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ELEMENTS 
 
 I 
 
 OF THE 
 
 iNflHITE 
 
 SIMAL 
 
 Calcu 
 
 LUi 
 
 BY 
 
 GEORGE H. CHANDLER, M.A., 
 
 P lOFESSOR IN THE FACULTY OF APPLIED SCIENCE, 
 MCGILL UNIVERSITY, MONTREAL. 
 
 SniontreaP: 
 W. DRYSDALE & CO. 
 
 1892. 
 
1 
 
 Entered according to Act of Parliament, in the year 
 one thousand eight hundred and ninety-two, by 
 G. H. Chandler, in the office of the Minister of 
 Agriculture and Statistics, at Ottawa. 
 
 JOHN tOVELL A 60N, MINTER8, tS * tt 6T. NICHOLAS ST., MONTRgAU 
 
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 doc- 
 trine of Limits seems to be absolutely essential, but I have en- 
 deavoured to accustom the reader from the outset to those prin- 
 ciples iind operations which are always used in the practical 
 applications of the subject. 
 
 The order in which the subject is developed and many of the 
 explanations ofiered, though novel in some respects, are believed 
 to be well calculated to meet the difficulties ctnd secure the 
 interest of the student. 
 
 The Hyperbolic Functions are not introduced into the body of 
 the work, but in the Appendix I have stated some of their pro- 
 perties, and also given curves and tables which will, I hone, assist 
 in illustrating the nature of these functions. 
 
 The sign / for division has been freely used. To supplement the 
 ordinary Mathematical Tables I have added short tables which 
 are intended to facilitate curve tracing as well as the rapid 
 (approximate) calculation of integrals, etc. 
 
 a. H. ClIANDLKR. 
 
 Montreal, May, 1892. 
 
CONTENTS. 
 
 Chap. Paoe 
 
 I. Limits. Infinitksimai-s 1 
 
 II. DlFFERENTlALH H 
 
 TTI. Differentials OF Ai.oEBRAic Quantities !) 
 
 IV. Tan(;ents AND Normals 13 
 
 V. MuLTiPi-K Points 18 
 
 VI. Asymptotes 23 
 
 VIT. Differentials of Exponentials and Logar- 
 ithms 28 
 
 VIII. Differentials of the Trigonometric Ratios 
 
 (Circular Functions) 31 
 
 ' Differentials of Angles (Inverse Circular 
 
 Functions) 35 
 
 ^.. Functions OF more THAN One Variable 38 
 
 XI. Small Differences 42 
 
 XII. Derived Functions. Successive Differentials. 46 
 
 XIII. Direction of Curvature. Points OF Inflexion. 49 
 
 XIV. Amount of Curvature 53 
 
 XV. Maxima AND Minima 62 
 
 XVL Integration. Elementary Illustrations 70 
 
 XVII. Fundamental Integrals 1 74 
 
 XVIII. Fundamental Integrals II 77 
 
 XIX. Fundamental Integrals III 80 
 
 XX. Fundamental Integrals IV 84 
 
 XXT. Integration of Rational Fractions , . 87 
 
 XXII. Integ ration by Substitution 89 
 
 XXIII. Integration by Parts 1 96 
 
 XXIV. Integration by Parts II. (Successive Reduc- 
 
 tion) 99 
 
 XXV. Certain Definite Integrals 103 
 
 XXVI. Areas, Lengths, Surfaces, Volumes 107 
 
 XXVII. Polar Co-ordinates 117 
 
 XXVIII. AsfociATED Curves 134 
 
 XXIX. ExVVELOPES 140 
 
VI 
 
 CONTENTS. 
 
 XXX. 
 
 XXXI. 
 
 XXXII. 
 
 XXXIII. 
 
 XXXIV. 
 
 XXXV. 
 
 XXXVI. 
 
 XXX VII. 
 
 APPENDIX. 
 
 TABLES. 
 
 CUIIVK TllACING 145 
 
 CENTREi« OF GRAVITY 150 
 
 Moments of Inertia 166 
 
 Expansion OF Functions 165 
 
 Limits of Functions (Singular Forms) 172 
 
 Equations op Curves by Integration. The 
 
 Catenary 179 
 
 Solution of a Differential Equation. Change 
 
 op the Independent Variable 183 
 
 SucoEssiVK Differentials OP Functions of more 
 than one Variable. Extension of Taylor's 
 Theorem. Maxima and Minima from Taylor's 
 
 Theorem. Multiple Integr vtion 187 
 
 Note A. Exponential Series 197 
 
 Note B. Partial Fractions 198 
 
 Note C. Hyperbolic Functions 2C. 
 
 Note D. Elliptic Integrals 205 
 
 Powers, Napierian Logarithms, Etc. 
 Circular Functions. 
 Hyperbolic Functions. 
 Elliptic Integrals. 
 
 
ERRATA. 
 
 J 
 
 Page 11, Ex. 3,. /or 2 (a + 6) x read 2 (ax+b), 
 *• - 15, line 2 from bottom, /or axis read axes. 
 " ^29, line 4, /or x logm e reatZ -r- logi„ e. 
 
 n- I 
 
 *' ^0, E: 
 
 
 " '-46, Ex. 7, (6), rmcZ X'' - 12x = 200. 
 «* l/61, Ex. 4,ybr i read - J. 
 
 52, Fig. 40, join CQ instead of cp. 
 
 57, line 9, /or (x, y) read at (.r, y). ^ 
 
 58, Ex. 2,/or 4(x - 2a) read 4 (x - 2a)\ 
 68, Ex. 12,>r ^n read |rr. 
 
 77, Ex. 2,/or ^ax^ read la-x\ 
 
 78, Ex. 12, for (a+bx^) dx read (a + 6a;)3 dx. 
 
 81, Ex. 13, the - should follow the second = , not the iirst. 
 
 82, Ex. 41, for Ktan26/+ log cos tf) read J tsin'^6+ log cos 6. 
 
 84, Ex. 7, for log (-^) read log (-JL.\ . 
 110, line 2 from bottom, /or ?/''(a; - 1) ready'^(x - ^ f. 
 
 
 (( 
 
 203, for log(a; + Va;- + a^) reacZ log /^±V»l+_af\ 
 and for log (a + Vx^ - «-) re«fZ log (—^ ^^^ ~ ^"^ 
 
 a 
 
 ) 
 
 I 
 
CHAPTER I. 
 
 LIMITS. INFINITESIMALS 
 
 1. Limits. The reader is probably familiar with the idea 
 of a limit from his study of elementary matluinatics. For ex- 
 ample : 
 
 (1) As two points on a curve approach coincidence, the 
 secant pas-sing tiirough them approaches a limiting position which 
 is called the tangent, i.e., the tangent is the limit of the liecant. 
 
 (2) As three points on a curve approach coincidence, the 
 ei)cle passing through them approaches a limiting position called 
 the circle of curvature. 
 
 (3) The limit of the ratio of an arc to its chord is 1 as arc 
 and chord approach zero. 
 
 (4) The limit of the perimeter of a regular polygon inscribed 
 in a circle is the circumference of the circle as the number of 
 sides increases towards infinity and the length of each side 
 approaches zero. 
 
 (5) The limit of the sum ^ + ^ + ^ -!- yV H etc., is 1 as the 
 number of terms approaches infinity. 
 
 (6) If be the radian* measure of any acute angle, 6 lies 
 between sin o and tan 6. Hence, dividing each of these into sin e, 
 sin ^/f; lies between 1 and cos0. But the limit, of cose is 1 
 as 6 approaches zero, .*. the limit of sinfl/e is also 1. 
 
 Also, since tan d/d = (sin e/d) / cos e, the limit of tan e/d is 
 also 1. 
 
 t <l 
 
 • The radian (= 57°.2958 = 206265") will be always understood 
 to be the unit angle, unlees the contrary is manifest. 
 
LIMITS. INFINITESIMALS. 
 
 (7) '.-l -cos^ = 
 
 1 + cos 6* 
 
 , and tan 61 — sin ^ = 
 
 8in'(y 
 
 cos ^ (1 + cos ff)' 
 
 / sin H\ ^ 
 
 It. ofir^°-^..lt.of ^_Z2 -1 
 ti^ 1 4-nns« 2 
 
 and 
 
 J ftan^-8in_«^ ^^^ 
 
 (^) 
 
 cos^(l +cos^^) 2 
 
 2. Infinitesimals. An infinitesimal is a quantity which is 
 extremely or indefinitely small. It may be thought of as a quantity 
 which is just on the point of vanishing (i.e., of becoming zero), 
 or one which has just emerged from aero. The same thing is 
 meant when it is said to be " infinitely " small or " less than 
 any assignable magnitude," 
 
 3. It must not be supposed that all infinitesimals are equal 
 
 M 
 
 Fig. 1. Fig. 2. Fig. 3. 
 
 because they are infinitely small. In fig. 1, which represents the 
 straight line y = x, suppose that MP moves up towards o ; then 
 when M is about to coincide with o, OM, MP, and op, all become 
 infinitesimals ; the first two are equal, but the third is V2 times 
 each of the others. In the parabola y = x^, fig. 2, when a; = ^, 
 y = ^] when x = ^, y - ^, etc. ; hence, when m is about to 
 coincide with o and OM and MP become infinitesimals, the latter 
 is only an infinitesimal part of the former. Again in the cubical 
 parabola y = x', fig. 3, when m is about to coincide with o, om 
 and MP become infinitesimals, but the latter is only an infinites- 
 imal part of an infinitesimal part of the former. 
 
INFINITESIMALS. 
 
 3 
 
 4. Quantities are said to be of the same order when the ratio 
 of each to the other k finite. If a; be regarded as an infinitesi- 
 mal of the first order, x\ x^, ..., aro irnown .is infinitesimals of 
 the second, third, ..., orders. Also, a being finite, ax^ is an 
 infinitesimal of the second order, and bo also is any quantity 
 whose ratio to x^ is a finite quantity ; and similarly for infinites- 
 imals of higher orders. Thus MP becomes, in comparison with 
 OM, an infinitesimal of the first order in fig. 1, of the second in 
 fig. 2, and of tlie third in fig. 3. 
 
 It should be observed that each infinitesimal of a higher order 
 is an infinitely small part of every one of a lower order. Hence 
 the product of two infinitesimals is one of a higher order than 
 either of the given ones. 
 
 5. Whenever an expression approaches a limit on account of 
 the vanishing of some quantity involved in it, we may, in any 
 investigation in which that limit is to be used, assume the value 
 of the expression just before the vanishing quantity has become 
 zero to be equal to the limit, since any infinitely small error 
 involved in this supposition must disappear when we proceed to 
 the limit. 
 
 TIjus regarding an infinitesimal as a quantity which is about 
 to vanish, when e is infinitesimal we may put (see §1) 
 
 cos^ = 1* ; 
 sin fi/0 = 1, or sin^ = ^; 
 tsine/6 = 1, or tsiud = d: 
 (1 - cose) /^^ = 1/2, or 1 - cos ^ = ^ ^; 
 (tan e - sin o)/t>^ = 1/2, or tan e- sin ^ = ^ ^. 
 Similarly, since the limit of the ratio of an arc to its chord is 
 1, we may assume that any infinitesimal arc is equal to its chord 
 i.e., that any infinitesimal arc may be regarded as a straight 
 line. 
 
 St' 
 
 * This is equivalent to saying that the two sides of a right-angled 
 triangle which contain an infinitesimal angle may be regarded as 
 equal. 
 
4 INFINITESIMALS. 
 
 6. It will be seen from the above that when u is regarded as 
 an infinitesimal of the first order, sin d and tan d are infinitesimals 
 of the first order, 1 - cos an infinitesimal of the second order, 
 and tan — s\n6 one of the third order. 
 
 Fig. 4. 
 
 These results may be represented graj^hically by taking (fig. 4) 
 an arc ap of a circle of which the radius is the unit of the^^scale 
 of measurement, and drawing the perpecidiculir pb and the 
 tangent aq. Then cos 6, sin (i, 6, tan ft, 1 — cos ft^ tan 6 — sin ft, 
 are equal to the numbers which represent the lengths of OB, bp, 
 AP, AQ, ba, cq on the given scale. All of these numbers are 
 less thin 1, and all except the length of ob are small fractions 
 which be3ome smaller as 6 diminishes, while their squares and 
 cubes are much smaller fractions. From §5, UA isiuliimatdy 
 (when P is about to coincide with a) an infinitesimal of the 
 second order, and CQ one of the third order, in comparison with 
 AP. (The figure should be drawn carefully to a larger scale and 
 exact measurements made when 6 19 a small angle. It may thus 
 be verified that the length of ba is very nearly half of the square 
 of the fraction representing the length of ap, i.e., 1 - cos 6= ^ ff.) 
 
 EXEIICISES. 
 
 1, A being a given point in one side of an angle ft (fig. 5)^ 
 and ab, BO, CD, ... being perpendiculars, show that, ultimately, 
 when becomes infinitesimal, AB, AC, ad, ... become infini- 
 tesimals of the first, second, third, ... order. 
 
 [They = a sin 6, a sin^^^, a sin^fi, ..., where a = OA.] 
 
 2. Prove that ultimately oa - ob :=^ ^ a i/^ = ^ ac. 
 
KXKKCISKF. 
 
 i; ig. 6. 
 
 Pig. 6. 
 
 3. AB (fig. 6) being a circular arc of radius a, subtending an- 
 angle 6 at the centre, BC perpendicular to oa, ad and be tan- 
 gents ; show that ultimately when 6 or the arc ab becomes in- 
 finitesimal, chord AB = aO = siTC ab, ca = ^ aff, bd = ^ a ^, 
 BD - AC = ;| aii^, DE - EA = I a (f\ (ae + be) - chord AB - 
 I (ad-bc) =^ a(j\ ^ 
 
 [ae = a tan ^^'j chord AB = 2 a sin ^0.] 
 
 4. To show that the tangent at any point of an ellipse is 
 equally inclined to the focal radii of the point of contact. 
 
 F!g. 7. 
 
 Fig. 8. 
 
 Let p, Q (fig. 7) be two points on the curve which arc about 
 t<i coincide. Let pa, qb be perpendiculars to fq, f'p. Then 
 (§5, note) FA = FP, f'q = f'b ; and since the sum of the focal 
 radii of any point on the ellipse is constant, the increase of fp 
 {i.e., aq) is equal to the decrease of f'p (i.e., BP) ; and the 
 hypotenuse pq is common to the right-angled triangles, /. the 
 angle aqp = the angle bpq. Hence, when p and Q coincide, 
 the angle cpf becomes equal to dqf' (fig. 8). (Strictly 
 speaking, we should say that the triangles, and lience also the 
 angles, cpf and dqf' approach equality as the points p and Q 
 approach each other, i.e., as DC approaches the tangent.) 
 
 t 
 
CHAPTER II. 
 
 DIFFERENTIALS. 
 
 7. Variables. Functions. Any quantity is said to be 
 variable or constant, according as it is or is not supposed to 
 change in value. When a variable quantity does not depend 
 upon any other quantity for its value it is called an indepen- 
 dent variable ; when, however, its variations are dependent 
 upon those of some other quantity, it is called a function of 
 this latter quantity , e.g., '3x^ — 2x + 2, log (a + cc), Va^ - x^ tan x, 
 are functions of x. So also in any equation involving two 
 variable quantities x and y, e.g., y'^ - 4 ax, ay? f hy^ + c = 0, 
 we may regard 3/ as a function of x, or a; as a function of y, since 
 either quantity must change when the other changes. 
 
 8. The expression /(x) is used as a general symbol for a 
 function of x, f{a) being the result of substituting a for x in the 
 function; e.g., if/(x) = 1 -x^/(0) = 1,/(1) = 0,/(2) = - 3. 
 
 For a similar purpose F(a;), <^{x), etc., may be used, and 
 fix, y) for a function of x and y. 
 
 9. Increments. Any change in the value of a quantity 
 is called an increment or a diflFerence of that quantity. An 
 increase being considered as a positive increment, a decrease is 
 to be considered as a negative increment. The symbols ax, 6x, 
 are often used for increments or differences of x. 
 
 10> Differentials. A differential is an infinitesimal increment. 
 The expression " differential of x " is abbreviated into dx. 
 Powers of the differential of x are written dx^, dx^, etc. 
 
DIFFEHENTIATION. 7 
 
 11. Differentiation. The prioiary object of the Differential 
 Calculus may be stated thus ; 
 
 A variable quantity having received an infinitesimal increment 
 (differential), to find the infinitesimal increment (differential) of 
 any function of that quantity ; or, n'ore generally, the variable 
 quantities in a function having received infinitesimal increments 
 to find the infinitesimal increment of the function. 
 
 12. This operation (called differentiation) differs from the 
 ordinary operations of algebra and trigonometry in this respect, 
 viz., that from the result we reject all infinitesimals except those 
 of the lowest order. The effect of this omission will be pointed 
 out presently (§15). 
 
 Examples. 
 
 1. The variable ac having become x + dx, to find d^x^), i.e., 
 the infinitesimal increment of x^. The Jictual change in £c^ = 
 
 (x + dxf - x^ = 2x dx-\ dx^, 
 
 and rejecting the rfx^, which is infinitely small compared with 
 the 2x dXf we have 
 
 d{x^) ^'jLx dx. 
 
 2. Similarly the change in x^ = (a5+ dxY — x^ 
 
 = 3 x^ dx + 3 X dx^ + dx^, 
 and retaining the first term only we have 
 
 d{x') =Sx^dx. 
 
 3. d{2 x' ^ 8 x'^ + 4) = 2 d{x') - 3 dix") * 
 
 ■ = ij X? dx ~ ^ X dx = Q x {x - \^ dx. 
 
 4. d{x^ + 2 ax) = 2 (x + a) dx, a being constant. 
 b. d{\ x^-\ax^ + x + \) - {x^-ax + l) dx,. 
 
 6. d{a - xf = 2 (x - a) dx. 
 
 • The constant 4 is not regarded in differentiating, since from its 
 nature it can have no differential. 
 
8 
 
 DIFFERENTIAL COEFFICIENT. 
 
 13. Genersl Method. Difierential Coefficient Let ,y stand 
 for /'(a;), any function of x. When x increases by the amount 
 rfa", U't th»? change in y be dy. Then 
 
 dy = df{x) =f(x + dx) -f{x). 
 
 When the last member of this equation is simplified, and 
 the second and higher orders of infinitesimals are omitted, we 
 have da" with a coefficient which we may call/'(.x). 
 
 /. dy = d f{x) = f'(x) dx, or dy/dx =f'(x). 
 
 The new funCuion/'(.T) is called the differential coefficient or 
 the derivative of /(x). Thus. if/( r) ^ x\ /'(re) = 2x ; if /(.x) 
 = x\ f'(x) ^ 3x- ; if ./"(x) = X- i 2../..', f'{x) := 2(x + o), etc. 
 
 14. It must be noticed, however, tiiat/'(:c) will be merely a 
 very close approximation to dy/dx, as long as dy and dx con- 
 tinue to be actual quantities, however small. In reality 
 
 dy/dr. =/'(x) + terms containing infinitesimals. 
 
 If dx be taken smaller and smaller the right-hand side 
 approaches /'(x) as a limit; hence the approximation, /'(x), 
 actually sidopted for dy/dx, is really the limit which the fraction 
 constantly approaches as dx diminishes towards zero. 
 
 16. The omission from any expression of all infinitesimals 
 except those of the lowest order is equivalent to assuming that 
 the higher infinitesimals vanish just before the lowest ones do. 
 This supposition is, of course, not correct ; but it may be adopted 
 as a convenient working hypothesis. In reality the higher infin- 
 itesimals are omitted, not because they vanish before the others, 
 nor because they are infinitely small compared with the lowest, 
 and therefore of no appreciable value, but because their omission 
 leaves an expression from which the limit of the ratio of dy to dx 
 (or of the other infinitesimals involved) may be at once written 
 down. 
 
 The reader will have no difficulty in seeing that the principle 
 here adopted is equivalent to that of §5. 
 
CHAPTER III. 
 
 DIFFERENTIALS OF ALGEBRAIC FUNCTIONS. 
 
 16. Tt isjobvious from tho nature of differentials and the way 
 in whicli they are obtained, that the differential of the algebraic 
 sum of any number of terms is equal to the alizebraic sum of the 
 diflferentials of the several tirms ; also that a constint factor in 
 a term or function appears as a factor in the differenti:il, and 
 that a constant term disappears in differentiating. 
 
 We now proceed to consider a few ireneral formuhc wliieh will 
 assist in the process of differentiating. 
 
 17. Differential of a power. Let v be any variable quantity, 
 to find £Z(w") in terms of dv, n being any constant. 
 
 We have for the change in w", 
 
 (y -f dvY ~ w" = (f" + Mw""' do + hi^-her powers of do) ~ v'* 
 
 = no'^^ dv + higher powers of do. 
 Hence, neglecting the higher powers, we have 
 
 </(?;«) = nv"-* dv. (A) 
 
 This formula is true for .tU values of u, whether positive or 
 negative, whole or fractional. The case in whicii n = h deserves 
 special mention ; (A) then becomes 
 
 dv 
 
 2V^' 
 
 d(^v) 
 
 (B) 
 
 Wl 
 
 len n 
 
 1 , we have 
 
 
 C.) 
 
 Ill 
 
 m 
 
 
10 
 
 DIFFERENTIAL OF A POWElf. 
 
 Examples. 
 
 1. d(x') = 5.TS/;r. 
 
 2. ./(3x^"-f-2) = 3rf(x^).15x*rfe. 
 
 3. d(3x* ~ LV + 6) = 3.4.r;' d.r - 2.2.i; dx = 4,r (3.x^ ^ 1) rf«. 
 
 ^- «^(^.) = d(x-^) = - 2.x- dx = - ~. 
 
 «. r/( a^ + xy = 3 (a^ + x'y d(a' + x') 
 
 = 3(a2 + xy 2x dx = 6x(a' -i- x^)^ dx. 
 In this example . = c*^ + x\ .-.nd /i . 3, a boin^^ constant. 
 
 7. c;__i__ 
 '^(ax + bx^i' 
 
 = -l:ax+ bx'yl-^ d{ax + hx') = - I (ax + ix^)-:; (« + 26x) dx. 
 
 8. ^ V^^^^^= "^("Izi^ bv rB) _ zl'L^ __ xdx 
 
 >X^)'' 
 
 9. 
 
 V«^ - x^; (a^~:^y ^^ ^^ ' ^' = jyT 
 
 2x c?e 
 
 ^ 18. Differential of a product. Let u and . be any variable 
 quantities, du and dv their differentials ; to find the differential 
 01 uv. 
 
 The change in uv / 
 
 = {u + du) (» + dv) -uv^u dv + vdu + du dv. 
 
 The last term is the product of two infinitely small quantities, 
 and IS therefore omitted ; ' 
 
 .'. diuv) -udv + vdu. (0) 
 
 Similarly d(;uvw) = uv dw + vw du ou dv. (Cj ) 
 
 • ■ 
 
 Example, 
 
 dCx" ~ 1) (4x + 3) ^ (x^- 1 ) d(i4x + 3) + (4x + 3) d(x' - I) 
 = (a;* - 1) 4dx + (4x 4- 3) 2x dx = 2(6x^ + 3x-2) dx. 
 
 Jr 
 
T 
 
 DIFFERENTIAL OF A FRACTION. 
 
 a 
 
 19. Differential of a quotient or fraction. Let^ = u/v. Thun 
 V = yv, :. (by C) du = ydo + vdy. Solving for dy and then 
 iubstituting u/v for j/, we liave 
 
 o= 
 
 d r^ 
 
 V du — u dv 
 
 V 
 
 (D) 
 
 Example. 
 
 d 
 
 \x'+l) - 
 
 l^^ _ (x" + 1) djx" -])-{x'-l) djx" + 1) _ 4xdx 
 
 Exercises. 
 / 1. d(a' - x'y = - 6x (a" - x^f dx. 
 
 V 2. d yli + x:^ =-- X dx / -Jl + x^ ^ 
 
 I 3. If/(x) = ax^ + 26.x + c, f'(x) = 2(rt f- 6) s^. 
 
 " 4. y = 'Jx^ - a\ dy/dx = 3xV2 Va;" - d\ ^-^ 
 
 ^ 5. d[a» (x^ - I) (x + 1)] = a(x + 1) (4x^ - x - 1) rZx.^ 
 
 •^7. d(l + x) Vr^ = (1 - 3x) :?x / 2 Vl-a;. 
 r 8. y = 3(a + 6x^)^, <^y = 106x (a + 6x^)t rfx. 
 
 t 9. y = 'Ja^+ (b - xy, dy = (x - 6) (Zx / ^Jd^ +(b- x)'\ 
 / 10. </ = (a^' - x')-\ ^Z^/</x = 3xV («' - xy. 
 ^ 11. y = xV (f* + xy, dy/dx = (3rt + x) xV (« + •'c)^ 
 / 12. ^ = V<*^ - 35'^, '^Z^ = —xdx/ \lii^ -it'. 
 
 13. ^ = V2ox - x^, dy = (a-x) dx/ V2ax - x'^. 
 
 14. y = xV (a' - x^), <?y / dx = (3a^ - x^) x^ / (a' xiy. 
 
 1 5. t/[x" / (1 + x)"] = rix"-^ <fx / (1 + x)'»*\ 
 
 16. d{a^ - xy = 2x dx / (a^ - x^)^ 
 
 17. ^ = xV VlTx^. (/jz/cfx = 2x / (1 + x*)t. 
 
 18. y = (x-a) / V*, cify/cfx = (a + x) / 2 Va;". 
 
 19. /y = x/ V2ax - x^, rf^/dx = ox/ (2ax- x'*)^. 
 
12 
 
 KXEUCISK8. 
 
 21 . 3/ = X / Vti* + x', % = a- dx / {d^ -)- x'f:. 
 
 22. y = X (a^ f- x-= ) Va^^l?, ./y/,f.r. = (a* + a' x' - 4a;*)/ V'/~T'. 
 
 23. /(x) = Vx + VI +x"^ /' (X) ^ V:^TviT^ / 2 VlTn?^. 
 
 x' I/'' 
 
 <? "*" 6^ "^ ■^' Difforentiatin;]? each ttirm, 
 2x rfx 2y dy ,/y ^2,. 
 
 " />" dx ny 
 
 25. //2 =. 4ax, <^,/y/(/x = (a/x) ■ = 2a/ j/. 
 20. .r,V + 62x-a^y = 0, %A/x = {h' + 'lx!,) / (,r-x^). 
 
 27. x' + 3/» = 3r*.Ty, ,V'^.'- = - (x' - ay) / \y^ - ax). 
 
 28. x^ ,/ - xf = a», dy/dx = {f - 2xy) / (x'^ _ 2xy) 
 
 dx 
 
 29. l{' y =. -, show that —'—_=+— ^;^ :^0 
 
 dx 
 
 dy 
 
 ^/a'-x:" ' V«'-/ 
 
tv^ 
 
 ^r 
 
 ^'" 
 
 CHAPTER IV. 
 TANGENTS AND NORMALS. 
 
 20. In th(! Intinitesimal Calculus a curve may be considered as 
 made up of straight lines (" length-elements ") each of infinitesi- 
 mal length. The extremities of these lines are called consecutive 
 points on the curve ; and any one of them is, when [iroduced, 
 regarded as a tangent to the curve. 
 
 21. Let the co-ordinates of a point P on a curve (fig. 9) of 
 which the equation is given be (x, y) ; then x - OB, y = BP. 
 Let X increase by the infinitesimal amount dx,i.e., bc or PR; 
 and let the corresponding change IIQ in // be called <Ii/. Then Q 
 is to be regarded as a point on the curve consecutive to P. PQ as 
 a straight line, and PQ produced as the tangent at P ; it being 
 understood that dj/ is to be obtained by difi'ercntiating the equa- 
 tion of the curve by the methods already explained. 
 
 Let the tangent make with the axis of x at T the angle </». 
 Then tan ^ = tan rpq = rq:pr - dy/tlx. Hence, the value of 
 dy/dx at any point is equal to the tangent of the angle which 
 the tangent at that point makes loith the axis of x* 
 
 * The fiction of §20 is adopted as a matter of practical convenience. 
 The tangent at p is not, strictly speaking, pq produced, but the limit- 
 ing position which pv continually approaches as q moves down to 
 coincide with P. The tangent of the angle which this true tangent 
 makes with the axis of x is .*. not, strictly speaking, rq : pb, but the 
 limit which this ratio approaches as pr approaches zero ; but this is 
 precisely the value adopted for dy/dx (§14). Hence, when we say 
 that tan ^ = dy/dx, <p is the exact angle which the true tangent at p 
 makes with the axis of x, provided that (as is always the case) dy/dx 
 is obtained by the ordinary process of differentiating. 
 
 On any infinitesimal portion of a curve being regarded as a straight 
 line, see §5. 
 
14 
 
 TANOENTS^AND NORMALS. 
 
 Fig. 0. 
 
 22. Let 8 E the length of the curve measured up to p from 
 any assumed point; then PQ - ds. Hence, in addition to tan ^ 
 = dt//dx, we have cos ^ = dx/dx, sin ^ = dy/dn, ds^ = dx^ + dy'\ 
 
 Also, 
 
 dx 
 
 the subtangent ib = y^ the tangent tp = v V 1 + (—\ , 
 
 dij \dyj 
 
 the subnormal bn = y -^, the normal np = y \ \+ (-^\ 
 
 dx ' \dxj 
 
 the intercept ot - x — // 
 
 dx 
 
 the intercept os = - ot tan ^ = ^ — x -^. 
 Also, 
 
 .'/-.'/i 
 
 - (j-^ (x-xjisthe equation of the tangent, 
 
 (/X< 
 
 and y —y^ = — ( -r- ) (x-x^) is the equation of the normal 
 
 at a point whose co-ordinates are (x j , y j ), the parentheses ( ) 
 
 indicating the particular value which the enclosed quantity hag 
 when Xj and y^ are substituted for x and y. 
 
 ^ 
 
^ 
 
 TANGENTS AND NORMALS. 
 
 Examples. 
 
 1. The curve a" y = x (x' - a»), fig. 10. 
 Differentiating we Imve tan o - dy/dx = (3x'- a") / a^. 
 At the origin'x = 0, ,'. tan </, = - 1, and /. ^ = - 45°. 
 At A or B, X = Hh a, :. tan ^ = 2, and f> = 63° 26'. 
 When x = ± a / v'S, tan v» = 0, /. 0=0. 
 The equation of the tangent at any point (xj, .Vi) is 
 
 _ /3xi'-a' 
 
 15 
 
 
 Fig. 10. Fig. 11. 
 
 2. The common parabola //'' = 4«x. 
 
 Differentiating each term, 2^ dy = 4a f/x, ,*. dy/dx = 2a/y. 
 
 •'y-yi = (2«/yi) (a^-a^i) is the equation of the tangent at 
 
 (xj, yi), and reduces to y^ 3/ = 2a(x H-Xj). The subnormal 
 = y dy/dx = 2a, a constant, 
 
 3. The equation x^ + y^ = a^ represents a four-cusped hypo- 
 cycloid (fig. 11), i.e., the locus of a point in the circumference 
 of a circle which rolls inside the circumference of a fixed circle, 
 the diam(?ter of the latter being 4 times that of the former. 
 Differentiating the equation, we get |x-i dx + ^y^ dy = 0, whence 
 dx/dy = - (x/y)^. The intercepts of the tangent on the axes 
 will be found to be a* x^ and a^ yK Squaring, adding, and 
 taking the square root we find that the partof the tangent inter- 
 cepted between the axis is of constant length, viz., a. Hence, if 
 a straight line of length a slide with its extremities on two given 
 
 ■ 
 
 
 m 
 
 i 
 
 
16 
 
 EXAMPLES. 
 
 lines at right angles to one another it will constantly touch this 
 curve, 
 
 4. To find tan (i> at any point of the curve 
 
 x^ y~- x}/' ■= 2, 
 Differentiating each term oy (C) 
 
 x^ dy + 2,xy dx — 2xy dy — y^ dx = 
 .-. dy/dx = (y'-2xy)/ (x^- 2xy). 
 
 5 . Find the equations of the tangents at the points ( — 1, 1), 
 (2. ! ) on this curve. Ans. x—y\-2 = 0, x = 2. 
 
 (t. Of the rectangular hyperbola xy = A;^ show that 
 
 (1) the tangent at (Xj, y^) is x/x^ +y/yx --- 2, 
 
 (2) the normal at (k, k) h y = x, v 
 
 (3) the subtangent always = the abscissa, 
 
 (4) the tangent makes with the axes a triangle of constant 
 area, viz., 2k^. 
 
 7. Show that the tangent to the curve 
 
 (x + ((y y = a^ X 
 
 is parallel to the axis of x when x = a, 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^ = x^. Ans. 3x — 2?/ = a, 2x i-'B?/ = 5a. 
 Also show that the subtangent = ^«, the subnormal = |tt, 
 the tangent = ^a V13, the normal =^a \fl6. 
 
 9. On the curve ;c^ y + b'^ x = a^ y, show that tan^ ^ h^/a^ 
 when X = 0, 2^1^ /^a"^ when -r. = \a, and f)h^/^a^ when x = 2a. 
 
 10. Find the equations of the tangents of the following curves 
 at th«^ given points : 
 
 (1) xy = 1 ^-x" at (1, 2). Ans. 
 
 (2) 3/(l+x)=x«at(-2, 8). 
 
 (3) x« + / = x''at (2,2). 
 
 (4) xN-// = x*at (2,2). 
 
 x-y + 1 = 0. 
 
 4x+y = 0. 
 
 2x-.y = 2. 
 
 5x — 3y = 4. 
 
 
 
EXAMPLES, 
 
 17 
 
 (5) x« +y'' = x^-' at (2, 2) . (n + 2)x- ny = 4. 
 
 (6) 3(«2_x) + 2(3/ + l)2 = 0at(l,-l). x = 1. 
 
 (7) o'y = «'at (a;i,3^i). 
 
 3xi'''x - a^y = 2x,'. 
 
 (8) 3/'' = 3x + lat(a;i,yi). '6x-2y^ ^ + 3x^+2 = 0. 
 
 x' 
 
 (9)l,+|. = lat(x„j,,). 
 
 a' 
 
 
 11. ^xy = A + x^] show that at (2, f) the subnormal = |, 
 the subtangeLt = 2, the normal = y^, the tangent \ . 
 
 12. y(l+x) = x^; show that at (1, ^) the Rubnormal = |, 
 the subtangent = |, the normal = ^ \^U, the tangent = y\^ V41. 
 
 13. x^+y* = x^; show that at (f, f V2) the subnormal = yj, 
 the subtangent = f, the normal = f V33, the tangent = ^)j V66. 
 
 14. y* = 3x + 1 ; show that at (5, 4) the subnormal = f , the 
 subtangent = 4/, the normal = ^ V73, the tangent = | V73. 
 
 15. a' +3/3 = as, fig. 11 ; show that ds = (a/x)i dx. 
 
 16. Find an expression for the length of the perpendicular 
 from the origin on the tangent at any point (x, y) of any curve- 
 
 Ans. (ydx-xdy)/ds. 
 
