dy/dx and find what values of x cause it to become zero or infinite. To distinguish the maxima from the minima we must determine whether dy/dx changes from + to — or from — to + as x passes through the critical value. In the former case y will be max., in the latter a min. It may happen, however, that dy/dx does not change sign, although it becomes or 00 (e.g., at G, Fig. 45), in which case y is neither a max. nor a min. Ex. 1. y=x*-6x 2 + 9x + l. Here dy/dx =3x 2 -12x + 9=3(x-l)(x -3). When x is a little less than 1, x — l is — and x—3 is — , ,\ dy/dx is +. When x = l, dy/dx is 0. When x is a little more than 1, re — 1 is + and x—3 is — , .'■ dy/dx is — . Hence dy/dx changes from + to — through and .'. y is a max. when x = l. Substituting 1 for x in the given function we find the max. value of y to be 5. Similarly x =3 makes y a min., viz., 1. 2. y = (x — l) 3 , dy/dx=3(x — l) 2 ; .*. dy/dx =0 when x = l, but does not change sign f when x passes through this value, /. y is neither a max. nor a min. * It will be remembered that dy/dx = tsm , and is therefore -f- or — according as <£ is + or — . f (x — a) n changes sign with x—a only when n is an odd integer, or a fraction whose numerator and denominator are both odd 77-79.] MAXIMA AND MINIMA. 77 2 3. y*=2 + (x — l)a, dy/dx = -r- —7; .". dy/dx becomes —00 o \X 1 J 3 and changes from — to -f as x passes through the value 1, hence x = l makes y a min., viz., 2. 78. The sign of d 2 y/dx 2 ( = the z-rate of dy/dx) tells us at any time whether dy/dx is increasing or decreasing. If then the value of x which makes dy/dx equal to also makes d 2 y/dx 2 plus, we infer that dy/dx is increasing when it passes through 0, i.e., that dy/dx changes from — to +, and hence that y is a min.; whereas, if the value of x which makes dy/dx equal to also makes d 2 y/dx 2 minus, we infer that dy/dx is decreasing when it passes through 0, i.e., that it changes from + to — , and hence that y is a max. Hence to distinguish the maxima from the minima we may find d 2 y/dx 2 , and in it substitute the values of x which make dy/dx equal to 0. Then for every -f- result y is a min., and for every — result y is a max.* Ex. 1. In Ex. 1, § 77, d 2 y/dx 2 =6x — 12, which is — when x = l, and + when x=S. Hence x = \ makes y a max. and x=3 makes y a min. 2. y=x 3 —7x 2 + 8x + 30, dy/dx =3x 2 — 14z + 8. For a max. or a min. 3x 2 — lix + 8=0, .'. x=\ or 4. Also d 2 y/dx 2 = Qx — 14, which is — when £=§ and + when x =4; .*. x =f makes y a max. and x =4 makes y a min. 79. It should be noticed: (1) That max. and min. values must occur alternately in a continuous function, i.e., between two successive max. values there must be a min., and between two successive min. values there must be a max. Also of two values of x which make y a max. or a min., if one makes it a max. the other must make it a min. (2) When y has a turning value, y n (n a positive or nega- tive integer) has a turning value. Thus a square-root sign * If d 2 y/dx 2 is or 00 it gives no information as to the turning values, and the test of § 77 must be applied. 78 INFINITESIMAL CALCULUS. [Ch. XVII. affecting the whole of the variable part of a function may be disregarded in differentiating. (3) A constant factor may be omitted from the function before differentiating, since it cannot affect the values of x for which the derivative is or oc . Ex. y = 7r£V / a 2 — x 2 . This =nVa 2 x 2 — x 4 , and .'. y will be a max. or a min. when a 2 x 2 — x 4 is a max. or a min.; hence 2a 2 x -4a; 3 = 0, .'. s = 0, and x= ±a/V2. 8o. In the practical applications of this subject it will be necessary to form the function which is to have a turning value. It will frequently be obvious from the nature of the problem whether the result corresponds to a max. or a min. Ex. 1. Of all arithmetical fractions, which one exceeds its square by the greatest quantity? Let the fraction be x. Then x — x 2 is to be a max. .*. 1— 2x = 0, and hence x = J. 2. How to make with a given amount (area) of material a cylindrical box (with lid) which shall have the greatest possible volume. We have the total surface of the cylinder given, call it s, and assume h for the height and x for the radius of the base. Then s = 2nx 2 + 27ixh, .\ h = s/(2nx)—x. . The volume V = 7ix 2 h = ^sx — nx 3 . .". dV /dx = \s — 37nr 2 = t for a max. .*. x = \ / s/67i ) whence h = 2\ / s/67z. Hence the height must = the diameter of the base and each = 2Vs/6tt. [Observe that in these examples the function which is to be a max. or a min. must be expressed in terms of some one variable with or without constants; in this case the function is 7:x 2 h, where both x and h are variable, but there is a relation connect- ing x and h from which h may be obtained in terms of x; this when substituted in nx 2 h gives a function with one variable.] SO] MAXIMA AND MINIMA. 79 3. To find the greatest isosceles triangle that can be inscribed in a given circle. Let ABC (Fig. 46) be an isosceles triangle inscribed in a circle of radius a and centre E. Let DC = x. Then AD = VaE^-DE 2 = Va 2 ~(x-a) 2 = V2ax-x 2 . .'. area of ABC=LC . AD = xV2ax-x 2 = V'Zax i -x\ This will be a max. when 2ax 3 — x { is a max., §79(2), i.e., when 6ax 2 — 4:X 3 = 0, .*. x = §a. The triangle is easily shown to be equilateral. 4. One corner A of a rectangular piece of paper ABCD (Fig. 47) is folded over to the side BC. Find when the crease EG is a min. Let AB = a, AE = x, EG = y, AGE = 6. Then BEF = 2d. .'. BE:EF=(a-x)/x=cos20 and AE\ FG=x/y = sm 6. Eliminating d by the relation cos 26 = 1—2 sin 2 0, we find y 2 = 2x 3 /(2x-a), from which y is found to be a min. when x = \a. Fig. 46. Similarly it may be shown that the area of the part folded over is a min. when x = fa. 5. To cut the parabola of greatest area from a given right circular cone, Fig. 48. Let AB = a and FB = x. The area = f ED . FG. Now EF 2 = AE . EB= (a-x)x, and ED is prop ortional to x. .\ area varies as xV( a -x)x or Vax 3 -x 4 , whence x = \a for a max ~.--- 80 INFINITESIMAL CALCULUS. [Ch. XVII. Examples. 1. x 2 — 3x + 4:, min. when £ = §. 2. x 5 — 5x 4 -\- 5x 3 -\- 1 , max. when x = \ y min. when x = 3. 3. a + 6(c— oO^, no turning value. 4. x 3 — 2x 2 — 4:X + l, max., when #= — §, min. when £ = 2. 5. (x — l) 3 (x + 2) 4 , max. when #= —2, min. when z = — f. 6. (1 +3z)/\/4 + 5£ 2 , max. when x = ± 5 2 -. 7. (:c + 2) 3 /(a; — 3) 2 , min. when £ = 13. 8. sin d + cos (9, max. when &=\n, min. when Q = \n. 9. sin 0/(1 +tan 0), max. when = \7i, min. when Q = ^n. 10. sin # sin(a — d), max. when # = -|«. 11. sin 2 # cos 3 #, max. when sin 6= ±Vf, min. when = 0. 12. Min. value of a tan d + b cot 6 = 2\/~ab. 13. Min. value of a 2 sec 2 # + & 2 cosec 2 # = (a + 6) 2 . 14. Min. value of ae nx + be~ nX = 2\ r ab. 15. Max. value of log x/x = l/e. 16. What is the longest ordinate of the curve a 2 y 2 = x 2 (a 2 — x 2 ), (Fig. 69)? Arts. £a. ' 17. Find the max. ordinates of the curves (y-x) 2 = x 3 , Fig. 30, and (y-x 2 ) 2 = x\ Fig. 31. Ans. 2 2 /3 3 , 4 4 /5 5 . 18. Find the max. ordinate of the curve x 3 + y 3 = 3axy, Fig. 28. Differentiating the equation and making dy = we have x 2 = ay; from this and the equation of the curve we find the max. ordinate to be at the point (a ^/2, a ^4), the latter coordinate being the required value. 19. Find the max. ordinate of the curve y 3 = x 5 + Sax 2 , Fig. 38. Arts. i/~ia. 20. How could you cut out four equal squares from the corners of a given square so that the remaining area (the edges being turned up) would form a rectangular box of greatest volume? Arts. Each side of the little squares = \ of a side of the given square. 21. Find the breadth and depth of the strongest beam that can be cut from a cylindrical log of diameter d, assuming that the 80.] MAXIMA AND MINIMA. 81 strength varies as the product of the breadth and the square of the depth. Ans. Breadth = i^3 d, depth = J^6 d. 22. To cut out from a given sphere the cone of greatest volume. Ans. Ht. of cone = f diam. of sphere. 23. How could you cut a sector out of a circle so that the re- mainder of the circle would form the lateral surface of a cone of max. volume? Ans. Leave Vf of circumference. 24. What is the shortest distance of the line y = x + 2 from the parabola y 2 = 4:X? Ans. ^V2. 25. Assuming that the work of propelling a vessel in still water varies as the cube of the speed, what is the most economical rate of steaming against a current of speed v ? The expense for a given distance varies as x 3 and the time, and the latter varies inversely as x — v. Ans, \v. CHAPTER XVIII. CURVATURE. 8 1. Direction of curvature. Let it be supposed that the tangent of a curve rolls round the curve in such a way that the abscissa of the point of contact P continually increases. Let the tangent make an angle with the x-axis. Then a + value of d 2 y/dx 2 (the x-rate of dy/dx) at P implies that dy/dx or tan <£, and therefore also (f>, is increasing with x, or that the tangent is turning in the positive direction as x increases. In other words, the curve bends upward, or is concave upward, when d 2 y/dx 2 is + , and bends downward, or is concave downward, when d 2 y/dx 2 is — . 82. Point of inflexion. A point where a curve has ceased to bend upward and is about to bend downward, or vice versa, is called a point of in- flexion. At such a point d 2 y/dx 2 must change sign, and must therefore become 0, 00 , or — 00 .* FlG - 49, A tangent at a point of in- flexion is sometimes called a stationary tangent, for, if the * It is assumed in the above that x is the independent variable. If y is the independent variable, d 2 x/dy 2 must change sign. If neither x nor y is independent, the quantity which must change sign is (§70) (dx d 2 y—dy d 2 x)/dx 3 . 82 81-83.] CURVATURE. 83 tangent is supposed to roll round the curve, it comes to rest at such a point and reverses its motion. 83. If P, Fig. 50, is a point of inflexion, the secant through P and a point Q near P also passes through another point Q' near P. As the secant approaches the position of the tangent at P, Q and Q' approach coincidence with P at the same time. Hence the inflexional tangent is sometimes said to pass through three coincident points of the curve. A tangent at an ordinary point on a curve of the nth degree cannot meet the curve in more than n — 2 other points; the tangent at a point of inflexion cannot meet the curve in more than n — 3 other points, and in not more than n — 4 other points if the point of con- tact is also a double point (as in Fig. 27). o Fig. 50. Ex. 1. y=(x-l) s , d 2 y/dx 2 = 6(z-l). This is - when x<\, when x = l, + when x>l; hence, as x increases, the curve bends downward until x=l, and upward afterwards; .*. there is a point of inflexion where x = l. Since y and dy/dx are also Fig. 51. Fig. 52. Fig. 53. when £=1, the axis of x is the tangent at the point of inflexion (Fig. 51). 2. y={x-\)\ d 2 y/dx 2 = 12(x-l) 2 , which is when x = l, but is never — hence there is no point of inflexion (Fig. 52). j ' 84 INFINITESIMAL CALCULUS. [Ch. XVIII. Arts. (0, 0). (-1,0). (2a, fa). (1, 0). (a, 0). 3. y 3 = x, or y = x$, d 2 y/dx 2 = — faHl, which becomes oo and changes from + to — when x = 0, .'. the origin is a point of inflexion (Fig. 53). 4. y = 3x*-4x*-6x\ d 2 y / dx 2 = 12 (Sx 2 -2x-l). Putting this = and solving for x } we get x= —^, x = l, which determine the points of inflexion. Find the points of inflexion on the curves: 5. a 2 y = x(x 2 -a 2 ), Fig. 17. 6. xy= 1 +x 3 . 7. (x + a) 2 y = a 2 x. 8. y = x(x-l)(x-2), Fig. 70. 9. x* — axy = a 3 . 10. (a 2 + x 2 )y = a 2 x. (0, 0), (±aV3, ±Ja\/3). 11. ?/ 3 = x 3 + 3a:r 2 , Fig. 38. (-3a, 0). 12. z 3 + ?/ 3 = a 3 . (a, 0), (0, a). 13. x = y 3 + 3y 2 . (2, -1). 14. 2/ 2 = z 2 (2;c-l), Fig. 33. (f, ±|V3). 15. Show that at a point (x, y) a curve is con- vex or concave to the axis of # (i.e., with reference to the foot of the ordinate) according as y d 2 y/dx 2 is + or — . 16. Show that the curves y = sinx, y = tsaix f meet the axis of x in points of inflexion. 17. Where are the points of inflexion of the curve y = cos a;+| cos Sx ? Ans. Where x = \%n ) n any integer not divisible by 4. 18. On the witch y 2 (a — x) = a 2 x (Fig. 54), show that the points of inflexion are (a/4, ±a/v3). Fig. 54. 84. Centre, radius, and circle of curvature. Let P and Q be two points near one another on a curve APQE, Fig. 55, at which tangents and normals are drawn, the latter meeting in D. The limit of position C which D approaches as Q moves towards coincidence with P is called the centre of curvature of the curve at P, PC is 84-86.] CURVATURE. 85 Fig. 55. called the radius of curvature, and the circle with C as centre and PC as radius is called the circle of curva- ture. The extremities of an infini- tesimal arc are called consec- utive points of the curve.* The normals at consecutive points are consecutive nor- mals. Hence the centre of curvature is the limit of the point of intersection of con- secutive normals. 85. Let the length of PC be R. Let the tangents at P, Q make angles , ) (§17) =ds/dcf>. .*. R=ds/d(f>. 86. Imagine the tangent to be rolling round the curve, the point of contact having arrived at P. Then dcf>/ds is the s-rate of <£, or the rate, in radians per unit length of the curve, at which the tangent is turning. This rate is taken as the measure of the curvature of the curve; hence 1/R measures the curvature at P. Since all normals of a circle intersect in the centre and are equal to the radius, the curvature of the circle of curvature is constant and = 1/P. * The point consecutive to P is the point which is next considered and supposed subsequently to approach coincidence with P. 86 INFINITESIMAL CALCULUS. [Ch. XVIII. 87. The circle of curvature generally crosses the curve at the point of contact, since in the circle the curvature is the same on both sides of the point of contact, which is not the case in the other curve except possibly at certain points, e.g., at the vertex of a conic section, where the circle of curva- ture does not cross the curve. 88. Length of the radius of curvature. We have seen that R = — (1); we also have -^ = tan<£. (2) Differentiating (2), 2 dx d 2 y — dy d 2 x , , , fds\ —^ -BecV #= [jj d4>, j. dx d 2 y — dy d 2 x •"• «9= —j^t > ( 3 ) ds s dx d 2 y — dy d 2 x We may generally take x as the independent variable and therefore make d 2 x = 0; also ds 2 = dx 2 + dy 2 . R _ (dx 2 + dy 2 f _ l 1+ \dx) J dx d 2 y * d 2 y dx 2 The sign of R when found from (5) will be + or — accord- ing as d 2 y/dx 2 is + or — , that is, according as the curve is concave upward or concave downward (§ 81). If x and y are given in terms of a third variable m which is taken as independent, (4) may be expressed in the form rfe% /dy\ 2 -|f _ L ^dml ' \dmJ J ( „. dx d 2 y dy d 2 x dm dm 2 dm dm 2 87, 88.] CURVATURE. 87 Ex. 1. To find the radius of curvature at any point (x, y) of the ellipse — , + \, = 1, Fig. 56. a 2 b 2 By differentiating the equation of the ellipse we have b 2 x d 2 y b* a 2 y dx 2 dy dx a 2 y 3 Substituting in (5), we have R = 4M a 4 6 which gives R in terms of x and ?/.* A more convenient expres- sion may be found by substi- tuting y 2 from the equation of the curve. Fig. 56. Then R = (a 2 — e 2 x 2 ) ab where e is the eccentricity \/a 2 — b 2 /a. It is known that (a 2 — e 2 x 2 )^ = the semi-diameter parallel to the tangent, or perpendicular to the normal, at (x, y). Calling this b x we have R=- ab' (7) 2. To find R at the origin of the curve ay 3 — 3ax 2 y = x i , Fig. 36, for the branch which touches the x-axis. Let y = mx,\ then x = a(m 3 —3m), y = a(m i — 3m 2 ). Thus x and y are known in terms of a third variable, and we require R from (6) for m = 0. Differentiating, dx/dm = a(3m 2 — 3)= —3a for ra = 0. d 2 x/dm 2 = 6am = for ra = 0. dy/dm=a(4:m 3 — 6m) = for m = 0. d 2 y/dm 2 = —6a. ■* The sign of R will be + or — according as y is -J- or — . f See foot note, p. 53. 88 INFINITESIMAL CALCULUS. [Ch. XV II. Whence, from (6), R = %a. Similarly ft = 24a for the other branches of this curve at the origin (ra = V3). 89. Coordinates of the centre of curvature. Let the co- ordinates be a and /?. Then, Fig. 57, dx d 2 y — dy d 2 x a = x — R sin cf> J? dy_ dy ds 2 — X xL ~ — X as {3 = y + R cos d — sin cf> dR. But dx=ds cos (f>=R dcf> cos . .'. da= —sin

the least of the fractions. 93 94 INFINITESIMAL CALCULUS. [Ch. XIX. 93. Let F(x) be a function of x, fix) its derivative, and suppose F(x) and f(x) to be continuous from x = a to x = b. When x changes from a to b the change in F(x) is F(b) — F(a), or, in symbols, [F(x)J = F(b)-F(a). Suppose that x changes by the successive addition of infinitesimali ncrements. When x has the increment dx] the corresponding increment of Fix) is, § 42 (2), fix) dx + I, where / stands for the higher infinitesimals. Hence F(b) — i<\a) = the limit of the sum of all such terms as fix) dx, while x changes from a to b, dx approaching its limit and the number of terms being infinite. Let this sum-limit be ex- pressed by fix) dx. Then J a \ b f(x) dx=[F(x)~f = F(b)-F(a). J a L -I a Hence, f(x) being a function of x which is continuous from x=a to x=b, to find the limit of the sum of all such terms as f(x) dx when x changes from a to b we must seek the func- tion F{x) of which the differential is f(x) dx, substitute therein b and a successively for x, and subtract the second result from the first. This process, which is analogous to summation,* is called integration (the making of a whole from infinitesimal parts) ; F(x) is called the integral of fix) dx, and fix) dx is called an element of the integral; a and b are called the limits f . f ft of the integration; f{x) dx is read "integral from a to b J a (or between a and b) of fix) dx" * Historically, the symbol / is the old form of the letter s f the initial letter of the word sum. t This meaning of the word limit is not the same as that employed elsewhere. It here signifies a value of the variable at one end of its range. 93, 94.] INTEGRATION. 95 It should be noticed that dx is here regarded as an in- finitesimal increment of x, and that the element or differ- ential f(x) dx is (§ 42) the increment of the function F(x) b x Fig. 62. when the higher infinitesimals are omitted or disregarded. The practical applications of integration depend upon the fact that the element can be written down when F(x) is unknown. Illustrations. 94. Areas of curves. Let y = f(x) be the equation of a continuous curve CD. Let OA = a, OB = b, OM=x, MP=y, MN=dx, RQ = dy, and let it be required to find the area ABDC. When x has the increment dx, the increment of the area is ATZVQP = rectangle MR + PRQ. MR = y dx, and PRQ) = cos v dv d(cos v) = — sin v dv d(tan v) = sec 2 v dv d(cot v) = — cosec 2/ y dv (F), ••• (G), ■•• (H), ••• (I), (J), e v dv = e v dv , — = log v v cos v dv = sin v (/) (9) (h) (K), sin v dv = — cos v (i) '. sec 2 ?; dv = tan v (j) '. cosec 2/ y dv= —cot v (k) LOFC 100 INFINITESIMAL CALCULUS. [Ch. XX. Differentials. Integrals. d(sec v) = sec v tan v dv (L), /. sec v tan vdv = secv (I) d (cosec t v) = — cosec v cot v dv (M), .*. d( sin x - ) = V a,/ dv ,V dUan' 1 - ) = V a 2 — v 2 a d# a 2 +v 2 , / . v \ a dv d sec -1 - ) = — \ a / Wt) 2 -a 2 To these may be added: * dv (N), (P), (Q), cosec v cot i? dv = — cosec v (m) dv Vfi2-,2 = sin x a av 1 -v -tan -1 — a a Zj rv^ a (n) (p) dv vvv 2 — a 2 a :=-sec- 1 -(a) a 1 I Vv 2 ±a' dv a 2 — v 2 r=log (v + Vv 2 ±a 2 ), 1 1 /a + v\ , . and 1 a — v, f a-\-v 1, = 2^ l0g (^j' *l>l a ' dv 1 log v vVa 2 ±v 2 a \a+Va 2 ±v 2 We add the hyperbolic equivalents of (r) ; (s) ; and (t). dv . , 1 v == = sinh 1 — > v 2 + a 2 a (r) («) (0 dv v (r') = cosh 1 —i Vv 2 -a 2 a i * Formulae (r), (s), (0 should be committed to memory with the others, as they are of fundamental importance. It will be seen later that they may be deduced from the preceding formulae. Compare carefully (n) and (r), (p) and (s), (q) and (t). Notice that / J v- — a 2 /dv -= s and is therefore known from (s). a 2 —v 2 97.] FUNDAMENTAL INTEGRALS. 101 and f dv 1 , , v , , \-z 9 =— tanh 1 —, v\<\a, Ja z -v z a a 1 4. U -1 V Kl = — coth l — , v\ > \a, a a' ' ■ ' J dv v V a 2 + v 2 — sinh x — a v 1 u-i v — cosecn x — > a a v V a 2 — v 2 — cosh l — = sech 1 — a v a a (»') (O CHAPTER XXI. FUNDAMENTAL INTEGRALS. II. Examples. Formulae (a) to (g). • 1. \ax 3 dx = \ax*, 2dx .r I (ax 3 + b) dx = \ax* + bx, 2x~ 2 1 Cdx 1 = \2x- 3 dx = X 2 ' X 2 X 2. 3. 4. 5. 6. I (#' — a 2 ) 2 dx = (x 4 — 2a 2 x 2 + a 4 ) dx = \x* — %a 2 x 3 + a*x< (x 2 -2Y*xdx = i Ux 2 -2)ld(x 2 -2) = \(x 2 -2)§. xdx [d(a 2 -x 2 ) ,— Va 2 — x 2 J2Va 2 — a: 2 xdx Cd(ci 2 — Xj 2 _„2 = ~~i\ „ 2 „2 = ~2 log (a 2 -£ 2 ), by (#). r( i-x 2 ) 2 a; c?x f 1 -2x 2 + a; 4 a; dx r/i X 2rc-f x 3 )e?.r 7. \e- 2X dx=-h ■I = log £ — x 2 + ia; 4 . L (a + 6x + cx 2 )x dx = T V(6a +'46 + 3c). *It is implied in § 92 that a and b are assigned values of x. In this example J is to be understood as the limit of / -r when 6 is J\ x infinite. 104 INFINITESIMAL CALCULI'S. [Ch. XXI. 35. x dx 1 o \ o . -> ••> • [a*-x 2 y ba- 37. I" ^^dx = h{log2y. 39. 41. (a —x) 2 x± dx = yW #-• a {a—x^dx o \ 2ax— x 2 = a. 36. ' a x dx o\ cr+ar '00 = fl(\ 2-1). r — ' Jo V2a-, >2rt dx ,— 40. = 2\^2a. 42. 3 3J "Jx = 26/log27. CHAPTER XXII. FUNDAMENTAL INTEGRALS. III. Examples. Formula (h) to (rn). •| si sin30d0 = * sin30d(30) = -J cos 30. 2. cos 50 cos 30 d0 = ^ (cos 80 + cos 20) dd 1 / sin 80 sin 20 N 8 3.* d0 sin cos ' d0 5. sin dd cos sec 2 ddd tan d0 2 sin ^0 cos %0 ' dfo + d) d tan tan log (tan 0). sin \d cos \d 175 = log (tan£0), by Ex. 3. sin (\n + 0) or = log tan (in + id), by Ex. 4, = log (sec + tan 0).t * Integrals 3-11 deserve special attention on account of their fre- quent occurrence. f This important integral may also be treated as follows: f sec- Odd i sec dd J \/tan 2 + 1 log (sec 0+tan 0), by (r). The integrals of Exs. 5, 4, 3 may also be expressed thus (see foot- note, p. 36): J cos J sin J i dO X{2d-hn). sin z ' v/; J sin cos 6/ Numerical valuas of X(6) are given at the end of the book. 105 106 INFINITESIMAL CALCULUS [Ch. XXII. 6. 7. 8. 9. 10. 11. tan0d0 = cot dd = sin 2 ddd = i cos 2 0d0=4 tan 2 0d0 = sin Odd a~ = — log cos 6 = log sec 0. cos ° ° cos dd = log sin 0. sin (1 -cos 2d)dd = ±(6-i sin 20). (1 + cos 2d)dd = i(d + i sin 20). (sec 2 0-l)d0 = tan 0-0. sin 2 cos 2 d0 = i sin 2 20 dd = \ 12. 13. 15. 17. 18. cos d0 sin 5 (sin 0)~ 5 d(sin 0) = (1 -cos 40)^0 = i(0-i sin 40). 1 4sin 4 0' cos(30-l)d0 = J sin (30-1). 14. sec 2 40d0 = itan40. r sin 3 7/i , T7,d0 = itan 4 0. u COS' 16. cos 2 n0 dd = ^6 + i(sin 2n0)/n. d0 sin + cos = iV2l gtan (i7r + i0)=*£V2 J(0-i*). Vl + cos dd = 2 V2 sin 40. 19 d0 V2 log tan 1(tt + 0) = V2 A(£0). J Vl+cos 20. J \/l ±sin d0 = 2(sin ^0T cos £0). 21 -Ji dO -f COS = tan 40. 22. d0 l+sin0 = tan (id -in). 97.] FUNDAMENTAL INTEGRALS. 107 23. [sin 50 cos 30 d0 = -J (cos 20 + J cos 80). 24. J sin 30 sin 20 dO = \ (sin -\ sin 50). 25. 26.* J sin 5 0d0 = (l-cos 2 0) 2 sin 0d0 = -cos + f cos 3 0-i cos 5 0. sin 4 dO l-cos20\ 2 d0 ~~ 4 [1-2 cos 20 + i(l + cos 40)] = f - J sin 20 +& sin 40. 27. 28. 29. 30. 32. 33. 34. 36. 38. I tan 3 0d0= (sec 2 0-l) tan d0 = i tan 2 + log cos 0. sin 3 d0 (l-cos 2 0)dcos0 cos 2 cos 4 0sin 3 0d0 = dO = sec -f- cos 0. cos 2 cos 4 0(l~cos 2 0) dcos0= -icos 5 0+|cos 7 0. sin 2 cos 2 = tan — cot 0. 31. "sin ^0 sin ^- d0 = log tan Ktt + 0) = K¥). fsin 2 d0 j- = log tan(i?r + iO) - sin = ^(0) -sin 0, cos d0 cos 4 (l+tan 2 0) sec 2 0d0 = tan + Jtan 3 0. sin OdO = 2. 35. cos 0d0 = 0. ?l* dO cos 2 = 1. J J f*w sm 2 0d0 = i7z = 37. tan 2 0d0 = l-i?r. *oob"»*.80. ^™^ = V2-1. cos 2 * Compare the met beds in Exs. 25 and 26 according as the index is odd or even. 108 INFINITESIMAL CALCULUS. [Cii. XXII 40 •I ^ dd 42. In u tan = \ log 2 = '347*. 41. fi* sec 3 6>tan ddd = 2\. qoVO di9 = i(l-log2)=-153. j i* Jo tan 4 0d0 = '119. o 44. 45. 46. = \ log tan A 7r = " 658 = $ Ki^). cos 20 sec Odd = -521. Ml+cos^ /J^ = . 201> + sin 6> 7T + 2. * Use the tables at the end of the book. CHAPTER XXIII. 1. dx FUNDAMENTAL INTEGRALS. IV. Examples. Formula (n) to (/')• fir 1 In, 1 , /a + x\ a 2 — x 2 ]2aLa + x a — xj ' 2a S a — x, It also = n - 1 — : \dx = log ( ) . \2a[_a + x x-aj 2a to \x-aj 2. 3. 4. 5. 6. 8. This is (s). dx 1 f d(aV3) 1 V4-3x 2 V3J Vl-x 4 ' dx \ / 2ax — x 2 dx V 2 2 -(xx / S) 2 V3 d(x 2 ) sin -1 ( x V3> Vl-(o; 2 ) 2 d(x — a) = \ sin _1 (^ 2 )- -sin -1 x — a Vx 2 ±2ax dx 1 2ax — x 2 2a dx Va 2 -(x-a) 2 \ a d(x±a) _ , /— — - — , = log (x ±a + Vx 2 ±2ax). dx \ // (x±a) 2 — a'' x \ log X \/x 2 — a 2 \2a — x) ' dx 7. x 2 + 2ax = ^ log ( x x 2 + 2a' *M- rr '5. X' \£ 1 . la ~= sin -1 — 2 a \x 1 x *The integral is also -sec -1 -. These apparently different re- sults differ only by a constant (in this case n/2a), and therefore have the same differential. 109 110 INFINITESIMAL CALCULUS. [Ch. XXIII. In a similar manner deduce (f) from (r) and {V) from (r'). 9. x 2 dx l+~xQ = ^ tan~ x . x dx 13 f x dx . ' (x 2 \ f dx . — ==: = sec~ 1 e x . \Ve 2x ~l 10. 12. 14. fx 2 dx x 2 dx tl /l+x 3 \ do; dx Vl-e = tan _1 6 :r . = — sech -1 e*. 2a: 15. dx W dx _ 1 ~ 2 J(x 2 -x + i)+i 2 = tan~ 1 (2x-l). d(x-*) (*-i) 2 + (i) 2 16. 17. dx Vl — x— x 2 dx Vl+x + x 2 sin - = sinh _1 2x + l \ , V5 / 2x + T V, dx Vx 2 — a 2 18. = \/x 2 — a 2 — a sec -1 — . J x a 19. 20. 21. 22. 23. 24. [Rationalize the numerator.] dx \/a 2 — x 2 x dx a — x = Va 2 — x 2 — a seen -1 — . a x Va 2 — x 2 + a sur -1 — a + x a dx \x + a x x — v = sec -1 — -fcosh" 1 — x \ix~a a a dx x V4:X 2 -9 dx =i sec-^fx), 1 \/5x 4 -3x 2 VS sec-Mx r- x*dx V5-4x 3 d(2xi) V5-(2z§) 2 i- i/ 2 * § \ 97.] FUNDAMENTAL INTEGRALS. Ill 25. 27. 28. 30. 32. x dx 3 dx f l x dx — - = i tan- 1 3 = '624. 26. — = Jtt = -785. 4 + y.c- J Vl-x 4 2 + 5 - 2 = T 1 oV / 10tan- 1 (iv / 10) = '318. '* x dx , _ B _^ rt f 2 dx 1 - 4 = ilogl=*128. 29. 1— x 4 4 & 3 2 4^1=itan- 1 | = -161. 31. f-^ r a; 4 + 4 La; 2 = i*=-785. 3c /•OO ^ = itan- 1 | = -322. ! ar\/2a; 2 -l rl x* da; V8-4x 3 CHAPTER XXIV. INTEGRATION OF RATIONAL FRACTIONS. 98. An algebraic fraction is rational when it contains no surd expressions involving the variable. // the fraction is improper, it must first be reduced to a mixed quantity. Ex. 1. x- l+x : x 2 -l l+x 2* ' x dx 7- — 2 = Jx 3 — x + tan-^x. 1 + x 2 * x 5 dx l+x 2 (x 3 -x + J dx = fx 4 - |x 2 + J log(l +x 2 ). When the fraction is a proper fraction, it should, in gen- eral, be decomposed into partial fractions. See Appendix, Note A. Examples. 1. x 2 + 3x + l x(x-l)(x + 2) 1 x 2 + 3x + l x(x-l)(x + 2) ± 1_ 1 1 1 2 x 3 x-l 6 x+2 dx= -ilogx+f log (x-l)-^log (x + 2) = ilog (*-d { ! -i vV+2x 3 ' (l+3x)dx en _1_ ?\,| \x~r+x + (l+x) 2 /^ = logx-log(l+ a :)- r J-=log (j^) - I |^. x +2x 2 +x 3 112 98.] INTEGRATION OF RATIONAL FRACTIONS. 113 L j(*~ dx 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14., 15. a)(x — b) dx dx a — b \x — a x — bl a — b ° \x — bl 1+2^ l+3x + 2? =1 °s(tt!)- CO ^.5 8x 5 dx ^ = o; 4 -x 2 + ilog (l+2x 2 ). = log (x + 1) l+2x (2a: + l)dx x x(x + l)(x + 2) dx / x (x-f-2) ; x(l+x) 2 l0g ' dx 1 X 4 (l+X 2 ) x 3x 3 (3x 2 — l)dx 4-tan _1 x. 3x4-11 log (x-2)- 2 log (x-1), x 2 — 3x + 2 x 3 + 5x 2 + 8x + 4 0g ( l+x)+ 2+x' '(x 3 + l)dx 31 3 . , '(1 + x) dx x(l+x 2 ) 2x dx 1 +x + x 2 + x 3 2x dx tan~ 1 x + log x Vl+x 2 = log vTT X' + tan _1 x. 1+x l+x\* (l+x 2 )(3 + x 2 ) log \3 + x 2 ) 3x + 4 (x 2 + 3x + 2)~ x dx 3x + 4 / x + l \ 2 + 3x + 2)~ 2 = x 2 -f-3x + 2 + g U + 2/' CHAPTER XXV. INTEGRATION BY SUBSTITUTION. 99. To assist in bringing certain differentials under forms already considered various substitutions are employed, the most important of which will be mentioned in this chapter. Ex. 1. -5-. Let * = I then dx= - ax + bx n z z dz dx dz -j, and — = x Substituting, we have z n ~ 2 dz log (az 71 - 1 + b) + 6 = -a(n-V) ' by W' n-i 1 / X n ~ 1 \ 1 lo § h^h^ti-i ) > when z is ^placed by -. a(n — 1) & \a + bx n x 2. Making the same substitution and integrating by (a) we have dx x 3. 2z dz. (a 2 -x 2 )i a\a 2 -x 2 )i' dv dx = ± x {x 2 ±a 2 )i a 2 (x 2 ±a 2 )V Vv 2 ±a' i . Let VV ± a 2 = z. Then v 2 ± a 2 = 2 2 and 2v dv = dv dz d{y + z) z v v-\-z 'dv = log (v + z), or — =^=r=log (v + Vv 2 ±a 2 ). J vr±a 2 Thus formula (r) is deduced from (g). 114 99-103.] INTEGRATION BY SUBSTITUTION. 115 ioo. Binomial differentials. Any expression of the form (ax p + bx q ) r dx, the indices being positive or negative, integers or fractions, may be called a binomial differential. For con- venience in making the following statements it will be best to suppose the binomial differential to be given in the form x m (ax n + b) r dx. This can be integrated immediately in the following cases: (1) When r is a positive integer, expand by the Binomial Theorem. TYl ~\- 1 (2) When is a positive integer, let ax n + b = z. lb Tfb "4" 1 (3) When Yr is a negative integer, let ax n + b = x n z. lb io i. When the differential is a function of a + bx let a + bx = z n , where n is the L.C.M. of the denominators of the indices. Ex. dx (1+X)i+(1+X)a dz 2z dz , if l+x = z 2 , z 3 + z> = 2 1+2 = 2 tan- 1 ,? = 2 tan-^Vl+z. 102. In ■ — . let ax 2 + b = x 2 z 2 . (Ax 2 + B)Vax 2 + b 103. sin m # dd, jra odd and + , let cos 6 = z, sin m cos"fl dd, j .-. -sin dd = dz. cos m # dd, j m odd and + , let sin 6 = z, sin"fl cos m # dd, J .