 17. In the case of [the parabola y* = 4ax, show that this per- 
 pendicular = X V« / {(i + x). 
 
 n 
 
CHAPTER V. 
 
 MDLTIPLE POINTS. 
 
 28. Tangents at the origin. If the origin of co-ordinates is a 
 point on the curve, the cfjuation of the curve can contain no 
 constant term ; let it be 
 
 a^x+ h^y + a,pi^ -f- h.pcij + c^jf -t- . . . = 0. 
 
 If {i,j) be a point on the curve infinitely near the origin, the 
 equation reduces to a^l-'r b^j = when i and ; are substituted 
 for X and y, the other terms being infinitely small in comparison 
 with those of the first degree (see §15). Hence a^x + i>^i/ = 
 is the equation of the tangent at the origl::, for it is satisfied by 
 the co-ordinates of the consecutive points (0, 0), and {ij) on the 
 curve. 
 
 24. If, however, in the given curve there are no terms of the 
 first degree, the equation reduces to «a r + b^ ij + c^j^ = when 
 i and j are substituted for x and y. Hence in this case 
 a^ x^ + b.iXy + c^ ij^ = is the ecjuation of a pair of tangents at 
 the origin ; and gen'.'rally, when the origui is a point on tha 
 curve, the terms of the. lowest dcyree (^equated to zero) represent 
 the tangents at the origin. 
 
 25. 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 b./- ^a^^ Cg is + , the equation a^ x^ + b,^ Xh r c^ y^ ~ 0, 
 represents a pair of distinct lines, and the point is called a node 
 (fig.s. 12, 18). 
 
 When b^^ ' AlU^c^ = 0, the lines are coincident; the two 
 branches of the curve therefore touch one another and the tangent 
 may be considered as a double tangent. Such a point is called a 
 cusp, which is said to be of the first or second species according 
 
T^f.' 
 
 MULTIPLE POINTS. 
 
 19 
 
 as the two branches of the curve lie on opposite sides (figs. 14, 
 15) or on the samo side (fig. 16) of their common tangent; and 
 to be double or single according as the branches lie on both sides 
 (fig. 19) or on one side only (fig. 14) of their common normal. 
 A cusp is also called a stationary point; for, considerint; the curve 
 as the path of a moving point, at a cusp the point must come to 
 rest and reverse its motion. 
 
 When b^^ — ^a^c^ '^^-~i the lines are imaginary and the point 
 is called a conjugate point. The co-ordinates of such a point 
 satisfy the equation of the curve, but the point is isolated from 
 the rest of the locus which the equation represents. 
 
 — Examples. 
 
 1 . The lemniscate a" (f - x^) + (3/^ + x^ = 0, fig. 1 2. 
 The origin is a node at which the tangents arc y^ — x^ = 0, l.e.^ 
 y -X and y = — x. 
 
 Fig. 12. 
 
 Fig. 13. 
 
 2. The folium x'-f/ =-. Saxy, fig. 13. 
 
 The origin is a node, the tangents being given by xy ~ 0, i.e., 
 X - 0, y =0, the axes. 
 
 :\. The semi-cubical parabola ay"^ = x^, fij^. 14. ' 
 
 The origin is a cusp, the tangents being given by y^ = 0, i.e., 
 two lines coinciding with the axis of .x. Moreover, the curve is 
 symmetrical with regard to the axis of ac, and?/ is impossible ifx 
 be negative ; hence the cusp is single and of the first species. 
 
 / 
 
 J 'il 
 
 11 
 
 
 
 :;fi 
 
 ;, ij 
 
 
20 
 
 EXAMPLES. 
 
 OX M X o 
 
 Fig. 15. 
 
 Fig. 16. 
 
 4. In the curve (jj x)^ = ce', fig. 15, the origin is a cusp at 
 which the tangent is y = x ; also, since y = x ±x^, y >x on one 
 branch and <x on the other, hence the cusp is of the first species. 
 
 5. In the curve {y —x')'^ ~ x', or y = x^ (1 ±'Jx), fig. 16, 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. 
 
 6. In the curve ^ = x^ (2x + l), fig. 17, the origin is a 
 node at which the tangents are y~ ±x. But in the curve 
 y'' = x^ (2x-l), tig. 18, the tangents are 7/^= —x\ and are 
 
 .'. imaginary, and hence the origin is a conjugate point. 
 
 G^ 
 
 Fig. 17. 
 
 Fig. 18. 
 
 O 
 
 Fig. 19. 
 
 Fig. 20. 
 
 7. There are certain cases in which the origin is a coniugate 
 point even when the terms of the second degree are a perfect 
 square. Thus, in the curves y^=x* (2x4-1), fig. 19, and 
 y^ = x* (2x — 1), fig. 20, the origin is a double point and the 
 tangents are given by y'= 0; in the first curve the origin is a 
 double cusp ; in the second, a conjugate point, since y is imagi- 
 nary for any value of x less than ^. 
 
imagi- 
 
 EXAMPLES. 
 
 21 
 
 8. The curve a/-3ax'3/ = as*, fig. 21. 
 
 The origin is a triple point at which the tangents are 
 
 Fig. 21. Pig. 22. 
 
 9. In the curve ay^ — ax^ y^ = oj^, fig. 22, the origin is a quad- 
 ruple point, at which the tangents are y =={i^ y = {i, y= ±x. 
 
 26- To ascertain whether any point other than the origin is a 
 multiple point we may transfer the axes to that point. 
 
 Example. 
 
 The point (1, 1) is a cusp on the curve (a? -?/)■ = (x — 1)^> 
 for the equation referred to parallel axes through (1, 1) is 
 (x-yy = x\ 
 
 Exercises. 
 
 1 . Find the tangents to the following curves at the origin : 
 
 (1) (a? + x"^) / = (a* - x^) X*. Ans, y = ± x, 
 
 (2) a^y (x^-y)=x\ y^o,x+y=0. 
 
 (3) x(y-x)«=/. x = 0,y = x,y=rx. 
 
 (4) a(y-x) (/ + X-) +x* = 0. y = x. 
 
 (5) y (»/-«) = (^Oy^+^O- . x+y^O, 
 
 (6) (x-a) y =x(x-2a). y-2x. 
 
 (7) .'/^- (a5- 1) xl Imaginary. 
 
 ! 
 
 ! ' 
 
 
Iil-rii,— TTtrT 
 
 22 
 
 EXERCISES. 
 
 2. Show that the origin is a single cnsp of tlic first species on 
 the cissoid y* (a - x) = x\ fig. 27. 
 
 3. Show that there is a node at the point (1,2), on the curve 
 
 {y-2y = {x-iyx. 
 
 4. Show that the point (2a, 0) is a node on the curve 
 a)j''=(x-a) (x-2ay. 
 
 5. Show that the point ( - «, 0) is a conjugate point on the 
 curve ay^ - x (a +a^)^ 
 
 .r- 
 
CHAPTEE VI. 
 
 ASYMPTOTES. 
 
 27. If a curve has infinite branches, and the co-ordinates of a 
 point on the curve and at a very great distance from the origin 
 are substituted for x and 1/ in the equation of tlie curve, all the 
 terms except those of the highest degree become infinitesimal in 
 compari>on with those of the liighest degree. Hence (cf. § 23) 
 the terms of the highest degree (equated to zero) represent lines 
 drawn through the origin in the direction of the infinite branches 
 of th' curve. If the curve has no infinite branches, the lines 
 represented by the terms of the liighest degree will be imaginary. 
 
 28. Befinition. An asymptote is a line to which a curve 
 approaches indefinitely near as it goes ofi" to infinity. We may 
 also regard it as a line which touches a curve, the point of contact 
 being at infinity, • lile the line itself passes within a finite dis- 
 tance from the origin. Asymptotes may be rectilinear or curvi- 
 linear. From §27 it follows that the rectilinear asymptotes of a 
 curve are parallel to the lines represented by the terms of the 
 highest degree in the equation of the curve. 
 
 29. If the equation of the curve, when y is expanded into a 
 series of descending powers of a;, take the form 
 
 - c d 
 
 V = aa; + 6 + - + -„+ •••> 
 
 the straight line y = ax + h will be a rectilinear asymptote. For, 
 when X becomes very large the terms after h will be very small, 
 and the value of y will approach indefinitely near ax + b, i.e., the 
 
 ordiDale of the line y = ax + b. The sign of the term - will 
 determine whether the curve lies above or below its nsymptote. 
 
 *l 
 
24 
 
 ASYMPTOTES. 
 
 But if the equation take the form 
 
 ., , d e 
 
 X X 
 
 there will be a curvilinear asymptote, viz., the parabola 
 y = ax^ + hx + c. Similarly there might be an asymptote of the 
 third degree. 
 
 Examples. 
 
 1. y^ = x^ + 8ax^, iSg. 23. 
 
 We have i/ = x^ ( 1 + — ) , or y = x ( 1 + — ) , 
 
 ^ X ^ ^ X / 
 
 which by the Binomial Theorem 
 
 ^ .X x^ ''' "^ X 1 
 
 .'. y = x-\-a is an asymptote. The curve lies below the asymptote 
 as X approaches + co and above it as x approaches — oo . 
 
 Fig. 23. 
 
 Fig. 24. 
 
 2. xf + ^ay"^ = X*, %. 24. Here ^ = x - a is an asymptote. 
 For 
 
 ' ^ X -^ ^ X ^ 
 
EXAMPLES. 
 
 26 
 
 The line x + 3a = is also a a asymptote, as will be seen from 
 §30. 
 
 2 ,,'i 
 
 X' y 
 
 3. The hyperbola -,-ra=^- 
 
 Here, y=± — U - -7 ) =* — U-^ni ) 
 
 hx ah 
 
 "^ a + tx 
 
 .'. the asymptotes are y =± — . 
 
 Fig. tiT). 
 
 4. 4i/(x-l) =x\ fig. 25. 
 
 By division, 4y = =x^ + x + l + ^ • 
 
 05-1 x-1 
 
 When 05 is very large the last term is very small, and the 
 ordinate of the curve becomes nearly equal to that of the parabola 
 4y =05*^ 4- 05 + 1, 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 05 is + and 
 when a; is — . The line 05 = 1 is a rectilinear asymptote, as will 
 be seen from § 30. 
 
II 
 
 26 
 
 ASYMPTOTES. 
 
 i 
 
 I 
 
 i J 
 
 30. Asymptotes parallel to the axes. Let the equation of a 
 curve, arranged according to descending powers of x, be 
 
 fxiy) ^"'+A{y) aj^-'+ZaO/) x«*-» + ... = o, 
 
 whence, /^Oy) +/, (;/) ;-^ A C^) "^ + - = 0. 
 
 *'■* JO 
 
 When X becomes very large, the terms after the first are very 
 small, and as x approaches oo the equation constantly approaches 
 /^{y) =0. 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 axis of x. The asymptotes parallel to the axis of y may be 
 Ibund in the same way. 
 
 Examples. 
 
 1 . xY - 3xy^ -x'-^- 2/ = 0, fig. 26 . 
 Arranged according to powers of x the equation is 
 (y-!_l) x2-8/x + 2y = 
 and according to powers of y, 
 
 (x-1) (x-2)/-x«=0. 
 Hence, ?/ = + !, and x = l,x = 2, are asymptotes parallel to 
 the axes. 
 
 
 
 Y 
 
 ] 
 
 
 \ 
 
 
 
 1 
 
 v_ 
 
 
 
 -~_ 
 
 .^ 
 
 / 
 
 
 
 
 \ 
 
 
 
 -'A 
 
 
 X 
 
 
 
 
 
 r 
 
 Fig. 2C. 
 
 Fig. 27. 
 
EXERCISES. 
 
 27 
 
 L>. The cissoid j/.'(tt-x) =x^ 6g. 27. 
 
 Tho line a - x = , or x = a is an aHymptote parallel to the 
 nxibof y. 
 
 3. yx- + h'^x - ahj = 0, or (x» - rt'O y + 6'^x = 0, 
 The asymptutcH are y = 0, x = a, x = - (*. 
 
 Exercises. 
 
 1. Find the asymptotes of the following curves : 
 
 (1) (x - a) y = x(x - 2a). Ans. x= a,x-y '^ a. 
 
 (2) (x-tt) 2/^ = x'(x + «). 
 
 :^ = a, X -t- 2/ + a = 0, X - J/ + a = 0. 
 
 (8) x^' + 2/' + 3a'x = 0. x+.y=0. 
 
 (4) x'-27/ = 2x«. ' 3x-9j/-2 = 0. 
 
 (5)x^ = x/ + 3/. a;-y = l, X -1-3 = 0. 
 
 (6) y = x^^ + 2x1 X -3/ + 1 = 0, X + .»/ - 1 = 0, ^ + 2 = 0. 
 
 (7) x'^ = xy-(l-x)/. 3(x-v/) = l, 2x = ±V5-l. 
 
 (8) x' = yix-\). 
 (&) 2^ + x^=xl 
 
 (10) x*^ =x^ + x + 3/. 
 
 (11) x^ + / = «l 
 
 (12) axy = x' - a*. 
 
 cc = 1 , 3/ = X* 4- x- + X + 1. 
 
 X4-1 = 0, 2/ = x'^- x+ 1. 
 
 ^ = X, x = ±l. 
 
 x-\-y = 0. 
 
 ay=x''. 
 
 2. Show that x-i-3 = and x-2 =0 arc asymptotes of the 
 «wve x'Y - 2xy + (x-Q) y'^-x = 0. 
 
 1 i 
 
s 
 
 CHAPTER VIT. 
 DIFFERENTIALS OF EXi*ONEx\TIALS AND LOGARITHMS. 
 
 31. Differentials of the exponentials a" and e". Let a be any 
 constant, v any variable quantity. Then (sec Appendix, 
 Note A.) 
 
 ^..Wf _ ^« ^ f^v^^dv _ 1 ) = a\\<lv + I A*(/y* + . . . ) 
 
 .•.(i(a') = A«'" do, (where A - log« a). (E) 
 
 When a = eE2-71828..., a is 1, 
 
 :.d{e^)=e''dv. (F) 
 
 If, as is sometimes the case, e" is written exp y, we have 
 
 d exp V = exp v dv. (Fj ) 
 
 Examples. 
 
 1. d{e'') = e'-' d (3x) = 3e^ (fe. 
 
 2. ^/(eO = e' ti( - x) = - c"^ t/x. 
 
 3 . d oxp \/l - x' = exp VI - x' r/(Vl - x'O 
 
 = — aj (/;c exp VI -'-^^ /Vl — a''*. 
 4. J(2-') = (log-, 2) 2-' tZ( - x) = - (loge 2) 2"' t^x. 
 
 32. Differentials cf Logarithms. First, suppose the loga- 
 rithms to be Napierian (also called hyperbolic and natural 
 logarithms), the base being e = 2'71828. Let y E log,, v, 
 then V = e", .'. dv = e^dy = vdy 
 
 :. dy, or r/(log, u) = . 
 
 Secondly, let the base be any number <i. 
 M loge Vj where m = 1 / logj a, ' 
 
 dv 
 
 (G) 
 Then '/ log,, v = 
 
 ^/(log„ t;) = M 
 
 V 
 
 (Gi) 
 
EXPONENTIALS AND LOGARITHMS. 
 
 29 
 
 M is the modulus of the syHtem of logarithuiH with base a. 
 Unless the contrary is indicated, the logarithms are always 
 assumed to be Napierian. (N.B. — Napierian log. = commou log. 
 — j^ log,o c = common log. X 2'302585.) 
 
 Examples. 
 
 1. t/ log (ax^) = </(a£c')/acc'' = iiax'^dx/ux^ = ddx/x. 
 
 2. d(x log x) = X t;(log x) H (log x) c/x = (1 + log x) dx. ' 
 
 33. To differentiate u'", where both u and v arc variable 
 quantities. Let ij E w", then log y = w log m ; hence, differen- 
 tiating, 
 
 ^ = V ^"^ + (log u) dv, :. dy = ,j r/^" + (log u) c/i;] 
 
 /. dijiC) = V w"-i du + (log «) w" dv, (G^) 
 
 i.e., the differential is obtained by supposing u and v in turn to 
 vary while the other remains constant^ and adding the results. 
 
 34. When an expression is made up of factors it is often 
 simpler to take logarithms before differentiating. 
 
 Example. 
 
 2/ = (x+l)^(x + 3)V(x + 4)^ 
 "" log3/ = ilog(x+l) + |log(x + 3)-41og(x + 4), 
 dy 
 
 dx . dx 
 -4 
 
 y " " x^\ ^X+3 *x+4 
 lOx + 12 
 
 ..hence, ,i,= [M:]'= 2^1^ </-. 
 ' ^ L a;4.i J (x-^-4)^ 
 
 (X -^- 4)^ 
 Exercises. 
 dx 
 
 x^-a* 
 
 2. y = log (aj/Vl+o!'), dj =- rfx/[a; (1 + a;')]. 
 
 I 
 
 I: 
 
 , 
 
:. 
 
 i\ 
 
 fvV 
 
 30 EXERCISES. 
 
 ^ 3. fix) --. Vx-log (1 + Vx),/'(x) = i(\+^xy\ 
 
 «4. d log (a; + Va?* ± 't'O = dx / ^Jx^ ± d\ 
 ^. !j = log e'/(l + e^), c/y = dx/{l + e"). 
 L,r,. y = (e' + e-'j/Ce'^-e-), r/y/(l -/) = dx. 
 
 7, vy = c'", dji = ?/, e'^"«"-i (/x. 
 
 8. d log(V.'e - a + ^Jx — b) = ^dx / V(x — a) {x — b). 
 
 10. Differentiate (1.) e^'«^ (2.) 3</', (3.) (3a)', (4.) x~, 
 
 (5.) x^', (G.) x'"-; (7.) (logx)», (8.) .y""'-'^ (9.) log (log x), 
 (10.) log(c' + e-0, (11.) x^logx, (12.) eXl-x-). 
 
 11. d(x') = (1 +logx) x'' dx. ^ 
 
 12. 1/ = \+x c", dx = (2 -//) tZ,y/e". 
 
 13. 7/ = log (y/x), (\-y)x dy = y dx. 
 
 14. Differentiate y = uv, y = uvio, and y = u/v, after taking 
 logarithms, and compare the results with formulae (C),(Ci) 
 and (D). 
 
 15. Differentiate y = v" after taking logarithms, thus showing 
 that d(v'') = V v"-^ dv. 
 
 16. The quantities ■^(e^ + ^"'), ^{(f - e-') are called, respect- 
 ively, the hyperbolic cosine (cosh x) and hyperbolic sine (sinh x) 
 of X. Show that 
 
 f/(sinh x) = cosh x dx, d(cos\i x) = sinh x dx. 
 
 17. Show that the subtangent of the logarithmic curve y = a' 
 is constant and = log^ e. 
 
 18. Find the subnormal of the curve ?/ = a^ logx. 
 
 Alls, a? fix. 
 
 19. Find the length of the tangent and the normal at the point 
 C.X, //) on the catenary y = -^ ( c" H ^^ " ) . 
 
 Ans. Tangent = y-/'\Jy^ - a^, normal = y^/a. 
 
 jmsm 
 
 '-^ A'i vA.^;iirr'i?:.l!Ti,iM' 
 
■A 
 
 \ 
 
 m 
 
 OlIAPTER Vlir. 
 
 DIFFEKENTIALS OF THE TRIfiOXOMETRIC RATIOS 
 (CIRCULAR FUNCTIONS). 
 
 ^y^Q. 
 
 o 
 
 B 
 
 Fig. '-'8. 
 
 35. To differentiate sin^;. In fig. 28, sinff = cp/w, iim(8 + dt)) 
 = BQ/a, :. inc'-jase of sinf^ =RQ/rt = pq cos<^/ a ( V the angle 
 RQP = w), and PQ/a -. (/^, 
 
 /. <Z(sin 6) = cos 6 de, 
 
 38. The same result may be obtained as follows : 
 
 sin {0 + dti) - sin ^ = sin cos do + cos d sin do — sin o 
 = sin « (cos de-\) -{- cos e sin f/(?. 
 
 But coBdM- 1 is an infinitesimal of the second oixler"(§ 6), and 
 sxvide - dd (§ 5), 
 
 /. d sin w = cos n dO. (H) 
 
 Similarly, d cos fi =- sin H do. (I) 
 
 l,^ 
 
 s'j , 
 
32 
 
 CIRCULAR FUNCTIONS. 
 
 \: 
 
 
 37. The differentials of the remaining ratios may be found by 
 first expressini:; them in terms of sine aucl cosine. The results 
 
 are^ 
 
 dtsind = sec'd de, 
 dGOtt)= — cosec^w dn, 
 d sec = sec tan do, 
 d cosec = — cosec cot do. 
 
 Exercises. 
 
 (J) 
 (K) 
 
 (M) 
 
 1. 6? sin 1111 = cosnH d(ni)) = n cos 7i6 do. 
 
 2. d sin (tan 6) = cos (tan 0) (i(tan o) = cos(tan ^) sec''^^ do. 
 
 3. y = tan « - w, dij = tan^B do. 
 
 4. d{^o - J sin 2(y) = ain^O do. ' 
 
 5. d{^ + I sin 2^)) = cos^^y do. 
 
 6. rf(sec + tan fl) = (1 + sin O) dd / cos2fl. 
 
 7. y = JtanVy - tan + «, dy = tan"*^ rf<y. 
 S. ij = log tan ^/^, t/y = cosec o do. 
 
 9. // = log tan(J,T -r ^^0) ^V = sec d dd. 
 
 -•A , / cos ^ \ sin « (2 + c js 2,0) dd 
 lU. d' * ^ 
 
 (COSf^ \ 
 COS 2«/ ~ 
 
 cus2 2^ 
 
 11. d 
 
 sin ^^ \ (cos*^ - sin'w) do 
 
 (sm \ 
 1 + tan 0/ 
 
 (sin e + cosy) 5 
 
 12. f{x) = sin(logx),/'(x) = aj-i cGs(logx). 
 
 13. d 
 
 e cos X 
 
 = eX 
 
 cos X — sin X 
 
 ) dx. 
 
 14. d log sin « = cot o do. 
 \h. d log cos y = - tan (^ c^'^. 
 16. d log tan 9 = sec cosec ^ do. 
 VI. d log sec = tan u do. 
 
 * To these results may be added 
 
 d vers = d(l - cos 6) = sin dO, 
 d covers = d(l - sin^) = - cos B dti. 
 
^tl ■:/! 
 
 EXERCISES. 
 
 33 
 
 18. ri 2/(l + tan J«) = -f/«/(l + sin/?). 
 
 19. d log V(l - cos w) / (1 + cos ^/) = cosec 6 dft. 
 
 20. (Z sin nH sia"^ = ;i sia'*-if^ .sin(n + 1) « c?ff. 
 
 21. By differentiating sin 28 - 28in « cos(^, show that 
 
 cos2^ r= cos^^- sin2^. 
 
 . 22. If tan * ;/ = V(l^ ac)7(lTx), di/ = - dx/s/1 - x^ . 
 
 23. Diiferentiate (1.) sin''/y cos^^^, (2.) tan^^-sec^^, (3.) 
 e*'°*cosx, (4.) x*sin^x-x2, (5.) (sinx)''"', (6.) (logx)'''"% 
 (7.) sin(e^), (8.) log Vsin ^ +logVcos^. 
 
 24. The Cycloid. This is the curve traced by a point in the 
 circumference of a circle which rolls along a straight line, fig. 29. 
 
 Let 6 = the angle through which the circle (of radius a) rolls 
 while the tracing point moves from to P. 
 
 A 
 
 O M B X 
 
 Fig. 2S). 
 
 Then 
 
 X = CM = OB — MB = arc PB - PD = ai9 — a sin o, 
 
 ^ = MP = BC — DC = a — a cos 6. 
 
 From these two results (/ may be eliminated ; but as the result- 
 ing equation is not algebraic we shall suppose the locus deter- 
 mined by the simultaneous equations 
 
 X = a(6- sine) f 1/ = a (1 — cos 6). 
 
 Produce BO to meet the circle in E, then PE is the tangent and 
 PB th3 normal at P. For, at each instant we may suppose the 
 circle to be turning about the lowest point as an " instantaneous 
 centre of rotation," and the angle bpe is a right angle (an angle 
 in a semicircle), .". p is moving in the direction PE, i.e., pe is 
 the tangent to the curve. 
 
34 
 
 THE CYCLOID. 
 
 liii 
 
 This may also be proved thus : 
 
 dx = a(l- cos 6) dd = BB do, dy = a sin H de = DP dt^, 
 .'. if the tangent make an angle with the axis of x, 
 tan^ = dy/dx = dp: bd = tan dbp = tan dpe 
 .*. PE is the tangent. 
 
 Since CE = Cp, /. OEP = \e^ .'. the normal pb = 2a sin \{). 
 
 25. If the axes of the cycloid be taken as in fig. 30, its equa- 
 
 /^ 
 
 
 M 
 
 
 V 
 
 
 
 (^^ 
 
 "x 
 
 1 \ 
 
 
 /v 
 
 
 V 
 
 \ 
 
 
 
 Fig 
 
 X 
 
 . 30. 
 
 
 
 
 tions are 
 
 X = « (1 — cos fi), y = a (6 + sin 0), 
 
 I) being the angle through which the circle has rolled from o. 
 
 The locus of Q (of d in fig. 29) is called the " companion to 
 the cycloid." Its equations are 
 
 X = a (1 - cos 6), y - ad. 
 
 Show that in this curve tan </. = cosec w, and hence that ^ is least 
 when X = a. 
 
 26. At any point of the cycloid, show that 
 
 dy_J'^_a_^J^a)i-y^ 2 29, 
 dx ^ y y 
 
 -^=V 1= , fig. 30. 
 
 dx ' X x 
 
 27. In the cycloid, fig. 30, show that 
 
 ds/dx = V(2<t)/«. 
 
 i 
 
CHAPTER IX. 
 
 DIFFERENTIALS OF ANGLES (INVERSE CIRCULAR 
 
 FUNCTIONS). 
 
 lit: 
 
 Fig. 31. 
 
 38. To differentiate sin- 1 -> i.e., the ande whose sine ia x/a. 
 
 Let 6 = the angle. Then, (hg. 31) x is CP, x + dx is bq, 
 .', dx = RQ, and RQP = ^, .'. d^ = vq/a = rq sec RQP/a = 
 
 dx sec / a = [dx . a / V«^ - ^t] / ^> 
 
 X 
 
 .•.cisia-i- = 
 
 dx 
 
 « Va^* - »^ 
 
 « 
 
 It may also be obtained as follows : letsin-^.-^f?, .'.»=» sm0, 
 
 dx =-- a cos c?(/ = V^t' - «^ <^, .*. <^ or . 
 
 
 I ■ 
 
 J. 
 
 ^j ^ 
 
 
 
 
 ' ' i 
 i ■ 
 

 36 
 
 INVERSE CIRCULAR FUNCTIONS. 
 
 (fsin-^'^.- --^- (N), ordsm-ix^-^—. if«=l. 
 
 VI -x' 
 
 Similarly 
 
 d cos-i - = ^^=; (Ni ), or t? cos-i jc = - , if a = 1 ; 
 
 <2tan-^- = 
 
 a; atZx 
 
 VI -x' 
 
 <?« 
 
 a a^ + x^ 
 
 (P), or d tan-i x = — ^, if a = 1 ; - 
 
 ,icoti-=-4^,(Pi),orcicot-^x=-~, ifa = lj 
 
 a a^ + x^ 
 
 1+x^ 
 
 asec-^- = — ( Q), or d sec-^x = — , ifa = l 
 
 a X Va>^ - a* » V*"'' - 1 
 
 a cosec-i - = =:^ (Q , ), or rf cosec-* x = > 
 
 * X V** - a'^ 
 
 X V»'^ - 1 
 
 if a = 1. 
 
 1. c?sin-i(2mx2)= 
 
 2. (i(log tan-^ x) = 
 
 3. tZsin^ 3x^ z- 
 
 Exercises. 
 dJ(2mx'^) 4mx <^x 
 
 VI - (2mx»)^ VI - 4mV 
 c? tan-i X _ (^£c 
 
 tan-^x ~ (\^ x^) tan-^x * 
 6x dx 
 
 VI 9x* 
 
 . , , a a dx 
 4. dcos-i— = 
 
 « X Va5* - a* 
 
 R T • 1 X — a a c?x 
 5. asm-i. = 
 
 * X V2ax - a* 
 
 6. (^cos-i^Zl^^ — ^'^ . 
 
 * V2ax - x'' 
 
 a V«* - X* 
 
 8. dtujT^e = 
 
 e' + e- 
 
EXERCISES. 
 
 37 
 
 9. dtm-^ ^/l+x = dx/l2(2 + x) ^/i+x'\. 
 10. c?tan-i V»^- 1 = dx/(x ^Jx^-l). 
 
 11. (^sec-i Vl+»' = ^«/(l+a;'). 
 
 12. d tan- 1 (VI + x' - .x) = - ^dx / (1 + x^). 
 
 
 13. d tan-i (x / VI - x^) = dx / ^/l ~ x\ 
 
 
 14. ^sm-i(x/Vl+x^) = c^/(l+x2). 
 
 [• 
 
 15. ^ sin-i [(1 - xO / (1 + a)')] = - 2r/x / (1 + x^). 
 
 ■■ J 
 
 16. (£tan-i [2x/ (1 -x^] = 2t?x/ (1 +x2). 
 
 1; 
 
 17 avers ^ — 
 
 
 « V2ax - x^ 
 
 -J 
 
 ^o , 1 X t/x 
 
 18 ffl covers" ^ 
 
 [ 
 
 a = 1. 
 
 a V2ax-x2 
 
 
 19. 3/ = a sin-i (x / a) + ^Ja? - x^, c?y = f/x V(« - x) / (a + x). 
 
 
 20. 7/ = V»^ — «^ - ^^ sec-i (x / a), dy = dx ^Ix^ - d^ / x. 
 
 
 21 ^f,n.n-i X/^"^ 1- '*'^'' . 
 
 
 « 2x V2ax - a' 
 
 
 22 a sin"-^ V — - • 
 
 
 " b-a 2^(x-<i) (h - x) 
 
 
 23. ?/ = X lan-i ,„ _ \aa ^1 + x^ di/ = tan-i x c?x. 
 
 
 X J (l+x) dx 
 
 
 ^/x — a, 1 X r/w 2ax^ 
 25. // = log V + tan-i "- j -^ = -^ — -^ • 
 
 ' ° X + « a (/x X* - a* 
 
 \. 
 
 26. Differentiate (1.) sin-i (2ax-x^), (2.) .re*"" % (3.) 
 
 
 sin- 1 (cos /i), (4.) sin(co8-ix), (5.) sec'i (a/ Va^-x^), (6.) 
 
 
 log(tan-i x), (Y.) log (sin-i x), (8.) sin-i (VsinW). 
 
 
 27. If sin-i X -1- sin-i j/ = a, d>j / Vl - .y' + ^^ / VI - a;' = 0. 
 
 1 ;..■,.•.•. 
 
 i j ii I li I 
 
^ 
 
 CHAPTER X. 
 
 FUNCTIONS OF MORE THAN ONE VARIABLE. 
 
 39. Such functions may be differentiated by the formula) 
 already given. 
 
 Examples. 
 
 1. u = (x + y'^y. Here u is to be regarded as a function of x 
 and y, both of which are assumed to have differentials. We 
 have 
 
 du = S(x + i/y d(x + f) = S(x + fy (dx-\- 2i/ dij), 
 .*. du = 3{x + i/'^y dx + Gy(x + i/'^y dy. 
 
 d 
 
 2. ?; = sin-i 
 
 
 ydx — xdy dx x dy 
 
 y V//' - x' Vi/' - a;' y Vi/' - «' 
 
 40. Partial Differentials. It will be observed in these ex- 
 amples that the first term of the result is what we should have 
 obtained if we had differentiated u on the suj)position that x 
 alone varied, y being regarded as a constant ; let this be written 
 d^u. The second term is what we should have obtained if we 
 had differentiated on the supposition that y alone varied, and 
 , this we call dy u. Hence in these examples 
 
 da = dji + d,,n. 
 
PARTIAL DIFFERENTIALS. 
 
 89 
 
 The same thin' is true for all functions of two variables. 
 For, if wj dirterontiate by the ordinary methods, we shall in 
 every case get a result which may be written 
 
 du = V dx + Q dy 
 
 where P and Q may contain x and y, but not dx or dy. l^dy = 
 the right-hand .side reduces to P dx, which is therefore dj.u, the 
 differential of u on the supposition that// is constant (i.e., dy = 0) 
 while .c varies. Similarly Q (?/y it^ dyii, t\\o differential of u on 
 the supposition that y alone varies. 
 
 .*. du = djt + dyU. (1) 
 
 The differentials dji, dyU are called partial differentials of u 
 with regard to x or y, du being called the total differential ofu. 
 The total differential is therefore equal to the sum of the partial 
 differentials. 
 
 Similarly if u is a function of three variables x, y, z, 
 
 du = djU 4- dyU +• djii. (2) 
 
 It should be noticed that formulae (C), (Ci), (D), (Gg) are 
 particular cases of functions of two or more variables. 
 The result (1) may be put into the form 
 
 which brings out clearly the fact that the coefficients of dx and 
 dy are the partial differential coefficients of u with regard to the 
 variables ; as a matter of fact, however, it is usually written 
 
 /du\ /du\ 
 
 in which it must be carefully noticed that du is used with three 
 distinct meanings. 
 
 41. Implicit Functions. If the equation of a curve be given 
 in the form y^ — 4ax = 0, ax^ + by^ = c, etc., y is to be regarded 
 as a function of x (§ 7) ; it is called an implicit function of x. 
 If, however, we should solve for y in terms of x, we should 
 obtain y as an explicit function of x. We have already had 
 numerous instances in which the differentials of such functions 
 have been obtained by the ordinary methods. 
 
 Ill 
 
40 
 
 IMPLICIT FUNCTIONS. 
 
 
 ''I: 
 
 In general, if /(x, if) = c (where c is any constant, including 
 zero) be the equation of a curve, the first member of the equa- 
 tion is in reality a function of two variable quantities x and y ; 
 calling it u and differentiating the equation we get by (3) 
 
 ■ O"-©''^ =«.•••!= -0/(|). w 
 
 by which the inclination of the tangent may in some cases be 
 most easily found. 
 
 If this be substituted in the equations of § 22, it will be seen 
 that 
 
 Qdx) (^* ~ *i ) + (I7/) J (y - y 1 ) = ^' is ^^»« tangent, (5) 
 
 and 
 
 
 '(/(r 
 
 ihr 
 
 the normal 
 
 \ihjj 1 \dxj 1 
 at the point (x^, //j) on the curve. 
 
 Exercises. 
 
 1. w = sin(x^ -3/^), djix = 2x cos(a;^-i/^) c?x, 
 
 dyU z= — 2y cos(a;^-y) dy. 
 
 2. n= (x-y) / {x +y), du = 2(y dx-x dy) / (x + y)^. 
 
 3. M = (ax^ + hy^ + cz^y, 
 
 du = 2nH^zL {(IX dx + hy dy + cz dz). 
 
 4. Given » = r cos 0, y = r sin ft, show that 
 
 dx^ + dy"^ = dr^ + r^ di?^ 
 X dy —ydx = r^ dO. 
 
 5. If tan ft = y/x, {x^ + y^) dft = xdy~y dx. 
 
 6. If M = tan-i (x/y), du = (y dx — x dy)/(j:c'^ + y'^) . 
 
 7. ?t = x^, dji = yx"-^ dx, dyU = (logx) x" dy. 
 
 . , hi = y x"-^ dx+ (log x) x" dy, as in (Gg). 
 
 8. If w = log^x, ux ~+y '^ = 0, 
 
 the differential coefl&cients being partial. 
 N.B. logy a; = loge X / loge y. 
 
 (6) 
 
EXERCISES. 
 
 41 
 
 9. Show that the perpendicular from the origin on the tan- 
 gent to the curve u = c is 
 
 / (III (lu\ /. / /du 
 
 \^ /dii\^ 
 dj/J / ' \dx/ \d}j/ 
 
 10. In the case of the curve x^ +2/^ = a^ fig. H, show that 
 this perpendicular = ^Jaxij. 
 
 11. In the case of the parabola (x/«) ^^ + (y/6) ^ = 1 , show 
 that this perpendicular = [rt/>a:/y/(«a5 + %)]-. 
 
 12. Find the equation of the tangent at any point of the 
 curve 
 
 (a;/a)'"+0//6)'« = 2, 
 
 and show that {x/ii) + {y/h) = 2 touches it at the point (a, 6). 
 
 I 
 
 
 >Si 
 
 iilfllilt 
 
 t > 
 
 i 
 
 ! ; 
 
 i 
 j 
 
CilAPTEll XI. 
 
 SMALL IIIFFKIIKNCES. 
 
 43. By substituting differentials for small differences wc may 
 sometimes obtain useful results. 
 
 EXAMl'LKS. 
 
 1. Given sin 30^ = 1/2, cos 30° = V3/2, find sin 30° 1'. 
 Here the anL:;le increases by a small amount and it is required 
 
 to find the snjuU increment in the sine. 
 
 We have d sin ti ^ cos t) dii ; cos i) = V3/2, d') = C0/20()2G5 rdn., 
 .'. d Hin II = -0002519, /. sin 30° 1' = .5002519, which is correct 
 to the last decimal place. 
 
 2. How much must be added to log sin 30° to get log sin 30° 1' ? 
 Wc have d log sin ii = qohii dt)/ii\n tl = "0005038, which is the 
 
 increase of the Napierian log. ; the increase of the common log. 
 is <" ined by multiplying by the mddulus. 
 
 .-. -0005038 X -4342945 = -0002188 
 
 is the requir.'d increment. 
 
 The ditfe. . .^e columns in the mathematical tables are found 
 or checked 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 = ^,t/"^ //, .*. dv = ^t (2r/i dr -t- r^ dh) 
 = ^;r (2 X 3 X 4 X -006 - 3^ 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 
 (centigrade), what will be the increase in volume of ati iron ball 
 of 8 in. radius when the temperature is raised 25 degrees ? 
 
 The volume v = ^rrr^, .'. do = "inr^ dr 
 
 = 4,r >. 8^ X 25 X 000011 X 8 = 1.77 cub. in. 
 
 5. In a certain triangle, h - 445, c = 006, a = 62° 51' 33", 
 whence a is calculated and found to be 565 ; it is then noticed 
 
 i. 
 
 iiS 
 
DIFFERENTIALS OF TRIANGLES. 
 
 48 
 
 that A ^^Imukl have Ictn 02° 53' 31"; what i^ the correction 
 to nl 
 
 The change in A is 1' 58" - 118" = 118/20f)265 rdn. 
 
 Also, a^ - P + c^" 2hc cos a ; diffoivntiating this, supposing 
 h and c constant, we have 
 
 2(1 <hf = 2bc sin a d a, .'. iht = he sin A dA/a 
 = 445 X (50(1 X sin 02° 51' 33" x 118/2002G5 x 5(55= -243. 
 G. Find the relation connecting small differences of t and A in 
 the equation 
 
 sin h = sin 6 sin 6 + cos <;» cos rJ cos t, 
 9 and h being constant. 
 Differentiating and arranging the terms, wc get 
 
 dt=C^'>-^^'\dA. 
 
 \ sin t tan t/ 
 This is the " Equation of Equal Altitudes " in astronomy. 
 
 43. General formulae for ditferantials of triangles. Differen- 
 tiating a- = h^ + c^ — 2hc cos a, supposing «, i, c, and A to vary, 
 we have, after dividing by 2 and arranging the terms, 
 
 a da = (/> - c cos a) db -t- (c — 6 cos a) dc ■{- he sin A dA. 
 
 But if perpendiculars be drawn from B and c to the opposite 
 sides it will be seen that 
 
 Z> — c cos A =a cos c, and c—h cos A = a cos B ; 
 also, c sin A = ffl sin c, ,*. substituting and dividing by a, 
 
 da = cos c dh + cos B dc+ b sin c dA. (1) 
 
 Similarly, db = cos A dc + oos c da + c sin A rfB, (2) 
 
 dc = cos B da + cos v db + a sin b dc. (3) 
 
 The same formulae will apply to spherical tria jgles when 
 
 b, c, a, in the last terms are changed into sin b, sin c, sin a, 
 
 respectively. 
 
 From these formulae we may find the differential of a side 
 when the differentials of the two other sides and their contained 
 angle are given, also the differentials of the angles when those of 
 the sides are given. 
 
 \ 
 
 0: 
 
 m I > 
 
 S > : ■ 
 
44 
 
 SOLUTION OF EQUATIONS. 
 
 ! 
 
 i 
 
 \N 
 
 «l 
 
 The relation connectinL>; the differentials of two sides and the 
 
 ^.J 
 
 opposite angles may be found thus : take the logarithms of 
 a/6 = siu A/sin b and differentiate, 
 
 . (I.t db _ d\ (Ib 
 'a b tan a tan B 
 
 (4) 
 
 44. Solution of equations by approximation. Since 
 
 /(x + dx) =f(x) -vf'Qc) dx (see § 13) 
 
 .-. /(a + h) =f(a) H-/'(a) h, nearly, (1) 
 
 where a is any particular value of x, and h is a small increment 
 of that value. 
 
 Let /(x) = be any equation ; let a be a quantity which is 
 known (by trial or otherwise) to be an approximate value of a 
 root of the equation; a + h to ha that root, where h is a small 
 quantity. Then/((i + h) = 0. But /(a + h) =/(a) +fX<t.) h, 
 nearly; whence h = —J\d)/f\a). Hence if a be a first 
 approximation to the root, 
 
 (I — 
 
 fw 
 
 (2) 
 
 will be a nearer approximation. If this new value be substituted 
 for a in (2) a nearer approximation still will be obtained, and 
 with this a closer approximation, and so on. 
 
 Example. 
 
 x^ - 4x - 2 = 0. Here/(x) = x=' - ix - 2, and /'(x) = 3x- - 4. 
 Since /(2) =-2, and/(3) = 13, there is a root of the equa- 
 tion lying between 2 and 3. Let us then assume 2 as the first 
 approximation. Then 
 
 2-/ffi- = 2-— =2-25 
 /'(2) 8 
 
 is a second approximation. Again 
 a third approximation. 
 
EXERCISES. 
 
 45 
 
 ! . ! I 
 
 Since /(O) = - 2, and /( - 1) = 1, there is a root lying be- 
 tween and -1. Assume it to be —-5. Then the next 
 approximation is - "SSS. Again, taking — "54 as tliis second 
 approximation we find the next to be - '5392. 
 
 Similarly x =- l'(J76 is the third root. 
 
 Exercises. 
 
 1. Given loge 900 = G-80U, find log^ 901. 
 
 Increase of log x = dx/x = 1/900. Ans. 6*8035. 
 
 2. Given log,^ 1000 = 3, find log,, 1002. Ans. 3-00087. 
 
 3. Find tan 45° 1', also sin 30° 1'. Ans. 100058, -50025. 
 
 4. Find the relation connecting small differences of 6 and A 
 in the equation 
 
 sin (5 = sin </) sin h — cos cos h cos A, 
 (f, and h being constant. 
 
 Ans. Ja = cos (5 c?(5/(cos ^ cos h sin a). 
 
 5. The sides a and 6 of a right-angled triangle abc (c = 90°) 
 receive small corrections da and db ; what is the change in the 
 perpendicular j:? from the right angle on the hypotenuse? 
 
 Ans. -J- = - — I J or, dp = cos^ A da + cos ^ B dh. 
 
 if a^ b^ 
 
 6. Show that the change in the hypoten'jse = 
 
 sin A da + sin P dh. 
 
 7. Find roots of the following equations : 
 
 (1) «^4-a;*-13x-4 = 
 
 (2) £c=*- 9x-10 = 0. 
 
 (3) ic' - x^ - 2 = 0. 
 
 (4) x*-12x'+12x- 3 = 0. 
 
 (5) x" = 34. 
 
 (6) x''-12x = 200. 
 
 (7) x'' = 5. 
 
 (8) e"(l+x^) =40. 
 
 Ans. 3-303. 
 
 3-449. . 
 
 1-696. 
 
 2-858, -3-907. 
 
 2-024. 
 
 2-982. 
 
 2-129. 
 
 2-046. 
 
 U 
 
 t 
 
 
 « 
 
 
 i '4 
 
 i 
 
 
 
I! 
 
 S i 
 
 i\ 
 
 t! 
 
 CHAPTER XII. 
 DERITED FUNCTIONS. SUCCESSIVE DIFFERENTIALS. 
 
 45. Derived Functions. We have seen that the result of 
 differentiating /(x) may be written /'(x) dx. The differential 
 ciefficient /''(x) is a new function of x; lot its differential 
 coefficient be written /"(x), then df'(x) =/"(x) dx. Similarly 
 d/"(x) =f"'{x) dx, etc. 
 
 The several functions /'(x),/"(x),/'''(x), etc. , are culled 
 the first, second, third, etc., differential coefficients of /(x) ; 
 they are also called the first, second, third, etc., derived functions 
 or derivatives of/(x). 
 
 Examples. 
 
 1. fix) = 3x' - 2x -h 4, f\x) = 9x^ - 2, /" (x) = 18x, 
 
 f"'(x) = 18. 
 
 2. /(x) = sinx, /'(x) = cosx,/'''(x) = — sinx, 
 
 /'"(x) =-cosx, etc. 
 
 3. /(x) ^ log (1 + x), /'(x) = 1/(1 + X), 
 
 /"(x) = - 1/(1 +x)-, etc. 
 
 ^. /(•-«) = <,/'i^) = «',/"(«^) = e% etc. 
 
 46. Independent variable. In any equation connecting two 
 variable quantities x and y, e.g., 
 
 2 2 
 
 ?/ = 2x^ '^, + ')7,= l, ^' = logy, 
 
 we may regard either x or t/ as the independent variable (§ 7), the 
 
 other variable being regarded as the function ; or both x and y 
 
 may be regarded as functions of some other variable which may 
 
 x^ y^ 
 be considered as independent ; thus in -^i + ^ = I we may have 
 
 2:1 , 
 
SUCCESSIVE DIFFERENTIALS. 
 
 47 
 
 x = acoae, i/ = b sine, and may take as the independent 
 variable. 
 
 The differential of the independent variable is always sup- 
 posed constant, i.e., the independent variable is supposed to 
 increase by the successive additions of the same infinitesimal 
 increment. 
 
 47. Successive Differentials. If ?/ =/(x) we have di/ =f'(x) dx. 
 If now iny(x), x be supposed to receive tlie same infinitesimal 
 increment dx, we have {dx being by supposition constant) 
 d(dy) = {df'(x)'\ dx = l/"(x) dx'] dx =f"(x) dx\ 
 
 The differential oft?// is written d"^ y ; 
 
 .-. dhj =/"(x) dx\ or d'y/dx" =/"(x). 
 
 Similarly if d(d'^y) E c?'y, we have 
 
 (fy =f"'(x) dx\ or d^y/dx^ =/"'(^), etc. 
 
 Whenever such expressions as d^y/dx^ d^y/dx^, etc., are met 
 with it is to be understood that x is the independent variable.* 
 
 The expressions d^y, iFy, etc., arc called the second, third, 
 etc., diiferentials of y ; d^y i.i read d-two y (or second dy, or 
 t?-second y, but not r?-^quared ;/). 
 
 * We have di/ =f'(x) dx, and d'l/ =/"(x) dx''. (I) 
 
 This is on the supposition that the new increase \ x is the same 
 
 differential dx. If, however, it exceed the original increment dx by 
 
 an amount represented by d^x, we differentiate by (C) supposing dx 
 
 as well 0,8 J '(x) to vary, the result being 
 
 d^y =f"(x) dx'- +f'(x) d^x (2> 
 
 from which d~y may be found wlicn d'^x is assigned. 
 
 If we substitute dy / dx (or f'(x) and solve forf'Xx) we get 
 
 f"(x) = (dx d'y - dy d-x) / dx\ (3> 
 
 Hence, comparing (1) and (.3), the d-y obtained on the supposition 
 that the differential of a; is constant 
 
 = (dx d'hj - dy d'^x) / dx (4^ 
 
 obtained on the supposition tiiat the differential of x* is not constant. 
 (4) may be written d-y - d'^x dy J dx, or for a curve d^y - d-x tan 0, 
 
 \m 
 
48 
 
 I >< 
 
 successive differentials. 
 Exercises. 
 
 1. ?/ = sin x, dy/dx, = cos :c, d^y/dx^ = — sin x, etc. 
 
 2. y = ax^ + 6a5 + c, dy/dx = 2ax + 6, d^y/dx^ = 2a. 
 
 3. y = (a + x)', c?j//ffe = 3(a + x)^, d^y/dx/ = 6(a + x). 
 
 4. y = e"^, t?"^ - e" t/x". 
 
 5. 3^ = e-^, d^y = {- \y e-' r^x". 
 ' 6. y = x^ log X, d^y/dx^ = 2x-i . 
 
 1. y = cos ax, d'^y/dx^ = a^ cos ax. 
 
 8. X = sin-V, ^xA^y = (1 -/)-'% ^VcZ/ = y(l -/)i. 
 dy/dx = cosx = (1 - y^)^, d'^y/dx/ = — sin x = — .V. 
 
 9. Given xdy-y dx = r^ de (Ex. 4, p. 40) 
 show that X d^y - y d^x = 2»' (7r rZ^ + r^ d% 
 
 10. If ?/ = a cos Tix + 6 sin nx, d'^y/dx/ + w^^ = 0. 
 
 11. If ^ = ae'" + 6e-"^ d^y/dx'' - n^y = 0. 
 
 12. If // = 6""' cos X, d^y/dx'^ + 4^ = 0. 
 
 13. If ^ = sin-^ar, (1 - x^) d^y/dx? - x dy/dx = . 
 
 14. If y = tan-ix, (1 + x^) ^^^/cZx^ + 2x dy/dx = 0. 
 
 15. If y = e^ sin x, tZy^Zx^ - 2 ^/y/de + 2^ = 0. 
 
 16. By differentiating 
 
 dx = cos ^ ds, and cZy = sin (/> dJs, 
 show that 
 
 (<zv-)' + (d'yy = {df dsy + (dhy. 
 
 17. y^ = 4ax, 2y dy = 4a dx, or ytZy = 2a c^x; differentiating 
 again, y d^y + dy'^ = (dx being constant), 
 
 /. y d'y f (2a (7x/^)2 = 0, or d\ij/dx^ = - 4aV?/'. 
 
 18. Given the ellipse L-+^, = 1 show that 
 
 ..2 ^^ ' 
 
 dx 
 
 ¥x 
 
 a'y 
 
 a 
 
 d^ 
 dx^' 
 
 a^f 
 
 19. x^ + / = 3ax^, d^y/dx? = '2ia?xy/(ax-y'^y 
 
 20. a* + / = 2x^, (^Vc^x^ = aV(a; - yy. 
 
CHAPTER XIII. 
 DIRECTION OF CURVATURE. POINTS OF INFLEXION. 
 
 48. Geometrical meaning of d-y. Let the co-ordinates of p 
 (figs. 32, 33) be (x, y). Then oa = jc, ap = ^, ab = dx, SQ = dy, 
 and PQ produced is the tangent at P (§ 20). 
 
 Now, if X be again increased by the same increment dx (= bc) 
 the new dy is tr, the increment of dy is therefore VR (since 
 TV = sq), .'. d'^y = VR ; hence d^y represents the amount, mea- 
 sured paral'jl to the axis of y, by which the curve rises above 
 the tangent in passiug to the next consecutive point. 
 
 m 
 
 lig. yi. 
 
 Fig. 36. 
 
60 
 
 POINTS OF INFLEXION. 
 
 m\ ^ 
 
 49. In fig. 32, fPi/ and /. d^iz/dx" is + , in fig. 33 it is - ; 
 hence, when (Py/dx^ is + the curve is concave upward, and 
 when it is — the curve is concave downward. A point where, 
 as X increases, the curve ceases to bend upward and begins to 
 bend downward, or vice versa, is called a point of contrary 
 flexure, or a point of inflexion- At such a point d^y/dv? must 
 change from + to - or from — to + as x increases. But in 
 order to change sign, a quantity must pass through the value 
 or 00. Hence, at a point of inflexion d^y/do? is or oo and changes 
 
 sign 
 
 * 
 
 The tangent at a point of inflexion is sometimes called a 
 stationary tangent ; for, if the tangf^nt be supposed to roll round 
 the curve, it comes to rest at such a point and reverses its motion. 
 The tangent must be regarded as passing through three con- 
 secutive points of the curve; in other words, at a point of 
 inflexion, two consecutive length elements, and therefore two 
 tangents, are in the same straight line. The tangent appears to 
 cut the curve, or the curve to cross the tangent, 
 
 Examples. 
 
 1. y = (x- 1)'', d^y/d'y? - ^^{x - 1). This is - when 5C<1, 
 when a- = 1, + when x> 1 ; hence, as x increases, the curve bends 
 downward until o^ = 1, and upward afterwards; .'. there is a 
 point of inflexion where x = 1. Since 2/ and dy/dx are also 
 when a; = 1, the axis of a; is tlie tangent at the point of inflexion, 
 (fig. 36). 
 
 2. ?/ = (« - l)^ dhj/dx? = 12(x - 1)'^, which is when x = \, 
 but is never — , hence there is no point of inflexion (fig, 37). 
 
 In almost all cases d^y/dx^ is at a point of inflexion, but 
 occasionally the tangent at such a point is perpendicular to the 
 axis of X, in which case d'^y/dx^ will be go. 
 
 * It is assumed in the above that x is the independent variable. 
 If y be the independent variable, d^x/dy- must change sign. If 
 neither x nor y be independent, the quantity which must change sign 
 is (see note, §47) 
 
 (dx d'^y - dy d'^x) / cZa;^. 
 
EXAMPLES. 
 
 51 
 
 X 
 
 Fig. 30. Fig. 37. Fig. 38. 
 
 3. if = X, or y = X3-, dhj/tM = - fx-il^ which changes from 
 + to - through GO when x = 0, /. the origin is a point of in- 
 flexion (fig, 38). 
 
 4. ?/ = 3x^ - 4x' - 6x\ dhj/iM = 12(3x2 - 2x - ] ) . Putting 
 this = and solving for x, we get a; = J, a; =: 1, which determine 
 the points of inflexion. 
 
 5. Find the point of inflexion on the curves : 
 
 (1) a'i/ = x(x'-a-'). 
 
 (2) x>/ = l+x\ 
 
 (3) (x + ay^y = cL^x. 
 
 (4) y =x(x~ 1) (;i; - 2), fig. 60. 
 
 (5) x^ — axy = <f\ 
 
 (6) (a' + x')y = a'x. 
 
 (7) a;^ + / = ffxl 
 
 (8) x' + y' = a\ 
 
 (9) x = f + 3y\ 
 
 (10) r = «^'(2a;-l), fig. 18. 
 
 yl»s. (0, 0). 
 (-1.0). 
 (2a, -^rt). 
 
 (1.0). 
 (a, 0). 
 
 (0, 0), (+./V3, ±iaV3). 
 
 (a, 0). 
 
 («, 0), (0, (0. 
 
 (2,-1). 
 
 (§> ± fV3). 
 
 6. Show that at a point (x, y) a curve is convex or concave 
 to the axis of x (i.e., with reference to the foot of the ordinate) 
 according as y (Py/dx^ is + or - . 
 
 7. Show that the curves y = sin x, y = tan x, meet the axis of 
 x in points of inflexion. 
 
 8. On the witch y\a - x) = a^x (fig. 39), show that the points 
 of inflexion are ((f/4, ± a/V3). 
 
 \ 
 
 » 
 
 ii- 
 
 
 n, 
 
^ 
 
 ! 
 
 62 
 
 EXAMPLES. /C P^ ^' 
 
 f 
 
 Fig. 39. 
 
 Fig. 40. 
 
 9, If the ordinate MQ of a circle, tig. 40, is produced so as to 
 equal the arc OQ, the locus of its extremity P is called tlie com- 
 panion to the cycloid (cf. Ex. 25 p. 34 ). If the arc subtend an 
 angle at the centre of the circle we have for the co-ordinates of 
 a point on the curve 
 
 X = «(1 -- cos 6), y - all. 
 
 To show that there are points of inflexion at points imme- 
 diately above and below the centre of the circle. 
 
 Here dx = a sin d dii, d^x = a ens 6 dli^ + a sin 6 d% (1) 
 
 tZy = a de, d^y = a d^tf. (2) 
 
 If now x is taken as independent variable (i.e., if d is changed 
 in such a way that dx is constant), d^x always = 0; also, at a 
 point of inflexion d^y = 0, .', from (2), d^d = 0, /. from (1), 
 cos ti = 0, :, 6 = ^,T. 
 
 But if y is the independent variable, d'^y is throughout 0, 
 and .'. d'^O also = by (2) ; and at a point of inflexion d'^x = 
 if ^ is the independent variable ; hence, cose = as before. 
 
 Or, we may take B as independent variable ; then d^d = 0, 
 and the condition ax d'y — dy d^x = (note, § 49) will lead to 
 the same conclusion. 
 
CHAPTER XIV. 
 AMOUNT OF CURVATURE. 
 
 Fig. 41. 
 
 50. Let p and Q be consecutive points on a curve apqb, fig. 
 41, the tangents at P and Q being pt, qv, and the consecutive 
 normals being PC, QC meeting in c. The circle with centre c 
 and radius cp is called the circle of curvature, c is called the 
 centre of curvature, and cp the radius of curvature of the given 
 curve at the point p. Since the circle and the given curve have 
 two consecutive tangents in common, they must be considered as 
 having two length-elements in common, and hence the circle of 
 curvature passes through three consecutive points of the curve.* 
 
 * The three points i>, q, r of figs- 32, 33. From §21, note, it will be 
 evident that the circle of curvature is really the limit which the 
 circle through these points approaches as the three points approach 
 coincidence. 
 
 • I 
 
 i. 
 
54 
 
 EVOLUTE. INVOLUTE. 
 
 
 
 A 
 
 
 61. Let the longth-element PQ = tZs ; let the consecutive tan- 
 gents make angles (j, and ^ + d(t> with ox ; and let OP, the radius 
 of curvature = R. Then rZ</> = the angle between the consecutive 
 tangents or normals, and hence <hi) - ds/R, 
 
 ds 
 . . R = —J- • 
 
 - dip 
 
 52. Evolute. Involute. The locus of the centres of curvature 
 of a curve is another curve which is called the evolute (fig. 42 ) 
 of the given curve, and the given curve is called the involute of 
 the evolute. 
 
 Fig. 42. 
 
 Each normal of the involute is a tancrent of the evolute, since 
 it joins two consecutive points of the evolute. Hence, if one 
 of the tangents of the evolute were supposed to roll round the 
 curve, a tracing point in it would describe the involute. Thus 
 a given curve can have any number of involutes, but it can have 
 only one evolute. The involute might also be described by a 
 tracing point in a string which is kept stretched at the same 
 time that it is unwound (" evolved ") from the evolute, since 
 this string would at every instant be a tangent to the evolute. 
 
 As long as the radius of curvature of a curve continues to 
 increase or to diminish, the difference of any two values is equal 
 
 nl 
 
RADIUS OF CURVATURE. 
 
 55 
 
 to the length of tlio arc of the evoluto included between them, 
 since the difference of two consecutive radii forms a length- 
 element of the evoluto. 
 
 53. If the tangent at any point of a curve bo supposed to roll 
 round the curve, it will turn througli the angle dt;) while the 
 point of contact moves through the distance ds ; hence, the 
 value of d(,')/ds at any point expresses the rate, in radians per 
 unit length of the curve, at which the tangent is turning; this 
 is called the curvature of the curve at the point. Now 
 d<i)/ds = 1/R, .". the curvature is numerically equal to the reci- 
 procal of the radius of curvature, and hence the circle of cur- 
 vature has throughout its circumference the curvature which the 
 curve has at the given point. 
 
 54. The circle of curvature generally crosses the curve at tho 
 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 at certain p jii.is, e.g., at the vertex of a para- 
 bola, where the circle of curvature does not cross the curve. 
 
 55. Length of the radius of curvature. We have seen that 
 
 R= — (1), we also have -5^ = tan(4. (2) 
 
 d(j> ax 
 
 Differentiating (2), 
 
 dx d-y - dy d-x 
 
 ds' 
 
 dx' 
 
 = sec-f/) dtp = ( — - "^ d6, 
 \dx/ 
 
 .'. di^ 
 
 R = 
 
 dx d'^y — dy d-x 
 d^ ' 
 
 di 
 
 (3)* 
 
 (4) 
 dx d^y — dy d-x 
 
 We may generally take x as in the independent variable and 
 make d^x = ; also d.n'^ = dx^ + dy^, 
 
 * When d^x = 0, (3) gives d(j> = d^y cos / ds, which may be derived 
 immediately from fig. 32. 
 
 ip 
 
 N 
 
 1*1 
 
 
 
 
 ^ .' 
 
■^r 
 
 66 
 
 CENTRE OF CURVATURE. 
 {dx' + il;/')i _ L Vtfa/ -I . 
 
 dx d^y 
 
 dy 
 
 dx' 
 
 (5) 
 
 (4) should be used when x and // are given in terms of a third 
 
 variable. 
 
 The sign of R when found from (5) will be + or - accordmg 
 as dh/ is + or - , that is according as the curve is concave 
 upward or concave downward (§4^)- 
 
 56. Co-ordinates of the centre of curvature. Let the co-or- 
 dinates be called a and /?. Then, fig. 43, 
 
 dy 
 a = X - R sm <;& = « - R ^^> 
 
 ft = 7/ + R cos </• = y + R 
 
 dx 
 ds 
 
 (6) 
 
 CO 
 
 ' \ 
 
 '^: 1 
 
 p,&, K \ \ 
 
 Fig. 43. Fig. 44. 
 
 57. Illustration. To find the radius of curvature at any point 
 (x, y) of the ellipse ^ + '^ = 1, and the equation of the evolute 
 (fig. 44). 
 
 «' 
 
 
CURVATURK OF AN KLLIPSK. 57 
 
 By ditt'erentiating the equation of tlie ellipse we have 
 
 dx ah/ dx* <i\i/^ 
 Substituting in (5), we have 
 
 R = - 
 
 
 (8) 
 
 Taking ?/3 from the equation of the curve and writing e' for 
 (a^ - 6^)/<t', we get 
 
 R = - '"'-f'^^ (9) 
 
 (I h 
 
 It is known that (^^ - e^r'^)* = the semi-dianietcr parallel to 
 the tangent, or perpendicular to the normal, (x, y). 
 Calling this b^ we have 
 
 R = - -1 . 
 ab 
 
 (10) 
 
 Substituting in (6) and (7) it may be shown after some 
 reduction that 
 
 "= (-IT*-)" '"=-(-!?-)*• 
 
 Solving for x and tj and substituting in the equation of the 
 ellipse we obtain 
 
 for the equation of the evolutc. If (x, y) be used for (a, ,i) the 
 equation becomes 
 
 {>(x)'^-V(Jyy)^={<r-b')1^. 
 
 The radii of curvature at the extremities of the axes are by 
 (10), b~/a and (<V^>, respectively. Hence (§ 52) the whole length 
 of the evolute 
 
 = 4(a='-6'')A/?>. 
 
 The curve in which x = a cos i) and y =b sin 6 is also an ellipse, 
 as may be seen by eliminating 0, We shall find the same value 
 
 W 
 
 ¥ 
 
 -^^-i^m^K^iS^^''' 
 
 .■6:d£^m': .^ 'na-tTiiWtffl^mr/frgfmrmrfrgitffTiik^-fr'tfiTJ'fi'iitTirfi^ 
 
 BBitaJL-:^*^.: w. urn.-;. 
 
58 
 
 EXERCISES. 
 
 I 
 
 K 
 
 
 
 of the radius of curvature by taking as the independent 
 variable and substituting in (4). For this purpose 
 dx = — a sin ^ dii, dy = a cos dd, 
 d^x = —acosd dtf, dij = -a sin d d(i^. 
 
 Exercises. 
 
 1 . At any point of the parabola y"^ = 4ax show that 
 
 R=-2V(rV«), 
 where r is the focal distance (=a + «)of the point. Hence 
 it may be shown that R = twice the intercept on the normal 
 between the directrix and the curve. 
 
 Fig. 45. 
 
 2. Show that the evolute of the parabola is the semi-cubical 
 parabola 27«^' = 4(x - 2a), fig. 45. 
 
 3. Find the co-ordinates of c, c', and show that they are the 
 centres of curvature of b', b. 
 
 Ans. c, c' are (8a, ± 4V2rt) ; b', b are (2a, ± 2 V2a). 
 
 4. Show that the arc AC = b'c - oa = 2« (3\/3 - 1). 
 
 5. Show that the radius of curvature at any point of the 
 cycloid X = a (u -sinO), y ^ a (1 - cos ii) is 
 
 • t) . ' 
 
 — 4a sin ^ = twice the normal pb. 
 2 
 
 I 
 
EXERCISES. 
 
 69 
 
 Fig. 40. 
 
 6. Also that a = a (0 + sin O), ,3 = - a (1 - cos ft), and hence 
 (see fig. 30) that the evolute is an equal cycloid. 
 
 7. In the hypocycloid x^+j/^ = a^ (fig. U) show that 
 
 = 3 times the perpendicular from the origin on the tam-ent 
 (Ex. 10, p. 41 ). 
 
 Let X = a cos^fy, i/ = a i>m^i). 
 
 8. Show that the evolute of this curve is 
 
 and by turning the axes through 45° show that this is a, similar 
 hypocycloid, its equation becoming 
 
 x^ + y^ = (2a)t. 
 
 9. The equation QV + ^^y = l represents a common 
 
 parabola, the origin being a point on the directrix, and the axes 
 tangents to the curve. Show that 
 
 R = 2(tix + bi/)'^/ab. 
 Let X = a 008*61, i/ = h s'm^n. 
 
 10. Show that R = oo at a point of inflexion. 
 
 11. Show that R = (1 +a^)i/2h at the origin on the curve 
 
 J/ = ax + bx^ + cx^+ •" (1) 
 
 or, X = €17/ + b7f+cf+-" (2) 
 
 m 
 
 
 J 
 
 r 
 
 i ^^1 
 
 1 
 
 ■i 
 
 1 
 
60 
 
 EXERCISES. 
 
 L'^J 
 
 1 
 
 I 
 
 12. Also that the centre of curvature of (1) at the origin is 
 the point 
 
 __ a(l + o^) l + g" 
 ~2b~' '~2b~' 
 
 and tliat the circle of curvature is 
 
 h(x'^ + f) = (1 + a^) {y - ax). 
 
 13. Find r at the origin of the following curves: 
 
 (1) The parabola ?/^ = Aax, or x^ = 4«^. Ana. 2a. 
 
 (2) y^ = x^ (1 f 2.T.), fig. ] 7. We have 
 
 :y = x(l + 2x)l = x(l+x -^.r7 +...). ±V2/ 
 
 (3) / = x*(l+2x),fig. 19. ±^. 
 
 (4) {y-xy^x% fig. 16. |. 
 
 (5) {y-x)-' = x\ 0. 
 
 14. Find R at the origin of .x'+ y^ = L :.xy, fig. 13. 
 Consider the branch which touches the axis of x. We are 
 
 concerned only with points consecutive with the origin on either 
 side, and for these points //^ may be neglected ; the equation 
 then reduces to x^ = 3(rxy, or x'^ = 3^/^ and ,'. R is the same as 
 at the origin of the parabola x^ = ^'ty, v.z., %a = 00. 
 
 Note. — The point a, (^k, §((), is the intersection of the curve 
 with the line y = x which bisects the angle between the axes. 
 
 15. Show that the general value of R for the curve 
 x^ + y^ = Saxy is 
 
 "Id'^ry 
 
 and hence that R =-^-0 A at the point A. (SeeExs. 27, p. 12; 19, 
 p. 48 ). 
 
 16. As in Ex. 14, show that the radius of curvature of the 
 branch of the curve 
 
 ay'' - '^ax^y = x*, fig. 21, 
 
 which touches the axis of x is %n at the oricin. 
 
 17. On the other branches of this curve, show that R = 24a 
 at the origin. 
 
EXERCISES. 
 
 61 
 
 het 2/ = mx in the equation of the curve. Then 
 X = a(m^-3w), 1/ = a(m*-3m'). 
 Use formula (4; remembering that m = V3 in the present instance. 
 
 18. In a similar way show that r = 2V2rt on the branch of 
 the curve 
 
 ai/^ - ayyif = x^, fig. 22, 
 which touches ?/ = x at the origin. 
 
 1 9. Prove that in any conic section 
 
 R = normaP/Csemi-latus rectum)''. 
 
 20. To find R at any point of a helix (screw-line) on a cylin- 
 der of radius r. 
 
 The plane passing through three consecutive points of the 
 curve intersects the cylinder in an ellipse whose semi-axis minor 
 = r, and semi-axis major ^^ r sec a, a being the angle of the helix ; 
 
 .'. R = (/• sec «)-/>* = r sec^a = r (1 + t^) 
 where t = pitch / circumference of cylinder. 
 
 21. At any point of a curve 
 
 R = 
 
 '^(iVxy-^{d'^y-(d's)' 
 
 f' ! 
 
 See Ex. 16, p. 48. 
 
 fif 
 
 i 
 
 - \ 
 
 . ! 
 
 
CHAPTER XY. 
 MAXIMA AND MIMMA. 
 
 X' 
 
 o 
 
 X 
 
 Fig. 47. 
 
 58. Suppose y to be any f'uiiction* of x, and that x continually 
 increases. Then y will increase as long as dy h + , and will 
 decrease when dy is — . Hence as x increases (dx therefore 
 remaining + ) a function will increase or decrease according as 
 the differential coefficient dy/dx is + or - . When dy/dx 
 changes from -f 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 in- 
 crease, and is then said to be a minimum. Now, in order that 
 a quantity may change sign it must pass through the value 
 or 00 ; hence when y is a max. or a min., dy/dx is either or oo 
 and changes sign from + to - for a max. an(^ from - to + 
 for a min. 
 