\ cos 6 dd = dz. ) let tan d = z, s'm m ddd, cos m ddd 1 m even and-, .-. cos d=l/(l+z 2 )l, sin m dcos n ddd, ?n + n even and-, j sin d = z/(l-{-z 2 )t, J dd = dz/(l+z 2 ). 116 INFINITESIMAL CALCULUS. [Ch. XXV. 104. More generally, any rational function of sin or cos d becomes algebraic and rational when tanJ0 = 2. For cos d=(l-z 2 )/(l+z 2 ), sin = 2z/(l+z 2 ), dd = 2dz/(l+z 2 ). 105. Any rational function of tan 6 becomes algebraic and rational when tan Q = z. For dd = dz/{l+z 2 ). 106. Any rational function of e x becomes algebraic and rational when e x = z. For dx = dz/z. 107. On the other hand, certain algebraic surds are ren- dered trigonometric and rational by substitution. For if x = a sin d, (a 2 — x 2 )$ = a cos 0; if x = a tan 0, (x 2 + a 2 )$ = a sec 6; if x = a sec 0, (x 2 —a 2 )? = a tan 6; if x = 2a sm 2 0, (2ax—x 2 )% = 2a sin 6 cos d; if x = 2a tan 2 # ; (x 2 + 2ax)$ = 2a sec tan 8; if x = 2a sec 2 #, {x 2 — 2ax)% = 2a sec tan 0. Hyperbolic substitutions may also be employed. For if x = a sinh z, (x 2 + a 2 )$ = a cosh z; if x = a cosh z, {x 2 —a 2 )% = a sinh z; if x = 2a sinh 2 2, (x 2 + 2aa;)* = 2a sinh z cosh 2; if x = 2a cosh 2 £, {x 2 — 2ax)$ = 2a sinh z cosh 2. 108. Since ax 2 + bx + c = — [(2az + 6) 2 + 4ac— 6 2 ], the follow- ing general results may be obtained from previous integra- tions by the substitution 2ax + b = z. (1) dx 2 _J 2ax + b tan x ax 2 + bx + c x/^ac-b 2 \V4ac-b 2 if \ac—b 2 is + , and 1 . /2as + 6-V& 2 - -4ac log Vb 2 -4:ac \2ax + 6 + V 6 2 - 4ac, 104-109.] INTEGRATION BY SUBSTITUTION, if b 2 — 4ac is +. 117 (2) = =^\og[2ax + b + 2Va(ax 2 + bx + c)l J V ax 2 J rbx-\-c V a dx sin *■ 2ax— b (3) | V — ax 2 + bx + c Va \V4ac + ft 2 . ris _ 2(2ax + b) (ax 2 + bx + c> 3 (4 ac _ 62)Vax 2 + 6a; + c' )■ (4) # da; ax 2 + 6x + c 2a '(2ax + b)dx—bdx ax 2 + bx + c (5) # dx . <»J Vax 2 + bx + c x dx =— log (a£ 2 + 6z + c) — — — x — '- . 1 / — 7T7 1 — r~ & r ^ ==— Vaar+os+c— — I- c a 2aJ ' vax 2 + 6x -f c 2(te + 2c) (az 2 + fcr + c)* (4ac-& 2 )Vaa; 2 + fcr + c' dx or 109. If we put x = — in - 1 . f d£ —111 « J £ + & (x + ft) Vax 2 + 6x + c these integrals will be reduced to § 108 (2). Examples. 1. dx tan -1 — — 1. Let V2ax — a 2 = z. SI a _xV2ax~a 2 a 2. f X -^i5=(io:-l)2v / x + 2tan- 1 v / ^. Let J 1+x 3 f dx ' }(2+x)\/l+x x = z. = 2 tair^Vl+z. 118 INFINITESIMAL CALCULUS. [Ch. XXV. 4. 5. 6. 7. 8. 9. 10. 11. 13. 14. 15. 16. and dx x + a 1 m (2ax+x 2 )? a 2 \/2ax + x 2 * dx V(x-a)(x-b) dx = 2 log (Vx-a + Vx-b). \ = 2 sin - ! \x — a \jb~a (• Let x — a = z 2 . V(x-a)(b-x) x\l +x 2 )-2dx = e 1 1 i(7x 2 -2)a+x 2 )l. dx V x 2 -1 xWx 2 -l Sx 3 dx 1 (l+2z 2 ). (l+x 2 )Vi-x 2 V2 tan — i lx 2x' sin 5 # dO=-l cos'O + 1 cos 3 fl - cos 0. sin 3 # d# cos 4 # dd cos 3 cos 3 0* 12. d0 'sin_0 cos 5 = -|\/tan 3 0. sin 4 cos 2 f dO = -4 cot 3 0-2 cot + tan 0. l+sin0 l+tan^fl* sin — sin 2 1 + sin d0 d0 = 20 + cos + 1+tan W a + b cos Va 2 — b 2 2 Vb 2 ~ tan - tanh a — b \\ja + b ib — a tan w b + a i°), <*\>\b, ), a\<\b. tan hO ■ 17. tan 4 0d0 = itan 3 0-tan + 0. 109.] INTEGRATION BY SUBSTITUTION. 119 18. dx sec -1 x \/x 2 — l 19. x 3 \/x 2 -l 2 2x 2 dx 1 , x log x Vao; 2 + 6a; + c \Tc ^ bx + 2c + 2\ / c(ax 2 ~+bx + cj' [ dx 1 . , bx — 2c 20. — ■=— rrsin- 1 - a;Vaa; s + 6x — c Vc xV6 2 + 4ac CHAPTER XXVI. INTEGRATION BY PARTS. no. Since v du + u dv = d(uv), .*. v du+\u dv=uv, v du, u dv = uv— or u dv can be integrated provided that v du can be. Integration by this formula is called integration by parts. Ex. 1. 2. 3. log x dx= (log x)x —\x e — = (log *)*-}<**=* log*-*. sin -1 £ dx = (sm~ 1 x)x — x . = x siii- 1 ^ 4- v^l — x 2 . J VT^aT 2 x log x dx X log x . x dx = (log x)-~- — [x 2 ^ dx ~2 *~& x 2 . x 2 ~2 lo g x ~4' 4. Similarly; x n log x dx = , ., log x n + l (n + l) r in. It is often necessary to repeat the integration by parts before the complete integral is obtained, 120 110-112.1 INTEGRATION BY PARTS. 121 Ex. Again, x 2 cos x dx = x 2 . cos x dx = x 2 sin x — 2 x sin x dx = r x sin x dx. cos x dx Substituting, x . sin x dx = — x cos £ + = — x cos z-fsin x. x 2 cos x dx = x 2 sin # + 2x cos x — 2 sin z. ii2. Sometimes the integration reproduces the given ex- pression with a new coefficient. Ex. 1. Va 2 -x 2 = a 2 —x 2 Va 2 — x 2 Va 2 —x 2 dx = a 2 f cfcr . va 2 — a; 2 = a 2 sin -1 a x 2 dx . Va 2 — z 2 a: d( — \^a 2 —x 2 ) = a 2 sin -1 — hi^a 2 -^ 2 a va 2 — x 2 c?x. Transposing the last term to the left-hand side and dividing by 2, we have Va 2 —x 2 dx = -^r sin -1 — -frV^-x 2 . 2 a 2 a) * 2. Similarly, a- x Vx 2 ±a 2 dx= i-o-log Or + V;r 2 ±a 2 )+-Va; 2 ±a 2 . (2)t * (1) and (2) are of frequent occurrence and should be carefully noted. They are also easily obtained by the substitutions of § 107. t Or, yv a 4 ,x . x x 2 + a 2 dx=— sinh -1 - — \-—Vx 2J ra 2 , i // ■ (2 2 XX r vr- a 2 dx = — — cosh -1 — ^-Va; 2 - a- (3) 122 INFINITESIMAL CALCULUS. [Ch. XXVL 3. I see xan-6 dd = tan see 6 - I sec 3 dO t whence, since sec*0 = 1 +taa'0 and sec dd = \og (sec 6 +tan 6) I sec 0tan 2 0d0 = *see 0tan 0-*log (sec + tan0). EXAMPLES. 1 . I x cos j dx = x sin j — cos x. 2. I tan- 1 3. 4. o. 6. . . ~ l x dx = x tan _1 x — los vl + xK x tan _1 .r dx = h (1 -\-x 2 ) tan - 1 x— %x. x sec -1 x dx = h [x 2 sec _1 x — \ x 2 — 1). xe x dx= {x — l)e x . z 2 e*dx=(x 2 -2x+2)e* e*an .r rf.r = J6*(sin .r — cos x). 8. | . : an x dx = 2x sin x+ (2— x) 1 cos x. 9. x sec\r dx = x tan x + log cos J. e* sin 2x dx=-£ (sin 2x — 2 cos 2j). 11. Lr tan\r Jj* = x tan x — log cos x — hx : , ■ 112.] INTEGRATION BY PARTS. 123 f x 3 12. x 2 (\ogx) 2 dx = ^[(\ogx) 2 -% logrr+f]. m 13. sec 3 dd = i sec tan + \ log (sec + tan 0). 14. e a * cos wa; dx = -=-; — ,(a cos mx + m sin raz). e ax a 2 + rri' 15. 6 a:c sin mx dx = —£— — 2 (a sin mx —m cos mx). ,2 16. V^az — x l dx = —^—\ / '2ax — x 2 +—sm- 1 2 ' 2 V a x + a /- a 2 17. v / 2ax + x 2 rfx = — 7r-V / 2ax + a; 2 — ^-log (x + a + v / 2ax+a: 2 ). CHAPTER XXVII. SUCCESSIVE REDUCTION. 113. To integrate sin n # dd, n being a positive integer. I sin n # dd = J sin 71-1 ^ sin d dd = — sin n_1 (9 cos d + (n— 1) sin" -2 /? cos 2 d dd, and writing 1— sin 2 (9 for cos 2 #, = -sm n ~ 1 d cos 0+ (n— 1) sin"~ 2 dd— (n- 1) sin"0 dd. Transposing the last term and dividing by n, we get f . „ ,„ sin n-1 cos d , n—lC . „_ nrx nrt sin"0 dd= + sm n " 2 dd. (1) J n n J Writing n— 2 for n, we have f . __ ,_ sin"- 3 0cos0 , n-3f sm n-2 dd = - 1 J J n—2 n— 2J Thus by each integration the index is diminished by 2, and hence will in the end depend upon sin0d0= — cos d, or d#=0, according as n is odd or even. sm n ~ 3 cos d . n— 3f . . n _. sm n_4 d0. 124 3-115.] SUCCESSIVE REDUCTION. By a similar process cos n-1 sin . n— 1 / cos n fl dd = w " • " ""* v + - — ~ cos n - 2 ddd. n n . cos 2 # + sin 2 # _. fcos 2 0,„. f d# f ^ f I4 ' Jsm^-J sin n -d0 = Js-m^ + J s i n n-2^- The first term= Jcos *<*(~ (w _ ^n-ij ) __JLL *0 n-ljsi d# -J, (w— 1) sin n_ d# cos # n-2f dd ljsi] sin n (n— 1) sin" -1 /? n— lj sin n - 2 0' r which we may reduce to -r-a=logtani0, or j-r-— = — cot (9, J sm J sm 2 cording as n is odd or even. dd sin# . n-2f d# cos n_2 0' Similarly ' jc^ = (n-lTcos-- 1 + ^ : lI; ld |c^ = l0gtan(i7r + i ^' |c5^ = tan ^ 115. [tan*0 d0 = [tan"- 2 (sec 2 6>- l)d0 = ftan"- 2 d(tan 6) tan»" 2 d<9 _ r tan^fl ~J n-1 tan"" 2 d0, I tan 0d0 = log sec 0, <20 = 0. 125 (2) M'/ 126 INFINITESIMAL CALCULUS. [Ch, XXVII. f cnt n ~ 1 f) f Similarly, cot"0 dd=- _ - cot"" 2 !? dd, and [cot 6 dd = log sin 6, [dd = fl. n6. cos m dsm n ddd = /sin n+1 #> cos m - 1 0d( — ) \ n + 1 I cos m-1 # sin n+1 # m— 1 CO8 m-20 s[ n n+2g fid, n + 1 ft + 1. and writing sin 714 " 2 /? in the form sin n # (1 — cos 2 #), cos m ~ 1 6sm n+1 d m—1 n + 1 ft + 1 m— 1 'n + 1, cos m-2 # sin n # dd cos m # sin n # d0. Transposing the last term and dividing, cos m-1 <9 sin n+1 m— 1 cos m d sm n 6 d6 = m+n m+n m cos m-20 sm n^ ^ In a similar way we might have obtained cos m+1 0sin w_1 n— 1 cos m 0sin n 0d0 = m + n m + n_ 117. The following will present no difficulty*: cos m sin n ~ 2 d0. cos m sin n sin n cos m d0=- dd = cos m-1 fti— 1 (n— 1) sin n_1 ft— 1. ' cos m ~ 2 sin"~ 2 ' sin 7 *" 1 /? ft— 1 118. d0 (m— 1) cos m *0 7ft— 1 fcos 2 + sin 2 in do f sin n_2 cos m-2 d0. cos m sin n J cos m sin n d0 1 cos m-2 sin M + d0 cos m sin"~ 2 0" 11G-119.] SUCCESSIVE REDUCTION. 127 The first term = ' 1 d ( -1 \ cos m_1 # \(n— I) sin 71-1 !?/ 1 (n-- 1) sin 1 m- 1 f d# (n- 1) cos 7 " -1 !? sin 71-1 /? n- 1 J cos m <9 sin 71 " 2 ^ Substituting this, we get dO 1 ?n + n-2 cos^sin 7 ^ (n-l)cos m_1 l9sin 71 - 1 ^ n-1 dd cos m 6s'm n - 2 d' By treating the second term in the same way we might have obtained dd 1 ra-hn— 2 cos m #sin n # (m-l)cos 7n_1 i9sin n - 1 m— 1 d# cos m - 2 #sin n #' Hence the integration may be reduced to one of the fol- lowing : d6, dd cos 6 sin 0' ' dd sind' or dd cos d (Ch. XXII). 119. The following may be obtained from the preceding re- ductions by the substitutions of § 107; they may also be obtained directly (cf. § 112). 1 x n dx x n x Va 2 —x 2 a 2 (n—l) I Va 2 dx X' n Va 2 - w n n-2 x n 2 dx Va 2 - x nx/ a 2 - x 2 a 2 (n- l)x n ~ l a 2 (n~ 1)J x n-2^J a 2_ X 2' -X' dx x n dx ^ _1 V 2ax~ x 2 q(2n— 1) V2ax-x 2 n n x n dx _x n ~ l Vx 2 + 2ax a(2n— 1) V x 2 + 2ax n n x n ~ l dx \ / 2ax—x 2 ' x n ~ l dx I n n (a 2 —x 2 ) 2 dx x(a 2 —x 2 ) 2 a 2 n n + l + n + l V x 2 + 2ax £-1 (a 2 — x 2 ) 2 dx. 128 INFINITESIMAL CALCULUS. [Ch. XXVII. Examples. 1. Obtain the results of §114 by integrating sec n # dO and cosec n # dd. 2. 3. 4. 5. _ x m cos nx mx m ~ l sin nx x m sin nx dx= — + — n n' m(m — l) n^~ x m-2 s [ n nx J x> x m cos nx dx x m sin nx mx m ~ l cos no: n n n 2 j x m-2 CQS nx ^ a: n 6 a2: dx = 'e ax dx x n e ax n a a e ax x n ~ 1 e ax dx. [e ax dx a x> (n — l)x n ~ x n — \ x n -l • 6. \x m (\ogx) n dx £ m+1 (k)g x) n n ra + 1 ra + 1 z m (log£) n-1 dx. CHAPTER XXVIII. CERTAIN DEFINITE INTEGRALS.* 120. The first term of § 113 (1) is when # = and also when 6=\n) hence ' 2 • „* j« (n-l)(n-3) . .. o n(n— 2)... each set of factors being carried to 2 or 1, and a being — when n is even, and 1 when n is odd. Also, cos* 6 dd=- s'm n 6 dd. 121. By examining the results of § 116 it will be found that f mp r,fij* [(m-l)(m-3) ...][(n-l)(n-3) . . .] sm m # cos n # dO = - ~ ^7 — — — t^t 1 ■ «, (m + n)(?n + ?i— 2) ... 7T each set of factors being carried to 2 or 1, a being — when Zt m and n are both even, and 1 in all other cases. This reduces to if n=l, and to — if m==l. ra+1 n+1 122. Many integrals may be reduced to the foregoing. Ex. 1. a x n dx o^a 2 — x 2 = a n sin*fl dd. Let x = asind (§ 107). * For a collection of indefinite and definite integrals see Peirce's Short Table of Integrals (Ginn & Co.). 129 130 INFINITESIMAL CALCULUS. [Ch. XXVIII Jo x 2 ) dx = a n+l cos n +^ dd. o 3. ) x m (a n --x 2 )~dx = a m + n + i \ sm m d cos n+1 d dd. Jo Jo ■K ' 2a x n dx ■ Jo V2 = 2(2a> ax — X' sin 2n dd. o 5. 6. 2a x rn (2ax-x 2 )-dx = 2(2a) m + n + 1 \ sm 2m + n+l d cos^ddO. o " x dx A si J o (a 2 -^x 2 ) n a n-i cos»~ 2 # dd. a -z 7. x m {a-x) n dx = 2a rn + n + l \ si Jo Jo sin 2 ™* 1 /? cos 2n + l 0J0. 123. From § 93 it is plain that 'b . }(x) dx=— f(x) dx; b that is, interchanging the limits merely changes the sign of the definite integral. 124. It is possible in certain cases to arrive at the value of a definite integral when the indefinite integral is unknown. The following important integral is an illustration. roc To prove .-r2 dx \ K 2 ' From § 120 we have 7T f T ■ sin" Jo Odd . i - sm» +1 ddd = tt + 1 2 (1) 123, 124.] CERTAIN DEFINITE INTEGRALS. 131 Let s'm n d = e~ x2 , or # = sin -1 (,-). /. dd= 2X n | n Ix dx SJl-e n n ^_- \Je » -1 which, by the Exponential Theorem. Substituting in (1) we have f° 2 gj* rf.r |° 12 e 'dx -12 or i n /n + h *„-f 2 e J ~ dx X -*=(— ) \ n / IT" rfx _ n - ~n+14" Now let r»=oo : then D = 1. and — — [ = H — ) also- 1, n \ - n/ - q: e -*"" rfx pot 4 ' Examples. e -**dx= Jo \ - 1. *wa*0d0=2\ 2 wi*0d0 3 n>0. Jo Jo 2. cos n dd = 0, n odd, and =2 2 cos n d dO, n even Jo Jo x sin x cos rix dx = ( -l) w+1 : _, ; n an integer not 1, 132 INFINITESIMAL CALCULUS. [Ch. XXVIII. and x = — 7 if n = l 4 f * • i / i \ nn 4. x cos a; sin nx ax = ( — l) n 2 _i > n an integer. not 1, and 7T — -T- if n=l. 4 5. The Gamma Function. The integral '00 x n - 1 e~ x dx (n positive is called the gamma function and is represented by T(n). Inte- grate by parts * and show that r(n)=-r(ri + l), or r(n + l) = nr(n). n 6 Show that T(l) = 1, and deduce T(2) = 1 T(3) = 1.2, T(4) = 1.2.3,... r(n) = (n — 1)! if n is an integer. 7. Show that r(^) = v / 7r. Let x = z 2 in the integral. Deduce /*( n + £) = l .3.5... (2n-l)V^/2 n , n an integer. r(n) 8. /•CO 00 x n-i e -arc ^ = — ^-. | n and a> 9. x 2n - l er x2 dx = %r{n), n> 0. Let x 2 = £ Jo cfc = r(n), n> 0. Let # = e~ z . io -£( io ^) n_1 11. I a*- 1 (log -J dz =— ,mandn>0, * £ T x n e- x = for all values of n. »X = 00' For, x n €r x = 1 \n n 1 - iX x n 2n 2 -0 if n>0. The conclusion is obvious directly if n<0. 124.] CERTAIN DEFINITE INTEGRALS. 133 Many other integrals may be expressed by means of gamma functions. The following, known as the Beta Function, is an important illustration (see Williamson's Integral Calculus, § 121). I r (tyi) r (ti) x m-i(i _ x )n -i^ = v . V > m and n> 0. q 1 \JfYl ~T"Tl) Assuming this result deduce : ca r(m) r(n) 12. x m - 1 (a-z) n - i dx=a m + n - 1 r \ , / . J I (m + n) r«> a 7 *- 1 ^ r(m) r(n) J (l+z)™+"~ r( m + n) * /- W*) „, f 1 d£ Vtt W 14. Jo r{m) r{n) 1 Let l+x=— • z Vl-x n n r ' 2 7T Let sin 2 0=z. /m + n + 2\ \ ~2~ I 15. f sm m cos n ddd = - , wandn> -1. 2 „ /m + n -f 2\ J o i. J 2 sin"0 cW = I 2 cos« — r, ^a + 9x ds = Vdx 2 + dy 2 = — : — ^r— dx. OD = i 2Va 2\Ta L 27Va -J 13^13-8 27 a. 2. The area OBD = %a\ 3. The volume of OBD about OX = \na\ i < " OY=±xa 3 . 11 BD = &m\ 4. Find the surface of revolution of the cubical parabola a 2 y = x* about OX, x varying from to a. Arts. ~;(10\/l0-l)a 2 . 126. It is sometimes desirable to express both x and y in terms of a third variable. Ex. The equation x$+y$ = a$ (Fig. 18) is satisfied if we put x = a sin 3 0, y = a eos 3 #. Then dx = 3a sin 2 cos 6 dd, / . ydx = 3a 2 cos 4 sin 2 dd f Jo Jo which, § 121, Sa< 6.4.22" 32 na 2 . ,\ the whole area bounded by the curve = |7ra 2 . For the length, ds = VoV + dy 2 = 3a sin 6 cos 6 dd, f ** . .'. whole length = 12a sin 6 cos d# = 6a. Similarly it may be shown that the volume of the solid made by revolving the whole area about one of the axes = rVk^a 3 , and that the surface of this solid = J t 2 -7:a 2 . 126-129.] AREAS, LENGTHS, SURFACES, VOLUMES. 137 127. It will often be necessary to determine the limits of the integration from the equation of the curve. Thus in finding the whole area enclosed by the curve a 2 y 2 =x 2 (a 2 — x 2 ), it will be seen that the curve cuts the x-axis at (±a, 0) and that the general shape is that of Fig. 69. Hence the complete area Fig. 69. ra = 4 \ y dx = #a 2 . The volume of the solid of revolution about the a>axis = tVt& 3 > and about the 2/-axis = |7r 2 a 3 . 128. When y is negative the sign of ydx is — , and accord- ingly an area lying below the axis of x will be affected by the same sign. Hence in calculating an area, care must be taken that y does not change sign between the given limits. Thus in the curve y = x(x-l)(x-2), Fig. 70, y is + from # = to x=l, — from x=l to x = 2; it will be found that n y dx = J, I ydx=-l, y dx = Q, And generally the sum-limit given by a definite integral f(x)dx is that of the algebraical sum of the elements, 1 which will be equal to that of the arithmetical sum only when f(x) is of the same sign for all values of x between a and 6. 129. If y is infinite in a given interval of x, the area will have a limit (and will therefore remain finite) if the indefi- 138 INFINITESIMAL CALCULUS. [Ch. XXIX. nite integral ydx remains finite for the interval of x in question. For example, if 2/ 3 (x-l) 2 =l, Fig. 71, j/=oo when x = 1 , and Fig. 72. o x Fig. 73. Thus if the area is imagined as described by an ordinate which starts from the ?/-axis and moves towards the asymp- tote x=l, the area = 3, and this is what is meant by the area between the curve, the axes, and the asymptote. Simi- larly if the ordinate starts from the line x = 2 and moves towards the asymptote x=l the area =3, and the sum '2 of the area-limits = 6 = y dx as if y were continuous for o the interval [0, 2] of x. Similarly if y*(x- 1) = 1, Fig. 72, y dx L J o Ji Jo the algebraical sum of the area-limits. But if y(x— 1) 2 = 1, Fig. 73, the indefinite integral y dx= 1-x = oo when x£l, and the area=oo. In this case 130] AREAS, LENGTHS, SURFACES, VOLUMES. 139 2 ydx= — 2, which represents no part of the area for the o interval [0, 2] of x. On the other hand, the area for [2, oo]=l. 130. As x increases the volume-element ny 2 dx changes sign only with y 2 , i.e., when y becomes imaginary. Thus in Fig. 70, 2 16tT f 1 ny 2 dx = — ^ = 2 ny 2 dx. 1U5 Jo }; Examples. 1. The circle x 2 + y 2 = a 2 . Show that (1) Area = 7ra 2 . (2) Length = 2na. (3) Volume of sphere = f na 3 . (4) Surface of sphere = 4;ra 2 . 2. The witch y 2 {a — x) = a 2 x, Fig. 54. Let x = a sin 2 #, then y =a tan 6. (1) Area between curve and asymptote = -ma 2 . (2) Volume of this about asymptote = ^n 2 a 3 , (3) Volume of same area about OF = |^ 2 a 3 . 3. The cissoid y 2 (a — x)=x 3 , Fig. 41. Let x = a sin 2 #, then y = a sin 2 # tan 6. (1) Area between the curve and asymptote = \na 2 . (2) Volume of this about asymptote = \n 2 a 3 . (3) Volume of same area about OY =\iz 2 a 3 . 4. Find the area bounded by the rectangular hyperbola xy=l, and the lines y = } x = l, x = n. Ans. log n. 5. The curve y 2 (a 2 — x 2 ) = a 4 , or x = asm 0, y = a sec 6. (1) Area between curve, t/-axis, and asymptote x = a is xa 2 . (2) Volume of this about 2/-axis = 4^a 3 . (3) Volume of same area about asymptote = 2na z {n— 2). 6. The curve y = e~ x . (1) Area from x = to x = oo is 1. (2) Volume of this about x-axis = ^7r. (3) Convex surface of this solid = tt[V2+ log (1 + V2)]. 140 INFINITESIMAL CALCULUS. [Ch. XXIX. 7. The curve x 2 y 2 + a 2 y 2 = a 2 x 2 . The area between the curve and each asymptote = 2a 2 . 8. Find the area between the following curves and the a>axis: (1) {y-x) 2 = x\ Fig. 30. Ans. A. (2) {y-x 2 ) 2 = x\Yig. 31. A. (3) a 2 y = x(x 2 -a 2 ) } Fig. 17. \a 2 . (4) y(l+x 2 ) = l. (5) y = x(l-x 2 ). i. (6) ?/ = a; 2 (z-l). i_ 12* 4 TITS"- 9. Find the area of a loop of the curves: (1) y 2 = x*{2x + \), Fig. 34. Ans. (2) y 2 = x 2 (2x + l), Fig. 32. A. (3) ay 2 =(x — a){x — 2a) 2 . j\a 2 . 10. The parabola (-) + (-y =1. See Ex. 11, p. 91. (1) Area between curve and axes = ^a&. (2) Volume of this about OX=j^ab 2 . 11. The cycloid x = a(6— sin 6) } y = a(l — cos 6), Fig. 19. For a single arch: (1) Area = 37ra 2 . (2) Length = Sa. (3) Volume about base = 5^ 2 a 3 , (4) Surface of this solid = -\^-7ia 2 . (5) Volume of the area 37ra 2 about tangent at vertex = 77r 2 a s . (6) Show that in Fig. 20, s 2 = 8ax (s = OP, x = OM). 12. The curve # = a(l-cos 0), y = ad; Fig. 20. (1) Area = 27ra 2 . (2) Volume of this about OX = n(n 2 -4:)a\ (3) Volume about OY==5n 2 aK x 2 y 2 13. The ellipse — + — = 1, or x = a sin 6. y = b cos 6. Show that ^ a 2 b 2 (1) Area = 7ra?>. (2) Volume of prolate spheroid * = %izab 2 . * The solid formed by the revolution of an ellipse about its major axis. 130.] AREAS, LENGTHS, SURFACES, VOLUMES. 141 27tab (3) Surface of prolate spheroid = 2nb 2 + sin -1 e. (4) Volume of oblate spheroid * = §7ra 2 6. (5) Surface of oblate spheroid = 2na 2 + — log (- J . Note. — The eccentricity e = Va 2 — b 2 /a. x 2 y 2 14. The hyperbola — 2 — r- = 1, or x = a sec 6, y = b tan 6. Show that the area bounded by the curve, the z-axis, and the ordinate at the point (x 1} y x ) is i^i-ia&log (~ 1+ ^7 > and hence that the second term in this result is the area of the hyperbolic sector OAP, where is the centre, A the vertex, and P the point on the curve. 15. The parabola y l = lax 1 Fig. 74. If OA=x lf AB = y ly show that (1) Area 0AB = %x 1 y 1 . (2) Length OB = ^ Via 2 + y 2 4a + a log Vi+^+Jh*. 2a (3) Volume of OAB about OX = \ny l 2 x l . (4) Surface of this solid = ^ [(4a 2 + y l 2 )* -8a 3 ] 3a = — [normal 3 — subnormal 3 ]. (5) Volume of OAB about A5=A?rx 1 2 i/ 1 . (6) Volume of OBC about OY =\izx 2 y x . (7) Volume of OBC about BC = iny l 2 x l . * The solid formed by the revolution of an ellipse about its minor axis. CHAPTER XXX. SIMPSON'S RULE. VOLUMES FROM PARALLEL SECTIONS. THE PRISMOIDAL FORMULA. LENGTH OF A CURVE IN SPACE. 131. Simpson's rule. An area (Fig. 75) is bounded by a line which is taken as the z-axis, a curve, and two ordi- nates of length yi, y 3 , at a distance h apart, and y 2 is the ordinate midway between them. The area A = ^h(y 1 +4y 2 + ys), (1) # provided that the equation of the curve is of the form y = a+bx + cx 2 +dx 3 , (2) where a, b, c, and d are constants. (1) is the statement of Simpson's Rule. For convenience take the origin at ; the foot of the middle ordinate. Then the area A y dx= (a + bx + ex 2 + dx s )dx — \h J — i;h = ah+ xV c ^ 3 = $h (6a + \ch 2 ) . V\, V2> 2/3 are the values of y in (2) when x=~ \ln, 0, \ln\ •'• 2/i+2/3 = 2a + icA 2 , and y 2 =a. Hence (1). 142 131-133.] SIMPSON'S RULE. 143 The origin may be any point in OX, for the equation would remain of the form (1) if the origin were transferred to 0. 132. When the equation is not of the form (2), or is altogether unknown, the area may be divided into four, six, or an even number n of parts by equidistant ordinates, and (1) applied to each part; the result will be a more or less close approximation to the correct area. This de- pends upon the fact that y can, in general, be expressed as a series of powers of x, and that higher powers than the third may, for purposes of approximation, be neglected if x is small. Formula (1) now becomes h 5- [2/1+4(2/2+2/4 + . • .)+2(2/3+2/5 + . • • ) + 2/n+lL Oft , h being the whole base, n the number of parts, y 1 and y n+1 the extreme ordinates, y2, 2/4? • • • the even-numbered ordi- nates, y s , 2/5? • • • the remaining ordinates. Ex. If in Fig. 75 the base h were divided into three equal parts, show that the area = ih(y 1 +3y 2 + 3ys + y 4 ) *, where y x and y 4 are the extreme ordinates, y 2 and y 3 the inter- mediate ones. Fig. 76. 133. Volumes from parallel sections. Let a solid be cut by parallel planes at perpendicular distances a, x, x-\-dx, b * Another of Simpson's Formulae, 144 INFINITESIMAL CALCULUS. [Ch XXX. from a fixed point, and let A be the area of the section at distance x. Then if A can be expressed as a function of x, the volume of the solid between the extreme planes is J a A dx. For the volume of the slice of thickness dx is {A+i)dx, where i is infinitesimal, .'. the element of the integral is Adx. 2 2 2 Ex. 1. To find the volume of the ellipsoid — ,+^+-^ = 1. The a 2 b 2 c 2 equation may be written V + = 1, which, x being regarded as con- stant, is the equation of a section at a distance x from the origin. The area of any ellipse y 2 /a 2 + z 2 /fi 2 = l is 7m/?. Hence the area of the section of which DEF is a quadrant is Fig. 77. 1-* H'o^) nbc volume is ^9) dx = ^7zabc. x 2 \ a 2 )' the whole 2. Find the volume of the elliptic paraboloid y 2 /b 2 -\-z 2 /c 2 = 2x from x = to x = a. Ans. na 2 bc. 3. Find the volume enclosed by the plane x = h and the surface (1) y 2 /x 2 + z 2 /a 2 = l, (2) xy 2 + az 2 = ax 2 . Ans. (1) \Ttah 2 , (2) %7za*hl. 4. Find the volume of the tetrahedron formed by the cooidi- X V z nate planes and the plane " +7- H — = 1. r a c Ans. \abc. 133.] VOLUMES FROM PARALLEL SECTIONS. 145 5. Two cylinders of altitude h have one extremity, viz., a circle of radius a, in common; the opposite extremities touch each other. To find the common volume. A section of the common volume parallel to the plane CDEF (which contains the centres of the circles) and at a distance OA =x from that plane is a triangle GBH similar to EQF. The area of EQF is ah. GBH AH 2 a 2 — x 2 ah OF' a' GBH=-(a 2 -x 2 ). a a a (a 2 ~x 2 ) dx = ^-a 2 h. Fig. 78. 6. A square moves with the middle points of its sides on the circumferences of two equal circles at right angles to each other. To show that the volume and surface of the groin thus formed are each 4A times those of the inscribed sphere. Let BCDE (Fig. 79) be one position of the square, OA = x y AP = y. The volume and surface elements of the sphere are ny 2 dx, 2nyds; those of the given solid are (2y) 2 dx, 4o(2y)ds; hence the proposi- tion. The volume and surface are therefore ^fa 3 , 16a 2 . The solid is evidently the common part of two equal right circular cylinders whose axes intersect at right angles. 7. A right circular cylinder is sharpened to an edge coinciding with a diameter, the equal plane faces forming a wedge. Find the volume cut off. 146 INFINITESIMAL CALCULUS. [Ch. XXX. Let a length h be cut from opposite sides of the cylinder of radius a. Sections may be made by planes parallel to the axis and the diameter, or parallel to the axis and perpendicular to the diameter. Ans. ±a 2 h. Show that for any diameter of a right elliptic cylinder the result is 4 abh. 8. A parallelogram moves with its angular points on two ellipses which have a common axis. The semi-axes are a, b, c, and the angle between the curves is co Show that the volume is \abc sin co. 9. Show that the volume of any cone or pyramid = J base X height, assuming that the area of a section parallel to the base varies as the square of its distance from the vertex. 10. A straight line is parallel to a plane which contains a closed curve. Another straight line moves so as to intersect the curve and the fixed straight line and remain perpendicular to the latter. Show that the volume of the right conoid thus formed = % base X height. Fig. 80. Fig. 81. 11. Form of an inverted column of uniform strength. Let A be area of a horizontal section at a distance x above the base, which is also assumed to be horizontal and of area a. The prescribed condition is that A varies as the volume V below A ; hence dA varies as dV. .*.dA=kAdx, or dA/A=hdx. Integrating, logA=kx + c. But A= a when x = 0, .'. loga = c. .'. log (A/a) = kx, or A = ae kx 12c Such a column is to be cast in the form of a solid of revo 134.] THE PRISMOIDAL FORMULA. 147 lution, R and r being the radii of the extremities, and h the height. nh(R 2 — r 2 ) How much metal is required? Arts. Vol. = ^- M") 134. The prismoidal formula. The extremities of a solid are parallel planes of area A 1, As, at a distance* h apart, and A 2 is the area of a parallel section midway between them. The volume v = ih(Ai+±A2+A s ), (1) provided that the area A of any section parallel to the ex- tremities can be expressed in the form a + bx+cx 2 +dx s , (2) where x is the distance of the section from a fixed point. (1) is the Prismoidal Formula. Since the volume Ji. ClXj the proof is the same as for Simpson's Rule. The Pris- moidal Formula will give exact values of the volume of many of the common solids, such as cones, pyramids, prisms, spheres, ellipsoids, paraboloids, etc. It will apply to Exs. 1-9 of § 133. (In Ex. 7 it will apply to the second ' mentioned sections, but not to the first.) Ex. 1. The area of a section of a sphere at a distance x from the centre is n(a 2 — x 2 ), which is of the form (2), hence the Prismoidal Formula will apply. For the whole volume h = 2a, A 1 =A3 = 0, A 2 = 7ia 2 . ,\ V = %na§. 2. Find the volume of the greatest solid that can be cut from a sphere of radius a, the parallel sections to be regular polygons of n sides. Arts. %na 3 sin 27r/n. The volume of a sphere may be deduced. For o o . « / a sin27r/n . „ _ $na 3 sin 2^/n=|^a 3 — - — - — = |^a 3 when n = oo . 2n/n 148 INFINITESIMAL CALCULUS. [Ch. XXX. 135. Length of a curve in space. Ex. 1. To find the length of the curve of intersection of az = x 2 and 3a 2 y = 2x* from the origin to the point (x lf y lt z x ). (2x\ 2 ' 1+— J dx\ s = '*i / 2x 2 \ a 2 x x z lx = x l +^—£ = x 1 + y 1 . 2. Find the length of the helix x = a sin nz, y = a cos n&, from the origin to the point (x ly y u zj. Ans. z 1 ^l-\-n 2 a 2 . CHAPTER XXXI. POLAR COORDINATES. 136. Let be the pole or polar origin, OA the polar axis or initial line, (#, r), (d + Jd, r+Jr) the coordinates of P and Q, ([> the angle which the tangent at P makes with the radius vector OP. Take PR perpendicular to OQ. Then is the limit of OQP as Q approaches coincidence with P, and teaiOQP=PR/RQ. Fig. 82. But PR = rsinje = rJd + I lf (§16), and RQ = r + Jr— r cos J0 = Jr+r(l — cos Jd) = Jr + I 2 . {•*• •! 1 • f W/l/ . Li/I Similarly sinf = r^ ; cos = — . ds 149 150 INFINITESIMAL CALCULUS. [Ch. XXXL Squaring and adding, 1 = (r 2 dd 2 +dr 2 )/ds 2 7 or ds 2 = r 2 dd 2 + dr 2 . 137. Through the origin (Fig. 83) let a line be drawn perpendicular to the radius vector OP, meeting the tangent in T and the normal in N. Then TP is called the polar tan- gent, NP the polar normal, TO the polar subtangent, and ON the polar subnormal. The lengths of these lines in terms of r and S can be written down at once; e.g., TO = r tan = r 2 dd /dr. The tangent at any point P is easily drawn by calculating OT and then joining T to P. 138. Some of these quantities are more conveniently ex- pressed in terms of d and the reciprocal of r. Calling this u, we have u=l/r, du= — dr/r 2 ; hence the polar subtangent TO=~d6/du. The polar coordinates of T are [hn + d, - dd\ dul ' Let OG, the perpendicular on the tangent, =p. Then v OTP is a right-angled triangle and OG the per- pendicular from the right angle to the hypotenuse, we have 1 _1_ J_ J_ 2 AM 2 ■ m OG 2 OP 2 ~^OT 2 ' or p2 ^ + W ' W 139. The polar equations of some of the commoner curves are as follows: (1) r cos 6 = a, a straight line. (2) r = a cos #, a circle of diameter a (origin a point on the circumference, initial line a diameter). (3) r 2 cos 26 = a 2 , a rectangular hyperbola, Fig. 98 (origin the centre, initial line the transverse axis). (4) r 2 =*a 2 cos 20, a lemniscate, Fig. 27 (origin the centre, initial line the axis). 137-139.] POLAR COORDINATES. 151 (5) H cos \d= a$, or r(l+cos 6) = 2a, a parabola (origin the focus, initial line the axis). (6) r*=a* cos £0, or r=^a(l+cos d), a cardioid, Fig. 89. (7) r(l+e cos 6) = m, an ellipse, hyperbola, or parabola according as the eccentricity e<, =, or >1 (pole the focus, initial line the axis, m half the latus rectum). (8) r = n(l+e cos 0), a limagon, Figs. 88, 89, 90, according as e<, =, or >1. Fig. 84. (9) r = ad, a spiral of Archimedes, Fig. 84. (In Figs. 84, 85, 6 varies from a little less than —2k to a little more than 2tt). Fig. 85. (10) rd = a, a reciprocal or hyperbolic spiral, Fig. 85. Fig. 86. (11) r 2 d = a 2 , a lituus, Fig. 86 {6 is necessarily +, and varies in the figure from to a little more than 2tt, r is ± for a given value of 6). 152 INFINITESIMAL CALCULUS. [Ch. XXXI (12) r = a d , a logarithmic or equiangular spiral, Fig. 87 r = a when when 6 is Fig. 87. (r=l when # = 0, 6 = 1 radian, r< 1 negative). Since 6 may be supposed to increase or decrease without bound, each spiral ^consists of an infinite number of whorls or spires. 140. Equations (1) to (6) are all included under the form r m cos m6 = a m ; in (1), (3), and (5) m has the values 1, 2, ^, respectively; in (2), (4), (6), it has the values — 1, —2 — J. In all cases a is the intercept on the initial line. The equation r m sin md = a m represents the same series of curves, the initial line having been turned backward through the angle n/{2m). Similarly (9), (10), (11) are particular cases of the equation r m = a m 6 n . 141. The radius vector of the limagon, equation (8), is proportional to the reciprocal of the radius vector of a conic section, equation 7; hence the limagon is called the inverse of a conic section with regard to a focus. Since r = en cos + n, the radius vector is equal to that of a circle of diameter en plus a constant line n, and hence the curve is easily con- structed. (The construction or auxiliary circles are shown in the figures.) Fig. 88. Fig. 89. Fig. £0. When 6 = 1 the curve becomes a cardioid (eqn. 6), which is therefore the inverse of a parabola. When e = 2 the curve is 140-142.] POLAR COORDINATES. 153 called a trisectrix, the loop then passing through the centre of the circle. Examples. 71% 1. If r m = a m 6n i show that tan