 * Although the stixtcnients made in the text apply to all functions, 
 the reader should at first consider y to be the ordinate of a curve, the 
 abscissa a; being supposed to increase continually between the limits 
 - CO and + 00. 
 
Tt 
 
 MAXIMA AMD MINIMA. 
 
 63 
 
 59. Suppose, for example, that the curve fig. 47 is traced by a 
 point moving from left to right so that dx is + . Then ?/, the 
 ordinate at any point, or distance from x'x, decreases from A 
 to B and dy/dx is - , between b and c // continually increases 
 and dy/dx is + ; at B ^ ceases to decrease and begins to increase, 
 dy/dx changes from - to + through the value 0, and y is a 
 min. Similarly at c // is a max., and again a min. at D. At e 
 dy/dx changes from -I- to — through oo, andv/ is a max., and 
 similiirly y is a min. at F. It will be noticed that a max. is not 
 necessarily the greatest of all the values of // ; it is greater than 
 the values which immediately precede or follow it ; and similarly 
 a min. is not necessarily the least value of//. 
 
 60- To find the values of x which make a function y a max. 
 or min. we must obtain dy/dx and find what values of x make 
 it or OD. To distinguisli the maxima from the minima we 
 must determine whether dy/dx changes from being + just 
 before it is or oo to being — just after, or vice versa. In the 
 former case y will be a max., and in the latter a min. It may 
 happen, however, that dy/dx does not change sign in passing 
 through or OD (e.g., at G, fig. 47), in which case y is neither a 
 max. nor a min. 
 
 Examples. 
 
 1. y = x^ - Qxr + 9a; + 1. 
 
 Here dy/dx = .Sx^ _ 12.c + 9 = 3(a; - 1 ) (.x - 3). 
 
 When X is a little less than 1 , x - 1 is - and x - 3 is — , .'. 
 dy/dx is -f- . 
 
 When .X = 1, dy/dx is 0. When x is a little more than I, 
 x -\ is + and x - .3 is - , ,'. dy/dx is - . Hence dy/dx 
 changes from + to - through and .*. // is a max. when x = 1 . 
 Substituting 1 for x in the given function we find the max. 
 value of ?/ to be 5. Similarly cc = 3 makes y a min., viz., 1. 
 
 2. y = {x - \)\ dy/dx = 3(rB - 1)* ; .*. dy/dx = when x = 1, 
 but does not change sign when x passes through this value, .'. y 
 is neither a max. nor a min. 
 
 hi 
 
 I: 
 
 m 
 
64 
 
 MAXIMA AND MINIMA. 
 
 3. ij = 2+(x- 1)«, di//dx 
 
 1 > 
 
 .*. dy/dx changes 
 
 it! 
 
 30c -l)t 
 
 from - to + through co as x passes through the value 1, hence 
 X = 1 maker y a min., viz., 2. 
 
 61. The sign oid^y/dx^ (= the differential coefficient of (/^/(7x) 
 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^y/dx' 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'^y/dx'^ minus, we infer that dy/dx is de- 
 creasing wiien it passes through 0, i.e., that it changes from -f 
 to - , and hence that y/ is a max. Thus at A and c, fig. 47, 
 i/y/d'x^ is - , at B and D it is + , (cf. § 49). 
 
 Hence to distinguish the maxima from the minima we may 
 find d^y/dx\ and in it substitute the values of x which make 
 dy/dx equal to 0. Then for every + result ^ is a min., and 
 for every - result ?/ is a max.* 
 
 Examples. 
 
 1. In Ex. 1, § 60, (Py/dx;' = 6x - 12, which is - when x = 1, 
 and 4- when x = 3. Hence x = 1 makes y a max. and x = 3 
 makes y a min. 
 
 2. y = x^ - 7x'- + 8x + 30, dy/dx = Sx^ - 14x + 8. For a max. 
 or min. 3x''' - 14x + 8 = .-. x = § or 4. 
 
 Also d/y/d^ = 6x - 14, which is - when x - § and + when 
 X = 4 ; .'. X = f makes // a max. and x = 4 makes y a min. 
 
 62. It should be noticed : 
 
 (1) That max. and min. values must occur alternately, l.v,, 
 between a pair of max. values there must be a min., and between 
 a pair of min. values there must be a max. Also of two values 
 
 • At E and F, fig. 47, d'-y / dx'^' as well as dij / dx is oo. The former 
 is + on both sides of e anl - on both sides of f. 
 
 
MAXIMA AND MINIMA. 
 
 65 
 
 of X which make y a max. or a min., if one makes it a max. the 
 other must make it a min. 
 
 (2) In general, when a function is a max. or a min. a power 
 or root or logarithm of it will also be a max. or a min. Hence 
 in differentiating we may disregard an index or radical sign 
 affecting the whole of the variable part of the quantity, and may 
 differentiate the logarithm of a function instead of the function 
 itself. 
 
 (3) A constant factor may be omitted from the function before 
 differentiating, since it cannot affect the values of % which make 
 the differential coefficient or oo. 
 
 Example. 
 
 y -- :TX\l{a- - X-). This = TrV(<''V - X*), and .•. y will be a 
 max. or a min. when a^x^ - x* is a max. or a min. ; hence 2«^x 
 - 4x^ = 0, .-. X = 0, and x = ±n/'J2. 
 
 63. In the practical applications of this subject it will be 
 necessary to form the function which is to become a max. or a 
 min. 
 
 Examples. 
 
 1. Of all arithmetical fractions, which one exceeds its square 
 by the greatest quantity ? 
 
 Let the fraction be x. Then x - x'^ is to be a max. 
 
 .'. 1 - 2.x = 0, and hence x = ^. 
 
 2. How to make with a given amount (area) of material a 
 cylindrical box (with lid) whicli 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 = It^x"^ -f 27rx/i, .-. li = s/{2zx) - x. 
 The volume v = ttx'^Ii = hsx - ttx^ 
 
 .'. dv/dx = ^s - 377x2 = 0, for a max. 
 
 .-. X = V(«/6n'), whence h = 2V(s/67r). 
 
 5 
 
66 
 
 MAXIMA AND MINIMA. 
 
 Hence, the height must = the diameter of the base and each 
 
 = 2V(8/6rr). 
 
 [Observe that in these examples the function which is to be a 
 max. or min, must be expressed in terms of some one variable 
 with or without constants; in this case the function is nx^'h 
 where booh 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 ttxVi gives a function with one variable.] 
 
 3. The greatest isosceles triangle that can be inscribed in a 
 given circle is equilateral. 
 
 Let ABC (fig. 48) be an isosceles triangle inscribed in a circle 
 of radius a and centre E. Let DC = x. Then 
 
 AD = Vae' - DE* = 'Ja^ - (x - a)- = V2ax - x\ 
 
 area of ABC = DC. AD = x V2ax - x' = V2ax' - 
 
 x^ 
 
 This will be a max. when 2ax^ - x* is a max., § 62 (2), i.e., 
 when Qax^ - 4x^ = 0, .'. x = f «. From this ap = ^\'3a, .', CD : 
 AD = V3, .'. DAC = 60°, hence the triangle is equilateral. 
 
 E 
 
 Fig. 48. Fig. 49. 
 
 4. One corner a of a rectangular piece of paper abcd (fig. 49) 
 is folded over to the side bc. Find when the crease £0 is a 
 min. 
 
 Let AB = a, AE = x, EG = y, age = 6. Then bep = 2d. 
 
 .'. be ; EF = (a - x)/x = cos 2o, and ae : eg = x/y = sin0. 
 
 Eliminating by the relation cos 2^ = 1 - 2 sin^^, we find 
 
 f = 2xV(2x - a) 
 from which y is found to be a min. when x = |a. 
 
MAXIMA AND MINIMA. 
 
 67 
 
 Similarly it may be shown that the area of the part folded 
 over is a min. wheiix = ^a (Rice and Johnson, Dif. CaL). 
 
 5. To cut the parabola of greatest area from a given right 
 circular cone, fig. 50. 
 
 Let AB = (i, and eb = x. The area = § ed.po. 
 
 Now EJf'^ ~ AE.EB = (a - x)x, aud ED OCX. 
 
 •. area oc x V(<t - x)'^"^ oi* \/"''«^ - *'*, whence x = f<« for a 
 
 
 max. 
 
 Fig. 51. 
 
 6. When the triangle of greatest area is inscribed in any oval 
 curve (fig. 51), the tangent at each angular point must be 
 parallel to the opposite side (Edwards, Bif. Cal.). 
 
 For if ACB, ADB be consecutive triangles on each side of the 
 max. they are equal, .-. uc is parallel to ab (Euclid I, 39). 
 
 Exercises. 
 
 1. X* - 3x -r 4, min. when x = |. 
 
 2. 2x' - 9x' + 12x + 5, max. when x = 1, min. when x = 2. 
 
 3. 2x^ - 3x* - 12x, max. when x = - 1, min. when x = 2. 
 
 4. X® - 5x* + 5x' + 1, max. when x = 1, min. when x = 3. 
 
 5. a + h(c - x)^, no max. or min. 
 
 6. 4x^ - 15x* + 12x - 1, max. when x = ^, min. when x = 2. 
 
 7. x^ - 2x2 - 4x + 1, max. when x = - f , min . when x = 2. 
 
 11, ! 
 
68 
 
 MAXIMA AND MINIMA. 
 
 8. {x + «) (x + h)/x, max. when x =- '^(ah), 
 
 min. when x --J (ah). 
 
 9. (a; - 1)^ (a; + 2)*, max. when x = - 2, min. when « = - ^. 
 
 _ 12 
 
 10. (1 + 3x)/V-4 + 5x2, max. when x = 
 
 11. (x + 2)V(a; - 3)a, min. when x = 13. 
 
 12. sin 61 + cosf), max. when ^> = ^tt, min. when H = fjr. 
 
 13. sin H / {1 + tan fi), max. when ^ = ^k. 
 
 14. sin « sin (a - w), max. when ^y = ^a. 
 
 15. sin^^^ cos'^y, max. when sin a = ± Vf, niin. when o = 0. 
 
 16. Min. value of a tan fl 4- 6 cot = 2V('^fi). 
 
 17. Min. value of a ^ seca^y + b^ cosec''^^y = (« + 6)^ 
 
 18. Min. value of «e'^ + 6e-'" = 2'J{nh). 
 
 19. Min. value of x/logx = e. 
 
 20. Show that (x - (i^y + (x - a.^)'^ + ... + (x - a„y is a min. 
 when X - (a j + a.^ + . . . + a„)/n. 
 
 21. What is the longest ordinate of the curve a'^^ = x''(a- - x*) , 
 (fig. 59) ? ^ Ans. ^o. 
 
 22. Find the max. ordinate of the curve 
 
 x^ + i/'^= 3<(x^, fig. 13. 
 
 Difi'erentiating the equation and making di/ = we have 
 x' = ay, from this and the equati on of the curve we find th<' 
 
 max. ordinate to be at the point (a\/2, ((J/4:), the latter co-or- 
 dinate being the required value. 
 
 23. Find the max. ordinates of the curves 
 
 (?/ - x)^ = x\ fig. 15, and (?y - x'-y = x\ fig. 16. 
 Ans. (i) 2V3; (ii) 4V5\ 
 
 24. Deduce the conditions lor a point of inflexion from § 55, 
 considering that at such a point r/> is a max. or a min, 
 
 25. Where should a given straight line be divided in order 
 that the rectangle contained by the parts may be as great as 
 possible ? Ans. It must be bisected. 
 
EXERCISES. 
 
 69 
 
 26. Given the volume of a right circuhir cylinder find wIumi 
 the surface is a min. 
 
 Ans. When the altitude = the diameter of the base. 
 
 27. How could you cut out four equal squares from the cor- 
 ners of a given square so that the remaining urea (the edges 
 being turned up) would form a rectangular box of greatest 
 volume ? 
 
 Ans. Kach side of the little squares = ^ of a side of the given 
 square. 
 
 28. Given the volume of a right circular cone, find its form 
 when the surface is a max. 
 
 Ans. Altitude = V2 x diameter of base. 
 
 29. Describe about a given circle the isosceles triangle of least 
 area. Ans. Altitude = 3 x radius. 
 
 30. Inscribe in a given circle the greatest rectangle. 
 
 Am. The inscribed square. 
 
 31. Inscribe in a given right segment of a parabola the greatest 
 rectangle. 
 
 An.t. Area of rectangle = area of parabola / V3. 
 
 82. To make from a given sphere the cone of greatest volume. 
 
 Ans. Vol. of cone = ^^ of vol. of sphere. 
 
 33. What is the vertical angle of the cone of greatest volume 
 which can be described by a right-angled triangle of given hypo- 
 tenuse? Ans. 2tan-V2. 
 
 34. How could you cut a sector out of a circle of radius a, so 
 that the remainder of the circle will form the 'ateral surface of 
 a cone of max. volume ? Ans. Arc of sector = 2Tr<i (1 — V§). 
 
 r t|5 
 
 <l 
 
 ' '/, 
 
1 
 
 I 
 
 CHAPTER XVI. 
 INTKiiltiTIOX. KLEMKNTAHV II.IUSTR VTIOXS. 
 
 64. Let F(£r) be any function of x. Then as x changes from 
 ji ralnc // to a value h tlie cliarge * in f(x) is ¥(1) - P(«). 
 Suppose that x cliangcs from o to b by the successive additions 
 of infinitesimal Increments. When x has the increment dx the 
 (•(trrespondiiij; iricrenient of F(rB) is f(x) dx, where /(x) is the 
 differential coefficient of f(x). Hence f(/>) - F(a) = the sum 
 of all such products as/(x) dx when x changes from n to 6; let 
 
 this be expressed by f(x) dx, the symbol being a lengtii- 
 
 J (I J 
 
 ened forn. of s tlie initial letter o^ sum. Then 
 
 ^'j{x) dx = F(/>) - F(.0 = [f(.T)] ' (1) 
 
 Hence, f(x) being any function of x, to find the sum of all 
 such terms as/(x) dx when x changes from a. to 6 we must seek 
 the function f(x) of which the differential is /(a;) f?x, substitute 
 therein b and n successively for x, and subtract the .second result 
 from the first. 
 
 65. This process of suiiimatioo is called integration (the 
 
 making of a ndioh out of infinitesimal parts f) ; f(x) dx is 
 
 read "integral between « and /> of /(x) rfx " ; /(x) dx itself 
 is called an element of the integral ; a and b are called the 
 limits of the integration. 
 
 * It is assumed that tlie function is one which varies continuously 
 between the extreme values v{a) and f(6), ef. § 107. 
 
 t The result is, strictly speaking, the limit to whicli the sum 
 approaches as dx approaches and the nuni ber of terms approaches oo. 
 
INTKGUATION. 
 
 71 
 
 B X 
 
 Illustrations. 
 
 B X 
 
 Fig. M. 
 
 66. Areas of Curves. Lit y =/(^) be the iquation of the 
 curve CD. Let OA = a, ob = h, OM = x, »i P = ^, M N = f/cc, RQ = rfy, 
 and let it be required to find the area abdc. 
 
 When X has the increment dx, the increnient of tlie area it< 
 MNQP = rectantrle mr + ^ the rectangle SR = ydx 4- i dx dy. This 
 second term is infinitely small compared with the first, and must 
 be omitted. Hence the infinitesimal element of the area is ydx. 
 
 :. area abdc = ydx = /(x) dx =rF(a3)~| , 
 
 where h(x) is the function of which the differential is/(.'c) dx » 
 e.g., the function of which the diflcrintial is x'^dx is 
 
 n + 1 
 
 X 
 
 n+l 
 
 except when w = - 1 , in which case it is log x. 
 
 In other words we imagine the area to be divided into an 
 immense number o? exceedingly narrow strips by lines drawn 
 parallel to OY, express the area of one of these strips as a differ" 
 ential (all infinitesimals r>f an order higher than the first being 
 omitted) and then integral*^. 
 
 Examples. 
 
 I. To find the irea obd of the curve y = x^, fig, 54, the limits 
 of X being and 1. 
 
 area is onc- 
 
 The area = ydx = x^ dx = [l^^'\ = J, i.e., the 
 
 fourth of the square on OB. 
 
 2. The area of the parabola y^ = 4ax, fig. 55 ; from x = to 
 
 X = /i is 
 
 ni 
 
 I 
 
 1 1 
 
 If' 
 
 li 
 
 111 
 
72 
 
 EXAMPLES. 
 
 J 
 
 
 I 
 
 / 
 
 O 
 
 V 
 
 
 
 B X ^\^ 
 Fig. 54. 
 
 X 
 
 Fig. 55. 
 
 Big. 56. 
 
 ydx = ^/4a.x-^ cHx = I V4«.|a;- I = f V4rt./t^ - |/<.V4tt/i 
 
 = § OB.BD - two-thirds of the rectangle having the same base 
 and height. 
 
 3. The area of the curve i/ - sin x, fig. 56, from x = to 
 
 « = TT is 
 
 sin X dx = r - cos xA = 2, 
 i.e,, twice the square on tlie maximuu. ordinate. 
 
 67. Volumes of Solids of Revolution. Suppose the curve, 
 fig. 52, to revolve about ox and generate a solid. The rectangles 
 MR, MQ generate cylinders of infinitesimal thickness dx, and 
 radii y, ]J + dy, and therefore of soixxma -y'dx^ Ti{y + dyfdx. 
 But these /olumes are ultimately equal when infinitef'imals of the 
 second and third orders are omitted. The element of the volume 
 
 IS ,'. TTl/V 
 
 ix. 
 
 whole volume 
 
 = - ]f'^^- 
 
 In other words, we imagine the solid to b divided into an 
 immense number of very thin slices by planes perpendicular to ox, 
 express the volume of one slice as a Jifferential, and then in- 
 tegrate. 
 
 Examples. 
 
 1. The volume formed by the revolution of obd, fig. 54, 
 round ox = 
 
EXAMPLES. 
 
 73 
 
 2. When the area of the parabola jf - 4crrB from « =0 to x = A 
 revolves about ox the volume = 
 
 TT y^dx = TT iaxdx = 7rr4a.^x*~| = ^Tr(4ah)h = ^ttBD^Op, 
 
 i.e., one half of the cylinder having the same base and height. 
 
 Note. It may be thought that the omission of the little triangles 
 above the rectangles in fig. 53 causes the n iilt to be too small. 
 It should be noticed that these triangles would together be less 
 than a rectangle whose height = bd and whose width is equal to 
 that of the widest rectaiigle, but this width is infinitesimal, and 
 hence the sum of the triangles is also infinitesimal. But in 
 reality there is no error at all. For if \ o consider that the pro- 
 cess of differentiating causes the higher orders of infinitesimals 
 to be dropped (§ 12), then we must consider that integrating 
 (which involve" the undifferentiating of a differential) picks 
 up these small quantities which were dropped, and restoi^es them 
 to their proper place in the integral . We thus see that the 
 little tria? Ls may (or "ather must) b(^ omitted in forming 
 the area-element, and that the process of integrating makes exact 
 compensation for this omission. Similar remarks will apply 
 to all other case? in which integration is involved. 
 
 ■ 
 
 ■h! 
 
CUAPTP^li XVII. 
 FUNDAMENTAL INTEGRALS. I. 
 
 68. We have seen that 
 
 jyO«)cZx=[F(x)j' 
 
 p(x) being the y»rimitivc function of wliich/(.'r) dx is the differ- 
 ential. If the limits be not cxproi^sed * we maj write 
 
 fix) dx = f(x) 
 
 and hence may be regarded as u. symbol which indicates the 
 
 operation of going from the differential f(x) dx back to the 
 primitive function P (ic). By this operation we can discover 
 only the variable part of the primitive function ; e.g., 
 
 2x dx = x\ or x' + 1, or x'^ + c. where c may be any constant 
 
 I- 
 
 I 
 
 (any quantity indepcnder^t of x). To every primitive function 
 
 thus obtained from a differentia) there should .'. be added a 
 
 constant, the value of which must depend upon special data ; 
 we should then write 
 
 fix) dx - f(x) + c. 
 
 If, however, we are hereafter to substitute limits, the constant 
 c need not be expressed, inasmuch as it would disappear in sub- 
 tracting. We shall accordingly, as a rule, omit the constant, 
 but its presence is always understood. 
 
 * The integral is said to be definite when the limits arc expressed, 
 indefinite when they are not expressed. 
 
FUNDAMENTAL INTEGRALS. 
 
 75 
 
 69. In the Diflfcrcntial Calculus general formulae arc given 
 by which every function may be diflFerentiated. Unfortunately 
 there are no gene:;il methods in the Integral Calculus by wliich 
 every expression may be integrated ; the various processes by 
 which integrals are obtained consist almost entirely in so changing 
 the form of given difiFer entials as to make them appear as par- 
 ticular cases of the fundamental ones uiven below. 
 
 Differentials 
 
 
 Integrals. 
 
 
 rf(M'*) = nv''-hlo 
 
 (A), 
 
 J «+ 1 
 
 (a) 
 
 _ dv 
 
 2'^v 
 
 (B), 
 
 Cdv 
 
 '' 2^/v 
 
 (&) 
 
 't)-'t 
 
 (BO, 
 
 ' ' J V" V 
 
 (^>i) 
 
 d(a'') = A a'dv 
 
 (E), 
 
 • 
 
 (0 
 
 d{e!') = e'dv 
 
 (F), 
 
 ;. e'dv = e'" 
 
 • 
 
 (/) 
 
 d(\ogv) = — 
 
 (<>). 
 
 . ^dv , 
 .-. _ = logt; 
 
 J V 
 
 (^) 
 
 d sin ft) = cos 8 dti 
 
 (H), 
 
 .•. cos ft dfi = sin 6 
 
 « 
 
 (/O 
 
 d(cos h) =- sin h dh 
 
 (I), 
 
 • 
 
 .', sin ft di) - — cos ft 
 
 • 
 
 (0 
 
 d[ tan 6) = scc'^^ da 
 
 (J;. 
 
 • 
 
 ;. sec^/^ di) = tan ft 
 
 (./) 
 
 d{cot f)) = - cosec^^ d6 
 
 (K), 
 
 • 
 
 .", cosec^^y (fo = - cot ft 
 
 • 
 
 (k) 
 
 c?(sec 6*) = sec tan 6 dt> 
 
 (L), 
 
 » 
 
 ;. sec ft tan ft dft = sec ft 
 
 • 
 
 (0 
 
 f/(cosec 6) = - cosec o cot ft do 
 
 (M), 
 
 • 
 
 *. cosec ft cot ft dn 
 
 
 
 
 = - cosec ft 
 
 (m) 
 
 
 I 
 
 :' i 
 
1? 
 
 76 
 
 FUNDAMENTAL INTKGRALS. 
 
 d( sin-i- ) = - 
 
 \ a/ Jaf- _ a 
 
 a dx 
 
 ,/. ,«\ <idx 
 V a J a^ + X- 
 
 adx 
 
 J <// + x' a a 
 
 (Q), .-.J 
 
 ' xsjx^ — <i- <i 
 
 X>f/ 
 
 d I sec ^ I = 
 
 V «/ x^jx^ - a' 
 
 To these may be added ; 
 
 f_^ . i iooY-l::ii*\ , 
 
 J x' - <i^ la ' \x + a/ 
 
 jx^ — (i- la \a,-\-x/ ) 
 
 r dx 1 , /re -I- ft \ 
 
 J^r- X- 2(6 Vo; - a,/ 
 ? dx 1 , / rt + x\ 
 
 \-i 5 = 0- ^^-"C ).«<'«' 
 
 Jfr — a;- la \a - x/ 
 
 dx 
 
 1 
 
 = - sec 
 
 X 
 
 a 
 
 J x\Jd- ± x'^ ct \<t -I- V«" + x'-'^ 
 
 (n) 
 
 (9) 
 
 (0 
 
 (0 
 
 70. -Formiilaj (r), (s), (^), {il), should bo committed to me- 
 mory with the others, as they are of fundamental importance > 
 that they are true may be proved at once by differentiation. 
 
 Compare carefully (»i) and (t) ; (/j), (r), and (s) ; (5) and (»)• 
 
 Observe that («) fails when /i = — 1, but that in this case (^) 
 applies. 
 
 Since a constant factor may always be removed from one side 
 to the other of the sign of differentiation, the same is true of *he 
 sign of integration. 
 
 'i:t..j 
 
^n 
 
 CHAPTEK XVIII. 
 FUNDAMENTAL INTEGRALS. II. 
 
 Examples. Formula (a) to (g), 
 
 C2dx r^ 2 r-a 1 „ 1 
 
 -r= \2x-3dx= -^'^ = _1 [dx 1 
 
 X 3. :■ (x'- 2)1 xdx = (or ~2)i^(^z3 ^1^(^-2)" 
 .*. J(«' - 2)1 x dx = \ {^? - 2) 2 
 
 iC. 
 
 
 5. f-^ „ _ iKa^-x^) 
 
 J a' _ .^.^ - 2j -^rr^ - = - ^ Jog. (a^ - :^)^ 
 
 by (^). 
 
 0. J-2:i^'rf^.= ri-2x^+.^^ 
 
 X 
 
 ~'^^"\{x ~^^ + »')^^.'- 
 
 V 
 
 
 
 % 
 
78 
 
 EXAMPLES. 
 
 i 11. f-^ = log Vom:^ 12. f (. -I- 6x^)1 = ^^^-:tM:. 
 
 ia^->cx'- J 46 
 
 13. J 72ax - x^ ((fc - .x) dx - ^(2ax - x^) ^• 
 
 14. [e'dt = eVa. 15. f_^ = log('-l-Y 
 J Jtt-x \a— x/ 
 
 16 
 
 c?x 1 
 
 { dx 1 1- fx-2 , 2 , , OA 
 
 • 7 a = .VT T-3 • ^17. dx = — (x + 2). 
 
 Ja^-x^ ^ °W'-xV J^ " ^ X n + 1 
 
 20. f-4?^ =log(loo-x). "^21. rac?x = 
 
 -/JxloKa; Jo 
 
 a 
 
 22. f*3Va; cZx = 14. 
 25. ["'^ = 1. 
 
 Ja X 
 
 9Q v^ 
 
 23.^1 x-te = 2. 24. r^ = ^. 
 
 1 
 
 '•I 
 
 26. e-'^'dx^ 
 
 a 
 
 27. ["(ax + 1) dx = |«' - a - 2. 28. [ e" <?x = e - 1. 
 
 1*^ 1 
 
 29. x"(ix= ;-. 
 
 Jo w+ 1 
 
 (•20 3 
 
 30. \jx — a dx - §a2. 
 
 Ja 
 
 31 
 
 . (a ± x)''Jx => ^ 
 
 J -a 
 
 71+1 
 
 32 f" ^^ ^ ^ 
 
 * J«(a + x)" {n-l) (2a)'*-i' 
 
 33. I (a + 6x + cx^) xdx = J^ (6a + 46 + 3(;). 
 Jo 
 
 -sy 
 
 / pi" X <fo; 1 
 
 ^^' Jo («'-»')' "67^" 
 
 35. T-^^ =a(V2-l). 
 
 J Oa/(#" 
 
 »V'*^ + «' 
 
wmmmmm 
 
 f" 
 
 K 
 
 36. £3V.. = 26/(31og,3). 
 
 EXAMPLES. 
 
 37. j^x'e-^dx^l 
 
 79 
 
 oq f" dx __ 
 
 " ° visa - a; 
 
 = «. 
 
 41. I ^ax dx = !^h ^ah. 
 
 r^loo'a; , ,.2 1 
 
 Ji a; 
 
 
 i log 2. 
 
 iJ 
 
 1 
 
CHAPTER XIX. 
 FUNDAMENTAL INTEGRALS. III. 
 
 Examples. Formulae (h) to (m). 
 
 1 . [sin 'do do ^ jjsin 'So d{Sti) = - J cos 3//. 
 
 2. fcoi 5w cos 3(/ d) = h [(cos 8^^ + cos 2«) rf^^ 
 
 = i( 8- + -2")• 
 3 f-_i;i- = h'"'" '^" = f^i^-"^' = log(tan »). 
 J sin fy cos f' J tan 6 J tan ^ 
 
 4. f* 
 
 sin fy cos ti J tan « 
 
 ,4. 
 
 J sin ^ J 2 sin ^^^ cos -g 
 
 = f - Al^^ = log(tanl^), 
 
 s-^fy J sin ^H cos ^(^ 
 
 by Ex. 3. 
 
 5. \JL = _ \4^^-Z^. = - log tan(i. - |«), by Ex. 4, 
 Jcosfy Jsin(i-T-fO 
 
 = logtan(|-Tr + -|^). 
 
 «^IL^= _ log cos ^ = log sec «. 
 cos^ 
 
 G. [tan da = [ 
 
 ^ f , fcos6/d5'y , . ^ 
 
 7. cot f/ £^f/ = I — ~ — = loK sin t). 
 
 J sin i> 
 
 8. [sin^Wc^fy = ^[(1 -cos2(y) (^'^ = f (« - ^ sin 2f/) . 
 
 9. [cos^^ d3 = l\(\ + cos 2fy) (^f? = |(« + i sin 2«). 
 
4 ain^i 
 
 13. r^£l^^_ r 
 
 15. Jsec^'4^nfo = ^tan4'/. 
 
 Ij [sin 6 do 
 
 ' J"!^^ ='''°'^- 18. r^!^W/y^_ 
 
 iq fsm^'^y - 
 
 23. Jcos"^^ sin d de^.- £^^^ 
 
 24. JsiV/?cos^^^^y=!ii^', 
 
 81 
 
 cosec tt. 
 
 n + l 
 -1 
 
 25_ fcos 6 d'j 
 
 26. f!!l^"_ 1 
 
 6 
 
 :JI 
 
f 
 
 III 
 
 82 
 
 EXAMPLES. 
 
 29. {-. — — = 1 Vi log tan(i7 f h,>). 
 
 30. J VI +co8« (/^ = 2 V2 siu ^«. 
 
 31 . I VI - COS fy t^y = - 2 V2 cos ^yy. 
 
 32. J VI ± sill 1^ do = 2(8in ^t) ^ cos ^ft). 
 
 33. {-=jL=r = V2 loo- tan i(7r + ^y). 
 •' VI + COS 8 
 
 34. L "^^ -=tanJr^y. V 35. f— ^^ tana^y-4.7r). 
 
 Jl -f-COS« J 1 +s\un 
 
 36. sin 5/y cos 3fy tZ^y = - ^(cos 2yy + ^ cos 8/y). 
 
 3*7. sin 3n sin 2^y do = J (sin o-l sin 5«). 
 
 38. sin^w c^ = ( 1 - cos^^y) sin 6 d6 = ~Gos6 + ^ cos^fl. 
 
 39. sin'Vy fZ^ = - cos »-', '^ ooa^ti - ^ cos"'^. 
 
 40. cos^« do = sin w - ;^ sin^y/. 
 
 41. I tan-^y dn = \(sec^d - I) tdLU e do = ^ (tau^o + log cos 0). - 
 
 .o fsin'yy (^w f(l-cos^^y)(Zcos« ^, . 
 
 42. -— = _ ^b '- = sec fl + cos 6. 
 
 J cos^^ J 
 
 cos^e 
 
 43. fcos^^ sin'^fl de = - \cos^6 ( 1 - cos^^) dooae = - ^cos^^ + ^cos''^. 
 
 44 r_i 
 
 J sin'^ 
 
 cos^O 
 
 = tan — cot 0. 
 
 sin^^ 
 
 45. I !':'-2': do = log tan ^^ + '^). 
 siaff 
 
 . 
 
■H 
 
 EXAMPLES. 
 
 4- COS ii. 
 
 83 
 
 52. r^"_^ = 1 
 
 Jo cos^^ 
 
 4b. — — di) = W tan U 
 
 J am ft * 2 
 
 47. log tan(i;r + hi) - sin a. 
 
 J COSH t. v» ^ y 
 
 48. J —-^^^ =• J (^ + ^a"''') ««c'^y (/// = tan ii + }^ tan'^^ 
 
 49. sinVd^^ = |:;r = cos^^y c^^. 
 
 Jo Jo. 
 
 50. £'sin 0(1(1 = 2. 51 . [""cos^y dh = 0. 
 
 53. tan^f) dii = 1 - Itt. 
 
 Jo 
 
 ''•J. . . + sin. ' ^^K^T^) = -^^Q^- 
 
 55. f'V'^'- = iios2 = -346G. 
 
 56. sec^fl tan ^ da = 2J. 
 Jo 
 
 -57. I _ cot^'/y t^^ = ^(1 - log 2) = -1534. 
 
 58. r^"^-!^' = V2-l = -4142. 
 J cos^y 
 
 59. I " cot 6* cosec « c^ = V2 - 1. 
 
 I, 
 
 60. 
 
 cos 2^ 
 
 ^{t 
 
 = I log tan At = -6585 
 
 I 
 
IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 4-y c*. 
 
 
 1.0 S» 
 
 I.I 
 
 1.25 
 
 2.8 
 
 M 
 2.2 
 
 15. |12 
 
 ! '- IIIIIM 
 
 1.8 
 
 U 111.6 
 
 P^ 
 
 ^ 
 
 /a 
 
 
 /a 
 
 ^fJ^ 
 
 ^ > ^ 
 
 vs 
 
 
 '/ 
 
 W; 
 
 4^ 
 
 iV 
 
 4i>^ 
 
 '*\^- -" 
 
 
 i^ 
 
 
 1^ \^ "*"' 
 
f 1 
 
 Ui 
 
 ^ 
 
 CHAPTEE XX. 
 FUNDAMENTAL INTEGRALS. IV. 
 
 Examples. Formulae (n) to (w). 
 
 ix^-te yia \x-a x + aj 
 = ~ [(log^ - a) - log (X + a)] = I log(^^ «^ . 
 This is formula (/•). * 
 
 2. [-^,= f 1 r^ + -1-] dx = 1 log (ii±^) 
 
 )a^-x^ J 2a \_a + x a - xj 2a ^ \a~x^ 
 
 This is («). 
 
 3. f-^^ = JL [ ^(^V3) ^ 1 ^.^., , V3x 
 JV4-3x=^ V3Jv2^_(a;V3)^ V3 V 2/" 
 
 . f tB rfcB f d{x'') 1 • 1 / 2N 
 
 5. f — J-^ i^(« - ^) ^ sJD-i /« - ^\ 
 
 • A /x — a\ 
 or = sinM I. 
 
 7. r_^ iiog/'—* \. 
 
 " 
 
 • 
 
EXAMPLES. 
 
 85 
 
 ■ 
 
 8. f— ^^_ = llo../i^-"A 
 
 and r_^i_=±looY_^\ 
 h((,c-x' 2a '^\2a-x)' 
 
 dx 1 (• \x) 
 
 9 
 
 •' X Vx^ - a^ •" 
 
 X' 
 
 ©■ 
 
 1 • 1 /a\ * 
 12 f ■^^■^' 1 • , /»\ 1Q r ^a^ 
 
 14. |4^,= i-;tan-V'^'V 
 J 'f* + X* 2a2 V«V 
 
 Hint. Multiply numr. and denr. by V*^ -^^ 
 
 X 
 
 2 iX 
 
 \ n [dx V«^ - X^ , 
 
 -, 7 fff.c V2a,/; - x^ . /x - a\ _ 
 
 to; - X' 
 
 18. \dx \l'!LZl^a sin-i ^ + ^-^FZ:^\ 
 
 J a-hx (I 
 
 1Q [dx Ix + a X . , .-5 ,^ 
 
 J X X - a a "^ ^ 
 
 • The integral is also I sec-i ?. These apparently different re 
 
 suits only differ by a constant (in this case V2a), and therefore ha 
 the same differential. 
 
 ve 
 
 w 
 
i 
 
 86 
 
 20. 
 
 EXAMPLES. 
 
 I: 
 
 dx 
 
 X sj^x" - 9 
 
 = Jsec-i(§»). 
 
 21. r_^_ lsec-(^.v/-V 
 
 22. f.^* _ 1 r ^(2«^-) 1 . , /2.'cK 
 
 90 r^ dx ^ _ _ 
 
 Jo2T5^2 = TVV10tan-i(^V10) = -3184. 
 
 24.' j; 
 
 rfcc 
 
 .r V2«' - .1 
 
 25. r ^^ 
 
 Jo./«_ 
 
 = l^= -7854. 
 
 26 
 
 f 
 
 Jo 
 
 ''V8-4x^ 
 '' Sdx 
 
 = i^n = -2618. 
 
 4 + 9.^^ = ^*"°"'^ = •.^245. 
 
 orr r^ X dx ^ 
 
 27. . = ;^T = -7854. 
 
 xdx 
 
 29. f-^^ 
 Ji a; 
 
 + 4a; 
 
 28. J;i^. = ilog§ = -1277. 
 = i J'>g f = -1277. 
 
 J 1 05* + 4 * * 
 
 31. f-,-- = |tan-if = -3217. 
 
 ^ 
 
CHAPTER XXI. 
 
 INTEGRATION OF RATIONAL FRACTIONS. 
 
 71. An algebraic fraction is rational when it contains no surd 
 expressions involving the variable. 
 
 If the fraction is improper it must first be reduced to a mixed 
 qunutity. 
 
 Examples. 
 
 1 '^ ' 1 1 
 
 ^. = a;-- 1 + 
 
 1+x^ 
 
 
 JiT^^ = J(^-^ + r:ir^)^'»^ = i^*-i^^ + iiog(i+xO. 
 
 72. When the fraction is a proper fraction, it should, in gen- 
 eral, be decomposed into its partial fractions. See Appendix, 
 Note B. 
 
 Examples. 
 
 1. _ ^^ + 3iB + l 
 
 * x(x^)~(x + 2) ^ 
 
 1 
 
 X 
 
 1 
 
 ^ - ^ x-l 
 
 -* 
 
 x + 2 
 
 ■k 
 
 x^+^x + \ 
 
 ^^^^^y^^^C?X = -|l0gX + |l0g(x-l)-^l0g(..4-2) 
 
 ' Jx + 2«^ + x'~J V» 
 = logx-log(] +x) - 
 
 V»-* + 2x^ 
 2 
 
 1 + 
 
 X 
 
 (1+x) 
 
 1 + 
 
 = l0c 
 
 X 
 
 .)jx 
 
 1 +x 
 
 ' i 
 
F 
 
 V 
 
 88 
 
 RATIONAL IK ACTIONS. 
 
 
 a~h 
 
 I02: 
 
 
 Exercises. 
 J x^^^TS^TF = 3a: + 1 1 \o^(x - 2) - 2 log(x - 1 ). 
 
 y?dx 
 
 = log(l +a:) + 
 
 2 + a; 
 
 10 
 
 7. r 
 
 J «' 4- 5«2 + 8x + 4 
 q f(l +«) c?x , -V. 
 
 J 1 + X + .r 
 
 vr+x^ 
 
 
 ' + X'' 
 
 1+x 
 
 
 12 
 
 Jrx''+ 
 
 X d,x 
 
 ^ _3x + 4 
 
 (x''+ 3x4-2)''' r'-'+l^l 
 
 f 3 lo 
 
 •X4-1 
 
 "\x + 2/ 
 
 J 
 
1" 
 
 J 
 
 CHAPTEE XXII. 
 INTEGRATION BY SUBSTITUTION. 
 
 73. To assist in bringing certain differentials under forins 
 already considered various substitutions arc cmplojfcd; the most 
 important of which will be mentioned in this chapter. 
 
 . Examples. 
 
 ^- \-%-n' I^et * = 1. then rfx=-l^ and l^-.i^ 
 Substituting^ we have 
 
 + h -a{n-\) ' •^^^^' 
 
 1 n 1 
 
 
 X 
 
 2. Making the same substitution and integrating by (a) we 
 have 
 
 *'(«^-x^)t «^(a^-«;^)r J(^2+,,2)ii" "^ 
 
 X 
 
 74. Since ax'^ -i- /ax + c = } r(2ax + h)^ + 4ac - 62 1 
 
 the following may be reduced to ditferentials already integrated 
 by putting 2ax + b = z, 
 
 dx 
 
 X dx 
 
 dx 
 
 I 
 
 ax^ ^-hx + r' ax^ + bx + c' V^^^^^T^' 
 « ff^ dx X dx 
 
 -Jax^ +hx + c {nx^ +hx + c) §' {ax^ + hx -\- c) ^ 
 
 ^1 
 
 i !/l 
 
90 
 
 INTEGRATION BY SUBSTITUTION. 
 
 Examples. 
 
 dz 
 
 r iJx C dx /• 
 
 , if^ = 2a; + 3, 
 
 = log (. + ^z^-T^) = lo, (2a, + 3 + > V^^Ta^TI). 
 
 2. r ___j^^ ^ r 
 
 *' V2 + a; - .t2 J 
 
 rfaj 
 
 
 4 
 
 if2 = l-2a;. 
 
 = -sin-i =sin 
 o 
 
 C-^) 
 
 obfainrd." """''" ™^ ""^ ^""""'°°' S-^"^^"' '«^»"« »»y be 
 
 /, 
 
 ^* 1 '2«x-6 
 
 
 V4rtc + 68> 
 
 r ^ 
 
 or 
 
 J-Jf/'-4ac \2axi 
 
 
 (^ 
 
 2{2ux -f- h) 
 
 — ), if 6^- 4ac is + 
 
 [__J^jdx__ ^ _!_ |.(2ax + h)dx~ hdx 
 ^ax^+hx + c 2a] ^J^^Tb^^c 
 
 dx 
 
 ««* + bx + c' 
 
 -« ^ 2a J« 
 Similarly, f— _5_^___ _ 1 , b C 
 
 dx 
 
 ^/(ix^ -hbx + c 
 
 1 
 
 I 
 
J 
 
 ' 
 
 and 
 
 J: 
 
 INTEGRATION BY SUBSTITUTION. 
 
 91 
 
 {ax^ + 6x + c) ^ (4ac - //) Vaas" + hx + r 
 
 76. If we put X = _ in 
 
 dx 
 
 ^ X 'Jax^ ■\-hx-\-c 
 
 or 35 + A; =r _ in 
 
 dx 
 
 ^ (x + A;) ^jax^ + 6x + c 
 these differentials will be reduced to the third of § 74. 
 
 76. In 
 
 dx 
 
 77. - 
 
 (ax* + b) V«x' + 6 
 <fx 
 
 ;: let ux^ + h = x'zK 
 
 may be integrated when m + n -2 
 
 (x - «)»» (x - by 
 is a positive integer by the substitution 
 
 (x - a) / (x - b) = z. 
 
 78. Binomial Differentials. Any expression of the form 
 {axP+bx'^y dx, the indices being positive or negative, integer 
 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 
 
 «« (ax" + by dx. 
 
 This can be integrated immediately in the following cases : 
 
 (1) When ris a positive integer, expand by the Binomial 
 Theorem. 
 
 (2) When is a positive integer, let ax" + b = z. 
 
 n ' 
 
 (3) When -_ — + r is a negative integer, let ax" +b = x"«. 
 
 n 
 
 79. When the differential coefl&cient is a function of a + bx 
 let a+6x = 2" where n is the L.C.M. of the denominators of 
 the indices. For example : ' 
 
 T 
 
 i- II 
 
 •I X < 
 
 ill 
 
 i;lfi 
 
92 
 
 
 INTLGKATION BY SUH.STITUTION. 
 ^» f'Zz dz 
 
 •'(l+x)?+(l+x)i J^' + a 
 
 r2z dz .p. 
 
 X ) -' J z + z 
 
 = ^J rTp. = 2 tan- 1 z = 2 tarr' VI + x. 
 
 80. sin'»6 t/fl ■) 
 
 C w odd and + , let cos u = z. 
 
 ain^f) cos^ti do 
 
 I m odd and + , let sin h = z, 
 
 cos fy d6 = c?,v. 
 
 a\^u' — ;;r *^ ^^en and + , 
 
 let tan fi = z, 
 
 :. C0Sfl=l/(l.f-32)^^ 
 
 STn"*^ cos-^^y da, m -f- ;/ even and - , | sin « = -/( 1 + ^2)^ 
 
 de = dz/(\^-z'') 
 
 81. More o-enerally, any rational function of sin.orcos^is 
 made algebraic and rational by putting tan ^o = z. For 
 
 cos^/ = (1 _ ,2^)/^, ^ „,^^ ^.^^ ^^ ^ 2^^^ I _^ ^^^^ ^^^ ^ ^^^^^^^^ ^^^^^ 
 
 82. Any rational function of tan n is made algebraic and 
 rational by putting tan (i = z. For 
 
 dH = dz/(l +z^). 
 
 83. Any rational function of e' is made algebraic and rational 
 when e" = z, for dx ^ dz/z. 
 
 84. On the other hand, certain algebraic surds are rendered 
 trigonometric and rational by substitution. For 
 
 if X = a sin «, (^^ - 0^)1 = a cos 0, 
 ifx = a sec«, (x" - ft')l = „ tan 0, 
 if X = a tan ^;, (x" + <^)\ = f,, sec fy, 
 il'x-2asinV, (2^/x - x^)! = 2(i sin^cosry, 
 if « = 2a i^n\ (2ax + x')h = 2a tan « sec 6. 
 
 1 
 
 . 
 
 J 
 
1 
 
 . 
 
 J 
 
 INTEGKATJOxX BY SUliSTlTUTION. 
 
 KXERCISES. 
 1 f ilx 
 
 3 f ^^ 
 4. f- ^^* 
 
 93 
 
 VI - X - x^ 
 dx 
 
 /2x + 1 \ 
 V V5 ) 
 
 = .sin-i/2r.+ l 
 
 V5 
 
 03 + a 
 
 5. f — ^Iz u. + a 1 
 
 (^^«« + X ; i a* Vi^^ix + x' z 
 
 c- r dx 
 
 JJ'A 
 
 V3a; - a;2 _ 2 
 7 I dx 
 
 = sin-i(2a;-3). 
 
 '•J — ^ 
 
 (i - x=^) VI + x' 2V2 ° ^ 
 
 8. 
 
 J; 
 
 dx 
 
 ) VI - X 
 
 cfe 2 
 
 VI + CB^ - X V2 
 
 ZX' 
 
 - = -tan-iv/?^_l. Let2ax-a^ = ,!. 
 
 x V2rtx - a^ a 
 
 10. Jsin^^ Go^'o dft = :^ cos'ry - J cos^^. 
 
 1 1 . j^'m^e dff = -i co,^e + f cos'^ - cos ,1 
 
 12. J sin«^ cos^^ ^^ = I sin'^ - 1 sin^^y. 
 
 13. f?l5^=_jL_,i_L 
 
 14. UV^^fVi^^. 
 
 ill 
 
 ; 
 
 : i 
 
 
 li 
 
 _3, 
 
 11 J 
 
 "r 
 i. I 
 
n 
 
 94 
 
 15 ^ ^^ 
 
 10 
 
 EXEIICISES. 
 
 Jsin^6 
 
 -1 
 
 f cot 2n. 
 
 ^6 co.s^^y 3 cos sin'w 
 
 17. tan^/y r/w = J tanV^ - tao y + <y. 
 
 ^^- -2 2 rr^^r-= , tarr^ /^^ tan w^ 
 
 J «^ cos'« - i'^ sin't) ^ 2ab * V« - 6 tan w / 
 Hint. Multiply numr. and denr. by seo^^. 
 
 19. f f = itan-i(^tdnA^O. 
 
 20. f-_i^:!_ = ilo-!±^i^ 
 J3 + 5 cosft ° 2-tan^«' 
 
 21. r^ 
 
 J 1 -f- Sll 
 
 sin ft 1 + tan ^ti 
 
 oo fsin ti - sin^ft , , ,^ 
 
 22. — -, f^ft= 2ft + cosft + _ 
 
 J 1 + sm ft 1 
 
 + tan ^ft 
 23 f a^'^^ _ 2 (.x^ - -Ix - 8) 
 
 ' J(l4-a;)^~ 3VlT» * fr 
 
 24. p(l +xO' '^a; = ^ (7x^-2) (1 +x»)i 
 x^ dx ^^ + f 
 
 j* x^ (Za; ^^ + f 
 
 Vx'-l 
 
 :-ix' 
 
 (1 + 2x2). 
 
 27 
 
 28 
 
 ■t 
 
 X* Va;' - 1 
 
 VI +x* n Vl+«"+l 
 \—=^dx = 2(\-^x) VI -X. 
 
 •' Vl - X 
 
 < 
 
, 
 
 ' 
 
 EXERCISES. 
 
 ^^- J;^=2v^^+2iog(i-vx). 
 
 30. f-^^ ^Jo«^a+^) 3aU4abx 
 
 31. f__^li^ =^!:i?^' 
 •J: 
 
 32 
 
 db 
 
 _ sec- la; Vx^-1 
 
 x" V«^^ ^2"" "^ ^^2sF^ ' ^^^ ^ = 
 
 sec ^y. 
 
 34. f- 
 
 « + 2tan-i V*. 
 
 = 2 tan-i VI + 
 
 cc. 
 
 (2 + a;) Vl+x" 
 
 35. J — - -^-..^ ^- 1 loo. ^^ - 3 + V8(T3^- 
 •'(l+a;)Vl-a; V2 ° TT^^ ^• 
 
 36. 
 
 /: 
 
 dx 
 
 ^/x + a + ^/x+b 
 
 a — b 
 
 37. f ^ 
 
 I ax 
 
 X ^ax^ -{-bx + c" 'Jc 
 
 1 , 
 
 JC 
 
 38. L 
 
 •'a; 
 
 fjfe 
 
 Ax + 2c -f- 2 v^;(;^:?T6^:r^ 
 
 ^= ^ sii, 1 bx~2c 
 
 2 log (Vx - (/ 4- Vie - 6) 
 
 40. 
 
 J. 
 
 c?a; 
 
 95 
 
 V(« - a) (6 - a;^ 
 
 
 :;; }" Let as - rt = z». 
 
 11 
 
 
i I 
 
 IP 
 I 
 
 CHAPTER XXiri. 
 INTEGRATION BY PARTS. I., ; 
 
 85. Since udo + vdii = d(uu), .'. \ndv+ vdu ~. nVy 
 
 .*. udv = uv — \vdu, 
 
 or, udv can be integrated provided that vdii can be. 
 
 Integration by this formula is called integration by part?*. 
 
 Examples, 
 
 1. logx(/a3= (logx) x—{x.~- 
 
 . = (log x) X — \dx - X log X — X. 
 
 2. fsin-i xdx^ (sin-ix)x - \x ''^ = x sin-icc + V^^^'. 
 
 86. In applying the formula it is generally necessary to 
 separate the given ditt'erential into two parts, one of which is the 
 differential of some known function. 
 
 Examples. 
 
 I* I* /x?\ 01? ^x? dx 
 
 1. Ja;'r^xc?a; = J(logx)rf(_j - (logx) ^ -Jg" * ^ 
 
 •X^ 1 x^ 
 
 = — log X — - . 
 
 2 ° 4 
 
 2. Similarly, x" log x c^x = log x _ -— . 
 
 J n+\ (/1 4- 1 )' 
 
 87. It is often necessary to repeat tb :• method of integration 
 by par^s before the complete integral is obtained. For ex- 
 ample : 
 
 x^coi X c?x = x\l (sin x) = x^siu x — 2 x sin x dx. 
 
INTEGRATION BY PARTS. 
 
 Again, jx sin x dx =^(xd(- cos x) _: - x cos x + [( 
 
 97 
 cos X dx 
 
 = — X cosx + sma;. 
 
 Substituting, Jx^cos x dx = x^sin x + 2x cos x - 2 sin x. 
 
 88. Sometimes the integration reproduces the given expression 
 with a new coefficient. 
 
 Examples. 
 
 1. Since ^a^-x'^ = 
 
 a^ — x^ 
 
 '\Ja^ — x^ 
 
 .*. JVa'-x'-^ 
 
 dx = a^ 
 
 ^ dx 
 
 •'V«'-x2 JVa'-x^ 
 
 r x^ dx 
 
 = a2sin-i ^-Ja, j(-_^^._^2>j 
 
 = a^sin-i - + X Va^-x^ - fVa^^^ (7x. 
 Transposing the last term to the left-hand side and dividing 
 
 b^ 2, we have 
 
 W- 
 
 a 
 
 X X 
 
 x^ dx = ij-sin-i - + ^^0^- x' 
 
 Li Ob £i 
 
 2. Similarly, 
 
 Vx'±a^cte= +Jlog(x + Vx'±aO+|Vx^± 
 
 r x^c?x r 
 
 a 
 
 x'd Vx^ - a'^ = x^ V^J' 
 
 -af-^l 
 
 X Vi 
 
 =: X'^ V*^ 
 
 -a»-2j 
 
 x 
 
 '(7x 
 
 Vi 
 
 4-2a=^ 
 
 X* - a" 
 
 J Vx^^2 V 
 
 Vx''- 
 
 a' 
 
 'x" 
 
 -a^ 
 
 11 
 
 s 
 
 x^- 
 
 a» 
 
 Vx^ 
 
 -a" \ 
 
 = 3 VaJ* - a* + -3- log (x + Va^* - n^) 
 
 when the second term is transposed. 
 
 
98 
 
 i 
 
 I & 
 
 r: 
 
 INTEGRATION BY PARTS. 
 
 Exercises. 
 
 1. J 05 cos 05 c?x = « sin X + cos cc, 
 
 2. J taa-ix dx = x tan-ix - log VI f x\ 
 
 3. \x tan-ix cZa5 = ^ (1 +«0 taa-ix~^ x. 
 
 4. Jx sec-ix dx = \ (x* sec-^x - V»*^). 
 
 5. xe^ c?x = (x - 1) e*. 
 
 6. JxV (?x = (x- - 2x + 2) e^. 
 
 7. J e^ sin x (Zx = ^ e» (sin x - cos x). 
 
 8. J x"'^ sin xdx = 2x sin x + (2 - x') cos x. 
 
 9. X sec'^x dx = X tan x + log cos x. 
 
 10. I e' sin 2x (^x = - (sin 2x - 2 cos 2x). 
 
 11. J X tan'x cfo = X tan X + log cosx - Jx^ 
 
 12. |x«(logx)«cZx = I' [(logx)''-! logx + f]. 
 
 J V2ax -x-^ dx = ^^ V2ax - x' + ~ sin-i /^Z_^\ , 
 
 13. 
 
 x i-a 
 
 ■ J X i- a - a' 
 
 i«x + X f?x = -^ V2ax + X- - 2" lo;.'(x + a + V2ax + x'*). 
 
 14. |V2 
 
 15. e«'co8 mx dx = -_ (a cos wix + m sin mx). 
 
 16. e^'sin mxdx = (a sin mx - ?7i cos mx). 
 
 I 
 
CHAPTER XXIV. 
 INTEGRATIOrf BY PARTS. II, 
 
 (SUCCESSIVE REDUCTION.) 
 
 89. Many integrals m? y be obtained by successive applications 
 of the method of integration by parts. 
 
 90. To integrate sin'V^ de, n being any positive inteo-er. 
 
 J 8in«6' de = fsi n"- 1 « c?( - cos 6) = - sin»- ^ (? cos <? 
 
 + (n- I) I siti^^e cos^d dfi, 
 
 and writing 1 — sin^O for cos'^e, 
 
 = - sin'^10 GOSB+ (n - I)j8in"-2e dd- (n - 1) fsin^^^ dn. 
 
 Transposing the last term and dividing by n, we get 
 
 Writing n — 2 for n, we have 
 
 sin'»-3fl cos/9 . 71-3 
 71 — 2 n 
 
 + 
 
 3 f . 
 
 — SI 
 
 2 J 
 
 sin'^*e dd. 
 
 Thus I y each integration the index is diminished by 2, and 
 hence will in *^he end depend upon 
 
 P 
 
 sin 8 de = — cos i9, 
 
 ', or \dd = 6, 
 
 according as n is odd or even. 
 By a similar process 
 
 008"^ do = + cos'^^e d(L 
 
 J n n J 
 
 > ' I 
 
 m 
 
 i 
 
100 
 
 INTEGRATION BY PAATS. 
 
 V 
 
 m 
 
 91. f-^ = fcos^^H^sin^ ^^^ ^ rcos^ ^^^^ + f _J^ . 
 
 The first term = fcos ft d { i ^ 
 
 J V (w-1) sm"-i^y 
 
 _ cos^^ 1 C d6 
 
 . [ df^ _ cos ^ ?i - 2 r d!*^ ^ 
 
 by which we may reduce to 
 <% f^ = log tsmhi, (Ch. XIX. 4) ; or f.^ = _ cot «, Ck) 
 
 according as n is odd or even. 
 
 dft 
 
 cos^' 
 
 o- M 1 r <*^ sin fl w — 2 
 bimilarly, = -\ 
 
 Jco8^(^ (n-1) Gos^'^i) n-l 
 
 and f-^ = log tan (1 + I) , (Ch. XIX. 5) ; {-% = tan e,(j). 
 J cose ° \4 2/'^ Jcos'''<^ . '^*^^ 
 
 92. [tan^e c^e = | tan'^-a*^ (sec^rt -1) dH 
 
 = [tan"-2^ J(tan h) - [tan"-2/i da 
 
 j^:^::l!t.Unn-Hd,. 
 
 n-l J 
 and I tan e do = log sec 6, (Ch. XIX. 6) ; \dH = e. 
 
 Similarly, cbt"^ c?« = - ir^: -- coV-'^ftdn 
 
 J ?i - 1 . 
 
 and foot 6 de = log sin ^, (Ch. XIX. 7) ; \de = 6. 
 
 93. Jcos'"^ sin«/^ d^ = fcos"-! 9 d /'?15!!!^\ 
 
 cos"»-ifl sin"^if> m-1 
 
 - + 
 
 ri4-l 
 
 7i + 
 
 _ rcos«-2(^ sin"^2f^ dd, 
 
 i 
 
 1 
 
 
 1 
 
 "1^ 
 
INTEGRATION BY PARTS. 
 
 101 
 
 • 
 
 1 
 
 i 
 
 and writing sin''*^^^ in the form sin^w (1 - cos'fl), 
 
 - Ico&'^^O sin^t^ de - ^^-^ fcos'"<? siu''^? d6. 
 IJ »+l J 
 
 C08"»-i^ sin"-"!^ m— 1 
 
 = + 
 
 w + 1 n + 
 
 Transposing the last term and dividing, 
 
 cos'^^f? sin"*^^ m-1 
 
 f 
 
 co8'"« sin"/? f?^y = 
 
 + 
 
 f 
 
 cos"^^^ sin"^ do. 
 
 m + n m+ n 
 
 In a similar way we might have obtained 
 
 f cos"»(^ sin^d dti = - cos"*^'^^ ^i""''^ + ^^^l [cos^^^ sin«-^e do. 
 4 m + n m + n J 
 
 94. The following will present no difficulty : 
 
 cos"^if^ 
 
 ^i {'.2f::iide, 
 
 ■^6 n-1 J sin'*-=»0 
 
 Jsin^e (n-l) sin" 
 
 Tsin"^^ , _ sin*^!^ n-l fsin^-ay , 
 
 Jcos^y ~ (m-l) cos^^-^O m-lJcos'"-3^ 
 
 (m-1) 
 *cos3^ + sin2« 
 
 de 
 
 do 
 
 95. [ ^' =p 
 
 J cos'"^ •sm'^fl J 
 
 ~ J C06"^e sin"0 J cos"*^) sin"-^^ ' 
 
 The first term= f— -^ — d( zl ^ 
 
 Jcos'^10 \(n - 1) sin^if^/ 
 
 do 
 
 ■) 
 w — 1 
 
 (w - 1) cos"^if^ sin^-if^ n - I Jcos^e sin^-^d* 
 Substituting this, we get 
 
 I 
 
 d6 
 
 1 
 
 + 
 
 m + n — 2 
 
 + n-2 C 
 n — 1 J 
 
 de 
 
 cos'^y sm'^'.fl 
 
 co8"»w sin"^ (n-Y) co&^-'^O siu^^tl 
 
 By treating the second term in the same way we might have 
 obtained ' 
 
 J 
 
 do 
 
 + 
 
 m + n — 2 
 
 ^h 
 
 do 
 
 cos™^ sin"^ (m - I ) cos'^^W sin'*-^f^ m-1 J cos'"-^^ sin*^ 
 Hence the integration may be reduced to one of the following : 
 
 ii<i 
 
 >: ij 
 
102 
 
 INTEGRATION BY PAKTS. 
 
 1 
 
 U [ '^'. , {±, or f^. 
 J J cos a sin h J sin 8 J cos w 
 
 See 3, 4, 5, Ch. XIX. 
 
 96. The following may be obtained from the preceding re- 
 ductions by the substitutions of § 84 ; they may also be obtained 
 directly, (cf. § 88). 
 
 r dx 'Ja'^ -x^ n~2 C dx 
 
 Jx" V«^ - »' ~ "'(^^ - 1) x«-i "*" a\n - 1) J.^n-2 7^ 
 
 J: 
 
 X" <Zx _ x"-i ^-lax - x^ a (2« - 1) f x""^ <?x 
 
 1; 
 
 •f 
 
 V2ax-x'-^ « ^^ JV2ax-x^ 
 
 ■-' - n . 
 
 (a' - x-'ja tZx = —S --Z-. + (a-' - X-)-' cix. 
 
 J 71-1-1 « + 1 J 
 
 Exercises. 
 x'"cos rjx ?»x"'-isiu 7«x 
 
 1. [x^sin nxdx = - 
 
 n 
 m(ni — 1) 
 
 w 
 
 n- 
 
 x"'-2sin nxdx. 
 
 2. Jx- cos nx dx = ?!1'-^ + "^^""'^'^^^'^ 
 
 w 
 
 wi(m- 1) 
 
 w 
 
 x'"-2cos?ix dx. 
 
 m-1) r 
 
 ^^^ J" 
 
 3. {x^e"' dx = ^' - !!. fx^-i e"^ dx. 
 J a a J 
 
 ^ r e"' dx ^ e°-^ g fe"^ ^/x 
 
 ' J X" ~ (n-l)x^-^ n-\ J x"-i 
 
 5. fx«»(l0g X)" </x = «^"''G0g^a^)" ^ L« (logNn-l d^^ 
 
 , 
 
 I 
 
 .1 
 
i 
 
 .i 
 
 CHAPTER XXV. 
 CERTAIN DEFINITE INTEGRALS. 
 
 97. The first term of § 90, (1), is when 6 = and also 
 when (f= ^TT '^ hence 
 
 I 
 
 w being odd, {^ ^\r.n, rl.n _ (^ - '^) (w- 3) ...6.4.2 
 
 J" n{n-2) ...5.3.1 ' ' 
 
 w being even, sin "« rt'^ = 0^~ 1) (^- 3) ... 5.3.1 ^ 
 Jo w("-2) ... 6.4.2 2* 
 
 Also I " cos"^ dH = \^ sin"^ rf/^ 
 Jo Jo 
 
 98. By examining the results of § 93 it will be found that 
 
 '^ sin-^ cos"^ do = [(^-1) (m-3)...] \(n - 1) (ti - 3) ...] 
 (m + n) (m + w-2) ..". ** 
 
 each series of factors being carried to 2 or 1, and a being I^ttI^) 
 except when m and n are both even, when « = Jtt (W. Woolsey 
 Johnson, Int. Cal.). 
 
 99. Many integrals may be reduced to the foreffoino-. 
 In the following examples m and n are any integers. 
 
 Examples. 
 
 1- - == = <*" sin"^ de. Let a; = a sin B (^ 84\ 
 
 2. fV - aj')'^ rfa^ = a"-i f^ cos"-i^ <^^. 
 Jo Jo 
 
 3. x«*(a2 - 05-)^ (?x = a'"+"*i sin*"/? cos"*^ rf^. 
 Jo Jo 
 
 I if 
 
 ■s si 
 
 h» 
 
'i 
 
 n 
 
 
 104 
 
 DEFINITE INTEGRALS. 
 
 4. _ - zr = 2 (2a) » " sin2«^y f^^. 
 
 •'"V2aa;-x^ Jo 
 
 /•8a „ .T 
 
 5. J x'"(2ax - x-')^ dx = 2 (2a)'«+«+i ^ sin^'»+«*ifl cos"+ifl dd. 
 
 6. r_^__-_L.r^ 
 
 cos"- 2^^ ^^_ 
 
 100. From § 64 it is plain that 
 
 [f(x)dx = -rf(x)dx, 
 
 J a J 6 
 
 that is, interchanging the limits merely changes the sign of the 
 definite integral. 
 
 101. 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. 
 
 To prove e*^^ dx -~, 
 
 Jo -^ 
 
 From § 97 we have 
 
 r sin"6 do . P sin^+i^ dd = -l—E m 
 
 Jo Jo n + 1 2 ^^ 
 
 Let sin"^ = e-a:'-', or 6 = sin-i (en \ 
 
 x'i 
 
 :. dd = 
 
 en dx 
 
 n 
 
 1-e'^ 
 
 2x 
 n 
 
 dx 
 
 I 2x^ 
 
 V c"^ - 1 
 which by the Exponential Theorem (Appendix, Note A.) 
 
 -- V^ . ^, where D= v/l +-+- + -..• 
 
 n D 
 
 w 3/t« 
 
 ' 
 
 i: 
 
DEFINITE INTEGRALS. 
 Substituting in (1) we have 
 
 106 
 
 -r -a;* 
 
 p^2 e-a:'c/a; p ^2 e " ^ » ^ Ox _ 1 ^ 
 J«w D J«n D ~ n,-l 2 
 
 ' Jo D Jo D 
 
 c?x n 
 
 n 
 
 + 14 
 
 Now let w = CO ; then d = 1, and ?i^ (=1 + -\ also = 1, 
 
 n \ n/ 
 
 [J%-..<fe]' = |, .■.J><ix = 
 
 "2" 
 
 Exercises. 
 
 1. f sin"^ d6 = 2 r sin'*« t^f^. 
 Jo Jo 
 
 j»7r 
 
 2. cos"^ do = 0, n being odd, 
 Jo 
 
 = 2 1 cos**^ dfi, n being even. 
 Jo 
 
 3. fxe-^' dx = J. See Ch. XVIII, Ex. 8. 
 
 Jo 
 
 4. sC* e-aj'-^ tZa; = -_j!: x^-s e-^^dx, (by parts*), 
 Jo 2 Jo 
 
 n even 
 
 (71-1) (m- 3) ...5.3.1 , 
 
 2^ 
 
 or =[(!L^)('-L^)...3.2.l].J,»odd. 
 
 5. »4 e-^ (fx = Vtt- Letx = 2^ 
 
 • xmer"'^ s when a; = and also when a; - go . See Appendix, Note 
 A. Cor. 
 
 t* 
 
 ■ 1 , 
 
 -ii 
 
■< 
 
 I) 
 
 106 
 
 DEFINITE INTEGRALS. 
 
 6. j^x-e-^dx=: n£x-i e- dx, (by parts) 
 = w(/i-l) ...3.2.1. 
 
 •'0 2 Jo 
 
 _w(w-2)...3.1 
 
 /Hi 
 
 2" 
 
 V^. 
 
 8. ra.-/loc.h"rfa.= 12.3^11^. Leta:-e^ 
 
 ; 
 
 'i: 
 
 ^Mi 
 
 I' I 
 
 m. .Jii 
 
; 
 
 CHAPTER XXVI. 
 AREAS, LKXiiTHS, Si HFA( ES, V0L13IES. 
 
 r 
 
 H 
 
 Y 
 
 G I 
 
 ) 
 
 
 / 
 
 
 1 s 
 
 
 y 
 
 ^;> 
 
 
 i>! 
 
 v 
 
 
 
 
 R 
 
 
 G T 
 
 
 
 £ 
 
 C 
 
 ) / 
 
 V h 
 
 \t 
 
 ^ 1 
 
 3 X 
 
 Fig. 57. 
 
 102. Let the co ordinate's r (fiir. 57) be (x, i/). We have 
 seen (§§ 66, 67) that : 
 
 (1) The area abdc = \i/dx, 
 
 (2) The volume of the solid formed by the revolution of, this 
 
 area about ox = tt ly^dx, both integrations to be between assigned 
 
 limits. 
 
 Let the length ot the curve measured from some point up to P 
 be s, then pq = c?s, .*. 
 
 (3) The length of the curve = ids = ^dx- + djj'\ 
 
 (4) The area of the surface formed by the revolution of od 
 round ox = 27r 3/ (/s = 2- h/ (7s. 
 
 "' 1 
 
 i-, 
 
108 AREAS, LENGTHS, SURFACES, VOLUMES. 
 
 103. The student will Imvo no difficulty in verifying the 
 following: Jo" 
 
 (1) The area hcdp . jx d,/, and the volume formed by re- 
 volving this about oy = tTx^/^, 
 
 (2) If CED revolve about ce the volume = t [(y - ac)» dx, 
 
 (3) If it revolve about ed the volume = rr!(0B-xydi/ 
 
 (4) If it revolve about oy the volume = ^ f (ob'-* - x-) dy. 
 Other relations may be written down in a similar way. 
 
 
 li 
 
 I 
 
 Examples. 
 
 1. Find the length of the semi-cubical parabola mf = a;^ rfiff 
 58) from the origin to the point (a, a). ^* 
 
 From the equation we have dy = ~ xi dx 
 
 2va ' ' 
 
 
 . 
 
 ii^as&iiiau 
 
■;,*.'K<4;»**LJ/!I 
 
 AREAS, LENGTHS, SURFACES, VOLUMES. 
 
 109 
 
 I 
 
 . 
 
 
 J 
 
 
 
 2^|n 
 
 
 L 
 
 27 ^ 
 
 2. The 
 
 area obd = 
 
 ««*. 
 