= cot m#, i.e., that the angle between the radius vector and normal = md, and hence that GO A (Fig. 83) = (m-l)0. 3. In the logarithmic spiral r = a° show that $> is constant and = cot -1 (loge a). In Fig. 87. a = 1*318 cm.; show that ^ = 74° 33'. 4. To find the polar subtangent of a conic. From the equation 1 +e cos = m/r = mu we have — e sin 6 dd = mdu, and the polar subtangent = — dd/du = m/(e sin 6). 5. In any conic prove that 1 J2/1 l-e 2 \ p 2 m\r 2m / ' 6. In the curve r m cos md = a m prove that pr m - 1 = a m . 7. Changing the sign of m, show that pa m = r m+1 in the curve r m = a m cosmd. 8. Show that the polar subnormal of any curve = dr/dd. In what curve is the polar subnormal constant? 9. In what curve is the polar subtangent constant? 10. Show that the polar normal = ds/dQ. Asymptotes. 142. The position of any line is known when its direc- tion and one point in the line are known- We may there- fore determine an asymptote by finding a value of for which r=oo or ^ = 0, and then calculating the coordinates (§ 138) of T , the extremity of the corresponding polar sub- tangent, viz. ( %7z+d, —j , remembering that the asymptote and radius vector must be parallel. 154 INFINITESIMAL CALCULUS. [Ch. XXXI. 1. r = ad (Fig. 91), or u = Examples. 1 1 dd whence —=—ad 2 and du ad a r = oo or u = when 0=1. Hence the asymptote passes through the point (Jtt + 1, —a) or (1— J*, a) and is parallel to the line = 1. Fig. 91. Fig. 92. 2. Find the asymptotes of the curve (r — a)d 2 = r (Fig. 92). Ans. Lines through (^±1, ±\a) parallel to 0= ±1. 3. Find the asymptote of the reciprocal spiral rd = a (Fig. 85). Ans. A line through (%n, a) parallel to the initial line. 4. Show that the initial line is an asymptote to the lituus r*d = a 2 (Fig. 86). 5. Find the asymptotes of the curve r sin 40 = a (Fig. 93). Ans. Four pairs of parallel lines, each pair \a apart. (The numbers in figures indicate the order in which the branches are formed as increases from to 2n.) 6. Find the asymptotes of the curve r 2 sin 40 = a 2 (Fig. 94). Ans. Four lines passing through the origin. 7. Find the asymptotes of the curve r cos 20 = 2a. Ans. Four lines parallel respectively to = \n, d = \n, = |?r, = j7r, and passing through the points (|7r, — a), (|tt, a), 143, 144.] POLAR COORDINATES. 155 8. Shovi that the rectangular equation of an asymptote of the curve r~ l =f(6) is J'(a){x sin cc — y cos a) + 1=0, where « is one of the roots of the equation /(0) = O. VJl 15/ Fig. 93. Fig. 94. A. 143. In the curve Fig. 91, r Asymptotic Circles. ad a 1-0 1__ 6 a if = ± 00 , and hence the circle of radius a is called an asymptotic circle. The curve approaches the circle from the outside when increases from the value 1, and from the inside when 6 decreases from the value 0. Similarly r = a is an asymptotic circle of the curve r(6 2 —l) = ad 2 (Fig. 92). Points of Inflexion. 144. Whenever the extremity of the radius vector passes through a point of inflexion, the perpendicular on the tangent is a maximum or a minimum, and hence dp/dr changes sign. 1 /du\ 2 Differentiating -o = =u< 2 + (-Tn) (§ 138) we have 2 . rt , 2 du d 2 u n T / , d 2 u\ P dO 2 156 INFINITESIMAL CALCULUS. [Ch. XXXI. Also r=l/u, dr=—du/u 2 « dp o q / , d 2 u\ > d^u Hence at a point of inflexion u + -Tn- 2 changes sign. Examples. 1. Find the points of inflexion on the curve (r — a)0 2 = r or au = l- 6- 2 (Fig. 92). Ans. ( + VS } fa). 2. Find the point of inflexion on the curve r(l — 6) = ad (Fig. 91). Ans. is a root of the equation 3 -0 2 -2 = O, .*. (§50) = 1.696 rdn. = 97°% and .'. r= -2*437a. 3. Find the points of inflexion on the lituus r 2 = a 2 (Fig. 86). Ans. (4, ±aV2) 4. In the lemniscate r 2 = a 2 cos20 (Fig. 27) show that dp/dr = 3 cos 20, and hence that the origin is a point of inflexion on each branch. 5. Show that a curve is concave or convex to the origin accord- ing as u + d 2 u/dd 2 is + or — . Multiple Points. 145. The equation of a curve being r = /(#), the direction of the curve at the origin is determined by the values of d, Fig. 95. Fig. 96. which satisfy the equation /(0) = O. If this equation have two or more roots there will be a multiple point at the origin. 145, 146.] POLAR COORDINATES. 15' Examples, 1. In the lemniscate r 2 = a 2 cos 26 (Fig. 27) the equation cos 20 = gives 6 = ±\n for the directions of the tangents at the origin. 2. Find the tangents to the curve r = asin40 (Fig. 95) at the origin. Arts. 6 = 0, \n, \n, \n. These lines are also tangents to the curve r 2 = a 2 sin 40 (Fig. 96) at the origin. 3. Find the tangents to the curve r = a sin 36 at the origin. Arts. 6 = 0, \n, §7r. 4. Show that the curve (r — a)6 2 = r (Fig. 92) has a cusp at the origin. Curvature. 146. Let PD, QD be consecutive normals (see §84), and let the angle PDQ = J(f>. We shall first show that DP-DQ is an infinitesimal of at least the second order, PQ or A$ being of the first. Draw QF perpendicular to DP. Then DP- DQ = FP-DQ(l- cos Aj>). But 1 — cos dcf) is of the second order; so is FP, since it = chord PQXcos FPQ, and each factor is infinitesimal. Hence DP-DQ is of at least the second order. Let PD = n, then QD may be written n + L Let OP = r, OQ^r + Jr, OT = p, OT' = p + Jp. Then in the triangle OPD OD 2 = P0 2 + PD 2 -2PO . PD cos OPD = r 2 + n 2 —2rn sin (Jj = r 2 + n 2 —2pn. 158 INFINITESIMAL CALCULUS. [Ch. XXXI. Hence in the triangle OQD OD 2 =(r + Jr) 2 + (n + I) 2 -2(p + Jp)(n + I). Equating and simplifying, = 2r At— 2nJp + I ly .\ £n = £(r Jr/Jp). But £n = R, the radius of curvature PC, § 84. ■'■ # = *■?-• (1) dp - Hence also (§§ 138, 144) /du\ % R= ' P+Q] "'*■(» +^) * 8 ("+sp) Examples. 1. Find the radius of curvature at any point of r m cos md = a m or pr m ~ 1 = a m . -4ns. i2 = —-7 — — = — -z — . (m — l)a m (m — \)p If r m = a m cos m0, # = — : = — -. (m + l)r m - 1 (m + l)p 2. The equation r 2 = p 2 + a 2 represents an involute of a circle, find R. 3. In the logarithmic spiral r = a d , p = r sin ', and $ is con- stant, hence R = r/sm ^ = the polar normal. 4. Show that the evolute of the logarithmic spiral is an equal logarithmic spiral. [OC is a radius vector and PC a tangent to the evolute, and in this case the angle OCP=(/>, a constant.] 147.] POLAR COORDINATES. 159 5. Prove that in any curve „ [-(5)7 R = o ~ /dr\ 2 d 2 r* [ We have w = 1 /r, du=~ —dr/r 2 , d 2 u= — (r 2 d V — 2r dr 2 ) /r 4 , to substitute in (2) ] 6. In the spiral r = ad (Fig. 84), #= (a 2 + r 2 )§/(2a 2 + r 2 ). 7. In the spiral r0 = a (Fig. 85), R = r(a 2 + r 2 )%/a 3 . 8. If a curve touch the initial line at the origin, prove that R = the limit of \r/Q at that point; and hence show that the radius of curvature of the curve of Fig. 91 at the origin is half the radius of the circle in the figure. 9. Find R for the curve r = a sin nd at the origin. Arts. \na, 10. Prove that the intercept of the circle of curvature on the radius vector of any curve = 2p dr/dp. In the curves r m cos md = a m and r m = a m cos md show that these chords= — 2r/(m — 1) and 2r/(ra-fl), respectively. Areas, etc. 147. Let AOP=d, POQ = dd, OP = r y OQ = r + Jr. (1) The area-increment POQ lies between the circular sectors POD, EOQ, whose areas are \r 2 dd, %(r + Jr) 2 dd. .'. 8LTe&POD = ir 2 dd + L Hence the area between the curve and two radii vectores is J a r 2 dd. (2) The area bounded by two radii vectores and two given curves r 1 = / 1 (^) and r 2 = } 2 (0) is (rf-rftdd or %\\r 2 2 +n 2 ) dd a J a according as the curves lie on the same side or on opposite sides of the origin. 160 INFINITESIMAL CALCULUS. [Ch. XXXI (3) The length s= lds = Vr 2 dd 2 +dr 2 , taken between assigned limits. (4) The area of the surface formed by the revolution of the curve about the initial line is (§ 125 (4) ) 2tt r sin d ds. Examples. 1. The cardioidr = acos 2 i0 £Fig. 89). (1) The area = J a cos \Q dd = %na 2 . (2) The length-element ds = \ // r 2 dd 2 -\-dr 2 = a cos \6 dd, which does not change sign while 6 increases from — n to n 9 hence the whole length of the curve is a cos \d dd = 4a* (3) The surface of revolution about the initial line = 2tt r sin 6 ds = 2n = 27ia 3 a cos 2 J# . sin 6 . a cos J# dd cos 4 i# sin \ Jo \0d0 = \na % . (The volume = \na\ § 178, Ex. 6.) 2. The spiral of Archimedes r = ad (Fig. 84). (1) Let it be required to find the area included between the nth and (n + l)th spires. On the former r = a[2(n — l)n + ff] t and on the latter r = a[2nn + 0], hence the area between them (2n7t + d) 2 -(2(n-l)x + d) 2 dd = 87t 5 a 2 n, = W o L. and is .*. proportional to n. * A change in the sign of the length-element indicates a cusp, which occurs in this case when 0=7?. As increases the area-element \r 2 dd can change sign only with r 2 , i.e., when r becomes imaginary. Hence if we had integrated between the limits and 2n we should have obtained for the length, whereas the area would have been the same as above. 147.] POLAR COORDINATES. 161 (2) Show that the area of the first spire {0 varying from to2;r) =87rV/6. (3) The length of the curve from the origin to r = r x is 1 a Wa 2 + r 2 dr (see § 112, Ex. 2). o This is easily shown to be the same as the length of the parabola y 2 = 2ax from the vertex to y = r l . 3. The lemniscate r 2 = a 2 cos 2d (Fig. 27). (1) The area = a 2 . (2) Show that rds = a 2 dd. (3) The surface of revolution about the axis = 27ra 2 (2 — V2). (4) The surface of revolution about a tangent at the centre = 47ra 2 . [This tangent being taken as initial line, the equation becomes r 2 = a 2 sin 26.] f r dv 4. Prove that the length of any curve = ' . , and that the J v r 2 —p z _ i p ds = 2 pr dr area = i 5. To find the length and area of the logarithmic spiral r = a d (Fig. -87). (1) Let be the constant angle between the radius vector and the tangent. Then ds = dr /cos 0, whence .' where r x and r 2 are the radii vectores of the extremities of the arc. (2) For the area, ip ds = %r sin

. ;. area = itan^ r dr = \(r 2 2 —r± 2 ) tan $. 6. The length of the spiral r = e~ 6 from = to # = oo is V2*. 7. In the curve r 2 = a 2 sin4# (Fig. 96) show that the area of each loop = \a 2 . 162 INFINITESIMAL CALCULUS. [Ch XXXI. 8. In the curve r = a sin 40 (Fig. 95) show that the area of each loop = r^a'^i that of the circumscribed circular sector (centre the origin). - 1 - i i . 9. Prove that the length of the curve r n =a n cos — 6 is n n(n — 2) ... . 2aa, (n-l)(n-3) . . . where « is 1 or \% according as n is even or odd. 10. In the spiral r6 = a (Fig. 85) show that the area bounded by two radii vectores and the curve is ia(r 2 —r 1 ). 11. The polar equation of the cissoid (Fig. 41) is r cos 6 = a sin 2 0, that of its asymptote is r cos = a, that of the circle of diameter a is r = a cos 6; show that the area between the cissoid and its asymptote = f^a 2 , and that the area between the cissoid and the circle = (^7r — l)a 2 . 12. Find the area of a sector of the rectangular hyperbola r 2 cos 26 = a 2 (Fig. 98) between = and 0= a. Arts. \a 2 log tan ( \n + a). 13. Find the area of a sector of any hyperbola between 6 = and 6 = a, the centre being the origin and the transverse axis the initial line. , 71 /fr + atana:\ Ans. iab log ( r t I . 4 ° \b — a tan a) 14. Find the area of an elliptic sector between 6=0 and 6=a 9 the centre being the origin and the major axis the initial line. Ans. iab tan -1 (— tan.al . 15. Show that the area of the limagon r = n(l + e cos 6) is nn 2 (l+ie 2 ). 16. The chord which is drawn through the origin so as to cut off from a given curve a segment of maximum or minimum area is bisected by the origin. For d (area) = \r 2 d6-\r 2 d6 = 0, .*. r 1 = r 2 . 17. Find the area enclosed by the curves (1) r 2 = a 2 cos 2 # + 6 2 sin 2 #. Ans. in(a 2 + b 2 ). (2) r 2 = a 2 cos 2 #-& 2 sin 2 0. ab + (a 2 - b 2 ) tan" 1 (a/6). 18. The area of the common parabola r cos 2 %6 = a from 6 = to 6 = a is a 2 (tan ^a + J tan 3 Ja). 148.] POLAR COORDINATES. 163 19. If the conchoid r = a sec 6 — b has a loop, show that the area of the loop is aV / 6 2 — a 2 + b 2 cos~ l (a/b) — 2ab cosh-^b/a). 148. It is in general impossible to obtain the area exactly unless one coordinate can be expressed in terms of the other, or each in terms of a third variable. When the rectilinear equation of a curve consists of terms of two dimensions only, both x and y are expressible in terms of m, the slope of the line drawn from the origin to (x, y). We can sometimes obtain the area by taking m as the variable. If m = tan d, dm = sec 2 # dd, .'.%r 2 dd = \r 2 cos 2 # dm = \x 2 dm. .'. the area=i 2 j x 2 dm. The area included between two curves will be (x 2 2 ±xi 2 ) dm. Examples. 1 . The ellipse ax 2 + bxy + cy 2 = k. Substituting mx for y, we have x 2 = k/(a + bm + cm 2 ). Hence the whole area -I 00 kdm 2nk_ _na + bm+cm 2 ~~ V^ac-b 2 2. (1) The folium x 3 + y* = 3axy (Fig. 28). Here x = Sam /(l +m 3 ), . V area of the loop ,Q0 9a 2 m 2 dm (1+m 3 ) = %a 2 = WBAC. 2 ~2™ ~ 3 (2) On the asymptote x + y-ha = 0, m= — a/(l+m); hence the area in the second and fourth quadrants between the curve and the asymptote if 00 f a 2 9a 2 m 2 -1 2 Adding \a 2 , the area of the triangle ODE, we have the whole 164 INFINITESIMAL CALCULUS. [Ch. XXXI. area between the curve and the asymptote = fa 2 = the area of the loop. 3. Find the area of the closed part of the curve a 2 ir(y — x)+x'° = 0. Ans. Tta 2 . 4. Find the area of a loop of the following curves: (1) ay*-3ax 2 y = x'- 7 Fig. 36. Ans. IfVla 2 . (2) ay A -axS?=x\ Fig. 37. jha\ (3) x*+y 4 = ±a 2 xy. i~a 2 . (4) ax* ~-y 3 = axy. ^&a 2 . CHAPTER XXXII. ASSOCIATED CURVES. Inverse Curves. 149. If on the radius vector r of a curve, a distance r' is measured from the origin so that rr' = k 2 , where k is con- stant, the locus of the extremity of r f is called an inverse of the given curve. The radius vector of the inverse curve is proportional to the reciprocal of that of the given curve, and its polar equation may be found from that of the given curve by substituting k 2 /r for r. Thus (see § 139) the inverse of the equilateral hyperbola with reference to the centre is a lemniscate (Fig. 98), that of a conic section with reference to a focus is a limagon of the form Figs. 88, 89, or 90, according as e is < , =, or >1, i.e., according as the conic is an ellipse, parabola, or hyperbola. Examples. 1. Show that the inverse of a circle with reference to a point on the circumference is a straight line, and that with reference to any other point it is a circle. 2. The angle between the radius vector and the tangent at any point of the inverse is the supplement of the corresponding angle in the given curve. For, if OPQ, OP'Q' (Fig. 98) are consecutive (see § 84) radii vectores meeting one curve in P, P', and the other in Q, Q', the rectangles OP . OQ, OP r . OQ' are equal, .'.a circle may be described through P, Q, P', Q', .'. Q'P'P + PQQ' = two right angles; hence, supposing P' to approach P, the tangents at corresponding points 165 166 INFINITESIMAL CALCULUS. [Ch. XXXII. P ana Q make supplementary angles with the common radius vector. Otherwise thus : r = k 2 /r', .' . log r = log k 2 — log r' f .'. —dr/dd= — -dr'/dd, or cot , p) are the polar coor- dinates of the point on the pedal corresponding to (0, r) on the given curve. If then we can express p in terms of r, and cf) in terms of 0, the polar equation of the pedal will be easily obtained from that of the given curve. Examples. 1. The pedal of an equilateral hyperbola is a lemniscate (Fig. 98). For pr = a 2 (Ex. 6, § 141), and = d (Ex. 2, § 141), hence sub- stituting in r 2 cos 26 = a 2 we have a 2 cos2 = p 2 , or writing 6 and r for $ and p, r 2 = a 2 cos 26, the equation of the lemniscate. In a similar way it may be shown that the pedal of any curve of the form r m cos m6 = a m is r n cosn#=a n , where n=m/{l—m) J and that the pedal of r m = a m cos m6 is r n = a n cos nd, where n=m/(l +m). 2. The angle between the radius vector and tangent at any point of the pedal = that between the radius vector and tangent at the corresponding point of the given curve. For, in Fig. 97, let OP produced meet T'Q in S. Then 0, T, T', S are on the circumference of a circle since the angles at I 7 , T' are right angles, .'. OT'T = 0ST, and the limits of these angles are the angles referred to in the enunciation. (In Fig. 98, OPT = 0TV, if PT and TV are tangents.) 3. Prove that the pedal of a circle with reference to any point is a limagon of the form Figs. 88, 89, or 90, according as the point is inside, on, or outside the circumference. (These figures are the pedals with reference to of the circles with centres B and radii B A.) 4. Show that the pedal of a logarithmic spiral with reference to its origin is also a logarithmic spiral. 5. Find the pedal of a parabola with reference to its vertex. Arts, r cos 6 = a sin 2 #, the polar equation of the cissoid, Fig. 41. (The directrix of the parabola is the asymptote of the cissoid.) 6. Show that the pedal of the involute of a circle is a spiral of Archimedes. (It will be found that tan # is proportional to the radius vector.) 168 INFINITESIMAL CALCULUS. [Ch. XXXII. 7. Find the pedal of the ellipse with reference to the centre. Ans. r 2 = a 2 cos 2 # + b 2 sin 2 0. Polar Reciprocals. 151. The inverse of the pedal of a curve (both pedal and inverse being taken with reference to the same point) is called the polar reciprocal of the given curve. Examples. 1. Show that the polar reciprocal of a circle with reference to any point is a conic section. 2. Find the polar reciprocal of a parabola with reference to its vertex and with reference to its focus, of an ellipse with refer- ence to its centre and with reference to its focus. 3. Show that the polar reciprocal of a logarithmic spiral, r = a d t with reference to its origin is another logarithmic spiral. Roulettes. 152. When one curve rolls on another, the curve described by any point connected with the rolling curve is called a roulette. The simplest case is the cycloid, the properties of which have already been considered. Any involute of a curve may also be regarded as the roulette traced by a point in the tangent of the curve as it rolls round the curve. 153. The property of the normal of the cycloid holds for all roulettes, viz., the normal to the roulette at the tracing point passes through the point of contact of the fixed and moving curves, since at each instant the point of contact may be regarded as an instantaneous centre of rotation. 154. When a circle rolls on a straight line any point not on the circumference describes a curve called a trochoid, the equations of which are easily shown to be x = ad—b sin 0, y = a—b cos d, where a is the radius of the circle and b the distance of the tracing point from the centre (axes as in Fig. 19). 151-155.] ASSOCIATED CURVES. 169 155. When the circle rolls on the circumference of a fixed circle, the curve described by a point in its circumference is called an epicycloid or a hypocycloid according as the circle rolls on the outside or inside of the fixed circle. Corre- sponding to these curves we have epitrochoids and hypo- trochoids described by points not in the circumference. Fig. 99. For the coordinates of any point P (Fig. 99) on the epi- cycloid we have x=OB=OD cos 6-PD cos (0 + 6'). Hence, since arc PE = bd' = AE = ad, 'a-{-b y x= (a + b) cos 0— b cos ( — — ) d. Similarly, y=(a + b) sin 6—b sin ( — 7— ) d. 170 * INFINITESIMAL CALCULUS. [Ch. XXXII. The x and y of a point on the hypocycloid may be obtained n a similar way (or from the epicycloid by changing the sign of b), and are x= (a— b) cos + b cos (— r~)0- y=(a—b) sin 0— b sin ( — — J 0. The equations of the epitrochoid and hypotrochoid are of the same form, the coefficient b in the second term being changed into h, where h is the distance of the tracing point from the centre of the rolling circle. Examples. 1. Show that in any epicycloid ad b 6' ds = 2(a + b) sin-dd = 2{a + b)— sin-dd', 26 a 2 and hence that the length of the curve from cusp to cusp is 8(a + b)b/a r f 2. Show that the epicycloid is a cardioid when b = a. 3. Show that the hypocycloid is the curve x%+y% = a$ (Fig. 18) when b = \a. 4. When a circle rolls inside another circle of double its diam- eter, show that every point in the circumference describes a straight line and every other point an ellipse. 5. The radius of curvature at any point of an epicycloid 4(a + b)b ad 4(a + 6)6 . 6' 2(a + b) , , nn = — -. sin — = —7- sin — = — X chord EP a + 2b 2b a + 2b 2 a + 2b and is therefore proportional to the chord EP. For, if the tangent at P make an angle with OX, $ = 6 + ^6' £LndR = ds/d (§85). 6. Show in a similar way that in the cycloid x = a(6 — sin0), y = a(l—cosd), Fig. 60. ds = PB dd, 4>-h^-¥> and hence that R = 2PB. 156.] ASSOCIATED CURVES. 171 Envelopes. 156. Let f(x,y f a)=0 represent the equation of a curve (i.e. of any plane locus, including a straight line), a being a quantity involved in the equation, but independent of x and y for its value. As a may have any value the equa- tion may be regarded as representing a family of curves. Supposing a to have a certain value in one instance, let it receive an increment J a. The two equations f(x,y,a)=0 (1), f(x,y,a + Ja)=0 (2) then represent two curves of the family. Their points of intersection approach limits of position as Aa = 0. The locus of these point-limits for all values of a is called the envelope of the family of curves. The quantity a is called a variable parameter. Fig. 100. Ex. y=a 2 x + a represents a family of straight lines. Consider the two for which a is 1 and 1+Ja respectively. The lines y = x + l, y=(l + Ja) 2 x + (l+Ja) intersect in the point ( — , ). \ 2 + A a 2 + da/ The limit of this when Ja = is ( — J, J). This is therefore one point on the envelope of the family. 172 INFINITESIMAL CALCULUS. [Ch. XXXII. 157. Equation of the envelope. The points of intersec- tion of (1) and (2), § 156, lie on the curve f(x, y,a + Ja)-f(x, y, a) . since this equation is satisfied by any simultaneous values of x and y which make f(x,y,a) and f{x,y,a + Aa) sepa- rately =0. As Ja = the limit of (3) is 3/0, y, a) da = 0, (4) the differentiation being partial since only a varies. The point-limits of the intersections of (1) and (2) therefore lie on (4), and their locus, the envelope, is obtained by eliminating (a) from (1) and (4). Ex. Equation (4) for y=a 2 x + a is 2a:r + l = 0. Eliminating a, 4:xy=— 1 The envelope is therefore a rectangular hyperbola (Fig. 100). 158. Prop. The envelope touches every curve of the family. Let u stand for f{x, y, a), and suppose (x, y) to be a point common to the envelope and curve (1), § 156. For dy/dx, the slope of the tangent of (1), we have (§ 47), fx dX+ ¥y dy = °> a being constant. We may consider (1) to be also the equation of the envelope, a being a variable, viz., that function of x and y obtained from (4) . Hence for the envelope du , du du, o ax + —ay+— da = 0. ox oy ool But du/da = from (4). Hence dy/dx at (x,y) is the same for (1) and the envelope. Ex. In the example of §§156, 157, y=x + l and the envelope 4x2/ = — 1 touch at the point (-—J-, i). 157-159.] ASSOCIATED CURVES. 173 159. The given equation may contain two ol* more vari- able parameters, subject however to other relations con- necting them, whereby all except one may be eliminated from the given equation, Ex. To show that all ellipses having the same centre and area, and their axes in the same directions, touch a pair of hyperbolas of which the axes are asymptotes. We have to find the envelope of x 2 /a 2 + y 2 /P 2 = l, where a/3 = k 2 , a constant, whence Substituting, the equation becomes x 2 /a 2 + a 2 y 2 /k 4 = l. Differentiating with regard to a, -2x 2 /a 3 + 2ay 2 /k* = 0. Eliminating a, xy = ±%k 2 , j? IG ^ the envelope (Fig. 101). Examples. 1. Two sides of a right-angled triangle are given in position and the area is constant, find the envelope of the hypotenuse. Arts. A rectangular hyperbola. 2. Particles are projected in the same vertical plane with the same velocity v, but at different elevations; show that their paths all touch the parabola 2v 2 / v 2 \ X 2_ of which the point of projection is the focus. [In other words find the envelope of y = x tan a — gx 2 /(2v 2 cos 2 a).] 3. Show that the circles described on the double ordinates of the parabola y 2 = Aax as diameters touch the equal parabola y 2 = 4ia(x + a). . 4. Find the envelope of ua 2 + va + w = 0, where u, v, w are functions of x and y. Arts, v 2 = 4tuw. , 174 INFINITESIMAL CALCULUS. [Ch. XXXII. The result is the same as the condition that the given equation should have equal roots. Explain. 5. Find the envelopes of ucos m d + v sin m d = w, (1) u sec m # - v tan m = w. (2) Arts. (1) u n + v n = w n ) , 2 where n (2) u n — v n = w n ) 2— ra' Many examples may be reduced to these* by observing that a condition of the form ( — J + (t - ) = 1 is equivalent to the two relations a = a cosr d, /? = & sin?* d y while ( — ) — ( t-) = 1 is equiva- lent to a = a sec r d, /? = b tan r d. 6. Find the envelope of a line which moves in such a way that the sum of its intercepts on the axes is constant. x v We have — +— = 1, and a +/? = k. We may substitute the value of /? and then differentiate, or we may proceed as follows: Let a = k cos 2 d, /? = /csin 2 #; the line becomes #(cos 6)~ 2 + y(sin d)- 2 = k, hence (Ex. 5) the envelope is x* + y$ = kb, & parabola touching the axes. 7. A straight line of given length k moves with its extremities on two rectangular axes, find the envelope of the line. Arts. xi + y$ = ki, a four-cusped hypocycloid. 8. Given in position the axes of an ellipse and that their sum = 2k, show that the ellipse touches the curve x$ + y$ = k$. (x\ m ( y \ m — 1 ± I— \ =1 perpendiculars are drawn to meet the axes in A and B, find the envelope of AB. (x\ n /y\ n , . m Arts. [ — ) ± — ) =1, where n= -. W \6/ m + 1 x 2 y 2 10. To the ellipse or hyperbola —±— = 1 pairs of tangents are x 2 y 2 drawn from points in the ellipse — ^+— = 1, show that the chords 159.] ASSOCIATED CURVES. 