 
 
 The volume 
 
 of 
 
 OBD 
 
 about 
 
 OX 
 
 = h<^\ 
 
 <( 
 
 (( 
 
 (( 
 
 (( 
 
 <( 
 
 OY 
 
 = ^TTa\ 
 
 (1 
 
 >( 
 
 (( 
 
 • ( 
 
 (( 
 
 BD 
 
 = a^Tr<^ 
 
 104. It is sometimes desirable to cxpres? both x and ?/ in 
 terms of a third variable. 
 
 Example. 
 
 The equation a;^+y^ = «^ (fig. 11) is satisfied if we put 
 X = a sin^^y, y = a cos^/. 
 
 Then dx= 3a su\*e cos^ do, .'. ydx = 3a'| cow*/' sin*' do 
 
 Jo Jo 
 
 = 3a'»f^'^(eo8*// - cos*""^) dt) -= •^s^rra'' (§ 97) 
 
 Jo 
 
 .*. the whole area bounded by the curve = |7r«*. 
 For the length, ds = '^dx^ + dy^ = 3a sin a cos 6 do, 
 
 .*. whole length = 12a " sin ti cos dd = 6a. 
 
 Jo 
 
 Similarly it may be shown that the volume of the solid made 
 by revolving the whole area about one of the axes = ^^^^ rra^, and 
 that the surface of this solid - ^~a^. 
 
 105. It will often be necessary to determine the limits of the 
 integration from the equation of the curve. Thus in fiudino- 
 the whole area enclosed by the purve 
 
 a^y'^ = 03* (a' - «•), 
 
 it will be seen that the curve cuts the axis of x at ( i a, 0) and 
 that the general shape is that of fig. 59. Hence the complete 
 area 
 
 = 4 ydx = |a^ 
 Jo 
 
 If II 
 
 :ll 
 
 H 
 
n 
 
 Hi 
 
 II 
 
 t 
 
 I I 
 
 rl: 
 
 "I ! 
 
 110 AIIKAS, LENGTHS, SUHFACES, VOLUMES. 
 
 The volume of the solid of revolution about the axis of 
 
 J 
 1 
 
 X = A^a\ 
 
 Fife. 59. 
 
 Pig. 60, 
 
 106. When y is negative the sign of j/dx is — , and accordingly 
 an area lying below the axis of x will be affected by the same 
 sign. Hence in calculating an area, care must be taken thiit y 
 does not change sign between the given limits. Thus in the 
 curve 
 
 i/ = x{x-l) (a; - 2), fig. 60, 
 y is + from ai = to x = 1, - from x = 1, to a; - 2 ; it will be 
 
 /•I (•2 (•'i 
 
 found that ydx = ^, ydx = - J, ydx = 0. 
 
 Jo J 1 Jo 
 
 And generally in taking the sum of such terms as/(x)*<fo; 
 between assigned limits, the result is the algebraical sum, which 
 will be equal to the arithmetical sum only when/(x) is of the 
 same sign for all values of x between the given limits. 
 
 107. So also if y ov /(x) becomes imaginary or infinite for 
 a value of x between the given limits the definite integral will 
 not, in general, give us the true sum of the elements. For 
 example, if y(x — \y = 1 (fig. 61), y = co when x = 1, also 
 
 ydx = ( — — I = 00 , and = - 2, which .". does not re- 
 Jo*^ \\-x)o ' Jo 
 
 present the sum of the elements between the limits and 2 ; in 
 
 fact, each element is + and .'. their sum could not be — . 
 
 If, however, the indefinite integral does not become infinite 
 when y does, the above restriction does not hold unless y changes 
 sign in passing through the infinite value. Thus in y^(x - 1) = 1, 
 fig. 62, y does not change sign when it passes through oo , also 
 
 i I 
 
AREAS, LENGTHS, SURFACES, VOLUMES. Ill 
 
 i 
 
 O 
 
 ^'''''- ^^g«2. Fig.G3. 
 
 But in f(x - 1) . 1 (fig. r>3), .y (loos cluange sign. Also 
 
 108. As, r increases tl.c volume-clement nfdx changes sign 
 only with ,/', ;.,.., when ij becomes imaginary. Thus in fig. (JO, 
 
 • . „ 
 
 109. If it be required to find the area between two ..-jven 
 curves cpd and crd (fig. 64) the area-element evidently " 
 
 = (difi"erence of^/s).dx, 
 and the limits must be found by solving tlic equations to find 
 the points of intersection. 
 
 In calculating the complete area of the curve cqrd the limits 
 are the maximum and minim im abscissas of the curve. 
 
 Similar remarks apply to volumes, etc. Thus the volume of 
 the solid formed by revolving about the axis of x tlie area in- 
 cluded between the curves >/ = 2x and f = x + x^ is 
 
 ^lp^-(xrx')]dx=:^^. 
 
i 
 
 13 
 
 £ 
 
 1 
 
 112 
 
 AREAS, LENGTHS, SURFACES, VOLUMES. 
 
 Examples. 
 
 1. The parabola if - 4aa;, 
 If OA = a?!, AB = ?/i, show that 
 ^ (1) AreaOAB = fa;,?/i. 
 
 (2) Length ob = ll. Vi;^?^ + a lo- ^t^^oMj^^ ^ 
 
 (3) Volume of oab about ox = \'Ky^x^. 
 
 (4) Surface of this solid r. iL [(4(«« + y^) I - 8a''] 
 
 =:-^ [normaP-subnormaPJ. 
 
 oft 
 
 (5) Volume of oab about ab = f-^nx^yi . 
 
 (6) Volume of obc about OY = ^rra-i^//,. 
 
 (7) Volume of obc about bc = ^ry/,^xj. 
 
1 
 
 EXAMPLES. 
 
 2. The circle cc* +y* = a\ Show that 
 
 (1) Area = na\ 
 
 (2) Length = ^na. 
 
 (3) Volume of sphere = ^7ra\ 
 
 (4) Surface of sphere = 4n-rt^ 
 
 113 
 
 3. The ellipse ^+^- = 1. Show that 
 
 (1) Area = Tcab. 
 
 (2) Volume of prolate spheroid * = ^nah^. 
 
 (3) Surface of prolate spheroid = 2171"^ + — — sin-ic. 
 
 (4) Volume of oblate spheroid f = ^KO^b. 
 
 (5) Surface of oblate spheroid = 2nrt^ + 
 
 TTt 
 
 log 
 
 (^D 
 
 Note. — The eccentricity e = V"^ - b^a. 
 
 4. The witch f(a - x) = a' x, fig. 39. 
 Let X = a sin^^, then y = n tan ^. 
 
 (1) Area between curve and asymptote = na^, 
 
 (2) Volume of this about asymptote = ^Tv'^a^. 
 
 (3) Volume of same area about OY = InW. 
 
 5. The cissoid i/^(a—x) = x^, fig. 27. " 
 Let X = a sin^y, then y = a, sin^^ tan ft. 
 
 (1) Area between the '.jrve and asymptote = '^na^. 
 
 (2) Volume of this about asymptote = ^n'^a^. 
 
 (3) Volume of same area about OY = f Tr^'a'. 
 
 * The solid formed by the revolution of an ellipse about its major 
 axis. 
 
 t The solid formed by the revolution of an ellipse about its minor 
 axis. 
 
 8 
 
 i^ \ 
 
 
t ■ 
 
 I ■ 
 
 I 
 
 i I: 
 
 i; 
 "I- y 
 
 114 EXAMPLES. 
 
 (J. The cycloid x = a(6- sin 6), y = a(l - cos O), fig. 29. 
 
 (1) Area = Sna^. 
 
 (2) Lengtli = 8a. 
 
 (3) Volume about base = bn'W, 
 
 (4) Surface of this solid = %4;ra^ 
 
 (5) Volume of the area Sira^ about tangent at vertex = -hi^ 
 
 (6) Surface of this solid =^7ral 
 
 (7) Volume aboui axis = nci^ (f^^-|). 
 
 (8) Surface of this solid = Skci'' (rr-^). 
 
 (9) Show that in fig. 30, s" = Sax, (s = op, x = om). 
 
 7. The curve ij = a (1 - cos^), ij = ad; figs. 30, 40. 
 
 (1) Area = 27ra'^ 
 
 (2) Volume of this about ox = tt (tt^ - 4) a\ 
 
 (3) Volume about OY = 5-V. 
 
 8. The curve / (a^ - x^) = a*, or x = asin6,ij = a sec «. 
 
 (1) Area between curve and asymptote x = a is Ta^ 
 
 (2) Volume of this about axis of ^ =47ra\ 
 
 (3) Volume of same area about asymptote = 27ra^(- - 2), 
 
 9. The parabola Q " + (f )^ = ^' ^^^ ^^' ^' P' ^^• 
 
 (1) Area between curve and axes = ^ab. 
 
 (2) Volume of this about ox = ^^irah^. 
 
 2 2 
 
 10. The hyperbola ^ - f- = 1, or =asec6,i/ = h tan 6. 
 
 a b^ 
 
 Show that the area bounded by the curve, the axis of x and 
 the ordinate at the point (x„ 3/,) is 
 
 ^x,y,-^aMogQf|l): 
 
EXAMPLES. 
 
 and lieDce that the second term in this result is the area of the 
 hyperbolic sector oap, where o is the centre, a the vertex, and 
 P the point on the curve. 
 
 11. Find tlie area of the rectangular hyperbola x^ = 1 from 
 x = ltox =71. Ans. log >u 
 
 1 2. The curve i/ = «-"=. 
 
 (1) Area from cc = o to X ^ CXI is 1. 
 
 (2) Volume of this about axis of x = hw. 
 
 (3) Surface of this solid = 7r[V2 + log (1 + V2)]. 
 
 13. The probability curve ij = e-^^ 
 
 The area between the curve and its asymptote (the axis of x) 
 is V^r, (§ 101). 
 
 14. The curve xy + ay. = aV. 
 
 The area between the curve and its asymptote = 2a^ 
 
 15. Find the area between the following curves and the axis 
 of X : — 
 
 (1) (3/-x)^ = x^fig. 15. 
 
 (2) (2/-xO^ = x^, fig.l6. 
 
 (3) a'i/^x(x'-a'),^g. 10. 
 
 (4) 7/(l+X^)=l. 
 
 (5) y/=x(l-x^). 
 
 (6) 3/ = xXx-l). 
 
 16. Find the area of a loop of the curves : 
 
 (1) / = xX2x + l), fig. 19. Ans. 
 
 (2) / = x^(2x + l), fig. 17. 
 
 (3) a/=(x-a) (x-2ay. 
 
 17. Find the area included between the parabola y^ = 8x and 
 theliney = 2x-8. Ans. 36. 
 
 Ans -J 
 
 1 
 TIT* 
 
 * 
 
 i. 
 
 A- 
 
 4 
 
 A. 
 
 
116 
 
 EXAMPLES. 
 
 18. Find the area included between the parabolas y^ = 4a jc, 
 x^ = 4ay. Ans. ^^a'. 
 
 19. An area (see fig. 52) is bounded by the axis of x, a curve, 
 and two ordinates of lengths y,, y^, respectively, and at a dis- 
 tance h apart, and y^ is the ordinate midway between them ; 
 show that the area = 
 
 provided that any ordinate at a distance x from ?/, or ;/^ can be 
 expressed in the form 
 
 a + hx-\-cx'^ + dx^. 
 
 20. The extremities of a solid are parallel planes of area a,, 
 Ag, at a distance h apart, and a^ is the area of a section midway 
 between them ; show that the volume of the solid = 
 
 provided that the area of any section at a distance x from a, or 
 Aa can be expressed in the form 
 
 a + hx ■{■ cx^ + dx^. 
 
 * •' Simpson's Formula." 
 
 t The " Prismoidal Formula." By it the ^'olunies of a great 
 many common solids (cones, prisms, pyramids, spheroids, paraboloids, 
 etc.) may be obtained. 
 
 If 
 
CHAPTER XXVri. 
 POLAR CO-OBDIXATES. 
 
 Tangents, Normals, &c. 
 
 ^ 110. Let (fio. 66) be the pole, oa the polar axis or initial 
 line, (e, r), (d+da, r + dr) the co-ordinates of consecutive points 
 P and Q, ,], the angle opa between the radius vector and the tan- 
 gent at p. From p let fall pr perpendicular to OQ. Then 
 PR = r sin de = rde, OR = r cos dd = r, hence rq = dr ; /. rds/dr 
 = tan RQP ; but the limit of rqp is v as Q moves down'to coincide 
 with p, 
 
 .-. tanv = r^. 
 ^ dr 
 
 Pig. 66. 
 
 Similarly if pq ^ ds, we have 
 
 Fig. (j7. 
 
 sm i/; = »• cos \b = — . 
 ds ' ds' 
 
 d>? ^. r'' dtf" -h dr\ 
 
 ^i^-U 
 
I 
 
 118 
 
 POLAR CO-ORDINATES. TANGENTS, &C. 
 
 111. Through the pole o (fig. 67) let a line be drawn perpen- 
 dicular to the radius \ector op, meeting the tangent inT and the 
 normal in n. a pt is called the polar tangent pn the polar 
 normal ot the polar subtangent, and on the polar subnormal. 
 The lengths of these lines in terms of* r and d can be written 
 down at once; e.<jf., 
 
 OT = r tan i/* = r^ dn/dr. 
 
 The tangent at any point P is ea»ily drawn by calculating OT 
 und then joining t to P. 
 
 112. Some of these quantities arc more conveniently expressed 
 in terms of f) and the reciprocal of r. Calling this m, we have 
 u - 1/r, du = — dr/r^ ; hence the polar subtangent 
 
 OT = - d(i/du. 
 
 'Phe polar co-ordinates of T are (^■t + O, — -V 
 
 Let OG, the perpendicular on the tangent, =p. 
 Then ',• OTP is a right-angled triangle and OG the perpendi- 
 cular from the right angle to the hypotenuse, we have 
 
 1 
 
 1 
 
 OG' OP'' 
 
 + 
 
 OT 
 
 1 2 , /dii\' 
 „ , or — = ?r + / I 
 
 r 
 
 (1) 
 
 113. The polar equations of some of the commoner curves are 
 as follows : — 
 
 (1) r cose - a, a straight line. 
 
 (2) r = a cos^, a circle of diameter a (pole a point on the 
 circumference, initial line a diameter). 
 
 (3) r^ cos 2e = a^ a rectangular hyperbola, fig. 82 (pole the 
 centre, initial line the transverse axis). 
 
 (4) r^ = a^ cos 2e, a lemniscate, fig. 11 (pole the centre, 
 initial line the axis). 
 
 (5) ri cos^f) = a^, or r ( 1 -t- cos ^) = 2a, a parabola (pole the 
 focus, initial line the axis). 
 
POLAR CO-ORUINATES. EQUATIONS. 
 
 119 
 
 (6) A = J COS ^H, or r = *«(! +co8 h), a cardioid, fig. 73. 
 
 (7) r([+e cos t)) = wi, 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) y =w(l +eco8^), a lima9on, figs. 72, 73, 74 according 
 as e <, = , or > 1. 
 
 FiR. 68. 
 
 (9) r = ad, a spiral of Archimedes, fig. 68. 
 (In figures 68, 69, 71, 6 varies from a little less than - 2;r to 
 a little more than 27r), 
 
 Fig. 69. 
 
 (10) re = a, a reciprocal or hyperbolic spiral, fig. 69. 
 
 Fig. 70. 
 
 (11) 7^6 = a^^a lituus, fig. 70 (8 is necessarily + , and varies 
 in the figure from to a little more than 27r, r is ± for a given 
 value of 6), 
 
 f 
 
 4- 
 
120 
 
 POLAR CO-ORDINATES. EQUATIONS. 
 
 Fig. Tl. 
 
 (12) /• = a'^, a logarithmic or equiangular spiral, tig. 7 1 
 (r = 1 when a = 0, r = d when i) = I radian, r < 1 when fi is 
 negative). 
 
 As t) may be supposed to vary i'rom — co to -f co , each 
 spiral consists of an infinite number of whorls or spires. 
 
 114. Equations (1) to (6) are all included under the form 
 /•'"cosm^ = a'^ ; in (1), (3) and (5) m has the values 1,2,^, 
 respectively ; in (2), (4), (6), it has the values —1, — 2, -^» 
 In all cases a is the intercept on the initial lino. The equa- 
 tion r'^ sin 7nf) = a"* represents the same series of curves, the 
 initial line having been turned backward through the angle 
 7r/(2m). Similarly (9), (10), (11) are particular cases of the 
 equation r™ = «"' H^. 
 
 115. The radius vector of the limagon, equation (8), is pro- 
 portional 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 6/ + n, the radius 
 vector is equal to that of a circle of diameter en plus a constant 
 line w, and hence the curve is easily constructed (the construc- 
 tion or auxiliary circles are shown in the figures). 
 
POLAR CO-OIU)INATEH. TANGKNTS, &C. 
 
 121 
 
 Fig. 73. 
 
 Fig. 74. 
 
 When e = 1 tlus curve l)ecomes a cardioid (cmjii. (J), which is 
 ihcrefore the inverse of a parabola. When e = 2 the curve is 
 called a trisectrix, the loop then passing throuiih the centre of 
 the circle. 
 
 Examples. 
 
 m 
 
 1. If ?■'" = a'" If'" show that tan ./- = ^ 
 
 n 
 
 (Differentiate logarithmically.) 
 
 2. If »'" cos md = a"', or /•"* = a"' cos wM, show that 
 
 tan ^ = cotmfij i.e., that tlie angle between the radius vector and 
 normal = mii, and hence that qoa (fig. 67) - (m- \)h. 
 
 3. In the logarithmic spiral r = a^ show that i/^ is constant 
 and = cot-^(logea). 
 
 In fig. 71, « = 1-318 cm., show that ,b - 74° 33'. 
 
 4. To find the polar sub tangent of a conic. 
 
 From the equation \+e cos H = m/r = mu, we have — « sin y d^ 
 = nulu, and the polar subtangent = - dii/du = m/{<'. sin «). 
 
 5. In any conic prove that 
 
 r? m\r 2ni ) 
 
 6. In the curve r™ cos m9 = a'" prove that/>r"'-^ = a"\ 
 
 7. Changing the sign of »n, show that ;^'<"' = r'" ^ in the curve 
 
 ,y.m _ ^7/1 gQg ,^^^_ 
 
122 
 
 rOLAU CO-OUDINATES. ASYMPTOTES. 
 
 
 
 I i 
 
 ■i 
 
 
 8, Hhow that the polar subnormal of any curve a dr/dii. 
 In what curve is the polar subnormal onstant ? 
 
 9. In what curve is the polar 8ubtan<;ent constant ? 
 
 Asymptotes. 
 
 116. The position of any line is known when its direction and 
 one point in the line are Icnown. We may therefore determine 
 an asymptot'j by finding a value of fl which makes r = oo or u » 0, 
 and then calculating the co-ordinates (§ 112) of T, the extremity 
 
 of the corresponding polir subtangent, viz. Att + w, — V re- 
 membering that the asymptote and radius vector must be parallel. 
 
 1. r = 
 
 Examples. 
 
 (tig. 75), or u = whence _- = - ah', and 
 
 I — H (If) a ail 
 
 r = CO or ?4 = when ft =1. Hence the asymptote passes through 
 
 the point (^n- +1, - a), or (1 - ^t, a) and is parallel to the line 
 
 6 = 1, 
 
 Fig. 75. 
 
 Fig. 70. 
 
"Sf n*«^r ^ '-'" 
 
 POLAU CO-OKDINATES. ASYMPTOTES. 
 
 123 
 
 2. Find the asymptotes of the curve (r ~ a) ^^ - r (fig. 76), 
 
 Ans. Lines through (^^±1, ±^'0 parullel to (i = ±1. 
 
 3. Find the asymptote of the reciprocal spiral >•// = n ^fig. 69). 
 
 Alls, A line through (^t, a) parallel to the initial line. 
 
 4. Show that tlie initial line is an asymptote to the lituus 
 »'''^W=<r (fig. 70). 
 
 5. Find the asymptotes of the curve r sin 4'/ = a (fig. 77). 
 
 Ann. Four pairs of parallel lines, each pair \a apart. 
 
 ]i 
 
 /. 
 
 "A 
 
 y 
 
 "4 
 
 \Cn^^ 
 
 Fig. 77. 
 
 Pig. 78. 
 
 (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^ sin 40 = a^ (fig. 78), 
 
 Ans. Four lines passing through the pole. 
 
 7. Find the asymptotes of the curve r cos 2« = 2a. 
 
 Ans. Four lines parallel respectively to = :^t, ^ = f n-, 6 = |7r, 
 e = I^TT, and passing through the points (fir, — «), (fr, a), 
 
 (|^,-«'), (|-» «)• 
 
 8. Show that the rectangular equation of an asymptote of the 
 curve r-i =/(^) is 
 
 f'{a) (x sin a - y cos n) + 1 = 
 where a is one of the roots of the equation /(^) = 0. 
 
 [ 
 
 if 
 
■ft 
 
 ill 
 
 ■V 
 
 124 POLAR CO-OKDINATES. POINTS^ OF INFLEXION. 
 
 Asymptotic Circles. 
 
 117., In the curve fig. 7.5, r = 
 
 a 
 
 ae 
 
 1^0 1_^' 
 
 and hence as H 
 
 increases towards oo , the extremity of the radius vector is always 
 outside the circle r -- a but approaches indefinitely near it; the 
 circle is therefore called an asymptotic circle. As d approaches 
 — CO the same circle is asymptotic, the curve beiug inside the 
 circle. Similarly /• = a is the asymptotic circle of the curve 
 r (ff" ]) =ati^ (fig. 76). 
 
 Points of Inflexion. 
 
 118. 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 Jj^/dr changes sign. 
 
 Diiferentiatino; — = w^ -t- { — "^ (§ 112) we have 
 ^ p^ \dn/ 
 
 d(f 
 
 2 7 o J 2 (?« d^a 
 dp = Zit du + 
 
 P 
 
 do'' 
 
 2du(^u+.jy 
 
 Also r = 1/m, dr = - du/u'^, 
 
 . dp 
 
 dr 
 
 = V?- ]^ ( ?t + 
 
 / d^d 
 
 ) 
 
 Hence at a point of inflexion 
 
 u -I changes siiiu. 
 
 dM-' 
 
 Examples. 
 
 1. Find the points of inflexion on the curve {i' — a) <?* = r 
 or aw = 1 - « 2 ^fig 76). Ans. ( ± V3, fa). 
 
 2. Find the point of inflexion on the curve r (1 — 6*) = a^ 
 (fig. 75). 
 
 Ans. ff is a root of the equation «^ - /^ — 2 = 0, .'. (§ 44) W = 
 l-696rda. = 97°.2, and .'. r = -2-437a. 
 
 .3. Find the points of inflexion on the lituus r^fy = a^ (fig. 70). 
 
 Ans. (^, ±a V2). 
 
POLAR CO-ORDINATES. MULTIPLE POINTS. 
 
 125 
 
 4. In the lemniscate r^ = d^ cos 20 (fig. 12) show that dp/dr 
 = 3 cos 2fi, and hence that the pole is a point of inflexion on each 
 branch. 
 
 5. Show that a curve is concave or convex to the pole accord- 
 inf' as u + d^u/dB- is -|- or - . 
 
 Multiple Points. 
 
 119. The equation of a curve being r =/(«), the direction of 
 the curve at the pole is determined by the values of a which 
 satisfy the equation /(rt) = 0. If this equation have two or more 
 roots there will be a multiple point at the pole. 
 
 Examples. 
 
 1. In the lemniscate r^=a^coB28 (fig. 12) the equation 
 cos 2h = gives 8= ±^7r for the directions of the tangents at the 
 pole. 
 
 Fi 
 
 g- >'■ 
 
 Fig. 80. 
 
 2. Find the tangents to the curve r =: a sin 4*^ (fig. 79) at the 
 
 pole. A7iS. H = d, \,, ^rr, |t. 
 
 These lines are also tangents to the curve r^ = d^ sin 4^ (fio'. 
 80) at the pole. 
 
 3. Find the tangents to the curve r = <i sin Sy at the pole. 
 
 Am. = 0, i^, f;r. 
 
 4. Show that the curve (r - a) f? := r (fig. 7G) has a cusp at 
 
 If 
 
 the 
 
 origm. 
 
12G 
 
 POLAR CO-ORDINATES. CURVATURE. 
 
 i J 
 
 1 .: 
 
 ;1 •" 
 I 
 
 Fig. 81. 
 
 Curvature. 
 
 120. Let PC, p'c (fig. 81) be consecutive nciinals meeting in 
 C the centre of curvature. 
 
 Then oc^ = cp^ + op^ — 2cp.op cos opc 
 
 or, (since oP cos opc = op sin opt - ()t) ' 
 
 oc^ = R^ + r^ - 2r^9. 
 
 In passing to the consecutive point R and oc remain unchanged 
 while r and p vary ; 
 
 ;. =2rt^r-2RtZ/?, 
 
 dp 
 
 (1> 
 
 Hence also (§§ 112, 118) 
 
 R = 
 
 ['^Ol 
 
 ,3 „3 
 
 „ „ / dhi\ ,/ d'^a\ 
 
 "•^C^+dirO "("+<rO 
 
POLAR CO-OKDINATES. CURVATURE. 
 
 12T 
 
 or. 
 
 R = 
 
 [^^kl)T 
 
 dti' 
 
 Examples. 
 
 (2) 
 
 1. Find the radius of curvature at any point of r'" cos ni a = a'' 
 
 ov pr 
 
 <in-\ _ Qin 
 
 Ans. R = _ 
 
 ►.»«+ 1 
 
 (m-l)rt'« (■rn-\)p 
 
 If y'" = a'» cos me, R = 
 
 a'' 
 
 (m+1) r'«-i {m+l) p' 
 
 2. The equation y^ = ^^ ^ a^ represents an involute of a circle, 
 find R. 
 
 3. Ir the logarithmic spiral /• = a^, p = r sin v, and i, is con- 
 stant, hence R = r/sin li; = the polar normal. 
 
 4. Show that the evolute of the logarithmic spiral is an equal 
 logarithmic spiral. ' 
 
 foe is a radius vector and pc a tangent to the evolute, and 
 in this case the angle ocp = v, a constant.] 
 
 5. Prove that in any curve 
 
 dr^ 
 
 R = 
 
 KddJ di? 
 
 [We have w = 1/r, c^m = - dr/r'', dSi = - {r'ah- - 2r dr"^) / r\ 
 to substitute in (2)]. 
 
 6. In the spiral r = ae (fig. 68), r = {(i^ + r'^)i/(2d^ ■\-r'^), 
 
 7. In the spiral re = a (fig. 69), R = r (a^ + r^)^/a\ 
 
 8. If a curve touch the initial line at tLe pole, prove that R = 
 the value of -kr/e at that point; and hence show that the radius 
 of curvature of the curve fig. 75 at the pole is half the radius 
 of the circle in the figure. 
 
 1™ 
 
r" 
 
 % 
 
 128 POLAR CO-OllDINATES, AREAS, &C. 
 
 9. Find r for the curve r = a sin nd at the pole. Arts. \na, 
 
 10. Prove that the intercept of the circle of curvature on the 
 radius vector of any curve = 2^o dr/dp. 
 
 In the curves r'" cos mi) = a'" and r'" = a^ cos m'i show that 
 these chords = - 2r/(ni— 1) and 2r/(m-h 1), respectively. 
 
 Areas, &c. 
 
 121. (1) If. fig. 06 the elementary area OPQ = qpr -f- PQR 
 = ^ OR.RP + ^ PR.RQ = ^ r^ cZ^ + ^ rd.Ldr. 
 
 The second term is infinitely small compared with the first, 
 and hence the area comprised between the curve and two radii 
 vectores is 
 
 \\ v-'dB. 
 
 J a 
 
 (2) The length s = Jds = ^^7diFVd^\ 
 
 taken between assigned limits. 
 
 (3) The area of the surface formed by the revolution of the 
 curve about the aiitial line is 
 
 r sin fl ds. 
 
 (4) The area bounded by two radii vectores and two given 
 curves i\ =/i(^0 ^"^^ ''i ^fsiP) ^^ 
 
 lWr,'-r,')d^ or \[^ (r.^ ^-r') dO 
 J a J a 
 
 according as the curves lie on the same side or on opposite sides 
 of the pole. 
 
 Examples. 
 
 1. The cardioid r ^. a cos,^6 (fig. 73). 
 (1 ) The area = ^a _ cos* ^6 d'i = f n-a'. 
 
POLAR CO-ORDINATES. AREAS, ETC. 
 
 129 
 
 !S 
 
 (2) The length-element ds = 'Jr^db^ + dr^ = a cos ^B dii, which 
 does not change sign while 6 increases from - tt to -, hence the 
 whole leniith of the curve is 
 
 a ' cos^/7 di) = 4a*. 
 
 J -rr 
 
 (3) The surface of revolution about the initial line 
 
 = 2tt %• sin 6 cZs = 2jr a cos^ |/^.sin n. a cos ^l) di) 
 Jo Jo 
 
 = 2,Ta^ [" cos^^^y sin^ii dii = |Ta^ 
 Jo "* . 
 
 2. Tiie spiral of Archimedes r = til, (fig. 68). 
 
 (1) Let it be required to find the area included between the 
 71 th. and (n f 1) th. spires. On the former 
 
 r = aV2(n-l) - + i\^ 
 and on the latter 
 
 r = a r27i;r + ^n ; 
 
 hence the area between them 
 
 and is .'. proportional to w. 
 
 (2) Show that the area of the first spire (j) varying from 
 to 2t) = 8-V/6. 
 
 (3) The length of the curve from the pole to r = ?•, is 
 
 • A change in the sign of the length-element indicates a cuisp, 
 which oceurH in this case when ^ = 7r. As ^increases the area element 
 Jr^ do can change sign only with r^, i.e., when r becomes imaginary. 
 Hence if we had integrated between the liniits and 27r we siiould 
 have obtained Ofor the length, whereas the area would have been the 
 same as above. 
 
 9 
 

 I 1 
 
 i 
 
 130 POLAR CO-ORDINATES. AREAS, ETC. 
 
 - r^ 's/^FT? dr (see § 88, Ex. 3). 
 a Jo 
 
 This is easily shown to be the same as the length of the 
 parabola if - 2ax from the vertex to y = r^. 
 
 3 The lemniscate t^ = a^ cos 2d (fig. 12). 
 
 (1) The area = a?. 
 
 (2) Show that rds = <i^ do. 
 
 (3) The surface of revolution about the axis = 27ra* (2 — V2). 
 
 (4) The surface of revolution about a tangent at the centre 
 
 = 47^(t^ 
 
 [This tangent being taken as initial line, the equation becomes 
 r' = a" sin 2fl.] 
 
 4. Prove that the length of any curve = _—____, 
 
 and that the area = i » f?s = i ^ . 
 
 5. To find the length and area of the logarithmic spiral r = af^ 
 (fig. 71). 
 
 (1) Let i/; be the constant angle between the radius vector 
 and the tangent. Then ds = dr/ooa ^i, whence 
 
 ^2 dr 
 
 Jr, 
 
 r^-r^ 
 
 cos i/' COS ip 
 
 where r^ and r^ are the radii vectores of the extremities of the 
 arc. 
 
 (2) For the area, ^ pds = ^ r sin ij) dr/cos i/> 
 
 .'. area = J tan i/* rdr = ^ (r/ - r^^) tan ^p. 
 J r, 
 
 6. The whole length of the spiral r = e-^ from ^ = to « = x is V2. 
 
 7. In the curve r^ = a} sin 4:8 (fig. 80) show that the area of 
 each loop = \a}. 
 
POLAR CO-OllDINATEa. AKEAS, ETC. 
 
 131 
 
 f the 
 
 8. Show that the area of each loop of the curve r = a sin 4« 
 (fig. 79) is hall that of the circumscribed circular sector (centre 
 the pole). 
 
 9. The polar equation of the cissoid (fig. 27) is r cos h •= <i sixrti, 
 that of its asymptote is r cos n = d, that of the circle of diameter 
 a is r = <i C08<^; show that the area between the cissoid and its 
 asymptote = f-'«^ iind that the area between the cissoid and the 
 circle = (^n-— 1) <i'. 
 
 10. Find the area of a sector of the rectangular hyperbola 
 r' cos 26 = a^ (fig. 82) between /^ = and = a. 
 
 Ans. ^(i' log tan (^- + a) 
 
 11. Find the area of a sector of any hyperbola between « = 
 and b = ", the centre being the pole and the transverse axis the 
 initial line (see Ex. 18, p. 94). 
 
 A 1 / 1^ /^^ + '* tan a\ 
 \o — a tan a/ 
 
 12. Find the area of an elliptic sector between 0=0 and 6 - a, 
 the centre being the pole and tlie major axis the initial line. 
 
 Ans. h (lb tan-i ('- tan a\ 
 \b J' 
 
 13. Show that the area of the limayon r = /a (1 + c cos ti) is 
 
 14. The chord which is drawn through the pole so as to cut 
 off from a given curve a segment of maximum or minimum area 
 is bisected by the pole. 
 
 For d (area) = ^r,"^ dt) - ^r./ dn = 0, .-. r^ = r^. 
 
 112. It is in general impossible to obtain the area exactly 
 unless one co-ordinate 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 w, the tangent of the angle made witli the axis of x by a line 
 drawn through the origin to meet tht^ curve. We can sometimes 
 obtain the area by taking m as the variable. 
 
C~-ram 
 
 : i 
 
 \ i 
 
 ... s 
 
 ii 
 
 I 
 
 132 POLAR CO-OKDINATES, AREAS, ETC. 
 
 If m = tan H, dm = sec^/y dti, .*. ^r* do = j^r'^ cos*^ dti = ^x'^ dm 
 
 :. the area = J x* dm. 
 The area included between two curves will be 
 
 ifCx/ ± X,'') dm. 
 
 Examples. 
 
 1. The ellipse ax^ + hxy + cy^ = k. 
 
 Substituting mx for y, we have x'"* = />;/(« + Z;m + cm^). 
 Also, as the origin is the centre, half the area will be found 
 by varying m from — oo to + cc ; hence the whole area 
 
 k dm 2- k 
 
 -i: 
 
 (See p. 90, Ex.3). 
 
 a + Lm + cm' ^4ac - b'^ 
 
 2. (1) The f..lium x' + y^ = Saxy (fig. 13). 
 Here x = 3am/ (1 + m^), .'. area of the loop 
 
 1 f* 9a'' m'^ dm o « 2^^.^ 
 = h 71--— ^ = l« =fOBAC. 
 
 Jo {l + nr)^ 
 
 (2) On the asymptote x + y ■{- a = 0, m = — «/(l + m) ; hence 
 the area in the second and fourth quadrants between the curve 
 and the asymptote \ 
 
 = 1 r_^^^ ''"''''' 1 dm = a\ 
 
 ^J_x \-{\+mf {\+myA 
 
 Adding ^a", the area of the triangle ODE, we have the whole 
 area between the curve and the asymptote = |a^ = the area of 
 the loop. 
 
 3. Find the area of the closed part of the curve a^y* (y — x) 
 + x* = 0. Ans. J^Jrt^ 
 
 Exercises. 
 
 1. Find the area of a loop of the follwing curves : — 
 
 (1) ay^-Sax^y = x\ fig. 21. Ans. |f V3 a^ 
 
 1 
 
POLAR CO-ORDINATES. AREAS, ETC. 
 
 (2) a/--«xy=x^^fi^^ 22. 
 
 (3) x' + i/ = ^a''xy. 
 
 (4) ax^^y'^axy, 
 
 2. Prove that the length of the curve rl = «i cos ly is 
 
 133 
 
 15 U" • 
 
 «- 
 
 «(/t-2)... 
 — — . Jaa 
 
 (/i-1) 0*.-3)... 
 where r, is 1 or \z according as n is even or odd. 
 
 3. In the spiral r,i = « (fig. 69) show that the area bounded 
 by two radii vectores and the curve is 
 
 4. Find the area of a loop of the curve r = a sin «/?. 
 
 Ans. ^za^/n. 
 
 5. Find the area enclosed by the curves 
 
 (1) /^ = a^ cos'H + y sinV Ans. \- (a^ + i^). 
 
 (2) r^ = a? cos^^y - i^ sin^/^ ah + (a^ - i^) tan-i («/6). 
 
 6. The area of the common parabola r cos^ h\ = a from h = 
 to (y = a is 
 
 a' (tafi^a + J tan''4>«). 
 
 7. Show that the area of the conchoid r = a sec h + h from 
 6* = to « = a is 
 
 ailogtan (jT + ^„)+i62,,. 
 
 |i' 
 
CHAPTER XXVIIl. 
 ASSOCIATED CURVKS. 
 
 Inverse Curves, 
 
 ^ 
 
 -'■ ?j 
 ■ 4 i 
 
 123. If ou the radius vector r of a curve, a distance i^ be 
 measured from the pole so that rr' = /c^, where k is constant, the 
 locus of the extremity of r' is called an inverse of the <i;iven 
 curve. The radius vector of the inverse curve is proportional 
 to the reciprocal of that of the given curve, and its polar equa- 
 tion may be found from that of the '^Wnn curve by substituting 
 k^/r for r. Thus (see § 113) the inverse of the equilateral 
 hyperbola with reference to the centre is a lenmiscate (fig. 82), 
 that of a conic section with reference to a focus is a lima9on of 
 
 Fig. 82. 
 
 the form figs. 72, 73, or 74, according as e is < , = , or > 1, i.e., 
 according as the conic is an ellipse, parabola or hyperbola. 
 

 INVERSE CURVES. 
 
 1,S5 
 
 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. Tiie 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. 82) are consecutive radii vectores 
 meeting one curve in P, p', and the other in q, q', the rectangles 
 OP.OQ, op'.oq' are equal, ,". a circle may be described through 
 P, Q, p', q', .*. q'p'p + q'qp = two right angles; hence when p' 
 moves down to P the angle oqu becomes the supplement of OPD. 
 
 Otherwise thus : r = Jc^/r', .'. logr = log Ic^ — log r', 
 
 /. _ dr/dti = dr'/dn, or cot rj) = - cot \d\ :. t// = tt - ij>. 
 
 T r 
 
 3. Hence show that the inverse of a logarithmic spiral with 
 reference to its pole is a logarithmic spiral. 
 
 4. Show that the curves figs. G8 and 69, 77 and 79, and 78 
 and 80 are the inverse of each other. 
 
 Pedal Curves. 
 
 124. The locus of the foot of the perpendicular from a given 
 point on the tangent to a given curve is called the pedal of the 
 curve with reference to the point. For example, the pedal of a 
 parabola with reference to its focus is a straight line (the tangent 
 at th: vertex) ; the pedal of an ellipse or hyperbola with reference 
 to a focus is a circle (the auxiliary circle). 
 
 In fig. 81, T and t' are consecutive points on the pedal, cor- 
 responding to P and p' on the given curve ; and tt' produced is 
 ultimately the tangent to the pedal. If the angle aot = o and 
 OT = p, then (0, p) are the polar co-ordinates of the point on the 
 pedal corresponding to (6), r) on the given curve. If then we 
 can express p in terms of r, and ^ in terms of ^, the polar equa- 
 tion of the pedal will be easily obtained from that of the given 
 curve. 
 
136 
 
 PEDAL CURVES. 
 
 1 
 
 t 
 
 t? 
 
 Examples. 
 
 1. TIk! podal of an equilateral hyperbola is a lemniscate (fig. 
 82). 
 
 ¥or pr = a^ (Ex. G, p. 121), and <P = e (Ex. 2, p. 121), hence 
 substituting in r^ coa 2li = u^ we have <i^ cos 2(/) = p^, or writi- " 
 and r for ^ and jj, ?•' = rt'' co8 2/y the equation of the leuiuisca 
 
 In a isimilar way it may be shown that the pedal of any curve 
 of the form r'" cos mf) = a'" is r^cos nil = a'^ where ii = wi/(l — wi), 
 and that the pedal of r"* = a'" cos mi) is r" = a^ cos riH, whore 
 >t E w/(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. 81 let op produccid meet t'p' in q. Then o, T, t', Q, 
 are on the circumference of a circle since tlie an<»;les at t, t' arc 
 right angles, .*. ot't .- oq't, and these are ultimati'ly the an*^ 
 referred to in the enun.iiation, (!n fig. 82, opt = otv). 
 
 3. Prove that the pedal of a circle with reference to any point 
 is a limagon of the form figs. 72, 73, or 74, according as the point 
 is inside, on, or outside the circumference. (These figures are 
 the pedals with reference to o of the circles with centres B and 
 radii ba). 
 
 4. Show that the pedal of a logarithmic spiral witli reference 
 to its pole is also a logarithmic spiral. 
 
 5. Find the pedal of a parabola with reference to its vertex. 
 A71S. r cos H = a sin'^/y, the polar equation of the cissoid, fig. 27, 
 (The directrix of the parabola is tie 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 i/; is proportional to 
 the radius vector). 
 
 7. Find the pedal of the ellipse with reference to the centre. 
 
 Ans. r^ = c? Goa^O -1- b^ sin^^. 
 
-Tf- 
 
 POLAR RECIPROCALS. ROULETTES. 
 
 Polar Beoiprooals. 
 
 137 
 
 125. The inverse of the pedal of a curve (both podal and in- 
 verse hfina! 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 o 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 re- 
 ference to its centre and with reference to its focus. 
 
 3. Show that the polar reciprocal of a ogarithniie spiral with 
 reference to its pole is another logarithmic spiral. 
 
 Roulette3. 
 
 126. When one curve rolls on another, tlie 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. 
 
 127. 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. 
 
 128. 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 = a^ — 6 sin ^, y = a — h cos ft, 
 
 where a is the radius of the circle and h the distance of the 
 tracing point from the centre (axes as in fig. 29). 
 
 i 
 
i . 
 
 ■1 
 
 138 
 
 EPICYCLOIDS AND HYPOCYCLOIDS. 
 
 129. 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. Corresponding to these 
 curves we have epitrochoids and hypotrochoids described by 
 points not in the circumference. 
 
 Fig. 83. 
 
 For the co-ordinates of any point P (fig. 83) on the epicycloid 
 we have 
 
 X - OB = OC - PQ = OD cos ^ - PD COS (O + .'/)' 
 
 Hence, since arc PE = hn' = ae = aft, 
 
 X = (a + b) C0B6-h cos /__— J 6. 
 
 Similarly, y = (« + b) sin ^ - 6 sin ^- j ft. 
 
 The X and y of a point on the hypocycloid may be obtained 
 in a similar way (or from the epicycloid by changing the sign of 
 6), and are 
 
I 
 
 EPICYCLOIDS AND HYPOCYCLOIDS. 139 
 
 X = {a- h) cos « + Z> COS /'^5-Il-^ ti. 
 
 y = (^a-h) sin 6-b sin f—-—\o. 
 
 The equations of the epitrochoid and hypotrochoid arc 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 the case of any e .'cycloid 
 
 ds = 2 (a + b) sin % d6 = 2 (a + b)~ sin _ dt/, 
 ^ ^ 2b ^ II 2 
 
 and hence that the length of the curve from cusp to cusp i» 
 8 {ci + b) b/a. 
 
 2. Show that the epicycloid is a cardioid when b - a. 
 
 o o o 
 
 3. Show that the hypocycloid is the curve x^ 4- y 5 = «3 (fig. 11) 
 when b = \a. 
 
 4. When a circle rolls inside another circle of double its 
 diameter, 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 = 
 
 4iUi + b) b . ae 4(a + b) b . h' 2 (a + b) , , „ 
 — i^ — ■ — i- — sin — = — i — ■ — I — sm - =— A 1 X chord ep 
 
 a + 26 26 a + 26 2 a + 26 
 
 and is therefore proportional to the chord ep. 
 
 For, if the tangent at P make an angle with ox, ^ = W + ^ti^ 
 and R = ds/d0 (§ 51). 
 
 6. Show in a similar way that in the cycloid 
 
 ac = a (fl — sin fl), y = a (1 - cos «), fig. 46, 
 ds =i PB do, 9 = ^71 - y, and hence that r = 2pb. 
 
 If 
 
 I 
 
! 
 
 CHAPTER XXIX. 
 
 ENVELOPES. 
 
 130. Lct/(a;, y, «) - represent the equation of a curve (i.e. 
 of any plane <i;eometrical locus, includinira straight line), a being 
 a quantity involved in the equation but independent of x and 1/ 
 for its value. From this equation we m:iy obtain a series of 
 curves by giving diiferent values to a. Let it be supposed that 
 n changes by the successive addition of infinitesimal increments, 
 then the locus of the intersection of consecutive curves of the 
 series thus formed is called the envelope of the curves. 
 
 131. The envelope touches each of this series of intersecting 
 curvrs. For supposing A, b and c to be any three consecutive 
 curves of the system (see fig. 84), B and the envelope have two 
 consecutive points in common, viz., the intersections of A and B 
 and of B and c ; they therefore have a common tangent and hence 
 touch each other. 
 
 132. Equation of the Envelope. The points of inteisection of 
 the consecutive curves /(x, y, n) = and/(a;, y, « + dJ) = lie 
 on the curve 
 
 /(x, y, a + da) -f{x, y, a) ^ Q^ ^.^^^ ^^ df^x , y, a.) ^ ^^ 
 
 da da 
 
 since this equation is satisfied by any values of x and y which 
 make /(r«, //, «) and /(re, y, a + da) separately equal to zero. 
 Again, all the points of intersection of /(x, y/, «) = and 
 
 ■ •^^' ' "^' "'^ = lie on the curve whose equation is formed by 
 da 
 
 eliminating n from these equations ; this will not involve a and 
 
 will therefore be the equation of the required locus. 
 
 
ENVELOPES. 
 
 141 
 
 The quantity a is called a variable parameter. 
 
 The given oquatiou may contain two or more variable para- 
 meters, subject however to other relations connectini; them, 
 whereby all except one may be eliminated from the given equa- 
 tion. 
 
 Examples. 
 
 Fig. 84. 
 
 1. Two sides of a right angled triangle (fig. 8-4) are given in: 
 position and the area is constant, find the envelope of the hypo- 
 tenuse. 
 
 We have to find the envelope of the line ^ + ^ = 1 where a3 
 ^ constant, = /c^ say. Then /? = /cVa and the equation becomes 
 
 Differentiating with regard to a. 
 
 (I)- 
 
 •- -2 + 'L = ^» whence n = k y ? . 
 
 k 
 
 y 
 
 Substituting in (1) wc get for the envelope x\j = \h^^ a rec- 
 tangular hyperbola. 
 
 ? «) 
 
m 
 
 142 
 
 ENVELOPES. 
 
 2. Show that all ellipses having the same centre and area, 
 a,nd their axes in the same directions, touch a pair of hyperbolas 
 of which the axes are asymptotes. 
 
 3. Particles are projected in the same vertical plane with the 
 same velocity v, but at diflfercnt elevations j show that their paths 
 all touch the parabola 
 
 ,2v 
 
 X' = - 
 
 2«' 
 
 (-a 
 
 9 V 2g. 
 
 of which the point of projection is the focus. 
 [In other words find the envelope of 
 
 y = X «an a — gx'^/(2v'^ cos^a)]. 
 
 4. Show that the circles described on the double ordinates of 
 tht parabola y^ = 4ax as diameters touch the equal parabola 
 y^ = 4a(a3 + a). 
 
 5. Show that the circles described on the double ordinates of 
 the ellipse _ + -^^ = 1 as diameters are enveloped by the ellipse 
 
 6. Find the envelope of ua^ + vn + io = where u, v, w, are 
 any functions of x and y. Ans. v^ = 4iiw. 
 
 7. Find the envelope of itd^ -{-va + to = 0. 
 
 ins. 27im' + 4y' = 0. 
 
 8. Find the envelopes of 
 
 u cos'^t) + V Hin^d = v) (1) 
 
 u sec'"^ - V tan"»^ = w (2) 
 
 Ans. (1) u^ + v^ = w^^ 2 
 
 >■ where n - — 
 
 (2) «" - w" = w" ( ■" 2 - m 
 
 Many examples may be reduced to these by observing that a 
 
 condition of the form (-J + (r) = 1 is equivalent to the two 
 
ENVELOPES. 
 
 143 
 
 relations a = a cos"";^, (i = b sin'"^, while ( ~ ) - \j) = 1 is equi- 
 
 2 2 
 
 valent to n = a sec~<^, ,3 = 6 tan't^. 
 
 9. Find the envelope of a line which moves in such a way that 
 the sum of its intercepts on the axes is constant. 
 
 We have -+^ = 1, and a + ii=k. We may substitute the 
 
 value ot ,? and then differentiate, or we may proceed as follows : 
 Let (I = k cos*^, [i = k sin'w; the line becomes .x(cos^)-2 + 
 ^(sin «)-2 = 7c, hence (Ex. 8) the envelope is x^ + i/^ = ki, a 
 parabola touching the axes. 
 
 10. A straight line of given length k moves with its extremities 
 on two rectangular axes, find the envelope of the line. 
 
 Ans. x* + y3 = k^, a four-cusped hypocycloid. 
 
 11. Given in position the axes, of an ellipse and that their suni 
 = 2/c, show that the ellipse touches the curve x^ + y^ = k^. 
 
 /x\ '^ / w \ "* 
 
 12. From any point in J - 1 ± (j) =1 perpendiculars are 
 
 drawn to meet the axes in A and b, find the envelope of ab. 
 
 Ans. ( ~\ ± (r) = 1, wherew = — ^i_. 
 \a/ \o/ m+l 
 
 13. To the ellipse or hyperbola -;^±^^ = 1 pairs of tangents 
 
 are drawn from points in the ellipse - — 1- - = 1, show that the 
 chords of contact touch the ellipse 
 
 14, When the tangents are drawn from points in the hyperbola 
 
 —. — ^ = 1, shew that the chords of contact touch the hyperbola 
 or b' 
 
 ; 
 
 ■Wiiiiiiiilliiii 
 
144 
 
 ENVELOPES. 
 
 1 
 
 
 15. The evolute of a curve may be considered to be the en- 
 velope of its normals; find in this way the evolute of an ellipse. 
 
 The normal at (n, fi) is 
 
 /^ 
 
 or, writing a cos 6 for «, and h sin d for ,3, 
 
 X. a (gos tiy^ —i/.b {sin 8)-^ = a^ — b'^ 
 the envelope of which is (Ex. 8) 
 
 («x)3 + (%)*= (a«- 60^ 
 which is therefore the evolute (cf. § 57). 
 
 16. Show in a similar way tliat the evolute of the hyperbola is 
 
 i: 
 
 » 
 
CHAPTER XXX. 
 €URVK T'UCING. 
 
 134. lu order to trace a curve accurately from its e((uatiou we 
 must be abb to express one of the co-ordinates in terms of the 
 other, or both in terms of a third variable. When the rectan- 
 gular equation contains terms of two degrees only, we may sub- 
 stitute mx for y and solve for x, ;md in ihis way obtain both x 
 and ij in terms of m. In doing this we are solving for the points 
 of intersection of the curve and the line jj = mx drawn through 
 the origin ; e.g., in the curve rc' + y = Saxij (fig. 13) let ij = mx, 
 then X = Saw / (1 ^- w^), and hence both x and ij are known in 
 terms of w, where m is the tangent of the angle between the 
 axis of X and a line drawn from the origin to the point (x, y) on 
 the curve. By giving a series of values (e.g., '1, *2, "3, etc.) to 
 m we can get any number of points on tlie curve. 
 
 135. The following suggestions and remarks may be found 
 useful in curve tracing, in order to shorten or check the work. 
 
 (1) Examine the equation for symmetry. Wiien the equation 
 remains unchanged if -y is substituted for y the curve is sym- 
 metrical with reference to the line v/ - (the axis of x), for if 
 the co-ordinates (r/, 6) satisfy the equation, (a, — h) will also 
 satisfy it. This will always be the case if the equation contains 
 only even powers of ^. Similarly the curve is symmetrical with 
 reference to the line x = (the axis of .y) 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 
 — ?/ at the same time, every line drawn through the origin and 
 terminated by the curve is bisected by the origin ; for if (a, 6) 
 satisfy the equation, ( - «, - Z>) 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' (fig. 3), y - sin x, 
 etc. 
 
 10 
 
146 
 
 CURVE TKACING. 
 
 m. 
 
 The curve is symmetrical witli reference to the line y = x if the 
 equation is unaltered when x 's changed into y and ?/ into x, e.g., 
 x'' + }f = 3axi/ (fig. 13) ; and it is symmetrical with reference to 
 the line ^y = - x if we can change 1/ into — x and x into — // 
 without altering the equation, e.g., in x^ -y^ = 3axy. 
 
 If in polar equations the substitution of - 6 for dots not alter 
 the equation, the curve is symmetrical with reference to the 
 initial line (e.g., in figs. 72, 73, 74, 76); and if we may at the 
 same time change r into — r and into — 11 without altering the 
 equation, the curve is symmetrical with reference to a line through 
 the pole perpendicular to the initial line {e.g., in figs, 68, 61), 82). 
 The pole is a centre wlicn we can change /• into -r without 
 altering the equation (e.g., in figs. 70, 82). 
 
 (2) Find the tangents at the origin (if the origin lie on the 
 curve) and the shapu of the curve near the origin (§§ 130, 137) ; 
 also, if possible, the points of intersection of the curve and 
 the axes, and the directions of the tangents at these points ; the 
 points where the co-ordinates are maxima or minima ; the point 
 of inflexion ; the asymptotes rectilinear or curvilinear, etc. 
 
 (3) No straight line can meet a curve of the nt\i. 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 
 M — 2 points at a finite distance and no line parallel to an 
 asymptots in more than n — 1 points at a finite distance, no line 
 through a double point in more than n — 'l other points, etc. 
 
 (4) There are two branches running to 00 along each asymp- 
 tote. This is readily seen on examining the expansion of y in 
 descending powers of x (§§ 29, 138), 
 
 136. 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. 17, ,y = ± x (1 + 2x)^, and 
 taking first the + sign we have by the Binomial Theorem 
 7/ = X (1 + X ), or y = X + x? 
 
CURVE TRACING. 
 
 147 
 
 The term v? 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 4- and when a; is — ; in 
 fact the curve is, for points near the origin on the branch touch- 
 ing y ~ X, shaped like tiie parabola y =x + x^ Similarly on the 
 other branch // = - x - x^ + ... ; hence this branch lies below the 
 tangent on both sides of the origin. 
 
 Similarly we may show that in fig. 19 the curve near the 
 origin is shaped nearly like tiie parabolas y = x^, y = - x^ 
 
 137. When it is not convenient or possible to express one co- 
 ordinate in terms of the other we may proceed as in the following 
 examples. 
 
 Examples. 
 
 1. In the curve (i^ (y -x) (y +x) -- (y'^ + x^)-, fig, 12, con- 
 sidering first the branch which touches y — x = (§ 24) we 
 divide by a^ (y + x) and write the equation in the form 
 
 (i^ (y + x) 
 
 For points near the origin on the branch in question y is very 
 nearly equal to x, and the fraction in (1) must be a very small 
 quantity ; we shall get an approximation to its value by substi- 
 tuting X for y ; tiiis gives 
 
 y = x- 2xV«^ 
 which shows that the curve lies below the tangent when x is -f- 
 and above it when x is - . h'or the other branch we write the 
 equation in the form 
 
 (y' + x'Y 
 
 (1) 
 
 y = -x- 
 
 (2) 
 
 a\y + x) 
 
 and remembering that y is nearly equal to — x we substitute - x 
 for y in the fraction and get 
 
 y = -x-\- 2xV«^ 
 
 showing that the curve lies above the tangent when x is + and 
 below when x is — . 
 
 ! 
 
 i 
 
 iiMk>- 
 
T 
 
 il 
 
 148 
 
 CUKVE TUACliNG. 
 
 J 
 
 ' 
 
 
 2. In the curve 8axy = x^ + 1/'\ fij;. 13, tlic tangents at the 
 origin are i/ = and x = 0, Writing the equation in the form 
 
 // = 
 
 '6ux 
 
 we obrfervo that on the branch which touches ij = (the axis of 
 x) // is nearly near tlie origin, and substituting this for y in 
 the fraction gives //= xV3't for the approximate form of the 
 curve. For the other branch 
 
 X = 
 
 x' + 1/^ 
 
 and writing for x in the second member we get x = j/^/'^'< 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 
 ai/ {y - \l'6 x) (// + \/3 x) = x"*, fig. 21, near the origin. 
 
 Ans. C)a (_y — V3 .<•) = x", Cm (y + V3 x) = x*, 3a// = - x'^. 
 
 4. Also of ajj^ Q/ - x) (y + x) = x', fig. 22. 
 
 Ans. atf = - x', tn (7/ - x) = x\ 'la (y + x) = - X". 
 
 5. Sliow (by Ex. ^,1, p. 59) that the radii of curvature at the 
 
 origin arc \a and + 24« in tig. 21, and 0, ±2 V2 <.i in fig. 22. 
 (Cf. Exs. 17, p. 60; 18, p. 61). 
 
 138. To obtain the asymptotes of a curve we expand y into a 
 series of descending powers of x, (see § 27). When it is im- 
 possible or difficult to express one of the co-ordinates in terms of 
 the other we may proceed in a manner similar to that of § 137, 
 beginning however wioh the terms of the highest degree (^ 27) 
 instead of those of the lowest. 
 
 Examples. 
 
 1. x^ + if = 3axy, fig. 13. Here x +y is a factor of the terms 
 of the highest degree, and we may write the equation in the form 
 
 Zaxy 
 
 y r= - X + 
 
 X- - xy + y- 
 
 (1) 
 
 1 It 
 
GUi:VE TRACING. 
 
 149 
 
 Now tho infinite branch is in the direction of the line y = — x, 
 and therefore when x is very large, /y is nearly equal to - x ; 
 lience we shall get an approximation to the fraction in (1) by 
 substituting - x for /y ; this gives 
 
 y = — .T — n 
 
 which is the nearest linear approximation to the curve, and is 
 therefore the equation of the asymptote. W^'l'.iug —x — a for /y 
 in the fraction will give a second approximation, viz., 
 
 a' 
 
 from which it appears that the curve lies above the asymptote 
 whether x is + or - • 
 
 2. Find the asymptotes of the following curves : 
 
 (1) x' (l/-x) = a (y^ + X*). Ans. y = x-h 2(i. 
 
 (2) x/y- (y - x) - //' - 2x'y + x'. x = 1, /y = 1, // = x - 1. 
 
 (3) (x + 2/y) (x - yy = (W (x + y) . x + 2y = {),x-y = ± 2n, 
 
 Exercises. 
 
 1. Trace the following curves: * 
 
 (1) y=.x (x^- 1), (2) y (x^ - 1) = x, (3) y (1 -f- .x«) = x, 
 (4 ) y-' = x'' (x + 1), (5) / = x' (x - 1), (6) y' = x« (x - 1), 
 
 (7) X' - / = 3(fx?y, (8) x^-hy* = a'^xy, (9) x;' + y" = 2a^xy, 
 (10) x' + f = ax\ (11) x^4-/'= a\ (12) x (.y-x) = a/, (13) 
 X (y - xy = y\ (14) a-y (x+y) = x\ 
 
 2. Trace the following polar curves : 
 
 (1) r = n iiin2il, (2) r sin 2rt = <i, (3) r = a sin 3w, (4) 
 r sin 38 = a, (5) /' = rr sin 3/^, (6) r = (t tan ^^, (T) r"' =■ a'^d, 
 
 (8) r = a«^(9) rfl'=r/, (10) r(l +^) = afl, (11) r(o'-^ - 1 ) = a(^, 
 (12) n/- = «(^'- 1), (13) r(V+ 1) = afl^ (14) rfi = tanf^ 
 
 • Some of these examples are taken from B^rost's Curve Tracing* 
 to wliicli the student is referred for further information on this sub- 
 ject. 
 
 f 
 
CHAPTER XXXI. 
 
 (EXTRKS OF (>!I{AVITY. ' 
 
 140. In findinjj; the co-ordinates (x, y) of the centre of gravity 
 of a body, we suppose the body to be divided into parts of 
 weights W\, 10.2, •••> '^nd of which the centres of gravity are the 
 points (Xj, //j), (Xg, i/.j), ..., then put tlie sum of the momenis 
 of the weights equal to tlie moment of the sum of the weights 
 phiced at the centre of gravity. Thus supposing gravity per- 
 pendicuhir to the axis of x we have 
 
 ?0j Xi +MJa Xg + ... = (?nj -f- 11'.^ + ...) X 
 
 — Wj Xj + ?"2 Xg + ... 
 
 1 7/;x 
 
 r«j + ?tfg + ... i; w 
 
 the symbol i: being used to imply summation. 
 
 Similarly supposing the body and the axes turned round so 
 that gravity is perpendicular to the axis of ;y, we have 
 
 ?« J + Wg + . . . i; ?« 
 
 These formulae also hold when the points are not in one plane, 
 there being also a third co-ordinate found in the same way. 
 
 141. The division into parts and the limits of the summation 
 in the following cases are the same as if we were about to cal- 
 culate an area, volume, or length (§§ 66, 67, 102, etc.). 
 
 * Also called centroidp, centres of masp, centres of inertia, etc. 
 
CKNTKKS OF GHAVITY. 
 
 151 
 
 142. An Area. To fiiul the c.^. of a thin plati; or hiiuina of 
 the form abdc, tig. 52, wo have the area of the strip I'N = i/Jx) 
 and its e.g. at (x, ^//) ; weight of strip = icj/ dx, where w = the 
 weight per unit area ; moment of weight about o = (h?// dx) x 
 when gravity is perpendicuhir to the axis of x, and = (jtv/ dx) hi/ 
 whin gravity is perpendicular to the axis of ^; hence dividing 
 the sum of the moments by the sum of the weights (§ 140) we 
 have 
 
 f 
 
 ',x 
 
 \x^ dx ^'<'^ 
 
 ,^ - __ , y = ^ ^ » 
 
 tx 
 
 ji/dx " Lda 
 
 wlien w which is assumed to bj tiie same in all parts of the plate 
 is cancelled. 
 
 It will be noticed that the denominator - the area. 
 
 143. A Solid of Revolution about OX. Volume of slice 
 = -i/-dx, weight of slice = lor^i/hlx, w being the weight per unit 
 volume, moment of weight - (lo rrifdx^x \ 
 
 ,x 
 
 jx/, 
 
 Ix 
 
 I/- dx 
 
 U = '». 
 
 r|i 
 
 The denominator = volume 
 
 / TT. 
 
 144. An Arc. Proceeding as above we have for the e.g. of a 
 material line in the form of the curve CD. 
 
 X ds _ // dfi 
 x = ^— , ?/ = •'' 
 
 din ds 
 
 145. A Surface of Revolution about OX. 
 thin curved surface of this form. 
 
 For an indefinitely 
 
 
 r/.s 
 
 » .'/ 
 
 0. 
 
 .'/ 
 
 ds 
 
 HI 
 
152 
 
 CENTHES OE GlfAVlTY. 
 
 U 
 
 I 
 
 
 146. An Area in polar co-ordinates- The weight of the ele- 
 mentary triangle oPQ, fig. 6G, is proportional to its area ^r^ dd, 
 its e.g. is at the distance §7- from the pole ; hence if the initial 
 line be taken as iixis of x and the pole as the origin, the x and i/ 
 of this e.g. are §?• cos 6 and §)• sin (i, respectively. 
 
 h'^ dd . §?• cos II r'' cos t) dn 
 
 ■ X ^ -lL = * •L_ 
 
 do 
 
 and 
 
 lir' 
 
 ['' si 
 
 = f 
 
 sin 6 do 
 
 P 
 
 P 
 
 dH 
 
 147. Pappus's (or Guldin 's; Properties cf the Centre of Gravity. 
 From § 144 we have ly f/.s = // ds, and multijilying both sides 
 by 2., 
 
 J2.y.t/.s= (j./6-).2.,; (1) 
 
 Similarly from § 143, 
 
 jrrf-dx^ (^J^c/x^.2-y. (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 
 oT the revolving curve multiplied by the length of the path of 
 the e.g. of the curve (i.e. of the arc). 
 
 (2) Tlie volume of a solid of revolution is equal to the revel- 
 ing area multiplied by the length of the path of the e.g. of this 
 
 area 
 
 Examples. 
 
 1. The parabolic area oab, fig. 05. .4ns. x = fx^, y = ^y^. 
 Of the solid of revolution round ox, x = ^x^. 
 
CEKTliES OF GRAVITY. 
 
 153 
 
 2. The quadrant of an ellipse. 
 
 . - 4« U 
 
 A718. X =: ^a. 
 
 3. Half of a prolate spheroid 
 
 4. The circle x- -(- 1/^ = 2c^x between x = and x - /t revolves 
 about the axis of x, find the e.g. of the volume of the spherical 
 segment thus formed. 
 
 8a — 3h\ h 
 
 \3a-hJ 4 
 
 For a hemisphere this = |-a. 
 
 Show that for the surface of the segment x = ^h. 
 
 5. A circular arc. 
 Ans. Distance from centre of circle = chord x radius / arc. 
 
 For a quadrant this = 2 V2<'/t, and = 2a/r for a semi-circular 
 arc. 
 
 <). A circular sector. 
 
 Ans. Distance from centre = § chord x radius / arc. 
 
 For a quadrant this = 4 ^2 a/ 3t, and = 4rt/3T for a semi- 
 circle, 
 
 7. A circular seiiuient. 
 
 Ans. Distance from centie = chord' / (12 x area of segment). 
 
 8. Surface of a right circular cone. 
 
 An ft. Distance from vertex = § axis. 
 
 9. Area OMP of the curve a//- = x\ fig. 14. 
 
 Ans, x = h Oil, _// = -^jj MP. 
 
 10. The area between the curve y = sin .x (fig. 56) and the 
 axis of X, from re = to .x = -. Ans. x = ^r, y - ^rr. 
 
 11. The cycloid, fig. 2J). Ans. Distance from base = |rt. 
 
 12. A quadrant of the curve x^ + ?yt = as (fig. 11). 
 
 Ans. X = 25H a / 315 - = ^. 
 Of the arc, x = '4(t - n. 
 
154 
 
 CENTRKS OF GRAVITY. 
 
 13. The area between the curve (x/a)-^ + {y/h)'^ = 1 aud the 
 axes. (See Ex. 9, p. C9). Ans. x = j;a, y = \h, 
 
 14. A quadrant of the whole area of the curve d-y^ =■ x-(a^ - x""*), 
 (fig. 59). Am. X = fjrna, y = \a. 
 
 15. The area between the curve .y' (w* — .<'') = a* and the 
 asymptote x = a. Ana. x = 2a/-, y = 0. 
 
 16. The area between a'\i/ = x' (a - x) and the axis of x. 
 
 Ans. X = ^a, y = -:^-a. 
 
 Of the solid of revolution round ox, x= fa. 
 
 17. The area between tlie eissoid and its asymptote (fig. 27). 
 
 Ans. X = ^a- 
 
 18. The area between the witcli and its asymptote (fig. 39). 
 
 Ans, X = |«. 
 
 19. The area between the straight line // - mx "■nd the para- 
 bola y^ = 4:(ix. A71S. X = 8a/5m*, y = 2a/m. 
 
 20. Finding the e.g. of the frustum OCD (fig. 85) of a right 
 circular jylinder. 
 
 Fig. 85. 
 
CENTRES OF GRAVITY. 155 
 
 Let the angle COD = a, OB = x, BS = 3/, oc = 2a, 
 
 (•2a 
 
 _ 2y.x taua.dx.x 
 
 ^ _ Jo 
 
 Theo X = ?!a 
 
 /•2a 
 •Jo 
 
 /•2« 
 
 x^^J'Zax- 
 Jo 
 
 xV2 
 
 Jo 
 
 ) where ?/ = V2aa; - x^, 
 
 x tan a . dx 
 
 tJO CixJu 
 
 = fa. (See Ex. 5, p. 104). 
 
 ax — 7? dx 
 
 If z = the height of the e.g. we have 
 
 z = ^x tan a = ^a tan a. 
 Hence a; = f oc, z = y^ CD. 
 
 21 Hence show that for a frustum whose iirreatest and least 
 heights are h and h respective!}' 
 
 - 5H + 3/i - 5H2-10H7i + 2U2 
 
 x = .a, z = ,a. 
 
 4(H + /i) l(i(H + /0 
 
CHAPTER XXXII. 
 
 MOMENTS OF INERTIA. 
 
 148. The Moment of Inertia is a quantity which is often re- 
 quired in eonueetion with the motion of a body about an axis. 
 The following is an illustration. 
 
 149. 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. 
 
 Lot the perpendicular distance of a particle of mass m^ from 
 the axis be r^ and let w = the angular velocity of the body about 
 the axis. Then the kinetic energy of the particle = 
 
 A- (mass) X (linear velocity)' = ^m^ (w^'i)^ = i^^ ^'h^'i^' 
 
 and the whole kinetic energy of the body 
 
 - hf (nil r,M- vL^r./ + ...) = W f, 
 
 where i ^ m j i\^ + m.^ r.r + ... 
 
 The quantity i is called tlie moment of inertia of the body 
 with reference to the axis ; lience the following definition : 
 
 The Moment of 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 squar > of its distance from the axis. 
 
 The calculation of this (^aantity must, in general, be performed 
 by integration. Since both factors of the product ?nr^ are essen- 
 tially + , the moment of inertia is always + , and the moment 
 of inertia of a body about any axis is always equal to the 
 arithmetical sum of the moments of inertia of its parts about 
 the same axis. 
 
 I 
 
MOMENTS OF INERTIA. 
 
 157 
 
 150. Prop. The m.i. of a body about any axis = the m.i. 
 about a parallel axis through the centre of gravity + Mh\ where 
 M is the mass of the body and h is the distance between the 
 parallel axes. 
 
 Fig. 80. 
 