175 of contact touch the ellipse (f) '+(£)'-'• 11. When the tangents are drawn from points in the hyperbola x 2 ii 2 ~— f- = l, show that the chords of contact touch the hyperbola a 2 b 2 (?)-(f)-i. 12. The e volute of a curve may be considered to be the en- velope of its normals ; find in this way the evolute of an ellipse. cl 2 x b 2 y The normal at {a /?) is = a 2 — b 2 , a (j or, writing a cos 6 for a, and b sin 6 for /?, x . a(cos 0)- —y.b (sin 0)- 1 = a 2 — b 2 , the envelope of which is (Ex. 5), (ax)i + (by)$ = (a 2 -b 2 )i, which is therefore the evolute (cf. § 89). 13. Show in a similar way that the evolute of the hyperbola is (ax)t-(by)$=(a 2 + b')i. 14. Parallel rays of light are reflected from the circumference of a circle. Find the envelope of the reflected rays. Take the centre for origin and the x-axis parallel to the incident rays. The equation of the ray reflected from the point (a cos 0, a sin 0) is x sin 20 — y cos 20 — a sin = 0, whence the envelope is x=-{a(3 cos — cos 30), ?/ = |a(3 sin 0-sin 30), an epicycloid formed by a circle of radius \a on a circle of radius £a. CHAPTER XXXIII. CENTRES OF GRAVITY. 160. In finding the coordinates x, y of the centre of gravity of a body, we 'suppose the body to be divided into parts of weights W\, w 2 - . -, and of which the centres of gravity are the points (xi, yi), (x 2 , y 2 ), • • • , then equate the sum of the moments of the weights to the moment of the sum of the weights if placed at the centre of gravity. Thus supposing gravity perpendicular to the z-axis we have WiXi+W 2 X^ + . . . = (wi+w 2 + . - - )$y -__WiXi~\-W2X 2 -\-. . ._IWX W1+W2 + . . . " Iw ' Similarly supposing the body and the axes placed so that gravity is perpendicular to the y-axis, we have - = wiyi+w 2 y2 + - • - __ 2wy W\ + w 2 + . . . Iw ' These formulae also hold when the points are not in one plane, there being also a third coordinate, ~z = Iwz/Iw. If the parts referred to are infinitesimal the sign of inte- gration replaces that of summation to indicate the limit of a sum. 161. The division into parts and the limits of the sum- mation in the following cases are the same as if we were about to calculate an area, volume, or length. The bodies are assumed to be homogeneous (of uniform density) and 176 160-1G4J CENTRES OF GRAVITY. 177 hence weight is proportional to volume. For such bodies the centre of gravity is also known as the centroid. 162. An area. To find the e.g. of a thin plate or lamina * of the form ABDC, Fig. 61, we have the element of area = y dx; element of w T eight = w . y dx, where w = weight per unit area; e.g. of element at (x+i, \y), where i is infinitesimal; hence element of moment = wy dx . x when gravity is perpendicular to the x-axis, smd = wy dx . \y when gravity is perpendicular to the y-axis. Dividing the sum-limit of the moments by that of the weights (§ 160), xy dx y 2 dx x = V = h II dx y dx when w (which is assumed to be constant) is cancelled. ^It will be noticed that the denominator = the area. 163. A solid of revolution about OX. Element of vol- ume = xy 2 dx, of weight = w . ~y 2 dx, w being the weight per unit volume, element of moment = w . ~y 2 dx . x, . . *v — xy 2 dx = 0. y 2 dx The denominator = volume/7r. 164. An arc. Proceeding as above we have for the e.g. of a material line in the form of the curve CD,t x ds yds x y ds ds * Results for a lamina are limits for a uniform and infinitesimal thickness. f The results are limits for a body of uniform and infinitesimal cross-section. 178 INFINITESIMAL CALCULUS. [Ch. XXXIII. 165. A surface of revolution about OX. For a curved surface of this form and of infinitesimal thickness, x = xy ds y ds y = 0. 166. An area in polar coordinates. Element of area = \r 2 dd (§ 147) ; its e.g. is distant §r + i from the origin;* hence if the initial line is taken as a>axis, x = • w . 1 ■r 2 dd | -r cos 6 • r 3 ccs dd _ 2- 1 r 3 r • w . \r 2 dd a r 2 dd r 3 sin 6 dd V-i r r 2 dd and 167. The subject may also be considered from the point of view of geometry only. Let a be an area-element or volume-element which is infinitesimal in every direction, and which contains a point (x, y, z). Then the limits of lax lay laz la ' la ' la are the same as the x, y,z of § 160 for homogeneous bodies, and are the coordinates of a point which is called the cen- troid (or centre of gravity) of the area or volume. Or, a may be taken as a mass-element, in which case the point is called the centre of mass (or centre of gravity) of the body. 168. Pappus's (or Guldin's) properties of the centre of gravity. From § 164 we have yds = y J both sides by 2tt, ds, and multiplying * The e.g. of a triangle is assumed to be the point of intersection of the medians. 165-168.] CENTRES OF GRAVITY. 179 \2ny.ds=l ds) .2riy. (1) Similarly from § 163, ny 2 dx = ydx). 2ny. (2) These results are equivalent to the following statements, which are known as Pappus's or Guldin's Properties: (1) The surface of a solid of revolution is equal to the length of the revolving curve multiplied by the length of the path of the e.g. of the curve (i.e. of the arc), (2) The volume of a solid of revolution is equal to the revolving area multiplied by the length of the path of the e.g. of this area. N.B. The axis of revolution may touch but not cut the curve. Examples. 1. The parabolic area OAB, Fig. 74. Arts. x = %x iy y = iy 1 . Of the solid of revolution round OX, x = fa: 1 . 2. The quadrant of an ellipse. Ans. £ = — , y~= a. Ans. 4:7t 2 ab, 2n 2 a 2 b. 15. Find by Pappus's Properties the e.g. of the arc of a semi- circle and that of the area of a semicircle. CHAPTER XXXIV. MOMENTS OF INERTIA. 169. The Moment of Inertia is a quantity which is often required in connection with the motion of a body about an axis. The following is an illustration. 170. Kinetic energy of rotation. Let it be required to find the kinetic energy which a body possesses on account of its rotation about an axis. Let the perpendicular distance of a particle of mass mi from the axis be r x and let aj = the angular velocity of the body about the axis. Then the kinetic energy of the par- ticle = J (mass) X (linear velocity) 2 ^ \m± (a>ri) 2 = %co 2 miri 2 , and the whole kinetic energy of the body, = ^(rai7Y* + ra 2 r2 2 + . . . ) = i^ 2 /, where / = mir^ 2 + m 2 r 2 2 + . . . The quantity / is called the moment of inertia of the body about, or with reference to, the axis; hence the fol- lowing definition: The Moment 0} Inertia of a body about an axis is the sum of the products obtained by multiplying the mass of each particle of the body by the square of its distance from the axis. 181 182 INFINITESIMAL CALCULUS. [Ch. XXXIV. Since the particles of a body are infinitesimal portions of the body, the moment of inertia is obtained as follows: Imagine the body to be divided into parts which are infini- tesimal in every direction, and find the limit of the sum of the products of the mass of each part by the square of the distance of some point in it from the given axis. Since both factors of the product mr 2 are essentially + , the moment of inertia is always + , and the moment of inertia of a body about any axis is always equal to the arith- metical sum of the moments of inertia of its parts about that axis. 171. Prop. The m.i. of a body about any axis = them.i. about a parallel axis through the centre of gravity + MH 2 , where M is the mass of the body and h is the distance between the parallel axes. Take at a point P in the body a particle of mass m\. Let a plane through P perpendicular to the axes in question meet the one through the centre of gravity G in G' and the parallel one in H' , then G'H' = h. Draw PK perpendicular to G'E f and let G'K = x 1 . Then s 1 2 =r 1 2 +h 2 -2hx 1 (Euc. II. 13), .*. miSi 2 = m 1 ri 2 +mih 2 —2hmiXi. Similarly for particles ra 2 , m 3 , etc. .'.m 1 s 1 2 + ra 2 s 2 2 + . . . = (m 1 r 1 2 + m 2 r 2 2 + . . .) + h 2 (m 1 +m 2 + . . .) — 2h(miXi+m 2 x 2 + . . . ). The left-hand side = the m.i. about the axis through H; and of the three terms on the right, the first = the m.i. about the parallel axis through the centre of gravity, the second = 171-174.} MOMENTS OF INERTIA. 183 Mh 2 , and the third = (§ 160), since the centre of gravity is in the line from which xi, x 2 , . . .,are measured. 172. The proposition just proved is true for all bodies, but the following applies only to laminae. Prop. Let X'X, Y'Y be two lines in the plane of a lamina and meeting at right angles in 0, and let Z'Z be a line through perpendicular to the plane. Let 7i = the m.i. of the lamina about X'X, 7 2 = that about Y'Y, /-that about Z'Z) then iWi+/ 2 . For ri 2 = Xi 2 + yi 2 , .'. miri 2 = miXi 2 + m,iyi 2 . Fig. 103. The proposition is therefore true for a particle at P, and hence it is true for all the particles of the lamina. 173. When the m.i. is put into the form Mk 2 (M being the mass), k is called the radius of gyration; hence the radius of gyration of a body with reference to an axis is the dis- tance from the axis of a point at which a particle having the same mass as that of the body may be placed so that its m.i. may be the same as that of the body. If k = the radius of gyration with reference to an axis passing through the centre of gravity, and ki that about a parallel axis at a distance h, we have ki 2 = k 2 + h 2 , since (§171) Mk 1 2 =Mk 2 +Mh 2 . 174. In the following examples the density, i.e., the mass per unit volume, is represented by fi, and the bodies are assumed to be homogeneous, i.e., of uniform density, unless the contrary is specified. 184 INFINITESIMAL CALCULUS. [Ch. XXXIV. Ex. 1. To find the m.i. of a rectangular lamina whose sides are a, a, b, b, about an axis bisecting the sides a, a, Fig. 104. Divide the rectangle into parallel strips of length b and width dx, and measure x from the axis. The elements are as follows : area = b dx, volume = t . b dx, where t = the thickness of the lamina, mass = fi.tb dx, m.i. = ptb dx . x 2 , since every particle is at a distance x + i from the axis, i being infinitesimal. Integrating between and \a and doubling we have for the m.i. of the whole rectangle o x -a- Fig. 104. |*2 a 2 2 irtbx 2 dx = Tzfitba? = (v-tab)-— Jo 12 The quantity in parentheses is the whole mass ( = fx X volume), a' a .'. the m.i. =m-— , and hence the radius of gyration = ~7Yo' Simi- . . . . . b 2 larly the m i. about the axis bisecting the sides b, b is m — . 1.JU 2. The m.i. of the rectangle about a normal axis through the a 2 + b 2 intersection of the two axes of Ex. 1 is § (172) m — — — The same formula is true for any parallelogram (of which a and b are adjacent sides) about an axis drawn as in this case through the intersection of the diagonals at right angles to the plane. 3. The m.i. of the rectangle about a side b is (§ 171), a' m — +m . 12 \2 &- a' m 4. The m.i. of the rectangle about a normal axis through one angle = a- i-b' m- Fig. 105. 5. Any triangular lamina (Fig. 105) about one side BC. Let BC = a, the perpendicular OA = h. Then DE:BC::FA:OA, 174.] MOMENTS OF INERTIA. 185 .*. DE/a=(h-x)/h, .\ DE=(h-x)a/h. . . m.i. I"* a h 2 n(h — x)— . t . dx . x 2 = m—. oh o 6. A circular lamina of radius r about a normal axis through the centre. Consider the annulus between the concentric circles of radii x and x + dx. The elements are : area = 2nx . dx, mass = trt . 2nx dx, m.i. = pi . 27r# <£c . x 2 , since every particle of the annulus is at a distance x+i from the axis, i being infinitesimal, ,\ whole m.i = r r 2 2/intx 3 dx = hpxtr* = m~- o 2 7. A circular lamina about a diameter. Let the required m.i. =/. The sum of the moments of inertia about two diameters at right angles to each other = 27; it also 2 2 (by § 172 and Ex. 6)=ra- ,\ l = m-~. A 4 8. A circular lamina about a normal axis through the centre when the density is supposed to vary inversely as the distance from the centre. Let n = k/x, where k is a constant. Then m.i. -m ■ — ) . 2ntx z dx = %7ztkr 3 . But the mass m =\ { — ) . 2nx dx . t = 2xktr. -m ■ r 2 .". m.i. =ra— . o x 2 y 2 9. An ellipse — +— = 1 about its minor axis. a 2 b 2 The m.i. =4 a H . y dx .t . x 2 . Substitute y from the equation of o a 2 the curve and let x=*a sin 6. The result is m-. 4 186 INFINITESIMAL CALCULUS. [Ch. XXXIV. b 2 Similarly about the major axis the m.i. is m-r-. 10. A sphere about a diameter. Take the diameter as x-axis and the centre as origin. Consider the sphere to be made up of laminae perpendicular to the axis, and of thickness dx Then m.i. (see Ex. 6) r y 2 = 2 Jul . xy 2 . dx . ~2 and y 2 = r 2 —x 2 , whence the m.i. =m|r 2 . 11. A right circular cylinder of radius r about its geometrical axis. The cylinder may be considered as made up of circular laminae r 2 perpendicular to the axis, hence (Ex. 6) the m.i. =ra— . Similarly for a cube, a right prism, etc., about an edge or any parallel axis. 12. A right circular cylinder of radius r and length I about an axis bisecting at right angles the geometrical axis. As before, suppose the cylinder to be made up of circular laminae. The mass of the lamina at a distance x from the axis = fi . nr 2 dx, and its m.i. about a diameter in its own plane and r 2 parallel to the given axis = mass X— (Ex. 7), .*. its m.i. about (r 2 X the given axis = mass! — +x 2 J (§ 171); .\ whole m.i. = 2 ft 1 (r 2 \ (r 2 l 2 \ 175. As in Ch. XXXIII, the subject may be considered from the point of view of geometry alone. If a is an area (or volume) element which is infinitesimal in every direc- tion, and r the perpendicular distance of some point in it from a straight line, the limit of lar 2 is called the moment of inertia of the area (or volume) with reference to the straight line. The results are the same as those calculated for homo- geneous bodies, area (or volume) taking the place of mass. 175.] MOMENTS OF INERTIA. 187 Examples. Find the moment of inertia of 1. A triangle about (1) an axis through the centre of gravity parallel to the base, (2) a parallel axis through the vertex. h 2 h 2 Arts. (1) w— , (2) m-. a 2 b 2 2. A rectangle (a by 6) about a diagonal. Ans. m 6(a z +cr) 3. An isosceles triangle about a normal axis through the middle point of the base. . 4 alt. 2 + base 2 Ans. m — . 24 4. An isosceles triangle about a normal axis through the vertex. 12 alt. 2 + base 2 Ans. m- . 24 5. A circular annulus of radii R, r about a normal axis through the centre. A R 2 +T 2 'Ans. m . 2 R 2 +T 2 6. A circular annulus about a diameter. Ans. m . 4 7. A circle about a tangent. Ans. m^r 2 . 8. A circular arc of length s, radius r, and chord c, about an axis through its middle point perpendicular to its plane. Ans. m . 2r 2 ( 1 ). 9. A circle about a normal axis through a point in the cir- cumference. Ans. mlr 2 . v 2 10. A parabolic area, Fig. 74, about the z-axis. Ans. m— 5 11. The same about the y-axis. Ans. nijx 2 . 12. A spherical shell of infinitesimal thickness about a diameter. Ans. m^r 2 . 13. A right circular cone of radius r and altitude h about its geometrical axis. Ans. m-f^r 2 . 14. The same about an axis through the vertex perpendicular to the geometrical axis. Ans. 12h 2 +3r 2 m 20 188 INFINITESIMAL CALCULUS. [Ch. XXXIV. 15. The same about an axis through the centre of gravity per- pendicular to the geometrical axis. Sh 2 + 12r Arts, m — — . 16. An oblate spheroid about its geometrical axis. Arts. m%a 2 17 Any area ABDC (Fig. 61) about the z-axis. m Ans. — o ft dx > dx 18. The hypocycloid xs+y* = a? (Fig. 18) about the x-axis. Ans. m-Q^a 2 CHAPTER XXXV. SUCCESSIVE INTEGRATION. 176. Successive integration. Let T be a function of u, v, and Wj and suppose the following operation to be per- formed: T dw is integrated between the limits w\ and w 2 , u and v being treated as constants; the result is mul- tiplied by dv and integrated between v\ and v 2 , u being treated as a constant; the result is multiplied by du and integrated between u\ and u 2 . The whole operation is indicated by the notation T du dv dw. U X J V\J Wi The limits of w may be functions of u and v, the limits of v may be functions of u, but the limits of u are constants. Instead of three variables there may be two or four, or more. ! Ex. 1. 2. X 1, 3 Oj x 2 y dx dy = 9 X x 2 y dy) dx = C2 %x 4 dx = 3tV. 3. 4. 5. 1. •2 Ji- Jo. fz+2/ r3 rx (x — y)dx dy dz=\ (x 2 —y 2 )dx dy J0J0 = [ %x*dx = 13h Jo 6x 2 y dx dy = S5 •2x . Cad Cx+y dx dy dz = 9 J. r dd dr = in 3 a 2 . 189 190 INFINITESIMAL, CALCULUS. [Ch. XXXV. 6. C2t: , 2 J 2a cos 6 r 3 sin d cos dd dr = $7ia i . ' Applications of Successive Integration. 177. Plane area. Rectangular coordinates. Let P be the point (x, y), and let PR, PS be dx, dy, infinitesimal incre- ments of x and y. The rectangle PQ = dx dy may be taken as an element of area. To illustrate the method of finding the limit of the sum of such elements by succes- sive integration, let it be re- quired to find the area of the figure KM, which is bounded by the lines x = a, x = b, and the two curves KL or y = J(x) and NM or y = F{x). Fig. 106. (1) ( dy)dx=EG.dx, (2) V CE ' EG . dx=KM. a The first result is the sum of such rectangles as PQ, which make up the rectangle EH, which is equivalent (§ 15) to the strip EIJG of the given figure, the second is the sum of such strips for the whole area. The final result is the limit of the sum of such rectangles as PQ when both dx and dy = 0. The whole operation is indicated by the " double integral" '2/2 dx dy or Vi a F(x) fix) dx dy, yi and y 2 being the y's of the curves y = f(x), y = F(x). More generally, let u be a function of x or y, or of both x and y. The limit of the sum of such products as u dx dy taken for all parts of the area KM is CV2 u dx dy. J a J 2/1 177, 178.] SUCCESSIVE INTEGRATION. 191 Ex. 1. Find £1 y dx dy for a quadrant of the circle x 2 + y 2 = a' We are to obtain fa CVi J y dx dy, where y x is the y of x 2 + y 2 = a 2 . o J y\ y dx dy = ( y dy) dx = \l ' hji 2 dx = i(a 2 —x 2 )dx = ^a 3 . o This is the moment of the area with reference to the x-axis. 2. Find £1 y 2 dx dy for the same area. J V a 2-x 2 o y 2 dx dy = TG7za i . This is the second moment, or moment of inertia, of the area with reference to the x-axis. 3. Find £1 xy dx dy for the same area. Arts. |a 4 . This is the product of inertia of the area with reference to the axes. 4. Show that the volume of a solid of revolution about the x-axis is 2tt y dx dy. and deduce the formula of § 95. 178. Plane area. Polar coordinates. Let P be the point (0, r), and let POR = dd, PS = dr. With centre describe the arcs PR, SQ. Then the element of area PQ = ±(r + dr) 2 dd-ir 2 dd = rdddr + %dr 2 dd, the last term being a higher infinitesimal. .'. £1 PQ=£Irdddr. Let KLMN be a figure bounded by the lines 6 = a, 5 = /?, and the £ curves KL or r = }(0) andiVMcr Fia, 107. 192 INFINITESIMAL CALCULUS. [Ch. XXXV. r = F(d). Proceeding as in § 177 we find the area 'r 2 r dO dr, or r/3 a J n rF(e) aj Kd) r dd dr, r\ and r 2 being the r's of the given curves. The first integral %(r 2 2 —r 1 2 )dd = EGJF, Which is equivalent to the sectorial strip EGHI of the given figure; the second is the sum of such strips for the whole area. The final result is the limit of the sum of such figures as PQ when both dd and dr = 0. Observe that the element r dd dr=PR . PS as in Fig. 106. Ex. 1. The area (Fig. 108) between the circles r = 2acos#, r = 2b cos 0, (b>a)j is 2 26 cos 6 rdddr = 7t(b 2 -a 2 ) 2a cos d 2. Find the area (Fig. 109) bounded by tl\e curves d = r 3 i-r i d = r z — r r = L Fig. 108 Fig. 109. [Integrate first with regard to 0.] Arts. § . 3. The moment of inertia of the circle r = 2acos# with refer- ence to a normal axis through the polar origin = 9 ' 2 T2a cos 6 Jo r 3 d0 dr = %7ia 4 . 4. Find the moment of inertia of the lemniscate r 2 = a 2 cos2# (Fig. 27) with reference to a normal axis through the origin. Ans. jxa*. 179.] SUCCESSIVE INTEGRATION. 193 5. The potential of a particle of mass m at distance R being m/R, show that the potential of a lamina (thickness t, density /*) at a point B on the perpendicular to the lamina through the polar origin (OB = c) is 'C/trdOdr J J ^/c 2 +r 2 ' (1) Find the potential at B if the lamina is the circle r = a. Ans. 2nnt(Vc 2 + a 2 — c). (2) Find the potential of a circular lamina at a point in the circumference. Ans. \airt. 6. If a curve revolve about the initial line show that the volume -*■/. r 2 sin 6 dd dr. Find this volume for a complete revolution of the cardioid r = a cos 2 ^#, Fig. 89. Ans. \na*. 7. Find the moment of inertia of an anchor-ring (see Ex. 14, p. 180) about its axis. Take as origin the centre of the circle (radius a) to be revolved, and the perpendicular (length b) on the axis of revolution as initial line. The m.i = 2 o, a p. ,2n(b — r cos 0) .rdddr t (b — r cos d) 2 Ans. m(|a 2 + 6 2 ). 179. Volume of a solid. Rectangular coordinates. Let P be the point (x,y,z) and let PR, PS, PT be dx, dy, dz, infinitesi- mal increments of x, y, and z. The parallelepiped PQ = dx dy dz is taken as the element of volume, the limit of the sum of such elements being obtained by successive integration. Thus to find the volume of the solid, Fig. 110, bounded by the coordinate planes and a surface whose equation is given: ([ lv \ (1) ( dz) dx dy=IU. dx dy, \ ' (2) ( IV .dy)dx=VDG.dx, rOA (3) VDG .dx = COBA, Jo the required volume. 194 INFINITESIMAL CALCULUS. [Ch. XXXV. The first result is equivalent to the column standing on IK( = dx dy), the second to the slice between VDG and WEF and therefore of thickness dx, the third is the sum of such slices. Fig 110. The whole operation is indicated by the " triple integral" a f 2/1 J 0J J '21 dx dy dz, where Z\ is the z of the given surface, y\ is the y of the curve AGB in which the surface is cut by the plane z = 0, and a = OA. Ex. 1. To find the volume of the ellipsoid — ++— = 1. IV- V o. 2/1 J a 2 ' b 2 c 2 X s y 2 z 2 z x ^ X' ■ y z dx dy dz, where z x is the z of — 2 + T2+1 = 1, and o 2/iistheT/of — 2 +— = 1 ; i.e 6 c 2/,=-(a 2 -z 2 )* and z^-riyS-y 2 )*. 179.] SUCCESSIVE INTEGRATION. 195 Since •zi dz = z o • • 8 v ~ a J Vi #! dx dy = o a \V\\ \T(yi 2 -y 2 )My]dx , o Jo La J a izbc o 4a : (a 2 — x 2 )dx = nabc 6 -7T a6c. «i cfc cfa/ is equivalent to a column, - 2/i 2 dx to a slice, and is 46 6 the sum of the slices for an octant. 2. Find £I^(y 2 + z 2 )dx dy dz for an ellipsoid, i.e., the moment of inertia with reference to the #-axis. 8 o, '2/1 Zl [i{y 2 + z 2 )dx dy dz = &fJL o a '2/1 (2/ 2 £i +isi 8 )dz dy -£»■+•■> y A 4 dx = i%7zabcii(b 2 + c 2 )=m b 2 + c< o 3. Find the volume of the hyperbolic paraboloid az = x 2 — y 2 in the first octant, # varying from to h. x 2 —y 2 1 a 4 rfc Cx 0, a cfo dy dz = . 6 a 4. Find the volume of x 2 — y 2 = x 2 z 2 in the first octant, x varying from to h. Arts. \nh 2 5. In Ex. 3 find the moment of inertia with reference to the z-axis. Arts. m\ln 2 6. A rectangular parallelepiped which has its base in the xy- plane and its sides parallel to the other coordinate planes, intersects the hyperbolic paraboloid az = xy. Show that the enclosed volume = base X mean length of the vertical edges. 7. Find £1 xy dx dy dz and £1 xyz dx dy dz for the octant of an ellipsoid. Arts. (1) a 2 b 2 c (2) a 2 b 2 c' 1.3.5' v_/ 2.4.6 8. Find the volume enclosed in the first octant by the ccor- (x \ i / v \ i / z \ % — ) + I ,) ■+ ( — ) =1. Ans. abc 196 INFINITESIMAL CALCULUS. [Ch. XXXV. 180. Volumes. Polar coordinates. Let be the origin, OA the initial line, BAOC the initial plane. A plane re- volving through the angle about OA from the initial plane contains a point P, OP = r mak- ing an angle 6 with OA. The polar coordinates of P are <£, 0, r; r and 6 are the ordinary polar coordinates of plane geometry, and (§ 178) PM = r dd dr + Ii, where l\ is a higher infinitesimal. The increment dcj) brings P to T and PM to TQ, PT = PAd = r am d. Hence the element of volume PQ Fig. 111. = (r dd dr + /i)(r sin d + I 2 ) = r 2 sin d0 dfl dr + / 3 . /. 7 r 2 sin d<£ d# dr. (The element is, as in Fig. 110, PR. PS . P7 7 .*) Ex. 1. Volume of a sphere. (1) The origin being the centre, '2tt 7 = o J •2k r 2 sin dd = 2n fa 3 d = %7za 3 . The first integral ^a 3 sin 6 d dd = a, pyramid with vertex at the centre and base on the surface of the sphere, the second fa 3 d<£ = a wedge of which the angle is d$ and the edge a diameter of the sphere, the third f-;ra 3 =the sum of the wedges. (2) If the origin is on the surface, and the initial line a diameter, 7 2tt , 2a cos0 r 2 sin 6 d$ dd dr. ISO.] SUCCESSIVE INTEGRATION. 197 (3) If the origin is a point on the surface, the initial line a tan- gent, and the initial plane passes through the centre, 7 = 4 7C 7Z '2 "2a sin 6 cos d 0. r 2 sin 6 d dO dr. o 2. The vertex of a cone of vertical angle 2 a is on the surface of a sphere of radius a, and its axis passes through the centre of the sphere. Find the common volume. Arts. |-7ra 3 (l — cos 4 a). 3. The moment of inertia of a sphere about a diameter C2i J „ ixr 4 sin 3 # d dd dr=m fa 2 , and about a tangent line = 4 °~2 7C "2 • . J 2a sin cos jjtr 4 sin 3 # d<£ dd dr = m ia 2 . o 4. The potential of a solid sphere (density p) at a point on the surface = f2w J v J 2a cos 772- jj.r sin d<£ d# dr = f ^7ra 2 =~. o a 5. To find the potential 7 of a spherical shell of infinitesimal thickness (radius r, thickness t, density //) at any point A. Take 0, the centre of the sphere, as origin, OA as initial line, and let 0A = c. Then 7 = 7= r27r > jj.r 2 sin d ddd .t o 2nfj.rt. and 2nprt oV / c 2 + r 2 — 2cr cos [(c + r)-(c-r)], c>r, , where r is constant. and [(r + c) — (r— c)], c dr. Then V= zr d dr o ia 3 d = %na z . The first inetgral \a z d is a wedge whose edge is OZ and angle d, the second is the sum of such wedges for a hemisphere. 2. The axis of a right circular cylinder of radius b passes through the centre of a sphere of radius a(> b). Find the common volume. •2n Cb Va 2 -r 2 rd<}>dr = %n[a 3 - (a 2 - b 2 )*\ 181.] SUCCESSIVE INTEGRATION. 199 3. A right circular cylinder of diameter a penetrates a sphere of radius a, the centre of the sphere being on the surface of the cylinder. Find the common volume. TZ f] Jo J 'a cos Va 2 — r 2 rddr = l (37r — 4)a 3 . 4. The axes of two equal right circular cylinders of radius a intersect at right angles. Find the common volume. 8 71 J "2 /l — COS 3 ^ Va 2 — r 2 sin 2 0r d =-V 6 a 3 . 5. The volume between the surface z = e~( x2 + ^ 2 ) and its asymp- totic plane 2 = 0. This is a surface of revolution about the 2-axis zrddr, and x 2 + y 2 = r 2 . (1) F= • •2tt . roo zr d ^ .-. V = 2tt i (2) Also V = 4 • 00 n 00 L J |« 00 |«00 J V oo 2 dx dy e~y 2 dx dy = 4 '00 roo 6-2/ 2 C?2/ • /•OO er x2 dx* = 4: o 6~ x2 C?X ) . JO .\ From (1) and (2), '00 e~ x2 dx 1Z (Cf. § 124.) o * If the integration limits of each variable are independent of the other we evidently have J Jf(x)F{y)dxdy = Jf(x)dx JF{y)dy. 200 INFINITESIMAL CALCULUS. [Ch. XXXV. 182. Area of any surface. Let the parallelepiped. Fig. 42, which has the base dx dy in the #?/-plane and its sides parallel to the 2-axis, intersect the tangent plane at P{x, y, z) and the surface in sections of area A t and A s , respectively, which are assumed to be equivalent infinitesimals. Let a, /?, y be the direction angles of the normal at P. Then dxdy = A t cos y, .". A s = sec y dx dy + 1, where / is a higher infinitesimal. Hence the surface S = sec y dx dy= sec a dy dz = sec /? dz dx. uu uvl 3i^ cos a y cos B, cos y are proportional (§ 63) to ^— , ^-, ^— if r ' 9z ay 3-2 3z m=c is the equation of the surface, and hence to 1. — ^— , do; 3z if the equation is in the form z = f(x, y). Hence dy sec y or sKlt /3u\ 2 /9w\ 2 A9J/ 3m dz .U/9 Z\ 2 /32' \ 2 % Ex. 1 A right circular cylinder of diameter a penetrates a sphere of radius a, the centre of the sphere being on the surface of the cylinder. Find the surface of the sphere which is inside the cylinder. Let the equations be x 2 +y 2 +z 2 = a 2 (1), x 2 +y 2 = ax (2). For (1) sec r = ~ = / =:- z Va 2 -x 2 -y 2 182.] SUCCESSIVE INTEGRATION. 201 .\ S = 4 PT JoJo a Cv'ax-x* a dx dy Va 2 = 4a (x + a) sin -1 , x'-y 'a+x - = 4a sin -1 * I — ; — dx < 2 Jo \a+x * as a = 2(^-2)a 2 . -»o 2. In Ex. 1 find the volume of the cylinder which is inside the sphere. ^a a For (2), sec/? = — = ■ — . — , and (1) and (2) intersect in V 2Vax—x 2 z 2 = a 2 — ax. .\ £ = 4 Wa 2 -ax a dx dz J 2\/ax — x 2 = 2a a \2l «\x dx = 4a 2 . 3. The axes of two equal right circular cylinders of radius a intersect at right angles. Find the cylindrical surface enclosed. Let the equations be x 2 + z 2 = a 2 } y 2 + z 2 = a 2 . Arts. 16a 2 . *Lets/(a+aO=sin 2 0. ™^ CHAPTER XXXVI. MEAN VALUES. 183. Let the base b— a of the curve, Fig. 113, be divided into n equal parts, at the extremities of which ordinates 2/i? V2, • • « are drawn. The limit of 3/1+2/2 + - - , +Vn n for n infinite is called the mean value of y for the in- terval a to b of x. If dx = the length of the equal segments of b— a, n=(b—a)/dx. Hence Fig. 113." the mean value of y is rb £Zy dx y dx a a a) Since the numerator = the area A BCD, the mean ordinate is equal to the height of a rectangle which has the same base and area as the given figure. The result (1) may be regarded as the mean value, for the interval a to b of the variable, of any function which is single- valued and continuous for that interval. Ex. 1 The mean ordinate of a semicircle of radius a = ^-— =- 2a 4 = • 7854a. 202 183, 184.] MEAN VALUES. 203 2. The mean square of the ordinate of a semicircle a y 2 dx a (a 2 — x 2 )dx — a _J —a _ 2 n 2 2a 2a 3. Find the mean ordinate and the mean square of the ordinate of the curve y = a sin nx from x = to x = n. Arts. (1) 2a/ri7: } (2) a 2 /2n. 4. The arc of a semicircle is divided into equal parts, from the extremities of which perpendiculars are drawn to the diameter. What is the mean value of their length? Radii through the points of section divide the angle at the centre into equal parts. Hence . n re a £ la sin 6-7- — = — dd 7C sin dd = — . ?! 5. In Ex. 4 find the mean distance of the points in the cir- cumference from one end of the diameter. Arts. ^a/n. 6. A straight line A B of length a is divided into equal parts, P being a point of section. Find the mean value of AP . PB for all positions of P. Arts. \a 2 . 7. The northern hemisphere is divided into zones of equal area. Find their mean distance from the pole. Ans. Arc = radius. 8. A rectangle is divided into rectangles by lines which divide the sides equally. Find the mean square of the distance of the rectangles from one corner of the given rectangle. The sides at that corner being axes, ab 1 C a t h £ I ^ 2 + y 2 )^-^^ y = - b I (x 2 + y 2 )dxdy = Ua 2 + b 2 ). 184. Let dX be an element of any quantity (length, area, volume, mass, time, etc.), and u a variable which is taken any number m times per unit of X. Then £1 um dX £1 u dX £1 m dX = £1 dX 204 INFINITESIMAL CALCULUS. [Ch. XXXVI. expresses the limit of the sum of the u's divided by their number, and is thus the mean value of u for the range in- volved in the summation. If the elements are unequal the result is still the same as the mean value of u taken once for each of the elements if they were equal, since 1 for each one-mth part of the unit is equivalent to m per unit. Ex 1. To find the mean distance of points within a circle of radius a from a given point on the circumference. In this case eU is an element of area, say r dd dr (§178), u is r, hence the mean value 2" f 2 Jo 2a cos r 2 d6 dr 32 - =— a. ■KO? 971 2. The plane base of a hemisphere of radius a is horizontal. Find (1) the mean height of points within the hemisphere (the element being one of volume), (2) the mean height of points on the curved surface, (3) the mean depth of points in the base below the curved surface. Ans. (1) fa, (2) \a, (3) fa. In (1) and (2) the mean height is the height of the centre of gravity,* in (3) it is volume/base. 3. Find the mean square of the distance of points within a sphere of radius a from (1) the centre, (2) a point on the surface, (3) a diameter. Ans. (1) fa 2 , (2) fa 2 , (3) fa 2 . * The point whose coordinates are the mean values of the rec- tangular coordinates of points in a body for equal elements of mass is easily seen to be the centre of mass (or of gravity) and therefore the centroid if the body is homogeneous. The centroid is sometimes called the centre of mean position. CHAPTER XXXVII. INTRINSIC EQUATION OF A CURVE. THE TRACTRIX THE CATENARY. Intrinsic Equation of a Curve. 185. Let the tangent or normal of a curve turn through an angle X while the point of contact moves a distance s along the curve. The equation con- necting s and X is called the intrinsic equation of the curve. Ex. 1. The intrinsic equation of a circle of radius a is obviously s = aX. 2. To find the intrinsic equation of the semi-cubical parabola ay 2 = x 3 (Fig. 29), the Fig. 114. intrinsic origin being the cusp. The tangent at the origin is the z-axis. Hence , d y 3 / x \ i ax 2 \a/ .*. x=$ata,n 2 X, whence y = ?\a tan 3 X. Also ds = Vdx 2 + dy 2 = fa sec 3 A tan X dX. /. s = |a sec 3 A tan X dX, or s^-fca (sec 3 X — l), Jo the equation required. 3. Find the intrinsic equation of the common parabola y 2 = 4ax, the origin being the vertex If the tangent make an angle X with the positive direction of the ?/-axis, tan X = dx/dy. Ans. s = a sec X tan X + a\og (sec A + tan X). 205 206 INFINITESIMAL CALCULUS. [Ch. XXXVII. 4. Find the intrinsic equation of the cycloid x = a(0 — sin 6), y = a(l — cos 6) (Fig. 19), the origin being at a cusp (say, at x = 0, 2/ = 0). [Show that X( = YSP) = id.] Arts, s = 4a(l - cos X.) 5. Find the intrinsic equation of the four-eusped hypocycloid x = asin 3 d, y = a cos 3 # (Fig. 18), the origin being at a cusp (say at x = 0, y = a). [Show that X( = OST) = 6.] Arts. s = fa(l -cos 2X). 6. Show that an equation may be transformed to a new origin at (s f , X') by substituting s + s' for s and X + X' for X m the given equation. 7. Find the equations of the curves of Exs. 4 and 5 when the origin is a vertex (the middle point of the arc between successive 3usps). Ans. (l)s = 4asin^, (2) s = %a sin 2X. 186. Instead of expressing x and y in terms of X, we may be able to express s and X in terms of x or some other variable, and eliminate that variable. Ex.1. The catenary y = a cosh (x/a) (Fig. 117), the origin being the lowest point, and hence the initial tangent being parallel to the a:-axis. tan X = dy/dx = sinh. (x/a), and c?s = cosh (x/a) dx, whence s = a sinh (x/a). / . s = atan^. 2. The cardioid r = a(l— cos 6), the cusp being the polar and the intrinsic origin. ds = Vr 2 d6 2 + dr 2 = 2a sin £0 dd, .*. s^=4a(l-cos $0). Also X == 6 + , where ( § 136) tan cp = r dd/dr = tan \d. :. A=f0. Hence s = 4a(l — cos \X). Show that the equation is s = 4a sin \X if the origin is the point most remote from the cusp. 3. Show that the intrinsic equation of an epicycloid is 4b(a + b) / aX a when the origin is a cusp, and / a/ \ 1 -cos — — ) , V a + 26/' 4b(a + b) . aX s = — — - sin a a + 2b' when the origin is a vertex. 186-189.] THE TRACTRIX. 207 187. The radius of curvature. The curvature is the s-rate of a (§ 86) and hence R = ds/dX = f(X) if s = f(X) is the equation of the curve. 188. The evolute. Let be the origin of the given curve s = f(X), P any other point on the curve, C, Q the centres of curvature of and P, and let C be taken as the origin of the evolute. Then PQ = /'(A) and OC = /'(0), also CQ = PQ- OC (§ 90 (2)). Hence the equation of the evolute is s=/'W_./'-(0). Ex. 1. Show that the evolute (1) of a parabola is a semi- cubical parabola, (2) of a cycloid is an equal cycloid, (3) of a four-cusped hypocycloid is a similar curve of twice the size of the given curve, (4) of a cardioid is a cardioid of one third the size of the given one. 2. What is the intrinsic equation of the involute of (1) a circle, (2) a catenary, the involute beginning at a point on the given curve? Ans. (1) s = iaP, (2) s = a log sec L The Tractrix. 189. This is the curve in which the tangent is of con- stant length. Let {x, y) be the coordinates of a point P on the curve (Fig. 116), and let the tangent PT = a* From the figure dy/dx= — y/Va 2 —y 2 , from which the equation may be found by integrating, the result being x = a sech -1 (y/a) — V a 2 — y 2 . * The curve is the path of a body which is drawn along on a rough horizontal plane by a string of length a, the other end of which is moved along a straight line OX; whence the name of the curve. 208 INFINITESIMAL CALCULUS. [Ch. XXXVII Since ydx= — dyV / a 2 —y 2 , the element of the area of the curve = the element of the area of the circle of radius a, .'. the whole area between the curve and its asymptote (the #-axis) is the same as that of the circle, viz./7ra 2 . Fig. 116 The length of the curve from Y to any point whose ordinate is b may be found as follows: ds dy — ~~~" < • • & — ti> a y \o/ Let the tangent make an angle X with YO. Thens = a log sec A, which is the intrinsic equation (origin Y) of the curve. Hence the radius of curvature = ds/dX = a tan X, and the evolute is (§ 188) s = a tan J, a catenary (§ 192). The area of the surface of revolution * of the curve about the a>axis = 47ra 2 , and the volume = §7ra 3 . * This surface is known as the pseudo-sphere. 190-192.] THE CATENARY. 209 The Catenary. 190. This is the curve formed by a uniform chain hanging vertically. Let A be the lowest point, P any other point. From P draw PB vertically and equal to the length AP or s of the chain, and from B draw a horizontal line to meet the tan- ent at P in (7, and let BC = a. Y ?x ^^^^ ' \ B ^v A \K M Fig 117. N X 191. Mechanics of the figure. The portion AP of the chain is in equilibrium under the action of three forces, viz., the horizontal tension at A, the tension at P in the direction of the tangent, and the weight, which is vertical. Hence PBC is a triangle of forces for AP, and since the vertical force on AP is the weight of a length PB of the chain, it follows that the tension at P is equal to the weight of a length CP of the chain, and the tension at A to the weight of a length a of a chain, and therefore a is constant. 192. Geometry of the figure. Draw from A a vertical line and take QA=- a; take as the origin, OA as j/-axis and a 210 INFINITESIMAL CALCULUS. [Ch. XXXVII. horizontal line OX as z-axis. Then OM = x,MP = y. (By this choice of axes the constants of integration will = 0.) Since CB, BP, and CP are a, s, and Va 2 + s 2 , respectively, we have dx a ds x/ a 2 + s 2 K " ds Va 2 + s 2 From (3) dy=s ds/V a 2 + s 2 , .'. y=\ / a 2 + s 2 = CP, (4) .". the tension at any point P is equal to the weight of a length y of the chain. From (2), dx= ads/Va 2 + s 2 , .'. x=a sinb _1 (s/a), • /y* fi X X .\ s = asinh— . or — — (ea — e~~a), (5) a 1 which gives the length from the lowest point to the point whose abscissa is x. From (1) and (5), dy = sinh — dx. (j/ x a i — —\ .'. y = a cosh — , or =-feo+e a), (6) the equation of the curve. From (1), s = a tan , the intrinsic equation (origin A) of the curve. The normal. NP:MP: :CP:CB, .-. NP/y=y/a, or NP = y 2 /a. The radius of Curvature. R=ds/d = a sec 2 <$> = ay 2 / a 2 = y 2 /a, or the radius of curvature is equal to the normal. Let D be the foot of the perpendicular from M on PT '. Since MP = CP, .'. MD = CB = a and DP = BP = s. Hence the locus of D is the involute of the catenary. Also MD, the tangent at D to the involute, is of constant length, .*. the involute Is a tractrix. The intrinsic equation of the e volute is s = a(sec 2 1 the series has no limit or is non-con- vergent. 194. Let ^0+^1+^2+^3+ . . . +u n + u n +i+ ... (2) be an infinite series , and let s n be the sum of the first n terms. The series is then convergent if s n has a limit when n = 00 ; let the limit, if it exists, be s. The limit must as in all other cases be a definite finite quantity (§ 2) such that s—s n has the limit 0. Hence a series cannot be convergent unless u n = when n=oo, i.e. unless the terms tend to a limit 0. This is a necessary but not a sufficient condition for con- vergence. *O y i the subject of Infinite Series the student may consult Osgood's Introduction to In-finite Series (Harvard University), also Gibson's Calculus (MacmillanV 211 212 INFINITESIMAL CALCULUS. [Ch. XXXVIII. If a series is non-convergent it is either divergent or oscilla- tory, divergent if s n becomes infinite with n, oscillatory if s n remains finite but does not approach a limit. Series (1) is divergent if |#|>1 or z—1, and oscillatory if x= — 1, s n in the latter case being 1, 0, 1, 0, etc., as n increases, but never approaching a limit. In general, a series is of no practical value unless it is convergent. 195. Without expressing s n in terms of n it may be pos- sible to test a given series for convergence. For this pur- pose various methods are given in works on algebra; we recall a few results which are of importance in our work. (1) A series is convergent if (a) the terms are alternately + and — , (6) the absolute value of the terms constantly diminishes, and (c) the limit of that value is when n=oo. Ex. l-i+J-i + ... is convergent (see §2, Ex.3). The. limit of the sum lies between '69314 and '69315. It = log e 2 (§ 197 Ex. 1). The series 2 — f + £ — . . . satisfies conditions (a) and (6) but not (c). It is oscillatory (see § 2, Ex. 3). (2) The series 1+— +-—+... is convergent when c>l, Z c 6 C divergent when c 0, is divergent. For it > a+l a+2 * ' ° 6+1 6+2 + . . . , where 6 is an integer larger than a, and this is a part of the divergent series 1 + \ + \ + . . . (3) If a series is convergent when all the terms are posi- tive, it will be convergent if any of the'terms are negative. 195, 196.] INFINITE SERIES. 213 cos x cos 2x p t 2 : if all the terms were positive they would not be greater than 1 1 1 ill ^ / I i i ^ j^j i j Ex. -77- H — ^ — h... is convergent for all values of x } for the corresponding terms of — + Q ^ + . . . The contrary is not necessarily true, that is, a series may be convergent when some of the terms are negative, but not when those terms are made positive. Thus 1— i + J— ... is convergent, but 1 + J + J + . . . is divergent. A series is said to be absolutely convergent when the abso- lute values of the terms form a convergent series, i.e., when the series remains convergent if all the terms are made posi- tive. If not absolutely convergent it is said to be con- ditionally convergent. Thus 1 — i + i — 4"+ • • ■ -is absolutely convergent, 1— i + J— \+ ... is conditionally convergent. It is known that the convergence and limit of an absolutely convergent series are independent of the order in which the terms are taken, whereas the terms of a conditionally convergent series may be grouped so as to converge to a different limit (in fact to an assigned limit) or to diverge. For example, arrange the series 1— i+J— i+ ... as follows: (l~i) — i + ("5" — 6") — B" + (i — tV)"~ ••• This = i-i+£-i+ .•• =i(l-i+i-i+ ...), one-half of the original series. (4) A series uo + Ui + . . .+u n + u n +i +. . . is absolutely con- vergent if R, the absolute value of the limit of u n +i/u n when n= oo , is <1, divergent if R>1. It may or may not be convergent if R = 1 . 196. Power series. The most important infinite series are those of the form ao+aiX-\-a 2 x 2 + a s x 3 + . . . +a n £ n + a n+]L £ n+1 + . . . , (1) which is called a power series in x. The indices are posi- tive ascending integers, and ao, a\ } . . . are independent of x. 214 INFINITESIMAL CALCULUS. [Ch. XXXVIII. By assigning values to x any number of series may be formed from a given power series. Let the absolute value of the limit of a n /a n +i when n=oo be r. It follows from § 195 (4) that the power series is absolutely convergent if |#| — r and r. If x = r the series may be convergent or non-convergent. If a n /a n +i = co when n=oo,the series is convergent for all values of x. Since the limit for n + 1 = oo is the same as for n = oo , the value of r may be found equally well from a n+ i/a n+2 or from a n _i/a n , or any two successive coefficients. Ex.1. 1+X+—+- + . ..+—+.. . 2 3 n 1 o»n + l'l, dn = —, .*. — -= =1 + — =1 when n = oo, .\r = l. Hence the series is absolutely convergent if |x|1 there is no connection between the value of the function and the sum of the terms of the series. If we differentiate both sides of (1) as if the series were finite we have fix) =ai+2a 2 x + 3asx 2 + ... (2) Similarly, multiplying (1) by dx and integrating, F(x) =c + aox + \a\x 2 + \a 2 x z + . . . , (3) c being the integration constant, viz., the value of F(x) when x=0. It may be proved that these results hold true (i.e., the new functions are equal to the limits of the sum of the terms of the new series) for all values of x for which the new series are convergent. These values are — r — 1 and < 1, but it converges too slowly to be of much value for such calculation, unless \x\ is very small. Change x in (1) into —x and subtract from (1). Then lo S (fz|) =2(x + ix 5 +ix' + . . . ). (2) Let y = (l+x)/(l— x), then x= (y — l)/(y + l) Substituting in (2), This series may be used for the calculation of any Napierian logarithm, since (y — l)/(y + l) is necessarily a proper fraction when y is any positive quantity. Thus if y = 2, (y-l)/(y + l) = h .-. log e 2 = 2[Kia) 3 +i(i) 5 + ...]='693147. Similarly log e 3 = 2 [i + i(i) 3 + s(i) 5 + . . . ] = 1*098612. Also, loge 4 = 2 log e 2 = T 386294. Another series may be derived from (2) thus: put (l-\-x)/(l—x) = (l+y)/y, then x = l/(l+2y); hence, remembering that log [(1 +y)/y] = log (1 +y) -log y, log(l +y ) = log y + 2[ 1 -^ + j( 1 -^) 3 + |( f ^) 5 + ...]. 179.] INFINITE SERIES. 2l7 Thusif ?/ = 4, loge5 = loge4 + 2[i4-KI) 3 +*(i) 5 "f ..] = r609438. Hence loge 10 = log e 5 4- loge 2 = 2*302585, and hence the modulus* of the common logarithms (which = l/log e 10) is '4342945. The common logarithms may therefore be found from the Napierian logarithms by multiplying by '4342945, and, conversely, the Napierian from the common logarithms by multiplying by 2*302585. 2. Gregory } s series. Prove that for — 1<#<1, tan -1 x = x — Ja* 3 + Ja* 5 — . . . (1) This series gives the radian measure of an angle in terms of its tangent. Show that ! = i_ KW+ ...= 2 ( I L + _L + ...). This converges slowly, but by applying (1) to the relation Tt 1 — = 4 tan -1 \ — tan -1 — - a rapidly converging series for the calcu- lation of n is obtained. TT 1 . , Since tan~ 1 o* = — — tan -1 — , if b > 1 we have 2 x , TT 1 1 1 tan—a* =— ^7r^~^-E + ' • • 2 x 3a* 3 5x 5 3. Prove that 1 x 3 1 .3z 5 1 .3. bx 1 . . " and hence, making x = -|, that 1 3 24 + 640 H ' ^K 1+ 4 + ^ + 7i8 + --) =3 ' 14159 --' * If x = \og a y, then by definition of a logarithm, a x = y. Taking logarithms of this, the base being supposed e, we have x \og e a = \og e y, ,\ x or logay = 1 sc , or the logarithm of 2/ is changed from base e to base a by multiplying by = , which is called the modulus of the system of base a. If y = e. logo e =? , hence the modulus of the common system * ' & log e a J J is also equal to log l0 e. 218 INFINITESIMAL CALCULUS. [Ch XXXVIIL Show also that sec _1 a; = — — — r— - — * ' , — . . . , b]> 1. 2 a; 6x 3 2.4.5r ' v ' 198. Maclaurin's Series. If there is a power series * which =/(s), that series is /(0) + f\0)x + J -^x 2 + . . . For, suppose it to be ao + aiX + a 2 x 2 + . . . Then }(x) =do +CLiX-\-a 2 X 2 + CL3X 3 + CL4 : X 4: + ... (1) Differentiating successively, / / (x)=a 1 -! 2a 2 x+3asx 2 + 4:a4 : x 3 + ... (2) /" (x) =2a 2 + 2. 3a s x + 3 . 4a 4 x 2 + . . . (3) }'"(x) =2 . 3a 3 +2 . 3 . 4a 4 z + ... (4) etc. If (1) is convergent for |x| . . . Hence Maclaurin's Series takes the form j(h + x) = f(h)+f\h)x + } ^X 2 + } ^^x3 + . . . This is Taylor's Series. The conditions under which the function is represented by the series will be considered in Ch. XXXIX. 200, 201.] INFINITE SERIES. 221 jYlitYl — 1 ) Ex.1. (h + x) m = h m +mh m - 1 x + - — — h m ~ 2 x 2 + ... 2! sin h . cos ft X' 2. sin (ft + z) = sin ft + cos ft . £ Z I o I 3. If /(#) = x 3 — 2x 2 — x + 3, writedown f(x + h). X i" • • • Examples. 1. Expand (l+#) m , log (1+z), tan -1 x, by Maclaurin's series. a: 3 2x 5 llx 1 2. tan*-* + — +— +^- 9 + ... , z 2 5x 4 61z 6 3. sec x = l+- + — + — + ... a: 2 x 4 or 17ar 4. log sec x = ---\ 1 — . 6 2 12 45 2520 + . .. 5. cos 3 # = l 3x 2 7x 4 8 „ H X 2 X* T 6. e*™* = l+x+— —^-tz + -- 2i o 15 7. e x sec x = l+x-\-x 2 + %x 3 + . . . 1 x 7x 3 8. coseca;=-+— +t— — a: 3! 3.5! 9. Show that + x 2 x* X 3 X 5 cosh x = l +—+— + . . . , sinh x = x+— +— + . . . , 2! 4! o! 5! and hence that cosh x = cos ix, sinh #= — i sin tz, where i = v— T ^ A .111 7Z 2 t 10. Assuming — +—+— + . . . = — , show that V 2 2 3 2 6 1 1 "' p + 3i + 5- 2 + --=8' and 1 _1_ 1 p 2 2+ 3" 2 12' Also that f 1 ! 1 7T* J log (l+a:)dx = Y2. ri 11. Show that cfc _ _1_ 1 L .; _ o^l+x 4 2 5 2.59 12. When x = show from the series that (1) (sinaO/z=l, (2) (tanx)/s= 1, (3) (1-cos z)/:r 2 =ss J, (4) (tanx — sinx)/x 3 = J, (5) (e* — 1)A=1. 222 INFINITESIMAL CALCULUS. [Cxi. XXXVJII. 13. If a circular arc (radius a) subtends an angle 6 at the centre, show that when 8 is very small arc — chord = 4t a ^i nearly. 14. If 6 is a small angle, show that sin d = dVcos~6, ] . , }■ nearly. tan 6 = 6^/cos~'0 J From these formulae are derived the rules given in Mathe- matical Tables for finding the sines and tangents of small angles. 15. The chord of a circular arc is C, the chord of half the arc s c; show that the length of the arc is 2c + J (2c — C), very nearly. This formula (Huyghens's) will give J of the circumference of a circle of 100 feet radius with an error of less than 1J inch; it gives i of the circumference of the same circle with an error of less than -fa of an inch. CHAPTER XXXIX. TAYLOR'S THEOREM. 202. In Ch. XXXVIII the existence of a power series for j(x) or f(h + x) is assumed. We have now to consider under what circumstances this assumption may be justified. 203. Theorem of Mean Value. Let f(x) be a single-valued function, and suppose f(x) and its first derivative /'(#) to be continuous from x = a to x = b. The Theorem of Mean Value asserts that b— a =/'(^i), a) where xi is some value of x 'between a and b. a c B Fig. 118. B X Let (Fig. 118) 04 = a,0B=b. If PRQ represents the graph of f(x) from x = a to x = b, AP = f(a), BQ = J(b). Hence '-±-4 — -^- is the slope of the straight line PQ. At some b—a point R between P and Q the tangent is parallel to PQ, and the slope of the tangent at R is /'(#i), where x\ = OC. Hence the equality stated in (1). It should be noticed (Fig. 119) that X\ may have more than one value. 223 224 INFINITESIMAL CALCULUS. [Ch. XXXIX. The theorem may not be true if, between x = a and x = b, f(x) (Figs. 120, 121) or f(x) (Figs. 122, 123) has a finite or infinite discontinuity. Fig. 120. Fig. 121. Fig. 122. Fig. 123. 204. If, in (1), /(&) = /(a), then / / (a;i) = 0; i.e., if f(x) and f'(x) are continuous from x = a to x = b, and if /(a) = /(&), then f'(x) = for at least one value between x=a and x = b. This is known as Rollers Theorem. 205. In Fig. 118 let 4B = A and AC = 0h, O<0<1. Then (1) becomes f(a+h)-f(a) h = f(a + 0h), or f(a + h) = f(a)+hf'(a + Oh). This may be regarded as the beginning of an expansion of }(a + h) in powers of h. We have now to show that the expansion may under certain circumstances be continued to three or more terms. 206. Taylor's Theorem. Let f(x) and its first n deriva- tives be continuous from x=a to x=a+h. Let P be a quantity which is such that /(a + 7i)-/(a)-/ / (a)^-Q^/i 2 ~ . . . 2! " '" ,(n-l)! Consider also the following function of x: fix) f(n-l)( a ) f(a+h)-f(x)-f(x)(a+h-x) 2! -(a-t-h— x) 2 — . . . (n- 1) 1 {a+h-x) n - 1 -P(0 and <1, but its value generally depends upon a, h, and n, as well as the form of the function. Ex. log { a + h) = loga+~- 2 + --. ..+ n{a + eh)n - 207. If x is written for a, (5) takes the form f(z + h)=f($+r(x)h+^h*+ . . . + / (n) ^+ gft ) ftn, 208. The remainder. The last term of (5) is known as Lagrange's form of the remainder (R n ) after n terms. Another form of the remainder R n (Cauchy's), viz., (n-l)I U ; ' may be found by starting with Ph instead of Ph n in (2). 226 INFINITESIMAL CALCULUS. [Ch. XXXIX. B n is the amount of the error when the first n terms of the series are taken as the value of f(a + h). Thus for log (a+h) (§206) the numerical value of the error would h n h n lie between — - and — j-r-, the greatest and least values na n n(a+h) n of R n . The method of making small corrections explained in Ch. XI is equivalent to the use of the first two terms of Taylor's Theorem. In this case the error is therefore %f" (a+ dh)h 2 , where O<0<1. 209. Maclaurin's Theorem. Taking a=0 in (5) and writing x for h, we have /(*)=/(0)+/'(0)z+^W. . .+Rn, (7) where R n J^^l^ or =ffiM(i-^)n-i^ according as Lagrange's or Cauchy's form is adopted for the remainder. (7) is the statement of Maclaurin's Theorem. The expansion is therefore possible if f(x) and its first n derivatives are continuous from x=0 up to the value of x adopted in the series. 210. Maclaurin's Series. Taylor's Series. If j(x) can be expanded by means of the series (7), and if the values of x are such that £ n _oo R n =0, then or f(x) is equal to the limit of the sum of the terms of an infinite power series (Maclaurin's Series). Under similar conditions we obtain from (5) Ka+h)=f(a)+f(a)h+ f -^h2 + which is Taylor's Series, 209, 210.] TAYLOR'S THEOREM. 227 x 3 ar 5 Ex. 1 . From (7) sin x =x — — + — — . . . + Rn, where Rn (Lagrange's o! 5! (7T \ X X XXX 6x + n—)—-. Now — r — — . — . — . . . , and each f rac- 2/ n\ n\ 1 2 3 tion after a certain point is numerically <1, hence the limit of the product = for n = co . Also sin ( 6x + n— ) remains finite as n increases, since it cannot be >1 or <— 1. Thus £ n =oo Rn = for all values of x. Hence, for all values of x, X s x* m*z-x-f+ t X 2 x 4 Similarly, cos x = 1 — — +— : — . . . for all values of x. /y* it /v»0 2. log (l+x)=x— - + — — . . ,+Rn, where Rn (Cauchy's form) ( - I)"" 1 (1 - d)n-^ ( - I)"" 1 /l - d n (1 + Ox)* But l-0|<|- + if -1n, . . . The complete graph of the series con- ^ h\ -7T 7T Fig. 128. sists of the straight lines of Fig. 128 continued indefinitely in both directions, and the equation of all these lines is j/ = — (smz+f sin 3x + . . . ). K 214 125.] FOURIER'S SERIES. 233 Although the series is convergent, the series formed by the derivatives of its terms is non-convergent, and therefore does not represent the slope of the graph (the derivative of the func- tion) at any point. 2. # = 2(sin x — \ sin 2x+-$ sin 3x — . . . ). This represents x for 0 n"! • 215. The function represented by a Fourier series need not be a single function throughout the range iz of the value of x; the same series may represent one function for a part of the range and one or more other functions for the remainder of the range. Let }i(x) =A+di cos x + a 2 cos 2a: + . . . (1) for x=0 to x=a, and f 2 (x) =A+ai cos x+a 2 cos 2z + . . . (2) 234 INFINITESIMAL CALCULUS. [Oi XL. (the same series) for x=a to x=n. Multiply (1) by dx and integrate between and a, also multiply (2) by dx and integrate between a and n, and add the results. Then each term of the series is, on the whole, integrated between and n* Hence /i (x) dx + f 2 (x) dx=Anj .-. A = =—\ fi( x ) d> x + h( x ) d x V ^ L J J a J 2rf a f* ~i Similarly, a n =— /i (x) cos nz dx + /2OE) cos nx dx x*-J J a J for the cosine series, and 2 r f a ■ f* "I a n =— /1 (z) sin nx dx + / 2 (x) sin nx dx ttLJo J a J for the sine series. The series may not hold at the point or points where the change of function occurs. It may be noticed that A in the cosine series is always equal to the mean height of the graph from x=0 to x=n. -7T . f ¥ Fig. 130.* 7T Ex. 1. To find a cosine series which =1 for 0 an d An 71 2p 71 J 7T 2* 2 . 717T cos nx ax = — sin ~pr . nn 2 Hence the series is 1 2 — +— (cos x — \ cos 3a;+j cos 5x — . . . ). 2 71 * The electrician's make -arid-break curve. 216,217.] FOURIER'S SERIES. 235 It is true when x = and x = tt, but = ^ when x = ^tu. 2. Find a sine series for the same. 2 /sin x 2 sin 2x sin 3x sin 5x 2 sin 6x sin 7x \ 1 2 = 2 +— (sin 2x+i sin 6#+i sin lOx + . . . ). Arts, (See § 214, Ex. 6.) 216. The cosine and sine series. For values of x between — 7z and 7i j fix) =A +ai cos x + . . . +a n cos nx + > . . +6i sin x + . . . + 6 n sin n# + . . . It is easily shown, as in § 213, that the constants may be determined as follows: *=k fix) dx, a n =— fix) cos nx dx } 71 71 J —7C 71 f{x) sin nx dx. 2 sinh 71 /l _ z smn n /i cos W7r Ex. e* = ■ ( — + . . . H — - — - cos nx + . . . 7T \2 n 1J rl U COS 727T . \ — — — — — - sin nx — . . . J . 7i 2 + l / 217. By the following method a cosine series which will hold for values of x from to any number c (instead of n) may be obtained. If x=cz/tz, x=0 when z=0, and x=c when z=7i. Hence, in fix) change x into cz/tz, develop in terms of z, and change z into tzx/c. Similarly to obtain a sine series for values of x between and c, or a cosine and sine series for values of x between — c and c. In all cases the constant term is equal to the mean height of the graph (or the mean value of the function) for the interval in ques- 236 INFINITESIMAL CALCULUS. [Ch. XL. tion. In this way the series already obtained may be adapted to the intervals stated below. _ C 4C / TlX 1 6t:X \ r ^ _ Ex. 1. x =77-- i( c os — +— cos + . . .) , [0, c]. A 7Z \ Co C ' , c 2 4c 2 / 7tx 1 2nx \ _ - _ 4/z, / . 7r;r 1 . 37T£ \ 3. ft = — sin h— sin + . . .) , 10, c[. 7z \ c 3 c 1 2c [ . TZX 1 . 27:2 V 4. x =- sm — --sin K . .) , J — c, c[. 7i \ c 2 c / 7T . 7T:T 1 . 37TX _ 5. — = sin — + — sm h. . . , JO, c[. 4 c 3 c * 218. If a function of z is developed into a series for the interval — c to c, and if the values of the function are repeated periodically for every interval 2c of x, the series will con- tinue to represent those values as x increases or decreases. In other words, the periodic function of period 2c is developed into a series consisting of a constant term and harmonic functions of periods 2c, 2c/2, 2c/3, etc. Fourier's Theorem is to the effect that this development is always possible, the complete series being of the form TZX 2tzx , , , . TZX . 2tzx A+ai cos ±-a 2 cos h . . . +&i sm — + & 2 sm K . . • c c c c which is equivalent to (TZX \ (2tzx \ Voi\\ +A 2 sin( \-a 2 ) +. . . , where A n =Va n 2 +b n 2 and a w =tan"" 1 (o n /6 n ). As already stated, the function may consist of distinct functions for parts of the interval. 218.] FOURIER'S SERIES. 237 Examples. Develop the functions represented by the following figures : A 1. Fig. A. 2. Fig. B. 3. Fig. C. 5. Fig. E. 6. Fig. F. h 4ft ( --—(cos— + 2 TZ 2 \ C 8ft t . Ttx 1 . Stzx _( sm ___ sm _ + 2ft / . tzx 1 . 2nx — sm — sin h tz \ c 2 c ...). ...). h -c E ft h I . tzx 1 . — (sm h— si 2 tz \ c 2 F 2tt# sm f- tzx 1 4- 2 c 27T^ + — (sm h— sm 2 tz \ c ...). ...). 4ft / . 7TX 1 . 37nr — ( sm h— sm h TZ \ c 3 c • • . i • ft -*c G 7. Fig. G. 8. Fig. EL -c ft c 2c H A 2ft / . tzx 1 . Stzx \ - (sm- f-— sin - K . .) 2 7T \ ft 2ft/ 2 (cos— + 4 7T 2 \ CO 2 c 3 TZX c c Stzx 6TZX \ - +...). C / ft / . 7nc 1 . 2tzx \ -\ — (sm — sm K . .) . TZ \ c 2 c / 238 INFINITESIMAL CALCULUS. [Ch. XL. 9. Fig. J. (Parabolas. Latus rectum = 2c). 1 2nx \ + cos -- + ...). c 2c I tzx 3-A C0S 10. The displacements of a slide-valve actuated by a Gooch link were measured at eight intervals each of 45°, and found to be as follows, beginning with the crank on the inner dead-centre: 2*44, 1'65 ; 0, -T37, -L87, -T37, 0, T65. Assuming that the motion of the valve is compounded of two simple harmonic motions, one of double the frequency of the other, as represented by the equation y = k+a sin (0 + a) +6 sin (20+/?), where is the crank angle, find the values of k, a, a, b, /?. (Castle, Manual of Practical Mathematics.) There are various graphical or other practical methods by which the coefficients of a small number of terms of a Fourier series may be found, but in this example an algebraical solution will suffice. Assume y = k+a 1 sin J r b l cos d +a 2 sin 26 +b 2 cos 2d 7 substitute the given values of y for = 0, 45°, 90°, etc., and solve the equations. Arts. 2/ = *14+2 , 16 cos 0+*14 cos 20, or ='14-2-16 sin (0+90°) +'14 sin (20+90°). I CHAPTER XLI. APPROXIMATE INTEGRATION. ELLIPTIC INTEGRALS. 219. Approximate integration. If the general value of f(x) dx cannot be obtained it may be possible to find a sufficiently close approximation to the desired result. (1) If f(x) can be developed into a rapidly converging series, the integration of a few terms will give an approxi- mate value of the integral. (2) The curve y=f(x). may be plotted when f(x) is given. Its area obtained by Simpson's Rule (§ 131) or by the pla- nimeter (Appendix, Note D) will give an approximate value of f(x) dx between assigned values of x. (3) y 2 dx and y 3 dx as well as \y dx for a curve which has been drawn mechanically or otherwise can be obtained mechanically. The result, although theoretically exact, is affected by observation and instrumental error. On Mechan- ical Integration see Appendix, Note D. Elliptic Integrals. f dd , |Vl-m 2 sin 2 # dd, Jvi — ra 2 sin 2 220. dd and 1 (1+a sin^Vl — ra 2 sin 2 # 239 240 INFINITESIMAL CALCULUS. [Ch. XLT. are called elliptic integrals of the first, second, and third class respectively. The constant m, which is assumed to be not greater than unity, is called the modulus of the integrals. The lower limit is understood to be in each case, and, the angle varying from to 6, 6 is called the amplitude of the integral. The integrals are represented by the symbols F(m, 6), E(m, 6), and II "(a, m, 6), respectively; orby.F m (#), etc. When the limits are and \tz (i.e., when the ampli- tude is \iz) the integrals are said to be complete. If sin 0=x, the integrals become I dx and V (1 — x 2 ) (1 — m 2 x 2 ) dx l — m 2 x 2 , ax, x 2 J (l+ax 2 )V(l-x 2 )(l-m 2 x 2 )' and they are complete when the limits are and 1. 221. The values of the elliptic integrals cannot be expressed in finite terms, but may be approximated to by infinite series. • Thus by the Binomial Theorem = (l-m 2 sin 2 fl)-*d0 Vl — m 2 sin 2 # = (l +-m 2 sin 2 + y~i^ sin 4 # +■ ' ' fi m 6 sin 6 fl + . . .) dd, and each term may be integrated by § 113 (see Ex. 12 below). Taking the limits as and \n we have (§ 120) for the complete elliptic integral of the first class •nt ix rcf"i , /l \ 2 /1.3 9 \ 2 /1.3.5 a 2 i 221,222.] ELLIPTIC INTEGRALS. 241 Similarly for the integral of the second class we have Vl-m 2 sin 2 0d0 = (l-?n 2 sin 2 0)*d0 = (l- \m? sin 2 0- ^—rfi sin 4 0- * : 3 - m 6 sin 6 0- . . .) dd, \ 2 2.4 2.4.6 / and ttt /l \ 2 1/13 \ 2 1/135 \ 2 T sL^W -sfc™ 2 ) -5(2^4-6^) "•••J for the complete integral of the second class, E{m,\n). Three-figure tables of the integrals for certain values of the modulus and amplitude are given at the end of this volume. It may be noticed that also, E(l, 6) = JS(0, 0) ==F(0, 0) =0 (in radians); cos d0=sin 0, ' dd , . /7T cos =log tan (I +|)=;«?). 222. From the above expansions and the integral (§ 113) of sin n dd it may be shown that E(m, nn±6)=2nE±E{m J 0), F(m, nn±d)=2nK±F(m, 0), i? and if being the values of the integrals for the amplitude %tc, and n being any integer. Hence a table of the elliptic integrals in which the amplitude varies from to \n may be used for all higher values of the amplitude. Examples. x 2 xi 2 1. To find the length of an arc of the ellipse — + 7 -- = l. a 2 o 2 The complement of the eccentric angle being denoted by 6 we have x = a sin 6, and y = b cos 6. : . dx = a cos 6 dd, dy= —b sin 6 dd "■ 242 INFINITESIMAL CALCULUS. [Ch. XLI. whence ds 2 = dx 2 +dy 2 = (a 2 cos 2 6 +b 2 sin 2 6) dd 2 = [a 2 - (a 2 - b 2 ) sin 2 6>] dd 2 = a 2 (l -m 2 sin 2 /?) dd 2 , where m fc is the eccentricity of the ellipse. Hence the length of the elliptic arc measured from the end of the minor axis is a Vl -m 2 sin 2 dd = aE(m, X ), an elliptic integral of the second class. The length of the quad- rant of the ellipse = aE(m, %n). 2. Find the circumference of the ellipse x 2 +2y 2 =2. Arts. 7' 64. 3. Of the ellipse 3x 2 +±y 2 = 12 find (1) the length of the arc from x^O to x = l, (2) the length of the quadrant, (3) the middle point of the quadrant. Arts. T036, 2*934, (1*36, T27). 4. An arc of the lemniscate r 2 =a 2 cos 26. From ds 2 =r 2 d0 2 +dr 2 we have -J: dd Vl-2sin 2 Let 2sin 2 #=sin 2 0. Then 'fa d(f> a V2 =4^ o Vl-i s in 2 V2 \V2 ■> 0i), an elliptic integral of the first class. The length of a quadrant of the lemniscate is therefore a F (■ i^i — V2 \V2 If 6=30°, show that s=*584a. } 2 n) = l'311a. Cc 5. 6. 7. dx =—F a , sin — i V( a 2 -x 2 )(b 2 --x 2 ) a ^ dx 1 /vV-6 2 t c , =—F( , tan^r V(a 2 +x 2 )(b 2 +x 2 ) a \ a b Let x = b sin 0. dx F b V(a 2 +x 2 )(b 2 -x*) V a 2 + b 2 Wa 2 + b 2 >> co *- l y- 222. ELLIPTIC INTEGRALS 243 . I ™ =2F(m, sin- 1 ^). 8. I -y- V x{l —x)(l —m 2 x) 9 A simple pendulum of length I oscillates through an angle /5 on each side of the vertical. To find the time of an oscillation. When the pendulum makes an angle (f> with the vertical, the acceleration — g sin $ in the direction of the motion = d 2 s/dt 2 = I d 2 /dl\ d 2 g . '> Multiply by 2d and integrate. Then (S) =f( C0S ^~ C0S ^ = f( sin2 ^- sin H^). Hence solving for d£ and integrating, 2\|<7 w d4> (1) o Vsin 2 •§•/? — sin 2 J is the time of a half oscillation. Let sin £0='sin i/? sin 0. Then (1) becomes X \ J - f , *" = x &( S in tf, W . \<7Jo Vl-sinH/Ssin 2 ^ \9 ' Hence the time of an oscillation is T 4 ^(sin ift **). 10. Find the time of oscillation of a pendulum when a =60°. Arts. 3'372V77<7. Find the time through the lower half of the motion. Ans. 2^-i^sin 30°, sin-^^j =V102Vl/g. 11. If the arc s is small compared with the length I, show that the time of oscillation of a simple pendulum is approximately ^K l+ 6il)- 9 12. Show that F(m, 0) = 0+im 2 (0-sin cos 0) + "7^r"x = ^~, = TT w henx = l. .F(x) x 3 -l JP'(x) 3x 2 3 x — 1 1 The work may be conveniently expressed thus : When iil, x-l 1 *V-1 3x 2 Ji 1_ 3 ># c— x pX ±0— x pX _|_ />- 2. If z = 0, £ . " =— — sin x cos a; /Or) e*-l-log'(l+x) 3. TO X' when a: = 0. fix) F'(x) >z 1 ■=— when # = 0. 2x /"(*) F"(x) (l+x) ; *= 1 when x = 0. '. £ ° = 1 when # = 0. X' 225. The form 00/00. Let / (x)/F (x) =00/00 , first when £ = 00. Let the graphs of f(x) and jP(x) be PQ and P'Q', Fig. 131, and let OM=x. Let the limits of the tangents at 246 INFINITESIMAL CALCULUS. [Ch. XLn. P and P' be the asymptotes AS and A'S' when x = , and let A'A=c. Then MP=}(x), MP' =F(x), tan MTP=f'(x), O A' A T M Fig. 131. tan MT'P'=F'(x). Hence f(x) T Mfjx) F{x)~T'MF'{x)' But .TM AM £>T'M *A'M & V A' Mi when A'M = oo . Hence ,m nx) A: 77 /~A ^o 77f/ X (i) Secondly, let f(x)/F(x) =oc/oo when x = a. For s sub- stitute a + l/z. Then 2 = 00 when x = a. But by (1), if 2 = 00, ', ( . + i)-^ { „ + i)( -i) _ Vi)' or . /(*) x /'w 226,] SINGULAR FORMS. 247 Hence the result (1) holds in this case also. Thus when a fraction has the form oo /oo the limit of its value is found from the same differentiations as when it has the form 0/0. . TZX ,, s log cos — Ex. M = -V4 = -when^l. r (x) log(l— x) oo 71 TZX f(x) ~2" tan ^ n 1-x \ 7^t-t = = — . = — when x = 1. F'(x) 1 2 nx cot x tz „ 1 —x tz — 1 But (§224) -£ l-x 2 = 1 2 ^x 2 7i 7ro: cot — -- cosec 2 -- Hence the given fraction = 1 when x = 1 . 226. The forms 0.00 , 00 -00 . A function which assumes the form O.00 , or 00-00 may, by an algebraical or other change, be made to take the form 0/0 or 00 /oo . a Ex. 1. x(l —e x ) tends to 00 . when x = 00 . The limit is most easily found by using the exponential series. a For Jf( i-.-: ) .-,[i-(i-i + £-...)] a 2 = a— — +. . . = a when # = oo . 2x TZX 2. (1 — x) tan — tends to the form . 00 when x = 1. 1 —x . But it = , which = — when x = 1. TZX cot- — 1 2 .'. (§ 224) we have ==— when x = l. TZ TZX TZ --cosec'- 248 INFINITESIMAL CALCULUS. [Ch. XLII .\ £{l —x) tan — =— when x <~ 1. D a; 1 3. — -— , = oo—oo when x =1. x — 1 log x _ J a: 1 a: log a:— x + 1 . But r — ; = — ; tti = — wnena;=l, # — 1 log a: (a; — 1) log a; lo log sin 2x (o) ■; : = 1, log sin x a + x (8) a^ = e a , (3) ->%■ — o — X = 2, log (1 +x) 2 ? (7) a^loga; = 0, /r\ l°g SeC X 1 ( 5 ) 2 ' (9) x x = 1. s 3 -3a;+2 2. — — — — - = when x =\. x 3 +4x 2 -5 3. Es = oo, (1) — = oo, (2) 2* sin ^ = a. 4. Ha? = |-, (1) (sin a;) sec2 * =;—=., (2) sec a; (|-a; sin aA = 1. 5. (sin x) tan *= 1 when a; =0 or -. 1 +— ) = 6 a when a: = oo , and = 1 when x = 0. „ sec a; cos 3x n . 7r 7. — - = = — 3 when x = — . sec 3# cos re 2 ^ tan a; cos 3a: sin x , ^ 8. ; - = ■ ^-—-(-3)(-l) = 3 whena:~~ e tan 3a: cos a; sm 3x 2 9. (e*-l) tan 2 x /e* — 1\ /tan z\ 2 a;* - £r) (^) - 1 when x = 0. CHAPTER XLIII. SUCCESSIVE DIFFERENTIALS OF FUNCTIONS OF MORE THAN ONE VARIABLE. EXTENSION OF TAYLOR'S THEOREM. MAXIMA AND MINIMA FROM TAYLOR'S THEOREM. Successive Partial Differentials. 228. Suppose u to be ax 3 —xy 2 +y. We have as in § 45, supposing x alone to vary, d x u = (3ax 2 —y 2 )dx, d x 2 u=6ax dx 2 , d x 3 u=6adx 3 , d x 4 u=0, d y u = (— 2xy + l)dy, d y 2 u = — 2x dy 2 , d y 3 u =0. Again, d x u or (3ax 2 —y 2 ) dx contains y as well as x, and we may obtain its differential on the supposition that y alone varies. We then have d v d x u = — 2ydy dx, d v 2 d x u = — 2 dy 2 dx, d v 3 d x u =0. Similarly, d x d v u = — 2ydx dy, d x d v 2 u = — 2dx dy 2 , d x d y 3 u =0. 229. In comparing these results it will be seen that d x d y u=dyd x u, d x dy 2 u=d y 2 d x u, d x d y 3 u =d y 3 d x u; also, d x dyd x u=d x 2 d y u=dyd x 2 u; in other words, the succes- sive operations indicated by d x and d v may take place in any order. It will be shown that this is true generally. 230. Continuity of a function of two variables. Let u=j(x, y), and let dx and dy be infinitesimal increments of x 250 228-231.] SUCCESSIVE PARTIAL DIFFERENTIALS, and y. Then f(x, y) is continuous at x, y, if 251 £f(x+dx, y + dy) =f(x, y), when dx and dy approach the limit zero in any manner whatever. If in Fig. 132 OA=x, AB=y, AD or BG=dx, BH=dy, and BP=f(x, y), then EQ=f(x+dx, y+dy). The condition of continuity implies that £EQ = BP when E is any point near B in the plane XOY. In what follows it is assumed that the functions and their derivatives are continuous for the values of the variables under consideration. 231. Let A x indicate an increment produced by the incre- ment dx of x, y being regarded as constant, A y having a corresponding meaning. Then if u=f(x, y), A x A y u=A y A x u. For, A y u=J(x, y+dy)-f(x, y), A x A y u=f(x+dx, y+dy) — f(x + dx, y) -[/Or, y+dy)- f{x, y)] = }(x+dx ) y+dy)-f(x+dx, y)-f(x ) y+dy)+f(x, y). (1) 252 INFINITESIMAL CALCULUS, [Ch. XLIII. The symmetry of the result shows that it would also be obtained for A y A x u, Ex. In Fig. 132 let u be the volume COAB . P. Then (1) expresses that HBGE . Q = FODE .Q-CODG . I-FOAH .J -{-COAB . P, which is obvious from the figure, as is also the fact that HBGE . Q is A y A x u as well as A x A yU . 232. Since u and 'its derivatives are assumed to be con- tinuous at and near x, y, JyU=dyU+I 2 (§ 42), where I 2 is an infinitesimal of at least the second order, and A X AyU =A X {d y u + 1 2 ) = d x dyU + h J where ^3 is of least the third order. Hence d X dyU AxAyU dx dy dx dy ' Similarly ' ^=4^ ^^ x u=A x A v n. Hence, dxdy 2 u=dxdydyU=dydxdyU=dyd y dxU=dy 2 d x u J and similarly for any combination. These results may obviously be extended to functions of any number of variables. _. . d x 2 u d y 2 u d x dyU dyd x u d x 3 dyU The expressions -t-«-, -3-9, 1 — t-, 3 — ir> 7 o -, ? etc. are ^ aaH a?/ J ax dy dy dx dx 6 dy frequently written d 2 u d 2 u d 2 u d 2 u d 4 u dx 2 ' dy 2 ' dxdy' dydx' dx s dy , etc. 232, 233.] SUCCESSIVE TOTAL DIFFERENTIALS. 253 Ex. I. In Fig. 132, u being as before the volume of OP, dxdyu _ J x Ayu H BGE . Q . dxdy~^dxdy~^ HBGE' i.e., the limit of the mean height of the solid BQ, which limit is BP or z. Hence d x d y u = z dx dy, .'. the volume = tween assigned limits, as in § 179. z dx dy be- 2. Verify that d y d x u = dxdyU or d 2 u d 2 u dy dx dx dy (2) u = sinxy, (3) u = ta,n~ 1 (y/x). 3. If u= (2x — 3y) s , verify that d x dy 2 u = dy 2 d x u. . T - . 3 2 u du d 2 U rt 4. If u = r n sin no. r 2 — -+r 1 — — = 0. dr 2 dr dO 2 5. If u^ia-xy + ib-yy + ic-z) 2 ]-*, show that d 2 u d 2 u d 2 u if (1) u = x\ogy, dx 2 dy 2 dz' = 0. 6. If u = f(y + ax)+F(y — ax), show that indicating any continuous functions. dx 2 a' fi 2 u dy'' , / and F Successive Total Differentials. 233. To find d 2 u. We have (§ 45) du=d x u+d y it, or 7 du 7 du , au=^— dx-\-^— dy, dx dy * f du\ , du , rt , /du\ , du a) whence d 2 u=d l^—) dx-\-^- d 2 x-\-d (^— ) dy+^r- d 2 y. \dx/ dx \dy) u dy u Find d ( ^— ) and d (-7- ) by substituting ^— and ~- \dx/ \dyl J dx dy for u in (1). The final result is du dy d u d u d 2 u du du d 2 u=^—z dx 2 +2^ — ;=— ■ dx dy-\-^~— 5 dy 2 +^— d 2 x J r -^- d 2 y. dx z ox oy oy z ox oy d s u may be found in a similar manner. 254 INFINITESIMAL CALCULUS. [Ch. XLIII. Ex. If u = c is a plane curve, du = and d 2 u = 0. If also d 2 x = (i.e., if x is the independent variable), show that d 2 u /du\ 2 d 2 u dudu d 2 u /du\ 2 d 2 y dx 2 V yl dx dy dx dy dy 2 ^dx/ dx 2= ~ /^A 3 \dy/ Extension of Taylor's Theorem. 234. If in the formula of Taylor's Theorem, § 207, we write ~ — , j. 2 , ... for f(x), f'{x), . . . , we obtain as an equiva- lent form f( x+ h)=Kx)+ d J^h+^-^+... (i) Let f(x, y) be a function of x and y, and let x become x+h, y for the present remaining unchanged. Then, from (1), /(»+*, y) = f(x, y)+ ^f h+ ^l »+. . . (2) If now y becomes y + k, (2) becomes Kx+h> y + k ^f (x ^^^ and each term may be expanded by Taylor's Theorem as follows, using u for f(x, y): df(x,y+k) h _ L ' dy j —^ U h-L ^ 2u hkj- lv ^7 lb ^Z IV I ^Z ^T lilv ~\ • . . • x da; ore do; or/ 3 2 /(a-, y + k)h 2 d 2 [u + . . .] /i 2 3 2 w h 2 dx 2 2! 3a; 2 2! 3a; 2 2! |T~ • . • 9 234.] EXTENSION OF TAYLOR'S THEOREM. 255 whence (3) becomes f(x + h, y + k) = f(x, y) + [_^ x h + ^ y k ~] If DE=—h + ^k, the form of (4) is the same as ox dy f(x + h,y + k) = u+Du+ — J r-^j +. . . D 2 Z) 3 A similar result would apply to functions of three or more variables. Ex. 1. Euler's theorem on homogeneous functions. Def. A func- tion u or /(#, ?/) is said to be homogeneous and of the degree n when f(mx,my) = m n J(x, y), where m is any number. For example: 2x 3 + y 3 , x 2 — xy+y 2 , (x 2 +y 2 )/(x 2 — y 2 ), (x — y)/{x 2 +y 2 ), ax^ +by$. Letra = l+r. Then f(x+rx, y+ry)=(l+r) n f(x, y). Expanding the first member by Taylor's Theorem and the second by the Binomial Theorem, du dv\ ( Jd 2 u rt d 2 u d 2 u\ r 2 l du dv\ ( jd'u _ d'u d'u\ r £ u+ [x— +y — ) r-Y lx 2 —-^+2xy Y — -) — + . \ dx u dvJ \ dx 2 u dxdu du 2 ) 2\ ey* it = [l+nr+n(n — 1)— + . . .]u* Equating like powers of r, du du x — Yy — = nu t dx J dy d 2 U d 2 u d ^ii x 2 —+2xy—- + y 2 —=n(n-l)u. dx 2 dx dy dy 2 256 INFINITESIMAL CALCULUS. [Ch. XLIII. The results may evidently be extended to higher derivatives, and to functions of three or more variables. 2. If u is homogeneous, show that d 2 u d 2 u . ^.du x — -+y = (n — 1)— , dx 2 dx dy dx d 2 u d 2 u , ^du x \-y — =(n — 1)— . dx dy u dy 2 v dy Maxima and Minima from Taylor's Theorem. 2 35« By the aid of Taylor's Theorem we may verify and extend the conclusions- of Chapter XVII for maxima and minima. If a is a value of x for which any function f(x) is a max. or a min., and h any small quantity, it is plain that f(a + h)—f(a) and f(a — h)—f(a) must have the same sign, viz., + for a min. and — for a max. Now and }(a-h)-f(a)=-r(a)h+r(a)~-r(^ l + - ■ ■ 5 and by taking h small enough the sign of the right-hand side will depend upon that of the first term which does not vanish. Hence there cannot be a max. or a min. unless /'(a) = 0, and there will then be a max. if /"(a) is — and a min. if /"(a) is + . But if f"(a) also = 0, there cannot be a max. or a min. unless f'"(a) also = 0> and there will be a max. or a min. according as p v \a) is — or +. It will thus be seen that there cannot be a max. or a min. unless the first derivative which does not vanish is of an even order, and that /(a) will be a max. or a min. according as this derivative is — or + . For a similar reason, from § 234 (4), a function u or fix, y) of two independent variables is a max. or a min. 235.] MAXIMA AND MINIMA 257 for values a and b of the variables if a and b satisfy du/dx=0 and du/dy = Q, and at the same time f U 2h 2 + 2 Vu hk A2 Ox 2 dx dy dy 2 (1) is not zero, and is in sign independent of the values of h and k. These conditions are satisfied if / d 2 u \ 2 . \dx dy) d 2 ud 2 u I d 2 u dx 2 dy 2 \dxdy> is +. For, (l)^Ah 2 + 2Bhk + Ck 2 = (Ah + Bk) 2 +(AC-B 2 )k 2 A and .'. has the same sign as A if AC — B 2 is +. TT .- du _ du _ , d 2 ud 2 u ( d 2 u \ 2 . Hence it ~ = u, ~- = 0, and ^-^ ^—15 — I ~ — ^- ) is + , u is Xn r \ox dy/ dx dy dx 2 dy d 2 u a max. or a min. according as ~— 5 is — or +. or Similarly for a function of three independent variables we must have du/dx=0, du/dy=0, du/dz=0, to solve for x, y, and z. Ex 1. u = x +xy +y 2j rx— 2y +4. du/d£ = 2a;+?/-hl, du/dy=x + 2y — 2. Putting these = and solving for £ and y we get x which make u a min., viz., If. 2. The max. value of (2ax — x 2 )(2by — y 2 ) is a 2 b 2 . 3. The max. value of (x — l)(y — l)(x +y — 1) is ^V« 4. The min. value of x z +y 3 — 3axy is —a 3 . 5. The max. or min. value of ax 2 +2hxy + by 2 +2gx-\-2Jy + c is a h g = 5 31 h b f 9 f c a h h b 6. Find a point such that the sum of the squares of its dis- tances from any number of given points (a,, 6 2 ), (a 2 , 6 2 ), . . . may / 1 1 \ be a mm. Arts. ( — Ia y — lb) . the centre of mean position, \n n I 258 INFINITESIMAL CALCULUS. [Ch. XLIII. 7. Given r x = a x x + b x y + c u r 2 = a 2 x + b 2 y + c 2 , . . . , show that the values of x and y which make r 1 2 +r 2 2 +r 3 2 + . . . a min. are ob- tained by solving the equations x I (a 2 ) +yl(ab) + l(ae) = 0, xl(ab) +yZ(b 2 ) + l(bc) = 0. These are the normal equations in the method of Least Squares. 8. To make with the smallest possible amount of sheet metal an open rectangular box of given volume, show that the length and breadth must each be double of the depth. 9. To cut circular sectors from the angles of a triangle so as to leave the greatest area with a given perimeter, show that the radii must be equal. CHAPTER XLiV. DIFFERENTIAL EQUATIONS* OF THE FIRST ORDER. 236. A differential equation is an equation containing one or more derivatives. The derivatives are usually represented by the corresponding differentials. The order of a differential equation is the order of the highest derivative in the equation. The degree of the equa- tion is the degree of the highest derivative when the equation is free from fractions and radicals affecting the derivatives. d^y dy Ex. t4 + 2 - +y=0 is of the second order and first degree. dx 2 dx (dv\ 2 dy --) +2- - — H 2/ =0 is of the first order and second degree. Partial differential equations are those which contain partial derivatives; other differential equations are called ordinary. 237. Ordinary differential equations frequently appear in the statement of problems in Geometry, Mechanics, Physics, etc., but for our present purpose they may be supposed to arise from the elimination of constants. -n ^y y dy y Ex. 1. y = mx, ~=m = —. .*. -f=—. (J/X X (J/X X This is a differential equation of the first order obtained by differentiating, and eliminating the constant m. It may be * For further information relating to differential equations see Mur- ray's Differential Equations (Lorgmans), from which some of the examples of this and the following chapter have been taken. 259 260 INFINITESIMAL CALCULUS. [Ch. XLIV. called the differential equation of all straight lines passing through the origin. 7 dy d 2 y 2. y = mx + b, — =ra, -^— = 0. * ' dx dx 2 .*. d 2 y/dx 2 = is the differential equation of all straight lines. Two constants, m and b, have been eliminated. The equation is of the second order. o , dy rt d 2 y 1 dy 3. y = ax 2 + b, -- = 2ax, -^ = 2a = — . -f. dx dx 2 x dx d 2 y 1 dv .'. -7-^ = =-, an equation of the second order. dx 2 x dx The elimination of n constants requires n+1 equations viz., the original equation and n derived equations. Hence the order of the resulting differential equation is equal to the number of constants eliminated. Eliminate a, b, c from the following equations: 4. y = ae mx + be- mx . Arts. d 2 y/dx 2 = m 2 y. 5. y = a sin mx + b cos mx. d 2 y/dx 2 = —m 2 y, 6. y = ax 2 +bx + c. d 3 y/dx 3 *=0. 8. 2/ 2 =(x-c) 3 . 8(dy/dxY = 27y. 9. y = ax 2 +bx. x 2 -r±-2x~ + 2y = 0. 238. An integral or solution of a differential equation is a relation between the variables which satisfies the equation. Ex. ?/ = A cos x, y = B sinx, y = As in x+B cosx, y = a sin (# + 6), d\ dx' 1 d 2 y y = a cos (x +6), are all solutions of the equation x^+2/ = 0. The solution which contains a number of arbitrary constants equal to the order of the given equation is said to be a com- plete integral or general solution. Particular solutions are those which may be obtained from the general solution by assigning values to the constants, 238-240.] DIFFERENTIAL EQUATIONS. 261 Separation of the Variables. 239. In some cases no special method of solution is re- quired. An algebraical rearrangement of the terms will cause the equation to take the form h(x)dx+f 2 (y)dy=0, and each term may be integrated. Ex. 1. 2x 2 y dy=(l + x 2 )dx is the same as rt , /l+x 2 \ 1 dx , 2 y d y=\—-r) dx= -2+ dx - Hence, integrating, y 2 = —x~ l +x+c, where c may have any assigned value (see § 96). 1 7707 dy dx ?<, y dx—x dy = dx+ x 2 dy, or = — — . * J y-\ x 2 +x Integrating, log (y — l) = log x — log (x 4-1) +log c,* .*. y — l=cx/(x + l). P>. (x 2 + y 2 — y)dx + x dy = is the same as x dy—y dx k-;) -Jdx-'r x- = 0, or dx 4- (i) d *- 1 + © = 0. Hence x 4- tan -1 (y/x) = c, or ?/ = £ tan (c — x). 4. x 2 y dy+m dx = Q. Arts. y 2 = 2m/x+c. 5. (x — y 2 x)dx + (y — x 2 y)dy = Q. x 2 +y 2 = x 2 y 2 +c. 6. x dy= (x 3 +y)dx. y = %x 3 +cx. 7. (x 2 y+x)dy + (xy 2 — y)dx = 0. xy+\og(y/x) = c. 8. y dy=(^x 2 +y 2 — x)dx. y 2 = 2cx + c 2 . 240. The separation of the variables is sometimes assisted by a substitution. The following is an important case. * In order to simplify the final result the constant may be written in the form log c, or in any other form which permits of any arbitrary value. 262 INFINITESIMAL CALCULUS. [Ch. XLIV. Homogeneous equations. If the given equation is of the form /i(z, y)dx + f 2 (x, y)dy=0, where the functions are homogeneous in x and y, and of the same degree, let y=vx. In the new equation in terms of v and x the variables will be separable. In some cases the substitution x=vy may be simpler. Ex. 1. xy 2 dy= (x 3 +y 3 )dx. liy = vx, v 2 {x dv +v dx) = (1 + v 3 )dx, or v 2 dv = dx/x. .' . %v 3 = log ex, or y 3 = Sx 3 log ex. 2. (x 2 —2y 2 )dx+2xydy = 0. Arts. y 2 = —x 2 log ex. 3. (x 2 + y 2 )dx = 2xy dy. y 2 = x 2 +cx. 4. y 2 dx + x 2 dy = xy dy. x = y /log cy. 5. (x+y)dy + (x — y)dx = 0. Arts. tsni- 1 (y/x)+log\ // x 2 -\-y 2 = c. 6. Show that the homogeneous equation f x {x 9 y)dx+f 2 (x, y)dy = 0, or /,(1, v)dx+f 2 (l, v)dy = dx / 2 (1, v)dv becomes — +7-7^ — r^ — m — n = 0- x /x(l, t>)+v/ 2 (l, v) 7. Show that an equation of the form f 1 (xy)y dx + f 2 (xy)x dy = can be integrated by the substitution y = v/x. 241. An equation of the form (ax + by + c)dx+ (a'x + b'y + c')dy =0 (1) is not homogeneous, but may be reduced to a homogeneous equation by the method of the following example. Ex.1. (3x-y-5)dx + (x+y + l)dy = 0. Let x = X+h, y=Y + k. Then, substituting, {3X-Y+3h-k-5)dX + {X + Y + h + k + l)dY~0. 241,242.] DIFFERENTIAL EQUATIONS. 263 Take h and k so that Sh-k-5 = and h+k + l = 0. .'. fc=»l, &= — 2. The equation is now (3X - F)^x + (X + y )dr = o, which is homogeneous. Let Y = vX. Then 1+v , dX 3 +v 2 X 1 i» whence, ——tan -1 — = + i log (3 +v 2 ) +log X = c V3 V3 „ F V-fc V+2 Bnt -Z-i=A-i=i- — tan- 1 — 1±L_ + ilo g [3(a;-l) 2 + (2/+2) 2 ] = c. V3 \/3(x~l) Hence to solve an equation of the form (1), drop the c and c', solve the resulting homogeneous equation, and in the result substitute x — h for x and y — k for j/, where A and k are the roots of the simultaneous equations ax + by + c=0, a'x+b'y + c' =0. The method would fail if a'x + b'y=k(ax + by), k being any constant; but the equation could then be solved by the substitution v =ax + by and the elimination of y or x. Ex. 2. (3x-22/-5)dx + (2x-32/-5)di/ = 0. ^4ws. (x+y) 5 (x — y— 2)=c. 3. (2x-y)dx-(4x-2y-l)dy = 0. Ans. 3(x-2y)+log (3y-6x+2) = c. Exact Differential Equations. 242. The result of differentiating f(x,y)=c or u=c is (§ 45) M dx + N dy=0, where M=du/dx and N=du/dy. dM d 2 u . dN d 2 u* Hence t^— = ~ ~ ■ and dy dydx 'dx dxdy ., . 3 2 ^ 3 2 u >. OQO , . dM dN r>Ut ~ — ^-^ = ^ — ^-— (§ ZoZ). . . -^— * — - — . oy ox ox oy oy ox 264 INFINITESIMAL CALCULUS. [Ch. XLIV. Conversely, if M dx + N dy=0 is an equation such that dM/dy=dN/dx the equation is an exact differential equa- tion, i.e., one obtained directly by differentiating without further change. The re-integral of M dx contains all the terms of u except those which are independent of x. Hence to integrate an exact equation, integrate M dx with regard to x, integrate with regard to y those terms of N dy which contain y only, and put the sum of the results equal to a constant. Ex. 1. (4x~ + 6x 2 y)dx + (2x 3 -2y)dy = 0. Here dM/dy = 6x 2 , and dN/dx = 6x 2 . Hence the solution is x*+2x s y — y 2 = c. 2. (2-2xy-y 2 )dx-(x+y) 2 dy = 0. Ans. 2x — x 2 y — xy 2 — \y z = c. 3. (x 3 +y)dx+x dy = 0. ix 4 +xy = c. 243. Integrating factor. After forming a differential equa- tion the result can sometimes be simplified by dividing by a variable factor. Conversely, a differential equation nay sometimes be made exact by multiplying by a factor. Ex. 1. (1 +xy)y dx + (1 — xy)x dy = is not exact, since dM/dy = \-\-2xy and dN/dx = l— 2x. Multiplying by l/x 2 y 2 the equation becomes (— +-)-* 1. p, c?v 1 — 2x 7. / + — — 2/-1. Ans. i/ = Vex — x log #. 2/ = x 2 (l +ce x ~ l ). 266 INFINITESIMAL CALCULUS. [Ch. XL1\ . 8. (l+x 2 )dy = (a+xy)dx. Ans. y = ax + cVl+x 2 . _ dy _ . 9. -^-+ay = b smmx. dx Ans. ?/ = — -(a sin mx—m cos mx) +ce~ a:c . 245. Bernoulli's equation. An equation of the form where P and Q are independent of y, may be made linear by the substitution —r=z. It is best to divide through by yn-l & J if 1 before substituting. ^ ., dy , . 1 % 1 1 log x Ex. 1. x-f +j/ = t/ 2 log«, or - *+— .— =— — . ax y 2 ax x y x Let — y = z, then • • • 1 y 2 dz dx dy = z X = dz log X X ' which is linear. Solving, z = log X + 1 + ex. • * • 11 — 1 log 2 • + 1 + ex ' 2 (1- ax -#2/ = 3:r2/ 2 . Ans. y- 1 cVl — X 2 -3 3 dx ±xy=x l { 2/ 3 . y- 1 Vl+x 2 + ce x2 Equations of the First Order but not of the First Degree. 246. Let dy/dx be called p. If pcssible solve the equation for p. Ex. 1. p 2 — (x — Sy)p — 3xy = 0, or {p — x)(p+3y) = 0. The equation is satisfied if p — x = 0, or p + 3y = ; i.e., if -r-x = 0, or - 2/ + 3/y = 0; 9 dx dx ' * ' . 245-248.] DIFFERENTIAL EQUATIONS. 267 or, integrating, if y — %x 2 +c = 0, or y + ce~ 3x = 0* These equations may be regarded as the solutions of the given equation, or they may be combined into (y — %x 2 + c)(y+ce- 3x ) = 0. 2. p 2 -9p + l8 = 0. 3. p 3 = ax 4 . Ans. (y-6x + c)(y-3x+c) = 0. 343(1/ + c) 3 = 27ax 7 . 247. When it is not possible or convenient to solve for p we may be able to solve for y, then, differentiating through- out and substituting p dx for dy, obtain a new equation in p and x which we may be able to integrate and thus find the relation connecting p and x. From this result and the given equation we may be able to eliminate p and thus obtain the relation connecting x and y, or, if this elimination is not convenient or possible, x and y may be left in terms of p as a third variable. Ex. 1. p 2 x — 2py+x = 0, or 2y = px+x/p. Differentiating, substituting p dx for dy, and reducing, dp/p = dx/x, .'. p = cx. Hence, substituting in the given equation, 2y = cx 2 + — . 2. p 2 —py + 1=0. 1 1 Ans. x = —- '-f-log p+c, y = p + —. 2p V 248. Instead of solving for y we may be able to solve for x, then differentiate throughout and substitute dy/p for dx, and proceed as above. Ex. 1. p 2 -px + l- 2. p 2 y + 2px = y. 3. p 3 -p 2 x + 1=0. 0. Ans. x=p+—, y = hp 2 — log p+c. r y 2 = 2cx+c 2 . 1 p 2 p 2 2 2 p 268 INFINITESIMAL CALCULUS. [Ch. XLIV. 249. Clairaut's equation. Singular solution. The method of § 247 is applicable to Clairaut's equation, y=px+f(p). ■ (1) Differentiating and substituting p dx for dy, dp[x + f'(p)]=0. From dp—0 we have p=c, and substituting in the given equation, y=cx+f(c), (2) the general solution. The equation is also satisfied if x + f f (p) =0, and eliminating p from this and the given equation we have another solution which is not contained in the general solution and which does not contain any arbitrary constant. Such a solution is called a singular solution. The general solution (2) represents, for various values of c, a family of straight lines. The singular solution represents the envelope of these straight lines. For the envelope of the family of lines is obtained (§ 157) by eliminating c from y =cx + f(c) and 0=x + f'(c), the same equations (with c instead of p) as those from which the singular solution is obtained. Ex. 1. y = px + a/p. The general solution is y = cx+a/c. Also x J \-f'(p) = isx — a/p 2 = 0. .*. p 2 = a/x. Substituting in the given equation we obtain y 2 = 4:ax, the singular solution. 2. Find the singular solution of y = px-\-p 2 . Arts. x 2 +4y = 0. 3. Find the general and singular solutions of y = px + a\^l +p 2 . Arts. y = cx + aVl +c 2 , . x 2 +y 2 = a 2 . 4. Solve y= — xp+x 4 p 2 . Leto^z -1 . Arts. y = c/x+c 2 . r 249.] DIFFERENTIAL EQUATIONS. 269 Examples. 1. dy/dx+y cot x = 2 cos x. Arts. y = sin x+c cosec x. 2. x 2 dy + x 2 y 2 dx+4:dy = 0. y~ 1 = x — 2 tsur-^x + c. 3. p 2 = px — y. y = cx — c 2 . 4. px 2 + y 2 = xy. x = y(c +logx). 5. (2x 2 +4:xy)dx + (2x 2 — y 2 )dy = 0. 2x 3 + 6x 2 y—y 3 = c. 6. px + y = x 3 y Q . y- 5 = §x 3 +cx*. 7. (x 2 — y 2 +2x)dx = 2ydy. x 2 — y 2 = ce~ x . 8. (2ax+hy+f)dx + (hx+2by+g)dy = 0. ax 2 + 6i/ 2 + fon/ +fa+gy == c* 9. 2xydx + (y 2 — 3x 2 )dy = 0. x 2 — y 2 = cy 3 . 10. x dx dy = y dx 2 +2 dy 2 . cx = c 2 y + 2. 11. x 2 p 2 = 2xyp+3y 2 . (xy — c)(y — cx 3 ) = 0. 12. y 3 dx = (2x 2 + Sxy 2 )dy. 2xy+y 3 = cx. 13. (y — a)dx=(x 2 +x)dy. (x + l)y = a + cx. 14. x 2 {y — px) = p 2 y. Let y 2 = v, x 2 = z. y 2 = cx 2 -\-c 2 . 15. Find the curve in which the subnormal is constant and = a. The condition is that ?/ dy/dx = a. Ans. The parabola ?/ 2 = 2aa; + c. 16. Find the curve in which the subtangent is constant and = a. Arts. y = ce . 17. Find the curve in which the perpendicular on the tangent from the foot of the ordinate is constant and = a. Ans. The catenary y = a cosh (x+c) /a. 18. Find the curve in which the area bounded by the curve, two ordinates, and the #-axis is proportional to the length of the bounding arc. [y dx = dA = d(as) = a\ // dx 2 +dy 2 ]. Ans. The catenary y = a cosh (x+c) /a. 19. Find the curve in which log s = x. Ans. y = \/e 2X — 1 — sec~ 1 e x + c. 20. Find the curve in which — a) -1 £ n , n a positive integer, 1 / nx n ~ l n{n — l)x n ~ 2 n\\ = —lx n + + i- +. . .+— 1 -hce ax . a \ a a 2 a n I ,_ v , . a sin mx-\-m cos mx 6. (D— a)- 1 smmx= — \-ce ax . a 1 +m l ,_ . — a cos mx-\-m sin mx 7. (D — a) -1 cosmx = - = \-ce ax . a 1 +m 2 gnx 8. (D-a)- 1 e nx = h ce ax when n ^ a, n — a and =£e a z-l-ce a £, when n = a. Linear equation of the second order with constant coefficients. 255. The equation may be written where A and B are constants and X is a constant or a func- tion of x. 276 INFINITESIMAL CALCULUS. [Ch. XLV. First suppose X to be 0. The equation is now ^ + A^+By=0, or (D 2 + AD + B)y =0. Put D 2 + AD + B into the form of factors (D-a)(D-b). (Since a and b would be the roots of D 2 + AD + B =0 if D were a symbol of quantity, they may be said to be the roots of the auxiliary equation z 2 + Az + B =0.) The equation is (D — a)(D — b)y=0. .-. y = (D-b)~ 1 (D-a)- 1 = (D-b)- 1 ce ax =e bx ( \e~ bx . ce ax dx + c^\ (1) /ce (a—b)x v =^*( — -T- + c 1 j,if Ma, = -— T^ + Cie^. a — 6 Since c/(a — b) may equal any constant, it may be repre- sented by c. .-. y=ce ax + c 1 e bx y if Ma. (2) But if o =a, (1) becomes e a * f c dx + ci) . .'. y=e ax (cx + c 1 ). (3) If a pnd b are imaginary, let them be m + ni, m — ni, where £==V — 1. Then (2) becomes iy __^g(m+m)a; i q g(m—ni)x == ^mx(Q^nix _i_ q g—nix\ =e mx [c(cos nx + i sin nx) +Ci(cos nx — i sin nx)], § 200, =e m [(c + ci) cos ft£ + (ci — C\i) sin n^]. Since c + Ci and ci — C\i may equal any two constants, they may be represented by c and c\. /. y =e mx (c cos nx + ci sin n#). (4) . 256 257] DIFFERENTIAL EQUATIONS. 277 256. Hence to solve D 2 y + ADy + By=0 find the roots of z 2 + Az-\-B-=0. If the roots are unequal real numbers a and b the solution is y =ce axj rCie bx . If the roots are complex numbers m + ni, m — ni, y =e mx (c cos nx + Ci sin nx). If the roots are equal numbers a, a, y =e ax (cx + ci). Ex.1. -^ + 3-"-l(h/ = 0. Ans. y = ce 2x + c.e-^ . ax 2 ax 2. g-8*-0. y^c+c^. ax 2 ax 3. 2" 2 S +2/ = ' y=e x (cx+c 1 ). d 2 v dy 4. —? 2 +2-^+10y = 0. y = e- x (ccos3x+c 1 sinSx). d 2/ u 5. ~- 2 = a 2 y. y = ce aX + c x e~ aX or =c cosh ax-\-c x sinh ax. 6. —-•-= —a 2 v. y = c cos ax + c, sin ax. 7. (D 2 +4D+5)?/ = 0. y = e~ 2x (c cosx + Cj sinx). 8. (D 2 +4D+4)2/ = 0. 2/ ^-^(cz+cj. 9. (D 2 +4D+3)2/ = 0. ^/^cg-^+c^-^. 10. CD 2 +4D+2) 2 / = 0. i/ = 6~ 2:c (c6 V2:c + c 1 6~ V2:c ). 11. (6D 2 -5Z)-6)?/ = 0. ^cei^+c^-f*. 257. The same method may be extended to higher orders of linear equations of the form d n y A d n ~ l y dx n dx n ~ l * v > the coefficients A . . . K being constants. For every distinct real root of the auxiliary equation, z n + Az n ~ 1 + . . , + K=0, 278 INFINITESIMAL CALCULUS. [Ch. XLV there will be a term of the form ce ax in the solution. If a occur twice the corresponding term will be e ax (cx + c{) y and if it occur three times the corresponding term will be e ax (cx 2 + ciX + C2), and so on. Corresponding to a pair of imaginary roots m + ni, m — ni, there will be a term e mx (c cos nx + c\ sin nx), and if the same pair occur twice the corresponding term in the solution will be e mx [(cx + Ci) cos nx+ (C2X + C3) sin nx], and so on. Ex.1. CD-l)(D-3) 3 y = 0. Arts. y = ce x + (c 1 x 2 +c 2 x + C3)e 3x . 2. (Z) 2 +4) 2 ?/ = 0. Arts. y = fax + c^ cos 2x + (csX + c 4 ) sin 2x. 3. (D 2 -4) 2 2/ = 0. y=(c 1 x+c 2 )e 2x + (c 3 x + c 4 )er 2 *. 4. D 4 y = a 4 y. y ■= c x e aX + c 2 e~ ax + cz cos ax+c 4 sin ax. 5. CD 2 -4Z> + 13) 2 7/ = 0. Ans. 2/ = e 2x \{c x x + c 2 ) cos 3a; + (C3X + c 4 ) sin 3a;]. 258. Returning to the linear equation, d 2 y. Ay R Y suppose now that X is not zero. The equation is the same as (D-a)(D-b)y=X=0+X, ... y^{j)-h)-^{D-a)- l {)+{D-b)- l {p-a)- 1 X. The first term is the solution of the given equation when X is 0. This therefore forms a part of the required solution; it is called the complementary function, the remainder of the solution being called the particular integral. The com- plementary function will contain two arbitrary constants, and as the complete solution of an equation of the second order cannot contain more, we need not introduce constants in finding the particular integral. If they are introduced they will simply reproduce the complementary function. ■I 258,259.] DIFFERENTIAL EQUATIONS. 279 Ex. 1. (D 2 -a 2 )y = e aX . The c. f. is ce ax + ce~ aX . The p. i. is (D-a)- 1 (D + a)- 1 e" x =(D-a)- i e — ip—ax e ax % e ax& x e 2aX 1 = (D - a)-^-^ . = (D- a)~ l e^ x 2a 2a 1 = — — ax 2a 1 e-ax e ax(i x = Kze aX x. Hence the complete solution is y = ce aX + c y e~ ax + xe a x /2a. \{D + a)- l (D-a)- l e aX gives - e ax x--—e ax . The last term is 2a 4cr included in the c. f.; hence the results are equivalent.] d 2 v dy 2. ^.+2-^+y = e 2x . Ans. y = e~ x (cx + c 1 )+he 2x . ax ax 3. (D 2 -l)y = 5x+2. y = ce x + c 1 e~ x -5x-2. 4. {D-l) 2 y = x. y = e x {cx + c 1 )+x-\-2. 5. (D-a) 2 y = e aX . y^e^icx + c^+^e^. 6. (D 2 -4:D+3)y = x. y = ce x + c 1 e 3x +%x + i. 7. (D 2 -4D + 3)y = xe x . y = ce x + c 1 e 3x ~ie x (x 2 +x). 8. (D 2 +a 2 )y = e* x . The p. \.= (D + ai)~ l {D -aiy'e^, i^V^l, pax pax pax = (D+ai)~ l a(l-i) a(l-i) .a(l+i) 2a 2 ' .*. y = c cos ax + Ct sin ax+e aX /2a 2 . 259. The last equation, (D 2 + a 2 )y = e ax , may also be solved as follows : Differentiating, D(D 2 + a 2 )y=ae ax , and multiplying the given equation by a and subtracting, (D-a)(D 2 + a 2 )y=0, a linear equation with the second member zero. The solution of this equation is 2/=c cos ax + Ci sin ax + c 2 e ax . 280 INFINITESIMAL CALCULUS. [Ch. XLV. The first two terms are the c. f. of the given equation; hence c 2 e ax is the p. i., where c 2 is to be determined so that c 2 e ax may satisfy the given equation. Substituting c 2 e ax for y in the given equation we find c 2 =l/2a 2 . Ex. 1. (D 2 -l)y = x 2 . Differentiating three times, D 3 (D 2 — l)y = 0. The c. f. is {D 2 -l)- l = ce x + c 1 e~ x . The p. i. is D~ 3 = c 2 x 2 + c*x + c A . Substi- tuting this for y in the given equation we find c 2 = — 1, c 3 = 0, c 4 = — 2. .\ y = ce x + c 1 e~ x — x 2 — 2. The p. i. might have been found more quickly by treating (D 2 — 1) _1 as if it were developable by the Binomial Theorem. Thus (D 2 - 1)-^'= - (1 -D 2 )~ 1 x 2 = - (1 +D 2 + ...)x 2 =- (x 2 + 2). d 2 v 2. — 2 + a 2 y = b sin nx. Differentiate twice, eliminate the right- (J/ X hand member and show that b y = c cos ax + c x sin ax -f — ; sin nx. a 2 — n 2 If a = n, substitute —bx/2n for b/(a 2 — n 2 ). 3. (D 2 -2D+5)i/=l. Ans. y = i + e x (c cos 2x + c x sin 2a:). 4. (D — l) 2 y = x 2 . y = e x {cx J rc l ) + z 2 +4a; + 6. 5. (D 2 -2D+5)2/ = sin2a;. 2/ = tV (4 cos 2# -f sin 2# ) + e x (c cos 2 x + c x sin 2x) . Change of Variable. 260. Equations cf the second order, like those of the first order, are sometimes made integrable by a change of variable. Change of the dependent variable. _ d 2 v dy Ex. 1. x 2 - T Ji 9 +x/-y = x*. dx 2 dx Let y = vx, then dy = x dv + v dx, d 2 y = x d 2 v + 2dv dx. Substituting, the equation becomes d 2 v dv dx 2 dx 260,261.] DIFFERENTIAL EQUATIONS. 281 whence (§251) y = \x z + ex + c x x- x . d 2 y fdy\ 2 2. y— -+ I ■- I =1. Lety 2 = v. Ans. y 2 = x 2 + cx+c t . ax 2 \dxl 3. d 2 y/dx 2 = a 2 x — b 2 y. Let a 2 x — b 2 y = v. Ans. b 2 y = a 2 x + c cos bx + c x sin bx. 261. Change of the independent variable. First sub- stitute dx d 2 y — dy d 2 x dx s (i) for d 2 y/dx 2 (§ 70). The subsequent change will depend upon the quantity which is to be the independent variable. This may be y, or a third variable z, x or y being an assigned function of z. It must be remembered that the second differential of the independent variable =0. Ex.l. ^+2^+^=0. (2) ax' ax x' Substituting (1) for d 2 y/dx 2 , (2) becomes dx d 2 y - dy d 2 x dy ahj X "dx* + 2x dx + 1? ~ °- (3) Let x = l/z and take z as independent variable. Then dx = — dz/z 2 , d 2 x = 2dz 2 /z 3 . Substituting in (3) we obtain d 2 y - + a 2 y = 0, whence y = c cos az + c x sin az. dz a . a . \ y = c cos — 4- c x sin — X X _ d 2 y 2x dy y 2. -— + ——--+ 2 \2 i==Q - Letx = tan^. ax 2 l-hx 2 ax (l+x 2 ) 2 Ans. y=(c + c 1 x)/\ // l+x 2 . d 2 y /dy\ ^ /dv\ 3 3. -7-j — x ( • j + e^ ( — ) =0. Make 2/ the independent variable. d 2 x The equation becomes - T — + x = ey. whence dy 2 x = c cos y + c x sin ?/ + \e y . 282 INFINITESIMAL CALCULUS. [Ch. XLY. 4. (l-x 2 )- 1 X-x^ = 2. Let z = sin 2. dx 2 ax Ans. ?/= (sin _1 ^) 2 +c sin-^ + Cj. d 2 y jdy\ 2 (dy x 5 - {1 - y2) d +y {£) ~ x (-l) =0 - Let ^ sin *> * to be the independent variable. Ans. x = cy + c 1 \/l—y 2 . 6. Show that the "homogeneous linear equation" of the second order x2 B +a 4l +hy = }{x) d 2 y dy becomes "r: + (a-l)-f +by = f{e z ) dz 2 dz (a linear equation with constant coefficients) by the substitution x = e z . d 2 y dy c 7. x 2 - J {+2x---2y = 0. Ans. y = cx+-±. dx 2 dx x 2 d 2 y dy 8. x 2 -— — x -+?/ = log x. y= (cx + 1) log x + ^x+2. dx u dx 9. x 2 ^+x^+4:y = x 2 . dx 2 dx * y = c cos (log x 2 ) + c x sin (log x 2 ) -f %x 2 . 10. (2+^)4^ +3(2 +x) d ¥ + y = 0. Let2+z = e 3 . dx 2 dx Ans. y=(2+x)- 1 [clog (2+aO+cJ. Examples. 1. Find the curve in which the radius of curvature R is equal to and in the same direction as the normal N. D L W J (l+p 2 )i . A7 ds ., ,.. dx 2 dy Ans. The catenary y = c cosh (x+cj/c 2. Find the curve in which R= —N. Ans. The circle (x-\-c x ) 2 J r y 2 = c 2 . 3. Find the curve in which R = e x , it being given that p = when £ = qo. Ans. y = c — sec~ 1 e x . 261.] DIFFERENTIAL EQUATIONS. 283 4. An elastic string (or spiral spring) is fixed at one end and hangs vertically. A weight is attached at the lower end and descends a distance a to a position of equilibrium 0. It is then pushed down a further distance b( when x=l, 5 = 5(3), .*. B when rr=-2, -1 = C(6), .\ C= ~i " re(rc-l)(rr+2) a: re — 1 x + 2 1 5 2z 3(x-l) 6(rr + 2)" Case 2 . When the factors of the denominator are of the first degree, but two or more of them are equal. l+3x Ex. 1. x{x + l) 2 ' The denominator shows that the partial fractions have denomi- nators x and (rc + 1) 2 and (probably) rc + 1. We therefore assume 1+3Z A B C = + x(x+iy x x+i {x+iy .'. l+3x = A(x + l) 2 +Bx(x + l)+Cx. If z = 0, 1 = A, .'. A = l. If a? — — 1, -2=-C, .*. C=2, and B may be found by giving any value other than and —1 to x y e.g., if x = l we have ("." A = 1, and (7 = 2), 4=lX2 2 + #X2+2Xl, .*. £=-1, l+3rr 1 1 2 7 + ' x(x + l) 2 a; x + 1 (rr + 1) 2 * x + 2 ABC D 2. TT-; — = 7 + (x+i)(x-iy x+i x-\ (x-iy (x-iy .'. x+2 = A(x-iy+B(x + l)(x-l) 2 + C(x + l)(x-l)+D(x + l). If re 1, 1=A(-2)V .'. A=-%. Iix = l,3 = D(2), .-. D-|. PARTIAL FRACTIONS. 289 To gel B and C give x any two arbitrary values other than 1 and —1; thus (remembering that A and D are found) if x = 0, 2=\+B-C+% or B-C = h and if x = 3, 2B + C = ; hence from these two equations x+2 1113 (x + l)(x-l) 3 8(3 + 1) 8(o;-l) 4(z-l) 2 2(z-l) 3 ' Case 3. When the denominator contains a quadratic factor which cannot be conveniently factorized. We now assume the numerator of the fraction with a quadratic de- nominator to be of the form Ax+B. This is equivalent to assuming tw r o fractions with denominators of the first de- gree and constant numerators. _ 1+x Ax+B C . . _ _ „,_ , x Ex. — ■ -=- -+-. .*. l+x = Ax 2 +Bx + C(l+x 2 ). x(l+x 2 ) 1+X 2 X If 3 = 0, 1 = C, /.. C-l. Ifz = l, 2 = A+B+2, .'. A+B = 0, 1 .-. A=-l, If s--l, 0=A-B+2, .-. A-£=-2, J 5 = L 1+x -z + 1 1 ^ _ J — m x(l +X 2 ) 1 +X 2 X ' If the given denominator had contained the square of 1 +x 2 , we should have assumed an additional term — — . (1 +x 2 ) 2 Be ides the methods explained in the above examples others may sometimes be employed with advantage. For instance, in the last example l + x=Ax 2 + Bx+C(l + x 2 ). Since the left- and right-hand sides are to be identical, the coefficients of like powers of x on the two sides must be equal ; we .*. have 1 =C, 1=B, 0=A + C, which give the same results as before. 290 APPENDIX. Note B. Curve Tracing. 1. In order to trace a curve accurately from its equation we must be able to express one of the coordinates in terms of the other, or both in terms of a third variable. When the rectangular equation contains terms of two degrees only, we may substitute mx for y and solve for x, and in this way obtain both x and y in terms of m. See foot-note, p. 53. 2. The following suggestions and remarks may be found useful in curve tracing, in order to shorten or verify the work. (I) Examine the equation for symmetry. When the equa- tion remains unchanged if — y is substituted for y the curve is symmetrical with reference to the line y = (the x-axis), for if the coordinates (a, b) satisfy the equation, (a, —b) will also satisfy it. This will always be the case if the equa- tion contains only even powers of y. Similarly the curve is symmetrical with reference to the line x=0 (the y-axis) if its equation is not altered when x is changed into —x. If the equation is unaltered by changing x into — x and y into — y at the same time, every line drawn through the origin and terminated by the curve is bisected by the origin; for if (a, b) satisfy the equation, ( — a, —b) also satisfy it and the origin is the middle point of the line joining these points. The origin is then called a centre; e.g., in the curves y=x 3 , y=s'mx, etc. The curve is symmetrical with reference to the line y=x if the equation is unaltered when x is changed into y and y into x, e.g., x s + y s =3axy (Fig. 28) ; and it is symmetrical with reference to the line y = — x if we can change y into — x and xinto —y without altering the equation, e.g., inx s — y 3 =3axy. If in polar equations the substitution of — 6 for 6 does not alter the equation, the curve is symmetrical with reference to the initial line (e.g., in Figs. 88, 89, 90, 92); and if we may at the same time change r into — r and d into — with- out altering the equation, the curve is symmetrical with r CURVE TRACING. 291 reference to a line through the origin perpendicular to the initial line (e.g., in Figs. 84, 85, 98). The origin is a centre when we can change r into — r without altering the equation (e.g., in Figs. 86,98). (II) Find the tangents at the origin (if the origin lie on the curve) and the shape of the curve near the origin (§§3 and 4 below); also, if possible, the points of intersection of the curve and the ax^s, and the directions of the tangents at these- points; the points where the coordinates are maxima or minima; the points of inflexion; the asymptotes rectilinear or curvilinear, etc. (III) No straight line can meet a curve of the nth degree in more than n points, and therefore no tangent in more than n — 2 points besides the point of contact, no asymptote in more than n — 2 points at a finite distance and no line parallel to an asymptote in more than n— 1 points at a finite distance, no line through a double point in more than n — 2 other points, etc. 3. The work of tracing a curve from its equation is often considerably lightened by obtaining a preliminary idea of the shape of the curve at certain points. When the origin is a point on a curve we can find the shape of the curve very near that point by expanding y into a series of ascending powers of x. Thus in Fig. 32, y = ±x(l + 2x)^ y and taking first the + sign we have by the Binomial Theorem, y =x(l + x— . . . ), or y=x + x< • • • The term x shows that when x is very small (and .'. the third and higher powers of x may be neglected) the curve lies above its tangent y =x both when x is + and when x is — ; in fact the curve is, for points near the origin on the branch touching y=x, shaped nearly like the parabola y=x + x 2 . Similarly on the other branch y = — x— x 2 +. . . ; hence this branch lies below the tangent on both sides of the origin. 292 APPENDIX. Similarly we may show that in Fig. 34 the curve near the origin is shaped nearly like the parabolas y=x 2 , y = —x 2 . 4. When it is not convenient or possible to express one coordinate in terms of the other we may proceed as in the following examples: Ex. 1. In the curve a 2 (y — x)(y + x)= — (y 2 +x 2 ) 2 , Fig. 27, con- sidering first the branch which touches y—x = ® (§52) we divide by a 2 (y+x) and write the equation in the form _(y 2 +x 2 ) 2 V ~ X a 2 (y+xY (1) For points near the origin on the branch in question y is very nearly equal to x, and the fraction in (1) must be very small; we shall get an approximation to its value by substituting x for y; this gives y = x—2x 3 /a 2 , which shows that the curve lies below the tangent when a: isl- and above it when x is — . For the other branch we write the equation in the form (y 2 +x 2 ) 2 y=-x- \ \ , (2) a 2 (y. + x) and remembering that y is nearly equal to —x we substitute — x for y in the fraction and get y= —x-\-2x 3 /a 2 , showing that the curve lies above the tangent when x is + and below when x is — . 2. In the curve 3axy = x 3 +y 3 , Fig. 28, the tangents at the origin are y = Q and x = 0. Writing the equation in the form x 3 + y* Sax we observe that on the branch which touches y = (the rr-axis) y is nearly near the origin, and substituting this for y in the CURVE TRACING. 293 fraction gives y=-x 2 /3a for the approximate form of the curve. For the other branch x 3 +y 3 x Say and writing for x in the second member we get x = y 2 '/3a for the required approximation. Thus the curve is shaped near the origin like a pair of parabolas. 3. Find the approximations to the three branches of the curve ay(y-V3x)(y + V3x) = x 4 , Fig. 36, near the origin. Arts, 6a(y — \ /r 3x) = x 2 , 6a(y + V3x) = x 2 y 3ay= —x 2 . 4. Also of ay 2 (y-x)(y+x) = x 5 , Fig. 37. Ans. ay 2 =—x 3 , 2a(y — x) = x 2 , 2a(y+x)= — x 2 . 5. Show from these approximations that the radii of curvature at the origin are \a and ±24a in Fig. 36, and 0, ±2^/2a in Fig. 37. (Cf . § 88, Ex. 2.) 5. The asymptotes of a curve may be obtained by expand- ing y into a series of descending powers of x (see § 57) . When it is impossible or difficult to express one of the coordinates in terms of the other we may proceed in a manner similar to that of § 4 above, beginning, however, with the terms of the highest degree instead of those of the lowest. (See § 59.) Ex. 1. x 3 +y 3 = 3axy, Fig. 28. Here x + y is a factor of the terms of the highest degree, and we may write the equation in the form 3axy .. . y=-x + - -— . (1) x L — xy +y* Now the infinite branch is in the direction of the line y— — x, and therefore when x is very large, y is nearly equal to — x; hence we shall get an approximation to the fraction in (1) by substituting — x for y; this gives y = — x — a> which is the nearest linear approximation to the 'curve, and is therefore the equation of the asymptote. Writing — x — a for y in the fraction will give a second approximation, viz., a 3 294 APPENDIX. from which it appears that the curve lies above the asymptote whether a; is + or — . 2. Find by this method the asymptotes of the following curves: (1) x 3 (y — x) = a{y z +x*). Arts. y = x+2a. (2) xy 2 (y — x) = y 3 — 2x 2 y + x 2 . x = l, y=l, y = x — l. (3) (x + 2y)(x-y) 2 = 6a 2 (x + y). x+2y=Q, x-y= ±2a. Examples. 1. Trace the following curves: * (1) y = x(x 2 -l), (fe) y 2 = x 2 (x-l) y (11) x 5 + y* = a 3 , (2) y(x 2 -l) = x, (7) x 3 -y 3 = 3axy, (12) x{y-x) = ay 2 i (3) y(l + x 2 ) = x, (8) x* + y 4 = a 2 xy, (13) x(y-x) 2 = y\ (4) y 2 -x 3 (x + l), (9) x 5 +y f > = 2a 3 xy, (14) a 2 ?/(a;+2/) = z 4 . (5) 2/ 3t =x 3 (z-l), (10) a: 5 +2/ 5 = ax 4 , 2. Trace the following polar curves: (1) r = a sin 20, (6) r = atan0, (11) r(0 2 -l) = a0, (2) rsin20 = a, (7) r 2 = a 2 d, (12) r0 2 = a(0 2 -l), (3) r = a sin 30, (8) r = a0 2 , (13) r(0 2 + l) = a0 2 , (4) rsin30 = a, (9) rd 2 = a, (14) r0 = tan 0. (5) r 2 = a 2 sin 30, (10) r(l + 0) = a0, Note C. Hyperbolic Functions. (For definitions and graphs of these functions see Ch. VIII.) 1. The relations connecting the hyperbolic functions are similar to those connecting circular (trigonometrical) func- tions, and are easily proved by ordinary algebra, etc. Some of them are as follows : cosh 0=1, sinh0 = 0, tanh0 = 0, etc. cosh ( — x) =cosh:r, sinh (— x) =— sinhz, tanh (—x) = — tanhrr, etc. * Some of these examples are taken fiom Frost's Curve Tracing, to which the student is referred for further information on this sub- ject. HYPERBOLIC FUNCTIONS. 295 cosh 2 x — sinh 2 a; = 1 , sech 2 x = 1 — tanh 2 x, cosech 2 x = coth 2 x — 1 . cosh (x±y) = cosh x cosh 2/±sinh x sinh y, sinh (x±y) =sinh x cosh y±cosh x sinh y, cosh x + cosh y = 2 cosh \{x+y) cosh \{x — y), cosh x — cosh y =2 sinh ?(x + y) sinh §(^ — 2/)? sinh x + sinh y =2 sinh J (^ + 2/) cos h i( x ~~y)y sinh x — sinh ?/=2 cosh i(x + y) sinh §(# — y), cosh 2x=cosh 2 £ + sinh 2 :E, = 2 cosh 2 x —1 = 1 + 2 sinh 2 x, sinh 2x=2 sinh a; cosh x, tanh £±tanh y tanh (x±y) tanh 2x = l±tanh x tanh ?/' 2 tanh x 1 + tanh 2 #* 2. The differentials, integrals, etc., are as follows: d sinh x = cosh x dx, d cosh x =sinh x dx, d tanh x =sech 2 x dx, cosh x dx =sinh x. sinh x cte=cosha;. sech 2 x do; = tanh x. d coth x = — cosech 2 x dx, cosech 2 x dx = — coth x. rf sech x = — sech x tanh # dx, sech x tanh xdx^ - sech x. dcosech.r= —cosechxcothxdx, cosechxcothxdx= — cosechx. cfsinn l - - , f ^ = S inh-i* = log ^ +Va:2+a ' 2 ' a Vx 2 + a 2 JVx 2 + a 2 a °\ a 296 APPENDIX. a cosh" 1 — =■ a Vx 2 -a 27 - , = cosh 1 -=lo2; \/x 2 -a 2 a °\ a , 1 1 i i **^ a ax I ,, i atanh -= — „, x < \a, a a 2 — x 2 dx , , . N 1 , - a; 1 , fa + x ,*—, -s(^|< 1«) = — tanh i-=_log i a^ — X' a a 2a a — x a cotn - = — oj # > a, a a 2 — x 2; x a adx ax ^ i I v i ,i _i 3/ i i /x -r a\ ; -(x >a) = — coth 1 -=-—lo2; ( ) 5 -x 2V ' ] y a a 2a 5 \x-a/ d sech * — = — a xV a 2 — x 2 C dx. 1 , - x 1 , / x \ — - = seen i-=-log ( ) JxVa 2 -x 2 a a a Va+vV-x 2 / _ . - x a ax a cosech -1 — = — a xVa 2 + x 2 ' C dx 1 , , x 1 , / x v — = — cosecn" 1 - = - log ( ) JxVa 2 + x 2 a a a Va+va+x/ /v»0 /Y»5 sinh x = x + ^-: + ^-.+ . . . o! 5! cosh x=l + —. + — + . . . Z\ 4 ! tanh -1 x^ x + tt + ^ +. . . 6 5 . _ _J_x 3 1 . 3x 5 1 . 3.5 x 7 smh x-x 2 3 + 2.4 5 2 4.67 +### If isV-1, cos ix = cosh x, sin ix=n sinh x, cosh ix = cos x sinh ix =i sin x. — — HYPERBOLIC FUNCTIONS. 297 x 2 y 3. At any point of an ellipse — + ^-=1 (Fig. 136) we may a< put x =a cos u } y=bsmu } since cos 2 w + sin u=l. In this Fig. 136. case u=2 area AOP/ab (see Ex. 14, p. 162); it also = the eccentric angle AOQ. y At any point of a hyperbola -g— ^=1 (Fig. 137) we may put x = a cosh u, y=b sinh u, since cosh 2 ^ — sinh 2 ^ = 1 . In this case u=2 area AOP/ab =log ( — h^), (see Ex. 14, p. 140). fay)' Fig. 137. Fig. 138. If 6 = a the ellipse becomes the circle x 2 + y 2 = d 2 , and the hyperbola the equilateral hyperbola x 2 — y 2 =a 2 . Also u is in both cases the measure of the area of the sector AOP when \a 2 is taken as unit area. The circular and hyperbolic 298 APPENDIX. functions may be defined in terms of u, and correspond- ing to y/a=sinu, x/a=cos u, y/x=t&riu, etc., - for the circle ; we have y/a =sinh u, x/a =cosh u, y/#=tanh u, etc., for the equilateral hyperbola. 4, Gudermannian. If 2= log tan (\n + %0) or log (sec + tan 6), is called the gudermannian of z (gd z) and z =gd~ 1 0. Since e*=sec # + tan 0, .*. e~ z =sec — tan 0. Hence cosh 2=sec 6, sinh 2=tan 6, tanh 2=sin 0, etc. Thus if is tabulated for values of z the hyperbolic functions may be obtained from a table of circular functions. Differentiating one of the relations connecting and z, we obtain dd =sech z dz, or dz =sec dd. .*. d(gd z) =sech z dz, and d(gd -1 #) =sec dd. The inverse gudermannian is also written X{0) and called the lambda function, i.e., X{6) =log tan {\n+\0) =log (sec # + tan 0). Ex. Show that tanh ^ = tan \ 6. 5. In the equilateral hyperbola x 2 — y 2 =a 2 , Fig. 138, let u be, as in § 3 above, the area of the sector AOP in terms of |a 2 as unit area. From the foot M of the ordinate MP draw MB tangent to the circle x 2 + y 2 =a 2 . Then x/a =cosh u and also=sec 0. .-. 5 is the gudermannian of u. It may also be proved that (1) MB=y, (2) tanh w=tan AOP =sin 0, (3) the line through parallel to BP bisects both sectors AOB, AOP, and the chord AP. MECHANICAL INTEGRATION. 299 Note D. Mechanical Integration. 1. Sign of an area. Let a straight line AB of constant length move in a plane to any other position A r B r , thus describing or sweeping out an area. Let it be agreed that any portion of AB describes a positive or a negative area according as, when viewed from A, it moves toward the left or the right. Thus the whole area is + in Fig. 139, - in Fig. 140, while in Fig. 141, BOB' is + and AOA' is -. B' b a A' Fig. 140. Fig. 141. 2. Measurement of the area. AB can be moved to any other position A'B f (Fig. 142) by (1) a translation to A'D, during which the points in AB describe straight lines, and (2) a rotation about A', during which the points describe arcs of circles. The middle point M of AB moves first to F and then to M'. Take ME perpendicular to AB. The area of the parallelogram AD =AB . ME, the area of the sector DA'B' =A'D . FM r ) hence the whole area = AB(ME + FM'), i.e., ABXthe total normal displacement of its middle point. Suppose a wheel to be attached to AB at M with its axis in the direction AB, and that suitable graduations record the number of revolutions and parts of a revolution which the wheel makes. Let n be this number, i.e., the change cf reading of the recording circles between the time of starting and any subsequent time. Let c = the length of the circumference of the wheel; then en is the distance 300 APPENDIX. rolled through by a point in the circumference of the wheel. Take b for the length of AB. During the motion of transla- tion the wheel rolls over ME and slides through EF, during the rotation it rolls over FM'. Hence the total normal displacement of M =cn, and the total area described by AB=bcn. If, as in Figs. 139, 140, 141, A and B describe curves, imagine the motion to be a combined translation and rota- tion with infinitesimal displacements, any of which may be negative as well as positive. Then the rate at which the area is described is bXrate of normal displacement of M, Fig. 143. and hence the total resultant area =6 X total normal displace- ment of M =bcn. 3. Consider now the effect of putting the wheel at any point L in AB, Fig. 142. The distance rolled over by the wheel is now LG+HL', and hence the normal displace- ment of M =cn+FM'-HU=cn+(A'F-A'H)d=cn+hd, if LM=h. In a circle (Fig. 143) of radius h draw OP, OF' parallel to AB, A'B'. Then hd=PP'. Hence the area described by AB=b(cn + PP'), and if AB moves to any new position to which OQ is parallel, the resultant area swept out =b (en + PQ). If AB moves about, turns back, and finally returns to its first position, Q returns to P and the resultant area =bcn, as if the wheel were at M. But if AB makes a complete revolution and returns to its first position the resultant area =6(cn+2^/i). MECHANICAL INTEGRATION. 301 4. Closed curves. Let a straight line move so that its extremities describe any closed curves. Then in all cases the area swept out by the line will be equal to the arith- metical difference of the areas of the curves described by its extremities. When the areas are without one another, one will be described on the whole positively and the other on the whole negatively, while the area be- tween them, if swept out at all, will be swept both positively and negatively. When they inter- sect, the common portion, in so far as it is swept at all, will be swept . Fig 144 both positively and negatively; the rest as before. When one curve lies entirely inside the other, the portion of the foimer which is swept at all will be swept both positively and negatively. 5. Amsler's polar planimeter consists essentially of two bars, CA, AB, hinged at A, a recording wheel being attached to AB at any point L. C is fixed while B is moved round a curve. But if A is constrained to move along any line — whether straight or curved — without enclosing any area, Fig. 145. the area of the curve traced out by B is equal to the resultant area swept out by AB, and hence will be ben if AB returns to its starting place without making a complete revolution. But if C lies inside the curve described by B, AB makes a complete revolution, and the area of the curve described by B =b (en +2nh) + circle described by A tl/ , n is 07/ 2nbh + 7ia 2 \ = 6(cn+2^)+^a 2 =6cfnH ^ J. 302 APPENDIX The second term in the parentheses is constant (indepencU ent of n) and should be engraved on, or otherwise supplied with, the instrument. This number is then to be looked upon as a correction to n when the planimeter makes a complete revolution. 27ibh + iza 2 is evidently the area of the circle described by B when n remains =0, i.e., when the instru- ment is set so that the wheel slides without rolling, or when the perpendicular from C on AB passes through the wheel. By sliding the bar AB through a sleeve to which the hinge and the wheel are attached, its length may be altered and the instrument adapted to different units. Thus if the circumference of the wheel =c centimetres, and b is taken = 100/c, bcn = 100n, and hence the area is found in square centimetres by multiplying n by 100. Similarly if the circumference of the wheel is c inches and b is taken = 10/ c, the area ben/ = 10n square inches. 6. As we proceed from B to C by way of P, Fig. 146, I* x changes from OD to OE, and y dx is the area DBPCE; but if we proceed from C to B by way of P r , each element of area such as y dx is negative since dx is negative, and hence y dx is the area CEDB, but is negative. Hence if we sum the elements such as y dx in the order of proceeding clockwise round the curve, the result =DBPCE - DBP'CE = BPCP', the area of the curve. Let A = this area, M = the sum of the moments of the elements of the area w T ith respect to OX, 7= the momerrt of inertia of the area with respect to OX. Then A = y dx, M ' y__ 1 y ax . Q ~ o y 2 dx, I = y dx . W 1 3 y s dx. 7. Amsler's mechanical integrator. In this instrument one end of a sweeping bar FP traces a closed curve, while the other end is constrained to describe a straight line OX. Hence this part of the instrument is virtually a planimeter, and the area A = bcin, w r here b is the length of FP, c x the cir- MECHANICAL INTEGRATION. 303 cumference of the wheel W x which FP carries, and n x the change of reading of this wheel when the circuit of the curve has been made. FP also turns two arcs of centre F and radii 2a and 3a, which turn circles each of radius a, the circles carrying wheels W 2 and W s . The three wheels roll simul- taneously on the plane containing the diagram to be inte- grated. When the axis of W ± makes an angle with OX Fig. 146. the axes of W 2 and W 3 make angles \iz— 20 and 30, respect- ively, with the same line. For y substitute b sin 0. Then A = b sin dx, M =>\b 2 |"sin 2 dx, I = J6 3 [sin 3 dx. Now 2 sin 0=1 -cos 20 = 1 -sin (^-20), and sin 30 = 3 sin — 4 sin 3 0, or sin 3 = f sin — J sin 30. .-. M=W [1-sin (fr-2d)]dx= -\b sin {\n-2d)dx J since dx=0 for the complete circuit of the curve. Also, 1=W = i& 3 (fsin0-isin30)dx sin ddx—ihb 2 sin 30 dx. 304 APPENDIX. But A = bciUi f .*. sin 6 dx=cini, i.e., when the axis of a wheel makes an angle 6 with OX, sin 6 dx=C\U\. But the ■ axes of the other wheels make angles \tz — 2d and 3d with OX. /. sin (\n — 2d)dx = c 2 n 2 , and sin 30 dx C3713. .\ A =bcini, M=lb 2 c 2 n 2 , I =J6 3 cini — Y2^ Sc s n s- In the instrument under discussion the maker has taken 6=2 decimetres, ci= J dec, c 2 =cs =f dec. .-. A=n 1} M = fn 2 , /=m-|n 3 . The height y of the centre of gravity of the area above 0X = M/A, and the moment of inertia with respect to an axis through the centre of gravity and parallel to 0X = I-AP = I-M 2 /A. Results may be changed from decimetres to inches by multiplying y by a, A by a 2 , M by a 3 , and I by a 4 , where a=3*937, the number of inches in a decimetre; a 2 = 15*500, a 3 = 61'023, a 4 = 240'290. MISCELLANEOUS EXAMPLES. 1. Prove Leibnitz 's theorem for the nth differential of a product: 71(71 — 1 ) d n (uv) = (d n u)v + nd n ~ 1 u dv + — — — d n ~ 2 ud 2 v + . . ,+ud n v. [By induction from d(uv) =v du+u dv.] 2. If y 2 =a 2 +2xy, d 2 y/dx 2 =a 2 /(y — x) 3 and d 2 x/dy 2 = —a 2 /y*. 3. The maximum value of (1) X =1-202. 4. Show that the turning points of the curve y = sin x- 1 are where x=2/(nn), n being any odd integer. (The number when x = from any assigned value is therefore infinite.) 5. Given the volume of a right circular cylinder, show that the surface is a minimum when the altitude =the diameter. 6. The height of the greatest rectangle which can be inscribed in a given right segment of a parabola is two-thirds of the height of the segment. 7. Find the area of the rectangle circumscribing the loops of the curve ay 3 -Sax 2 y=x 4 (Fig. 36). Arts. 9a 2 . 8. 4 sin 4 0d0=*O45, 9. |tan 4 d0=*119. JO f 4 f °° dx 10. sec 4 0dfl=H. 11. -7— -i=-215. Jo Ji x +x 10 I" 1 dx f°°__^_ _3? 1Z ' ] l+x+x 2 ~ W ^ l6 ' J (i+*yi6- m M [ a x 2 dx , - „„ [s dx 14. ~m~ w*='174:. 15. =T317. J (x 2 +a 2 )* J cos a; n .„ f 3 dx . „ n ^ „„ [3 dx 16. — r =l'732. 17. — r =2*391. J cos 2 # J cos 3 a* 305 306 APPENDIX. 18. 20. 22. it 3~ dx cos 4 :r =3-464. 2 sec x dx='522. o 2 dx 2 (1 +cos x) 2 3 ' 7T -.. f2l — cos 3 # 7 24. — ^—dx=2. J sm 2 z 26. 28. 30. 32. 34. 36. 38. 39. 40. dx a^{x — a){b — x) = n. '2 dx 2 + cos x ='604. o 1 x dx (x + l)(x 2 + l) x%(l —x)%dx=-^^. 'cos \Q 128* sin dd = X (V)- foo 6 _a:r cos mo: dx o * / x 2 a 19. 21. 23. 25. 27. 29. •2 sin 2 x^=-002. ri e^cos x dx = 1*378. o 3~ eta* * sin 2x 1 ='275. dtf ^sin 2 ^ cos 6 T =•695. 2+ cos x dx ='219. 31. 33. sin £ + cos a; = 1*813. --=•810. x 2 (l-^)^='152. a a — x S :- X ) cos wx eta a 2 +m 2 1 35. .37. x dd dx=^7za. g + ^g-m+logCOBd. sec 1 dx a # 2 +2a; cos a + 1 2 sin a' m 2* d# 1 , ft n\ =-rtan- 1 1— tan ) . ab \a J a 2 cos 2 + b 2 sin 2 # dx 2 (x + a)% — (x + b)* Vx + a + Vx+b 3 a-b n n " ,+ 2ttV "* dx 7T r+7 2= 4" III/ IV 41 * £n = cc \ n 2 + l 2 + 7l 2 +2 2 + Let l/n=dx. 42. The area of the evolute of the ellipse =— : — n. 8 ab MISCELLANEOUS EXAMPLES. 307 43. A cycloid revolves about its axis of symmetry. Show that the volume of the solid is 7ra 3 (%x 2 — f) and that the convex surface is 87ra 2 (7r-|). 44. A hemispherical bowl of 1 ft. radius is filled with water which then runs out of an orifice at the bottom \ sq. in. in sec- tional area. Find the time of emptying, assuming that the velocity at the orifice = ^2gx, where x is the height of the surface of the water above the orifice. Arts. 1 m. 46 s. 45. Find the centre of gravity of the area between the curve (x/a)* + (y/b)% = 1 and the axes. Ans. ~x =ia } y =\b. 46. Prove that X(x) =x +— +— + . . . mm -, i O n= i°cosn7r Qosnx 47. a;sina; = l-i cos x=2 I — , for \ — n, n\. n=2 U 2 -l i . J 1 ^ ( - l) n n sin nx . . 48. x cos x= —i sm x+2 2 — -, for J — n, n[. n=2 % 1 49. x 3 =— +— 2 — cos?i7rH — 4 (1— cos W7r) cos no:, for [0, iz\. 4 n n =i Ln 2 n J 1 1 1 7T 4 1 1 1 ^ Hence prove that -^ + - 4 +j A +. . . =— , — +— +— + . . . =— . 50. Find (1) the area, and (2) the length, of one loop of the curve r n =a n cos nd. Wi+I\ r(l\ ■ m 2 ifp <2) ~Ta^r\ \nl \2n 2/ 51. Find the area in the first quadrant between the axes and «—(!)"+(!)"-• t ['•(i+i)]' Ans. -; — — ab. r(i+±) Show that the area = ab when n is infinite. 52. Find the area of x 3 +y 3 =a 3 in the first quadrant, and the whole area of x A + y 4 =a\ Ans. 0*883a 2 , 3*708a 2 . 53. Find the area of one oval of the curve y 2 =a 2 sin (x/b). Ans. 4*792a&, 2" dd 54. Show that Vsin 6 dd . o o Vsin 6 7Z. 308 APPENDIX. 55. The curve r m cos md=a m rolls on a straight line. To find the differential equation of the locus of the polar origin. Let the straight line be taken as z-axis, and the polar origin be at (x, y). Then r=the normal at (x, y), and 2/=the perpen- dicular on the tangent at (0, r). Hence u yVl +p*' y 1 o fdu\ 2 , —=u +\T d ) , a m u m =cosmd. Eliminate u and 6 and show that (1 +p 2 ) 1 ~ m = {y/a) 2m . 56. A parabola rolls on a straight line. Show that the focus describes a catenary. 57. Find the centre of gravity of the arc of the quadrant of an ellipse (semi-axes a and b, eccentricity e). Ans. 5-| (l +^(1 -e 2 ) log ~~j /E(e, fr), y =- (Vl-e 2 -f-sin-^j /E(e, in). 58. An ellipse (eccentricity e) and a circle have equal areas. Find the ratio of their circumferences. Ans. — — L -~ . 7r(l — e 2 p 59. Find the curves which make an angle a with the curves r n = a n cos n () j r n cos n Q = a n t Ans. r n = c n cos (nd + a), r n cos (nO — a) = c n . ™ ^. dx dy . ^ dx dy rt . , .- 60. Given — -~ +w = sm 2J, — +-/ +# = 0, show that dt dt * ' dt dt t t x = ce 2 + c x e 2 — f cos 2£, t t y = -(V2 + l)ce V2 + (V2-l)cx6 V2 +| sin 2* +f cos 2*. TABLES. PAGE 1. POWERS, NAPIERIAN LOGARITHMS, ETC 310 2. CIRCULAR FUNCTIONS, 1 312 3. CIRCULAR FUNCTIONS, II 313 4. HYPERBOLIC FUNCTIONS 314 5. LAMBDA FUNCTION 315 6. GAMMA FUNCTION 315 7. FIRST ELLIPTIC INTEGRAL 316 8. SECOND ELLIPTIC INTEGRAL 3*6 309 310 TABLES. 1 . POWERS, NAPIERIAN LOGARITHMS, ETC 1 e(io^) X x~ l X 2 X 3 X 1 (io*)£ X 5 log e # log 0*1 IO ■OI 'OOI 316 1 000 0-464 -2-303 o-oco 0*2 5 ooo •04 • 008 447 1 414 585 — I 609 6 93 o*3 3 333 ■09 '027 548 1 732 696 — I 204 I 099 0*4 2 500 l6 •064 632 2 000 737 — O 916 I 386 o'5 2 000 •25 125 707 2 236 794 — O 6 93 I 609 o*6 I 667 36 216 o- 775 2 449 843 — p- 5 J i I' 792 o*7 I 429 49 343 O' 837 2 646 888 — O- 357 I' 946 o-8 I- 250 64 512 894 2 828 928 — O' 223 2- 079 o-9 I- III 81 0< 729 O' 949 3 000 965 — 0- i°5 2- 197 10 I' 000 1 00 I 000 I ■ 000 3 162 1 000 O' 000 2« 3°3 ri 0- 909 i- 21 I' 33* I' 049 3" 317 I- 032 O' 095 2- 398 1*2 O- 833 1 44 I< 728 I' °95 3' 464 1 ■ 063 o- 182 2- 485 i'3 o< 769 1 69 2- 197 I- 140 3" 606 1 • 091 262 2- 5 6 5 i*4 o 714 1 96 2 744 I' 183 3' 742 i- 119 o- 33b 2 639 i'5 667 2 25 3 375 I 225 3' 873 1 !45 O' 405 2- 708 r6 625 2 56 4 096 I 265 4' 000 i' 170 O' 470 2- 773 i'7 o 588 2 89 4 9*3 I 3°4 4' 123 i- J 93 o- 53 1 2 ^33 r8 O' 556 3 24 5 832 I« 342 4' 243 i« 216 o< 588 2 890 rg o 526 3 61 6 859 I' 378 4' 359 i- 239 o- 642 2 944 2*0 o 500 4 •00 8 -ooo I 414 4' 472 i' 260 0- 6 93 2 996 2*1 o •476 4 -41 9 261 I 449 4' 583 I- 281 o« 742 3 045 2*2 o •455 4 84 10 648 I 483 4' 690 I- 301 o< 788 3 091 2 # 3 o •435 5 •29 12 167 I 517 4' 796 I- 320 O' 833 3 !35 2-4 o •417 5 76 13 824 I 549 4' 899 1 ■ 339 0' 875 3' 178 2*5 o 400 6 ,2 5 15 .625 I 58i 5 000 i- 357 o- 916 3 219 2*6 o 385 6 76 17 576 I 612 5 099 I- 375 o- 956 3" 258 2'7 37° •29 *9 .683 I 643 5' 196 1 392 O' 993 3 296 2-8 o 357 7 84 21 '952 I 6 73 5 292 1 41c I 030 3 33 2 2*9 o 345 8 4i 24 389 I 7°3 5 385 1 426 I 065 3 367 3'0 o 333 9 00 27 000 I ■732 5 '477 1 442 I 099 3 401 3'i 3 2 3 9 61 29 79 1 I 761 5 568 1 458 I 131 3 434 3- 2 '3*3 10 ■24 32 •768 I 789 5 657 1 •474 I 163 3 466 3*3 o '3°3 10 .89 35 •937 I •817 5 •745 1 ■489 I 194 3 '497 3 # 4 o 294 11 •56 39 •3°4 I ■844 5 ■831 1 •5°4 I 224 3 526 3*5 o 286 12 ■ 2C 42 •875 I .871 5 •9l6 1 .518 I ■253 3 '555 3'6 o 278 12 .96 46 .656 I •897 6 •OOO 1 •533 I •281 3 584 3*7 o -270 I 3 .69 50 '653 I .924 6 ■08 3 1 '547 I .308 3 611 3'8 o -263 14 .44 54 •£72 I •949 6 ' 164 1 ■561 I '335 3 638 3*9 o 256 15 21 59 ■3i9 I •975 6 •245 1 ■574 I •361 3 664 4*o o 250 16 OC 64 'OOO 2 •00c 6 '3 2 5 1 ■587 I ■386 3 ■689 4'i o 244 16 8l 68 ■921 2 -025 6 403 1 •601 I •411 3 714 4*2 o 238 17 -64 74 • 088 2 .049 6 481 1 •613 I '435 3 738 4*3 o 2 33 18 49 79 '5°7 2 •074 6 •557 1 •626 I •459 3 761 4*4 o 227 19 36 85 •184 2 '098 6 ^33 1 .639 I •482 3 •784 4*5 o 222 20 25 9i I2 5 2 121 6 708 1 651 I '5°4 3 •807 4'6 o« 217 21' 16 97 336 2 !45 6 782 1 663 I 526 3 829 4*7 o- 213 22 09 103 823 2 168 6 856 1 ■675 I 548 3 850 4*8 o- 208 2 3 04 no 592 2 191 6 928 1 687 I 569 3 ■871 4*9 o- 204 24 01 117 649 2 214 7 000 1 699 I 589 3 892 5*o O • 200 25-OC 125-000 2- 236 7-071 1 • 710 I • 609 3.912 e = 2*71828, log c 10=2'30259, log 10 e = 0'4342y. TABLES. 311 1 . POWERS, NAPIERIAN LOGARITHMS ETC. X XT 1 X 2 X 3 1 X 2 (io#)* 1 X* loge * loge(iox) 5'o O • 200 2 5 •00 125 •000 2 •236 7 •071 1 • 710 I 609 3 912 5'i O • I96 26 ■01 132 ■651 2 •258 7 •141 1 •721 I '629 3 93 2 5*2 O • I92 27 •04 140 •608 2 .280 7 •211 1 •732 I 649 3 95i 5*3 O •189 28 .09 148 ■877 2 .302 7 •280 1 "744 I 668 3 970 5*4 O .185 29 •16 157 •464 2 .324 7 ■348 1 '754 I 686 3 989 5*5 o •l82 30 •25 166 ■375 2 •345 7 416 1 •765 I 7°5 4 007 5*6 o •179 3 1 ■36 175 .616 2 .366 7 •483 1 776 I •723 4 025 5*7 o •175 3 2 •49 185 • J 93 2 •387 7 •55o 1 •786 I .740 4 043 5-8 o • 172 33 .64 195 - 112 2 •408 7 ■616 1 •797 I 758 4 060 5' 9 o • 169 34 ■81 205 '379 2 •429 7 •681 1 ■807 I •775 4 078 6-o o • 167 36 ■00 216 ■ooo 2 "449 7 .746 1 .817 I .792 4 094 6- 1 o • 164 37 •21 226 •981 2 ■470 7 810 1 -827 I 808 4 in 6'2 o •161 38 •44 238 •328 2 •49° 7 •874 1 •837 I •825 4 127 6*3 o •159 39 .69 250 047 2 .510 7 •937 1 •847 I 841 4 •143 6*4 o .156 40 .96 262 •144 2 •53o 8 •000 1 •857 I 856 4 •159 6*5 o •154 42 •25 274 •625 2 •55o 8 •062 1 •866 I •872 4 •174 6-6 o •i5 2 43 •56 287 •496 2 •5 6 9 8 ■ 124 1 •876 I •887 4 • 190 6° 7 o •149 44 .89 300 ■763 2 •588 8 ■18s 1 •885 I 902 4 •205 6-8 o •147 46 •24 3i4 •432 2 •608 8 •246 1 •895 I 917 4 ■ 220 6*9 o •145 47 •61 328 •5°9 2 •627 8 '3°7 1 •904 I •932 4 •234 7*o o •143 49 •00 343 •000 2 ■646 8 •367 1 •9 X 3 I .946 4 -248 7-i o •141 5° •41 357 .911 2 .665 8 •426 1 •922 I •960 4 •263 7-2 o •!39 5 1 .84 373 •248 2 •683 8 •485 1 ■93 1 I •974 4 •277 7'3 o •137 53 •29 389 •017 2 • 702 8 •544 1 .940 I •988 4 •290 7*4 o •135 54 .76 405 •224 2 • 720 8 •602 1 •949 2 •001 4 •3°4 7*5 o •!33 56 ' 2 5 421 ■875 2 •739 8 -66o 1 •957 2 •015 4 •317 7*6 o ■132 57 .76 438 .976 2 •757 8 •718 1 .966 2 •028 4 •33i 7*7 o •130 59 ■29 45 6 '533 2 •775 8 ■775 1 •975 2 ■041 4 •344 7'8 o •128 60 .84 474 •552 2 •793 8 •832 1 •983 2 •054 4 •357 7*9 o • 127 62 ■41 493 ■°39 2 ■811 8 •888 1 •992 2 •067 4 •369 8-o o ■125 64 •00 512 •000 2 •828 8 •944 2 •000 2 •079 4 .382 8-i o ■123 65 61 53i •441 2 •846 9 •00c 2 • 008 2 •092 4 ■394 8*2 o ■ 122 67 ■24 55i .368 2 •864 9 •°55 2 •017 2 • 104 4 •407 8-3 o ■ 120 68 .89 57i •787 2 •881 9 • no 2 •025 2 •116 4 .419 8-4 o •119 70 56 592 .704 2 •898 9 • 165 2 '°33 2 •128 4 •43i 8*5 o 118 72 •25 614 ■125 2 ■9i5 9 •220 2 •041 2 140 4 •443 8-6 o 116 73 96 636 ■056 2 •933 9 •274 2 •049 2 •152 4 •454 8-7 o 115 75 69 658 ■5°3 2 •95° 9 •327 2 •°57 2 163 4 •466 8-8 o 114 77' 44 681 •472 2 •966 9 •381 2 •065 2 .175 4 •477 8-9 o- 112 79' 21 704 .969 2 983 9 •434 2 •072 2 186 4 •489 9-o o- in 81 ■ 00 729 ■ooo 3 000 9 487 2 •080 2 197 4 •500 9-i o« no 82- 81 753 571 3 017 9 539 2 088 2 208 4 5ii 9*2 o- 109 84- 64 778 688 3 °33 9 592 2 095 2 219 4 522 9*3 o- 108 86- 49 804 357 3 050 9 644 2 103 2 230 4 533 9*4 o- 106 88- 36 830 584 3 066 9' 695 2 no 2- 241 4' 543 9' 5 o- 105 90- 25 857 375 3 082 9' 747 2 118 2- 25 1 4' 554 9-6 o- 104 92. 16 884- 736 3 098 9' 798 2 125 2« 262 4" 5 6 4 9*7 o- 103 94. 09 912- 673 3' 114 9" 849 2- *33 2- 272 4" 575 9'8 O' 102 96. 04 941- 192 3" 130 9' 899 2- 140 2- 282 4' 585 9*9 o« IOI 98. 01 970- 299 3' 146 9" 95o 2« 147 2- 2Q3 d- 4. then (approx ima ;tely) sin h x = COS h x- -h ?* = iti lie IS apj erian antilogarithm of x. TABLES. 315 5. LAMBDA FUNCTION. e M o-ooo e i5° KO) 0-265 6 30° X (0) 45° M e 6o° KO) \ 75° 0° o-549 o-88i i-3i7! 1° 0-017 1 6° 0-283 3i° "57° 46 0-906 6i° 1 '352 76 2° °'°3S ! i7° 0-301 32 59o 47° °-93 2 , 62 1 389 77° 3° 0.052J ! 18 0-319 33° 611 48 o-957, 63 1 4271 78 4° 0-070 l IQ° °'33% 34° 632 49° 0-984 64 1 466! 79° 5° 0-087 20° '356 35° 6.S3 50 I -OIIi 65 1 5o6j 8o° 6° 0-105 21° °'375 36° 674 5i° 1-038 66° 1 549 8i° 7° O- 122 22° o-394 37° 696 52° i- 066 67 1 592 82 8° o- 140 23° 0-413 38 718 53° 1-095 68° 1 638 83° 9° 0-I 5 8 24° 0-432 39° 740 54° 1 • 124 69 1 686 84 10° O.I75 25° 0-451 40 763 55° i-i54 70 I- 735 85 n° 0-IQ3 26° 0-470 4i° 786 56 i-i85 7i° 1 788 86° 12° 0-2II 27 0-490 42 o- 809 57° 1 -217 72 i" 843 87 13° 0-229 28 0-509 43° O' 833 58 1-249 73° i- 901 88° i 4 ° 0-247 29 0-529 44° o- 857 59° 1-283 74° I- 962 89 15° 0-265 30 o-549 45° 0.881 6o° i-3i7 75° 2-028 90 X(d) 2-028 2-097 2- 172 2-253 2-340 2-436 2-542 2- 660 2-794 2.949 3-I3I 3*355 3-643 4-048 4-741 CO /(0) = log e tan(l;r+J0) = loge (sec 0+tan 0), X(-6)=-X(d), 0=gdX(6). 6. GAMMA FUNCTION. n r(,) 'OO •01 .02 •03 •04 •05 -06 .07 .08. •09 •10 • n • 12 •13 •14 •15 •16 •17 •18 •19 •20 •9943 -9888 •9836, •9784 •9735 .9688 •9642 •9597 •9555 •95 J 4 •9474 •9436 •9399 •9364 •933o •9298 •9267 •9237 •9209 •9182 n n ) I-20 o- I-2I o- I -22 o- 1-23 o- 1-24 o- 1-25 o- 1-26 o- 1-27 o- 1.28 o- 1-29 o- 1-30 o- i-3i o- 1-32 o- 1 '33 o- i-34 o- i-35 o- 1-36 o- i-37 o- 1-38 o- i-39 o- 1 -40 o- 9182 9156 9 J 3i 9108 9085 9064 9044 9025 9007 8990 [ 8975| 8960I 8946, 89341 8922 891 1 8902 8893 8885 8879 8873 n 1-40 1. 41 1-42 i-43 1-44 i-45 1-46 1.47 1-48 1-49 1-50 i-5i 1-52 J -53 i-54 i-55 1-56 i-57 1-58 i-59 i- 60 r( n ) n •8873 i- •8868 i- •8864 1 • •8860 1 • •8858 1 • -88 S 6 i- -88^6 i- -88 S 6 i- •8857 i- •8860 1 • •8862 1 • ■8866 1 • •8870 1 • •8876 1 • •8882 i- •8889 i- -8896 i- •8905 i- •8914 i- •8924 i- •8935, 1 i- -6o •61 •62 •63 •64 •65 •66 .67 •68 .69 •7o •7i •72 •73 •74 •75 .76 •77 •78 •79 •8o r( n ) 0-8935 0.8947 0-8959 0-8972 0-8986 0-9001 0-9017 0-9033 0-9050 0-9068 0-9086 0-9106 0-9126 0-9147 0-9168 0-9191 0-9214 0-9238 0-9262 0-9288 0-9314 n r( n ) 80 81 82 S3 84 86 87i 88 89 90 9i 92 93 94 95 96 97 98 99 00 •93H •9341 •9369 •9397 •9426 •9456 •9487 .9518 •955i •9584 •9618 •96^2 •9688 •9724 .9761 •9799 •9837 .9877 .9917 •9958 fO0 r(n) = x n ~ l e-x dx, r(n+l) = nr(n). 316 TABLES. 7. FIRST ELLIPTIC INTEGRAL, F(m,6), w = sina. 6 a=o° a=i 5 ° o-ooo a=3o° «=45° o-ooo a=6o° o-ooo «=75° a=90° 0° o-ooo o-ooo o-ooo o-ooo 5° o •087 0-087 0-087 0-087 0-087 0-087 •087 10° o •175 0-175 0-175 0-175 0-175 0-175 •175 15° o •262 0-262 0-263 o- 263 0-264 0-265 •265 20° o •349 °'35° 0-351 o-353 o-354 0-356 '356 25° o •436 °*437 0-440 o-443 0-447 0-450 ■451 30° o 5 2 4 o-525 0.529 o-536 0-542 o-547 '549 35° o •611 0-613 0-620 0-630 0-641 0-649 •653 40 o 698 o- 702 0-712 0-727 0-744 o-757 763 45° o 7*5 0-790 0-804 0-826 0-851 0-873 881 5o° o •873 0-879 0-898 0-928 0-965 0-997 1 on 55° o 960 0-968 o-993 1-034 1-085 1 '^33 1 154 6o° I 047 1-058 1-090 i- 142 1-213 1-284 1 3 T 7 65° I- i34 1 -147 1-187 1-254 1*349 i*453 1- 5°6 70° I 222 1-237 1-285 1-370 1-494 1-647 i- 735 75° I' 309 1-327 I-385 1-488 1-649 1-871 2- 028 8o° I ■ 396 1-418 1-485 i- 608 1-813 2-134 2- 436 85° I- 484 1-508 I-585 i-73i 1-983 2-437 3- I3 1 90 i-57i 1-598 1-686 1-854 2*157 2-768 00 8. SECOND ELLIPTIC INTEGRAL, E(m, d), m = sma. e a=o° o-ooo a =30° o-ooo «=45° o-ooo a=6o° o-ooo a= 75 ° a=go° 0° o-ooo o-ooo o-ooo 5° 0-087 0-087 0-087 0-087 0-087 .087 ■087 10° 0-175 0-174 0-174 0-174 0-174 ■ 174 •174 15° 0-262 o- 262 o- 261 o- 260 o- 260 •259 ■259 20° o-349 o-349 o-347 0-346 o°344 •342 •342 25° o-436 o-435 o-433 0.430 0-426 424 ■423 30° 0-524 0-522 0-518 0-512 0-506 • 502 ■500 35° o-6n 0-608 0-602 o-593 0-583 576 574 40° 0-698 0-695 0.685 0-672 0-657 647 643 45° 0-785 0-781 0-767 0-748 0-728 713 707 50° 0-873 o-866 0.848 0-823 o-795 774 766 55° 0-960 0-952 0.928 0-895 0-859 o- 830 819 6o° 1-047 1-037 i-oo8 0-965 0-918 o- 881 o- 866 65° i-i34 1 • 122 i- 086 1 '°33 0-974 926 o- 906 70° 1-222 i- 206 1-163 1-099 1 -027 o- 965 o- 940 75° 1-309 1-291 1 • 240 1-163 1-076 o- 999 o- 966 8o° I-396 i-375 1. 316 1-227 i- 122 I- 028 o- 985 85° I-484 1*460 1.392 1-289 1-167 I" °53 o- 996 90° I-57I 1-544 1.467 J-35 1 I-2II I- 076 I- 000 INDEX. (The references are to pages.) Acceleration, 73 Amsler, planimeter, 301 integrator, 302 Anchor ring, 180, 193 Area, 95, 134, 159, 190, 191, 299 of any surface, 200 Argument of function, 11 Asymptotes, 57, 153 Asymptotic circle, 155 Barometric measurement of heights, 39 Bernoulli's equation, 266 Cardioid, 151, 160, 170 Catenary, 209, 269, 308 Cauchy, form of remainder, ZZb Centre of conic, 45 of any curve, 290 of curvature, 84, 88 of gravity, 176, 304 of quadric, 65 Centroid, 177 Circle, 139, 150 of curvature, 85 Circular functions, 28, 32, 311, 312 Cissoid, 55, 61, 139 Clairaut's equation, 268 Companion to cycloid, 31, 140 Compound interest problem, 269 Concavity and convexity, 82 Conchoid, 163 Conical point, 64 Conjugate point, 52 Consecutive points, 85 Constant of integration, 98 Continuity, 12, 250 Convergence, 211 Gurvature, 82, 157, 207 Curve of pursuit, 285 Curve tracing 290 Cusp, 52 Cycloid, 29, 91, 140, 307 companion to, 31, 140 Cylinder, 145, 198, 199, 200 . Damped vibrations, 284 Demoivre's theorem, 220 Derivative, 13 Differences, small, 47 Differential, 15 coefficient, 15 of area, 38 of exponentials, 25 of hyperbolic functions, 34 of logarithms, 25 of power, product, quotient, 16 partial, 41, 250 successive, 68, 250 total, 42, 253 Differential equations, 259, 272 exact, 263 homogeneous, 262 linear, 265, 275 Element of integral, 94 Ellipse, 70, 87, 88, 140, 151, 162, 241,308 Ellipsoid, 144 194. 195 Elliptic integrals, 239, 316 Envelopes, 171 Epicycloid, 169, 20G Epitrochoid, 170 Equations, differential, 259, 272 solution of, by approximat on, 49 Euler, exponential formulae, 220 theorem on homogeneous func- tions, 44, 255 Evolute, 88, 175, 207 317 318 INDEX. Folium, 53, 80, 163 Fourier, series, 228 theorem, 236 Function, complementary, 278 even, odd, 40 implicit, 11 single-valued , multiple-valued , 12 Gamma function, 132. 315 Graphs, 11 Guaermannian, 36, 298 Guldin, properties of the centre of gravity, 178 Gyration, radius of, 183 Helix, 67, 148 Huyghens, formula for circular arc, 222 Hyperbola, 141, 151, 162 rectangular, 23, 139, 150, 162 Hyperbolic functions, 34, 294, 313 Hypocycloid, 169 four-cusped, 22, 46, 92, 136 Indeterminate forms, 244 Infinitesimals, 4 equivalence of, 8 orders of, 7 Inflexion, point of, 82, 155 Integral, definite, indefinite, 98 double, 190 triple, 194 Integrals, definite, 129 elliptic, 239 fundamental, 99 particular, 278 Integration, 93 approximate, 239 by parts, 120 by substitution, 114 by successive reduction, 124 of rational fractions, 112 mechanical, 299 successive, 189 Integrator, 302 Interval, 13 Intrinsic equation, 205 Inverse curves, 165 Involute, 88 of circle, 158, 207 Kinetic energy of rotation, 181 Lagrange, form of remainder, 225 Lambda function, 105, 315 Lemniscate, 53, 150, 156, 157, 161, 242 Lengths of curves, 134, 148, 241 Limagon, 151, 162 Limit, 1 Lituus, 151, 154, 156 Logarithms, calculation of, 216 Napierian, 310, 311 Maclaurin, series, 218, 226 theorem, 226 Maxima and minima, 75, 256 Mean value, theorem of, 223 Mean values, 202 Moment, of area, 191, 302 of inertia, 181, 191, 192, 197, 302 Multiple points, 51, 56, 156 Node, 52 Normal, 21, 44, 150 of surface, 64 Pappus, properties of centre of gravity, 178 Parabola, 22, 70, 90, 141, 151 cubical, 136 semicubical, 53, 136 Paraboloid, elliptic, 144 hyperbolic, 195 Partial fractions, 287 Pedal curves, 166 Pendulum, 243 Planimeter, 302 Polar coordinates, 136 Polar reciprocals, 168 Potential, 193, 197 Prismoidal formula, 147 Product of inertia, 191 Radius, of curvature, 85, 158 of gyration, 183 Rates, 72 Rolle's theorem, 224 Roulettes, 168 Series, Fourier's, 228 Gregory's, 217 infinite, 211 logarithmic, 216 Maclaurin's, 218, 226 INDEX. 319 Series, power, 213 Taylor's, 220, 226 Simpson's rule, 142 Singular forms, 244 Solid of revolution, 96, 134 Sphere, 139, 196, 198 Spheroids, 141 Spiral, of Archimedes, 151, 160 hyperbolic, 151, 154 logarithmic, 152, 153. 158 Subnormal, 21, 150 Subtangent, 21, 150 Symmetry, 290 Tangent, 20, 44, 150 to curve in space, 65 Tangent plane, 62 Taylor, series, 220 226 theorem, 223 Torus, 180, 193 Tractrix, 207 Trajectories, 270 Triple point, 52 Trochoid, 168 Turning value, 76 Value of it, 217 Variable, 1, 11 change of, 280 independent, 11, 68 Velocity, 73 Volumes, 134, 144, 193, 196 Witch.. 84, 139 SHORT-TITLE CATALOGUE OP THE PUBLICATIONS OP JOHN WILEY & SONS, New York. 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Comstock's Field Astronomy for Engineers 8vo, 2 50 Craig's Azimuth 4to, 3 50 Doolittle's Treatise on Practical Astronomy 8vo, 4 00 Gore's Elements of Geodesy. 8vo, 2 50 Hayford's Text-book of Geodetic Astronomy 8vo, 3 00 Merriman's Elements of Precise Surveying and Geodesy 8vo, 2 50 * Michie and Harlow's Practical Astronomy .8vo, 3 00 * White's Elements of Theoretical and Descriptive Astronomy 121110, 2 00 BOTANY. Davenport's Statistical Methods, with Special Reference to Biological Variation. i6mo, morocco, 1 25 Thome and Bennett's Structural and Physiological Botany, i6mo, 2 25 Westermaier's Compendium of General Botany. (Schneider.). > 8vo, 2 00 3 I 50 3 00 3 OO 3 00 1 00 1 00 2 SO 2 00 3 CO 4 00 3 00 1 50 1 50 CHEMISTRY. Adriance's Laboratory Calculations and Specific Gravity Tables i2mo, i 25 Alexeyeff's GenerahPrinciples of Organic Synthesis. (Matthews.) 8\o, 3 00 Allen's Tables for Iron Analysis 8vo, 3 00 Arnold's Compendium of Chemistry. (Mandel.) 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W.) and Marx's i T achine Design 8vo * Titsworth's Elements of Mechanical Drawing Oblorg 8vo Warren's Elements of Plane and Solid Free-hand Geometrical Drawing. i2mo Drafting Instruments and Operations i2mo Manual of Elementary Projection Drawing i2mo Manual of Elementary Problems in the Linear Perspective of Form and Shadow i2mo Plane Problems in Elementary Geometry. . . . i2mo Primary Geometry i2mo Elements of Descriptive Geometry, Shadows, and Perspective 8vo General Problems of Shades and Shadows . . , 8vo Elements of Machine Construction and Drawing . . 8vo Problems, Theorems, and Examples in Descriptive Geometry 8vo Weisbach's Kinematics and Power of Transmission. (Hermann and Klein.). . 8vo Whelpley's Practical Instruction in the Art of Letter Engraving i2mo Wilson's (H. M.) Topographic Surveying 8vo Wilson's (V. T.) Free-hand Perspective 8vo Wilson's (V, T.) Free-hand Lettering. . , 8vo Woolf 's Elementary Course in Descriptive Geometry Large 8vo > 2 50 » 4 00 » 2 ro i 2 00 i 2 50 > 2 00 , I 50 , 3 00 > 3 00 , 5 00 » 4 00 , i 50 , i 50 , i 50 » 3 50 ► , 2 00 , 5 00 » 2 00 , 3 00 , 3 00 , 3 00 » 2 50 ,• 3 00 >J I 25 , I 00 , I 25 , I 50 , I 00 , I 25 75 , 3 50 » 3 00 , 7 50 > 2 3 50 , 5 00 > 2 00 , 3 50 > 2 50 , I 00 , 3 OG ELECTRICITY AND PHYSICS. Anthony and Brackett's Text-book of Physics. (Magie.) Small 8vo, Anthony's Lecture-notes on the Theory of Electrical Measurements. . . . i2mo, Benjamin's History of Electricity 8vo, Voltaic Cell 8vo, Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.).8vo, * Collins's Manual of Wireless Telegraphy i2mo, Morocco, Crehore and Squier's Polarizing Photo-chronograph. . , 8vo, Dawson's "Engineering" and Electric Traction Pocket-book. i6mo, morocco, 10 3 00 1 00 3 00 3 00 3 00 1 50 2 00 3 00 5 00 Dolezalek's Theory of the Lead Accumulator (Storage Battery). 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