 Take at the point P a particle of mass m^. Let a plane 
 through P perpendicular to the axes in question meet the one 
 through the centre of gravity in g' and the parallel one in ii, 
 then g'h = h. Draw a perpendicular from p to g'h and let 
 q'k = x^. Then 
 
 s^' = r,' + Jr-2hx, (Euc. II, 13) 
 .-. m, s,2 = m, r,2 + ?», A^ _ 2h m, a;,. 
 Similarly for particles m^, m^, etc. 
 .'. »n, s^^ -\- nu s/ 4- . . . = (m, v,2 +m,r.^+...)+Jr (m, + nu+ .. ) 
 
 The left hand side = the m.i. about the axis through ii; and 
 of the three terms on the right, the first = the m.i. about the 
 parallel axis through the centre of gravity, the second = M/i", and 
 the third = (§ 140) since the centre of gravity is in the line 
 from which x^,x^, ..., are measured. 
 
 151. The proposition just proved is true for all bodies, but the 
 following applies only to thin plates. 
 
 i 
 
 :■ iK'-: 
 
If 
 
 41 
 
 158 
 
 MOMENTS OF INERTIA. 
 
 ! 
 
 ill 
 
 Fig. 87, 
 
 Prop. Let x'x, y'y be two lines in the plane of a lamina and 
 meeting at right angles in o, and let z'z be a line through o per- 
 pendicular to the plane. Let ij = the m.i. of the lamina about 
 x'x, ig = that about y'y, i = that about z'z; then 
 
 i = ii-Hi,. 
 
 For n^ = x^ + y^-, :, m^ >•,'- = m, x^ -v m^ //,l 
 
 The proposition is therefore true for a particle at P, and hence 
 it is true for all the particles of the lamina. 
 
 152. When the m.i. is put into the form uk^ (m being the 
 mass),/*; is called the radius of gyration ; hence the radius of 
 gyration of a body with reference to an axis is the distance 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 A; = the radius of gyration with reference to an axis passing 
 through the centre of gravity, and h^ that about a parallel axis 
 at a distance A, we have h^ = Ic^ + Ji^, since (§ 150) 
 
 M&i^ = mJc^ -}- m/i-. 
 
 153. 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. 
 
 Bib..- at 
 
 m 
 
MOMENTS OF INERTIA. 
 
 Examples. 
 
 159 
 
 1. To find the m.i. of a rectangular lamina whose sides are 
 a, o, />, b, about an axis bisecting the sides a, a, fio-. 88. 
 
 Fig. 
 
 Fig. SI). 
 
 Divide the rectangle into parallel strips of length b and width 
 dx, and measure x from the axis. The area of a strip = b dx, 
 volume = t.bdx where t = the thickness of the plate, mass of 
 strip zz ti.tbdx, and its m.i. (since every particle of the strip is 
 at a distance x from the a^xis) = ^tbdx.x\ Integrating this 
 between and h^ and doubling we have for the m.i. of the whole 
 rectangle 
 
 2j ^ ^tbx" dx = j\ idba^ = (^utah) ^ . 
 
 The quantity in parentheses is the whole mass (=/z x volume). 
 
 2 
 
 .'. the m.i. = m _, and the radius of gyration = a / \/]2. 
 
 Similarly the m.i. about the axis bisecting the sides b, b is in^. 
 
 12 
 The same results hold good when the breadth of the rectande 
 becomes indefinitely small. Hence the m.i. of a material straight 
 
 line of length a about an axis bisecting it at riditano-les is m — • 
 
 12 
 2. The m.i. of the rectangle about a normal axis through the 
 
 intersection of the two axes of Ex. 1 is (§151) m 
 
 a^ + y 
 "T2~ 
 
if ■! 
 
 100 
 
 MOMENTS OK INEUTIA. 
 
 The same formula is true for auy parallelogram (of which a and 
 h lire adjacent sides) about an axis drawn as in this case through 
 the intersection of the diagonals at right angles to the plane. 
 
 '{. The m,i. of the rectangle about a side b is (§ 150) 
 
 a' 
 m 1- »i 
 
 12 
 
 G) ="':( 
 
 4. The m.i. of the rectangle about a normal axis through one 
 
 aiiiile = III 
 
 o^ + h' 
 
 I 
 
 5. Any triangular lamina (fig. 8!)) about one side BC. Let 
 BC = '«, the perpendicular OA - h. Tliun i)E:BO :: pa:oa, 
 .'.DE/d = (Ji-x)/h. .".DK = (Ji-x) (t/h. 
 
 m.i. 
 
 fi(Ji — x) -.t .dx.'xr = 111 — 
 J k t> 
 
 (5. A circular lamina of radius i' about a normal axis through 
 the centre. 
 
 Consider the annulus between the concentric circles of radii 
 X and X + dx. The area = 2-x.dx and volume = 2-x dx.t, and 
 '.* every particle is at the same distance from the axis, the m.i. 
 of the annulus = i.i.2tnx dx t.x^^ .'. whole m.i. = 
 
 2u-tx^ dx = hu-tr^ = m ~ . 
 
 f 
 
 Jo 
 
 7. A circular lamina about a diameter. 
 
 Let the required m.i. = i. The sum of the moments of inertia 
 about two diameters at right angles to each other = 2i, it also 
 
 >.2 
 
 (by § 151 and Ex. 6) = m , .'.i = wi 
 
 8. A circular lamina (of uniform thickness) about a normal 
 axis through the centre when the density is supposed to vary 
 inversely as the distance from the centre. 
 
 Let // = k/x where k is some constant. Then m.i. 
 
 -=[(^~\.2ntx'dx=-^-tkr\ 
 
 i| 
 
MOMENTS OF INERTIA. 
 
 1(31 
 
 But the mass m 
 
 r /7c 
 
 -\XD'' 
 
 nx dx»t = 27Tktr, 
 
 I 
 
 . . m.i. = m — . 
 3 
 
 9. An ellipse — + 'L = 1 about its minor axis. 
 a 6" 
 
 The m.i. = 4 fi.ydx.t.x^. Substitute y from the equation 
 Jo 
 
 of the curve and let x = a sin 0. The result is m — . 
 
 Similarly about the major axis the m.i. is wi ^. 
 
 4 
 
 10. A sphere about a diameter. 
 
 Consider the sphere to be made up of laminae perpendicular 
 to tbe axis, and take the axis as the axis of ;c. Then m.i. (see 
 Ex. 6) = 
 
 2 /z.777/^(/x.|-, and 2/^ = r^-a;^, /. m.i. =m frl 
 
 Jo -t 
 
 11. A right circular cylinder of radius r about its geometrical 
 axis. 
 
 The cylinder may be considered as made up of circular laminae 
 
 perpendicular to the axis, hence (Ex. 6) the m.i. = m *1. 
 
 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 
 = n . ttt^ . dx, and its m.i. about a diameter in its own plane and 
 
 parallel to the given axis = mass x L, (Rx. 7), .'. its m.i. about 
 
 the given axis = mass / -j-x^X (§150); /. whole m.i. = 
 
 11 
 
1G2 
 
 MOMliNTS OF INKUTIA. 
 
 !. ■ 
 
 (N^ 
 
 Exercises. 
 
 Find the uionunt of inertia of 
 
 1. A riglit angled triangle ABC (c = 90°), 
 
 (1) About the side b. 
 
 (2) About the side a. 
 
 (3) About tlie side c. 
 
 (4) About a noruial axis through c. 
 
 Ans. 
 
 a' 
 m — 
 
 G 
 
 Ann. 
 
 m — 
 G 
 
 Ans. in 
 
 a'b^ 
 
 He' 
 
 Ans. m 
 
 (5) About a normal axis tlirough the centre of gravity, 
 
 Alls, m 
 
 (<)) About a normal axis through the middle point of c. 
 
 G 
 
 18 
 
 c 
 Ans. m — - 
 
 12 
 
 (7) About a normal axis through A. 
 
 Ans. m 
 
 "" G ~ 
 
 a' L 
 
 z /.a 
 
 2. A rectangle (a by b) about a diagonal Ans. m 
 
 " ^ -^ ^ ° G(«^ + Z>^) 
 
 3. Au isosceles triangle about a normal axis throuuh the 
 
 middle point of the base. 
 
 Ans. m 
 
 4alt.^ + base^ 
 24 
 
 4. An isosceles triangle about a normal axis through the vertex, 
 
 12alt.'' + base2 
 
 Ans m 
 
 24 
 
 5. A circular annulus about a normal axis through the centre. 
 
 R'^ + r^ 
 
 Ans. m 
 
 2 
 
MOMENTS OF INERTIA. 
 
 163 
 
 ,i I ~a 
 
 ..2 
 
 To 
 
 the 
 
 • 
 
 .-tex. 
 
 ,2 
 
 itre. 
 
 12" 
 
 6. A circular annulus about a diameter. Aus. m . 
 
 4 
 
 7. A circle about a tangent. Ans. m ^r^. 
 
 8. A circular arc of length .s, radius r, and chord c, about ati 
 axis through itn middle point perpendicular to the plane of" the 
 
 arc. Aus, in . tr'^ l\ - ' V 
 
 9. A circle about a normal axis tiirough a point in the cir- 
 cumference. Aiu. m '^r^. 
 
 10. A parabolic area, fig. G5, about the axis of x. 
 
 11. The same about the axis of /y. 
 
 12. A spherical shell about a diameter. 
 
 Ans. wi •' ' . 
 5 
 
 Ans. m ^X;*. 
 Ans. ni ^r'. 
 
 13. A hollow sphere about a diameter. Ans. m f 
 
 r' - r' 
 
 «r3_^3 
 
 14. A rit^ht circular cone of radius r and altitude li about its 
 'omctrical axis. Ans. m y^r*. 
 
 15. The same, about an axis through the vertex perpendicular 
 
 to the geometrical axis. 
 
 Ans. m 
 
 20 
 
 16. The same about an axis through the centre of gravity per- 
 
 »J 7,2 I 1 92 
 
 pcndicular to the geometrical axis. a.,. ^ o t ^ 
 
 Ans. m 
 
 80 
 
 17. A paraboloid of revolution about its geometrical axis, 
 
 radius 2/1, altitude jc,. Ans. m 'sK, 
 
 3 
 
 18. The sam3, about an axis through the vertex perpendicular 
 
 to the geometrical axis. Ans. m ^' -^L . 
 
 6 
 
 19. A prolate spheroid about its geometrical axis. 
 
 Ans. m ^h^. 
 
 ffl 
 
«: 
 
 i^ 
 
 il 
 
 i 
 
 164 
 
 MOMENTS OF INEllTIA. 
 
 20. An oblate spheroid about its geometrical axis. 
 
 21. A helix about its axis. 
 
 22. Any area abdo (fig. 52) about the axis of x 
 
 Ans. m §«'. 
 Ana. mr^. 
 
 Ana, —. 
 
 
 23. Any solid of revolution about the axis of x. 
 
 A ni 
 Ana. -- . 
 
 2 
 
 \y^dx 
 
 m 
 
 {y"^ dx 
 
 24. The hypocycloid xKy^ = a^ (fig. H) about the axis of x. 
 
 Ans. mj^^a\ 
 
CHAPTEK XXXIII. 
 EXPANSION OF FUNCTIONS. 
 
 154. By the Binomial Theorem or by actual division, 
 
 "~ X ^ i// "p tC "■■ CC "f" • • • 
 
 1 +x 
 
 1 
 1 
 
 l+x' 
 
 1 
 
 = 1 4- a; -h x^ + »* + • • . 
 
 = 1 ~x'^ + x'^-'x'^'+ ... 
 
 , 2= 1 +a:2 + x^ + .V + ... 
 Multiplying by dx and integrating we have 
 
 X^ X^ T^ 
 
 log (1-t-x) =x -- + --• +... 
 ^64 
 
 a;'^ cc'' X* 
 
 ^ ^ 2 3 4 
 
 tan~i x = £c-- + ?l_^4. 
 3 5 7 ^ 
 
 1,../1 + 
 
 
 (1) 
 
 (2) 
 (3) 
 (4) 
 
 (5) 
 (6) 
 (7) 
 (8) 
 
 It should be noticed that (8) may also be obtained by addino- 
 (5) and (6). 
 
m 
 
 16G 
 
 LOGAIUTIIMIC SKIllES. 
 
 It is well known from elementary ^vlgebra that the scries (1), 
 (2), (3), (4) cannot be (supposed to be carried ad uijinifum 
 unless cc is a proper fraction ; (5), (6), (7), (8) are subject to 
 the sam(3 limitation, i.e., they are convergent^- if a; is a proper 
 fraction, otherwise they are divergent. Hence we miuht by (5) 
 find the Napierian logarithm of any number befween 1 and 2, 
 but not of any higher number. 
 
 155. Let // = (1 + x) / (1 - x), then x = (y - 1) / (// -t- 1). 
 Substituting in (8) we have 
 
 ''«-^[(i^)^Kf^y-^^(f^y-] 
 
 (9) 
 
 This series may be used for the calculation of any Napierian 
 logarithm, since (^- — 1) / (/y + 1) is necessarily a proper fraction 
 when i/ is any positive quantity. 
 
 
 Example. 
 
 Let3/ = 2, .•.0/-l)/(y-Hl).i 
 
 .-. log, 2 = 3 [i + ^ ay + i ay +...] = -693147. 
 Similarly log,3 = 2[Hi Q)' + 1^ (s)' + •••] = 1-098G12. 
 Also, log, 4 --. 2 log, 2 = 1.38(3294. 
 
 156. Another series may be derived from (8) thus : put 
 (1 + x) / (1 - x) = (1 -ry) /y, then x = 1 / (1 + 2y) ; hence re- 
 membering that log [(1 4-y) / ?/] = log (1 + I/) —logy, 
 
 log(l.,)=lo,,.2|-^^^.H-4(^).j(^J....] 
 
 (10) 
 
 • A pories is said to be convergent when the sum of the terms 
 approaches a finite limit as their number approaches infinity, other- 
 wise ilie series is divergent. On convero;ency and divergency see 
 works ii Algebra (tf.<7., Hall and Knight's Higher Algebra, Ch. XXI). 
 
TAYLOR'S THEOREM. 
 
 167 
 
 Example. 
 
 If/y = 4,lo,o:,,5 = lo,uv4 + 2[^ + J(i)-^ + i(i^)^H....j = l.G09438. 
 Hence log,, 10 = loo; 5 + log, 2 = 2-302585, and hcnc(3 the mo- 
 dulns* of the common loirarithms (which r-. 1 / log^ 10) is 
 •4342945, The common logarithms may therefore bo found by 
 multiplying the Napierian logarithms by -4342945. 
 
 157. Taylor's Theorem. Lot /(x) bo any function of x, then 
 f(x + h)or/(7i + x) is what /(x) becomes when h + xis sub- 
 stituted for X. If/(.c) = x^, we know that 
 
 
 (1) 
 
 by the Binomial Theorem. There are other functions wliich, 
 like this one, can be expanded into a series of the form 
 
 f(h + X) =-. A + B.X + CX^ + J)x^ -f- EX* + ... (2) 
 
 where the exponents of x are all positive integers and the cocfli- 
 cients A, b, c, etc., are finite quantities independent of x. We 
 shall now explain a general mothod by which t^ese coefficients 
 may be obtained in any case which admits ofsuch an expansion. 
 The successive lerived function.' of (2) are f 
 
 * If 3: = log,,?/, then l-v delinition of a loganthiu, a-= >/. Taking 
 logantliiiis of tliis, the buse being supposed e, we have x lo<o, c = ' <^ v. 
 
 \o^., 1/ ^ 1^^, or the logarithm of// is changed from base e 
 
 X or 
 
 toba-oabyiMultii.lyingby -, whicii is callea the mo.hihis of 
 
 t!.o svstein of base a. 
 
 ] 
 
 log,, a 
 
 If// - e, loL'„e = — , hence the modulus of the comiuon system 
 
 log,, u J 
 
 is also e(inal to logj „ e. 
 t The first is l/Li±f>-/'^^' + ^) <?(/' + •^) f.r,.^,. 
 
168 
 
 maclaurin's theokem. 
 
 
 
 I 
 
 /'(h + x) =B + 2cx + Sdx^ + 4fV + ... 
 
 /'Xh + x)= 2c +6dx 4-12ex^ + ... 
 
 f"'{h + x)= 6d +24EX+... 
 
 (3) 
 (4) 
 (5) 
 
 etc. 
 
 Now let X take the value in (2), (3), (4), etc., .'. f{li) = a, 
 fill) = B,f"{K) = 2c, f"'(Ji) = 6d, etc. Substituting these 
 values of a, b, c, etc., in (2) we get 
 
 fill + X) ^/(h) +fXh) X + 11^ x^ + /^ x^+... (6) 
 
 This is Taylor's Theorem. 
 
 The Binomial Theorem (1) is a particular case of this more 
 general expansion. For if 
 
 /(h) = h\f'(h) = ?i7t"-i,/''(/0 = «(n- 1) h^2^ etc., 
 
 whence, substituting in (6) we obtain (1). 
 
 Since it is immaterial what letters are used we may inter- 
 change h and x in (6), when tiie series takes the form 
 
 f(x + h) =/(x) +f'(x) h + 1^1 h^ + i!^ h^+,.. (7) 
 
 or, writing u for f(x) 
 
 /7,. rl2,. 7,2 J3„ 7,3 
 
 (8) 
 
 "^ ^ dx ^ dx^ l.-l dx^ 1.2.3 
 
 f ... 
 
 158. Maclaurin's Theorem- In the particular case in which h 
 is zero, (6) becomes 
 
 /(X)=/C0)+/'(0)X + Z^X= + :^^X^+... (1) 
 
 This special form of Taylor's Theorem is generally called 
 Maclaurin's Theorem ; it is easily established independently of 
 Taylor's Theorem and in the same manner. It must be remem- 
 bered that /(O), /'(O), ..., are what /(.c), f'(x), ..., become 
 when X has the value 0. 
 
 ff 
 
EXPANSION OF FUNCTIONS. 
 
 169 
 
 Examples. 
 
 1. To expand sin x. 
 f(x) = sin 05, 
 
 f'(x) = cos £C, 
 
 /"(x) =-sinx, 
 
 :. /(O) = sin = 
 '. fXO) =cosO =1 
 .-./"(O) =-sinO = 
 •./'"(0)=-cosO=-l, etc. 
 
 Substituting in (8) wc have 
 
 a;3 x-' 
 
 smx = X + _ 
 
 3! ^5! 
 
 (2) 
 
 which gives the sine of an angle in terms of its measure in radians. 
 The expansion of cos x may be found in the same manner, or 
 by differentiating (2) ; hence 
 
 a;2 a;* 
 
 cos»= !__ + _ 
 
 2! 4! 
 
 (3) 
 
 2. The expansions of e' and a' are particular cases of Maclau- 
 rin's Theorem. For 
 
 /(x) = e\ f'(x) = e\ fix) = e% etc. ; .-. /(O) = c^ = 1, 
 /'(O) = l,/"(0; = 1, etc. 
 
 x^ X? 
 
 r 1 .O ;t/ 
 
 (4) 
 
 S 
 
 A^ X^ A'^ X^ 
 
 imilarly, a' = 1 + ax + -^j- + ^ + . . . (a = log, a) (5) 
 
 159. Formulae derived from the Exponential Series. In 
 
 § 158, (4)^ write xi for x, where i E V^, (observe that i^ = ~-l^ 
 
 x^ X* 
 
 ••'^- ('-21+4!--)^' ("-^-^5!--) 
 
 or. 
 
 e ' = cos X + t sin x 
 
 (1) 
 
 .i*. 
 
f 
 
 : w' 
 
 170 
 
 EXPA^'SION OF FUNCTIONS. 
 Writing - x for x, we get 
 
 c"'' = cos X — I sin X 
 Adding and subtracting, wo liave 
 
 cos X = ^ (e'* + e-'») 
 1 
 
 21 
 
 sin X = -^ (e" - e-") 
 1 
 
 '■' a ■■''I . 
 
 ,'-'.'■/ '' 
 
 whence tanx = - ( -, ,\ = - {- L\ 
 
 (•1) 
 (5) 
 
 These are Euler's Formulae. 
 In (1) write nx for x 
 
 :. cos nx + i sin nx = c"" = (e"j« = (cos x + i sin xy 
 
 :. (cos X + i sin x)" = cos ?jx f i sin ?2X. (6) 
 
 This is Demoivre's Theorem. 
 
 Exercises. 
 
 1. Expand cos x, log (1 + x), tan-i x, by Maclauriu's Theorem. 
 
 2. Show that 
 
 (l)tanx = x-c- + ^^ + ^__,... 
 (Z) secx = 1 H — r+ 1 1- ... 
 
 4! G! 
 
 X/ X* x' 
 
 •2 0,4 „6 17.-^/ 
 
 (3) log sec x = — + — - + — H ^ + ... 
 
 ^ ^ "^ 2 12^45 2.520 
 
 (4) cos='x = l--^ + ---... 
 
 (5) 
 
 „Sill.T 
 
 8 
 
 ^ X* X* x^ 
 
 (6) c' sec X = 1 + X + x'* 4- f x^ + 
 3. Show that (see Ex. 16, p. 30) 
 
EXPANSION OF FUNCTIONS, 
 cosh x=l+-^-j — +... 
 
 lU 
 
 sinh cB = a; + ^ + 'p7 + 
 d! 5! 
 
 and hence that cosh x = cos ix, sinh x = - i sin /x, wliere / = V"^. 
 
 4. As in § 154, or othenvi>e, show that 
 
 2 3 2.4 5 ■ 2.4. G 7 ^ 
 and honce, making x = h, tliat 
 
 V 24 640 71(;8 ^ -•^•i-±iO^^... 
 
 5. If a circular arc (rad'iis a) subtend an angle li at the 
 centre, show that when // is very small 
 
 arc - chord = -^^(tif\ nearly. 
 
 6. If ti be a small angle, show that 
 
 s'lnti = ^cos ti 1 
 
 ,- , nearly. 
 
 tan (I = u K/coi^'ti ) 
 
 ^ From these formulae are derived the rules given in Mathema- 
 tical Tables for finding the sines and tangents of cMjall andes. 
 
 7. The chord of a circular arc is 0, the chord 0;' half the arc 
 is c, show that the length of the arc is 
 
 2c + ^(2(.'-C), very nearly. 
 
 This formula (Huyghens's) will give ^ of the circumference 
 of a circle of lllO feet radius with an error of less than 1^ inch ; 
 it gives 1 of the circumference of the same circle with an error 
 of less than t^V of an inch. 
 
 ! f 
 
 -fc«V . 
 

 
 
 CHAPTER XXXIV. 
 LIMITS OF FUNCTIONS. (SINGULAR FORMS). 
 
 160. For a certain value of the variable a function may assume 
 one of the forms 0/0, oo / oo, x oo, oo ± oo, 0", oo", etc. Such 
 forms are said to be singular or illusory, they are also known as 
 indeterminate forms. 
 
 Obtaining the true value of a function when it assumes one of 
 these forms means findini:; the limit which the value of the func- 
 tion approaches as the change in the variable causes the function 
 to approach the form in question. 
 
 161. The form 0/0. The fraction (x-l)/(a;^- 1) assumes 
 the form 0/0 vrhfin x = \. But 
 
 X - 1 
 
 x-l 
 
 (x - 1) (x^ + a;+l) x^ + 03-1-1 
 
 wliich latter fraction = ^ when a; = 1. Hence as x approaches 
 the value 1 the fraction (x — V)/(x^— 1) approaches the limit J; 
 or briefly, the given fraction = ^ when x = 1. 
 
 In this example the given fraction is compared with another 
 which is equal to it for all values of x. In the following method 
 the fraction is compared with another to wliich in general it is 
 not equal, but from which it differs by an infinitesimal cjuantity 
 when X has a value infinitely near the one which makes the 
 fraction = 0/0. 
 
 Let the fraction be /(x)/f(x), and suppose a to be the value 
 of X which causes the fraction to assume the form 0/0, or that 
 f(a) =0, and F(d) = 0; and that h is any small quantity which 
 is about to vanish. 
 
 Then by §157, (7). 
 
 
LIMITS OF FUNCTIONS. 173 
 
 1.2 1.2 
 
 Hence as h diminishes towards zero (i.e., as x approaches the 
 value a) the given fraction approaches /'(a)/F'(a) as a limit, or 
 
 i\«) F'(a)" 
 If /'(a) and F'(a) are also 0, it may be shown in the same 
 
 way ihat -^ =/!(!^, and so on. 
 F(a) f"(«)' 
 
 Examples. 
 
 • i(^ = ^^' P(x) -^3x^ = 3 ^^^'° ^ = ^• 
 
 . a; -1 1 
 
 • • -T. — zr = - when a; = 1 
 x^-l 3 
 
 The work may be conveniently expressed thus : 
 
 »''-lJi 3a;Ui 3' 
 
 2.^-^1 =!!±n =2^2 
 
 sine Jo cos 05 Jo i 
 
 o /(x) e^-l-log(l + x) ^ 
 'W)= ^ ^4»whena. = 0. 
 
 i 
 i I 
 
 F'(x) 
 
 e'_ 
 
 1 + 
 
 X 
 
 'Ax 
 
 = 5 when X r= 0. 
 
 e^ + 
 
 
 (X) 
 
 = 1 when a; = 0. 
 
 e'-l-log(l+a;) 
 
 X' 
 
 ],= 
 
174 
 
 LIMITS OF FUNCTIONS. 
 
 ■I I 
 
 162. The form 00/ oo. If'/(x)/F(a3) assumes the form co/oo 
 
 whea X = a, the equivaleut fraction 
 
 1 
 
 1 
 
 assumes the 
 
 form 0/0, and to this the method of § 101 will apply. Thus 
 
 1 V\a) 
 
 /(a^ F(a) 
 
 F(a) 
 
 
 I 
 
 L¥(<l)J 
 
 OOJ /\a) 
 
 whence, 
 
 /(«) /'(«) 
 
 p(a) F'(a) 
 
 Hence when a fraction approaches the form c/o / co the limit 
 of its value is found by the same method as when it approaches 
 the form 0/0. After the differentiation some algebraic or other 
 transformation will usually be needed, for the derived fractioQ 
 will also in general be of the form 00 / oc. For example : 
 
 nX 
 
 lo<>' cos 
 
 00 
 
 = — when x= 1. 
 
 f(x) loii'(l — a;) CO 
 
 -Ttan-^ 
 •^ ^ ^ -^ =-whenx = l. 
 
 ¥'{X) 
 
 But (§161) I. 
 
 1_ 
 
 \ — x 
 
 \ -x~ 
 
 9 
 
 cot 
 
 cot 
 
 17X 
 
 ttX 
 
 -1 
 
 - cosec- — 
 
 = 1. 
 
 Hence the given fraction = 1 when x = 1 . 
 163. The form 0.00. 
 
 Examples, 
 
 a 
 
 1. X (1 - e" '^) « 00 . when x = oc. 
 Butx(l-e")-[l-(i-^,^-...)] 
 
 I 
 
LIMITS OF rUNCTIONS. 
 
 o- H = «, when a; = CO. 
 
 2. (1 -x; tan Ii^'= 0. oo when x = i. 
 
 175 
 
 = a-.^ 
 
 1- 
 
 But it = ^ wliidi = wlieu X = 1 
 
 cot::^ 
 
 «> 
 
 
 
 -■. (§ IGl) wo have- 
 
 -^;- = when a; = 1 . 
 
 ~ '^ cosec^ - 
 
 2 '-> 
 
 X TT 
 
 .-. (1-x) tan 1^^? when cc^ 1 
 16i. The form ooioo. 
 
 EXAilPLK. 
 
 But 
 
 ^ I a; lo^'>- x-xi-] A , 
 
 «=-! log x" 7^^7171-^=^, when x.l. 
 
 Hence (§ J61) it = __J^^i^ ,,„,,^ o ,,^^ ^ __ ^ 
 
 1 - - + log X ^^ 
 
 1 
 
 A second differentiation gives ^ which = 1 when x = 1. 
 
 X- X 
 
 .'. A is the value of * ^ ,„i 
 
 S^ri-r^- wlien It takes the form 
 
 00— X. 
 
 165. The forms 0° ^o ia> t? 
 
 ,; , " ' ^ ' -^ • Functions which assume th-so 
 
 w/U or o)/ 00 by first taking logarithms. 
 
 "^-nw«w-. 
 
176 
 
 limits of fungtionb. 
 Examples. 
 
 1. f(x) = x'-'f"'"^ = 0" when ce = 0. 
 
 a 1 logx 00 
 
 log sin X 00 
 
 1 
 
 = a when x as 0. 
 
 But log/(a;) = , — "". — .logx = a . ^"^.'*' = — when x = 0. 
 lOiT sin X I 
 
 1 
 
 Differentiating (§1G2), a = a 
 
 cos X 
 sin X 
 
 X 
 
 :. log/(0) = a :. /(O) = t", tlic required result. 
 
 1 
 2. /(x) =xi -"^ = 1* when X = 1. 
 
 .'. log f(x) = log X = ~-s — = y when x = 1. 
 
 ^•^^ ^ 1-x ° 1-x U 
 
 log/(l) = 
 
 X 
 
 1 
 
 Ji 
 
 = -1, ;. /(l) = i 
 e 
 
 1 
 
 .'. x^~^ = - when it takes the form 1 ''. 
 e 
 
 Exercises. 
 
 1. When X = show that 
 
 (1) !i^ = l, (2) ^ = 1, (3) ^~5!"l = i, 
 
 (•i) 
 
 a; 
 
 e^ — e"'^ 
 log(l+a!) 
 
 X 
 
 = 2, (5) 
 
 X 
 
 « - sm X 
 
 tan X — sin X 1 
 
 1,(6) 
 
 sin'^x 
 
 
 (7) JLtan!i:^ = -, (8) -1, "L = ^*. (9) xMogx = 0, 
 
 '^ - 4cc 2 8 ^ ^ 2x' 2x tan ttx 6 ^ ^ ^ ' 
 
 ^^^^ cosx-cosmx ^l-m^^ ^^^^ ^. ^ ^^ 
 
 cos « — cos nx 1 —n'^ 
 
 (12) 
 
 a* -6^ 
 
 = 10i 
 
 X 
 
 Q, (13) 
 
 log sec X 1 
 
 aj- 
 
 = -, (14) X log gin X = 
 
LIMITS OF FU^XTIONS. 
 
 177 
 
 'I + X 
 
 (15) x'"^" =c", (IG) ^lll^ = w^ ri7) ^^.-'H+a:) , 
 
 sin a; ' ^ ^ ^ -=1> 
 
 (.8) (i)'""-l,(.9) f-i^)'^ 1,(20) ('l^)i_-y7, 
 
 iu^aiux cot rc + iou' .t: 
 
 -'O.N tan .X- sin x 1 i 
 
 <-■*> S. 5,(25) (I+x)- = c. 
 
 2. Wiicn X = 1 sliow that 
 
 (4) ( 1 - re) log (1 - X) = 0, (5) '^Lllzl - lo- / <^ \ 
 
 logx "^ * '• 
 
 (':) 
 
 3. When x = oo show that 
 
 (l)|=cc,(2).'sinl=»,(;j) („l_ ]), ^ ,„„.„_ 
 (4)2-..i..| = <,(5)l^iIi±fL) = 0,(6)(uiy__, 
 
 4. VVlien x = g, show that 
 
 (1 J soc X - tan x = 0, (2) (\c sin x _ j\ sec x = - 1, 
 
 (3) (sinx)8ec2a; _ g-3^ /^\ 1 - sin x + cos x 
 
 «in^+cosx- 1 "^ • 
 
 5. (sinx)t-=l when x = 0, and when X--. 
 
 w 
 
 6. (1 +1^ . .^ when X = CO, and = 1 when x = 0. 
 
 12 
 
"/). 
 
 V] 
 
 '^A 
 
 '# 
 
 
 
 /A 
 
 W 
 
 f 
 
 y 
 
 IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 1.0 
 
 I.I 
 
 1.25 
 
 S 1^ III 2.0 
 
 12.2 
 
 1.4 
 
 1.6 
 
 
 <^ 
 
 ^^'^'^^ 
 
 V 
 
 ."^ 
 
 ^9> 
 
 \ 
 
 
 c^ 
 
178 
 
 LIMITS OF FUNCTIONS. 
 
 7. 
 
 seox 
 sec3x 
 
 = _ 3 when aJ = ^ . 
 
 COS 3x^ 
 
 AVritc the expression in the form — ). 
 
 \ ^ cos X / 
 
 8. }-^- ^ cos3x stnx ^^_3)('_i)=3 ^hen x = ^ . 
 tan 3x cos x sin 3x 2 
 
 9. Show by the expansions of Ch. XXXIII, that 
 (1) sin^^t^!l^^i whenx = 0, 
 
 (2) 
 
 X X 
 
 tan X — sin X 1 
 
 X 
 
 - - when X = 0, 
 2 
 
 (3) - 1 = 1 whenx = 0. 
 
 X 
 
 10 (e'-l)i^xx'x ^ /e--l\ /tan^y = i when x = 0. 
 
 SC* \ X / \ X / 
 
' 
 
 CHAPTER XXXV. 
 
 EQUATIONS OF CURVES BY INTEGRATION. 
 THE CATENARY. 
 
 166. Integration is sooietimes used in finding the equations of 
 curves. 
 
 Examples. 
 
 1. In what curve is the subnormal constant? 
 
 "We have ydy/dx = a, a constant, /. ydy = adx ; hence 
 integrating, Jy^ = ax + 5, where h is Ihe constant of integration ; 
 
 /. y"" = 2nx + 2b 
 the equation of a common parabola. 
 
 2. In what curve is the subtangent consiiant ? 
 
 If ydx/dy = a, dy/y = dx/a, :. logy = (x/a) +b 
 
 or y = e" , the logarithmic curve. 
 
 3, In what curve is the tangent constant ? 
 
 Let (x, y) be the co-ordinates of a point p on the curve 
 (fig, 90), and let the tangent pt = a. 
 
 From the figure dy/dx = - ^//Va'-/, from which the equa- 
 tion may be found by integrating, the result bein"- 
 
n 
 
 180 
 
 THE TRACTRIX. 
 
 [if ! 
 
 rr 
 
 • Fig. 90. 
 
 The curve is ^aWed a tractrix.* 
 
 S'mcQ ydx = - dij ^/c."- — y'\ the clement 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 of x) 
 is the same as that of the circle, viz., tt'/A 
 
 The leniith of the curve from Y to any point whose ordinate 
 is h may be found as follows : 
 
 ds a . f' dy i /«\ 
 
 = --,..s = -a\ _:£ = alog(-). 
 
 ay y •'"3/ ^^^ 
 
 The area of the surface of revolution of the whole curve about 
 the axis of x = 4rra^, and the volume = §7ra^ 
 
 The Catenary. 
 
 167. This is the curve formed by a uniform chain hanging 
 vertically. 
 
 Let A be the lowest point, P any other point. From r draw 
 PB vertically and equal to the length ap or « of the chain, and 
 from B draw a horizontal line to meet the tangent at p in c, and 
 let BC = a. 
 
 • It is tlie patli of a body which is drawn along on a rough liori- 
 zontal piano by a string of length a, the otlierend of wliich is moved 
 along a straight line ox ; wlience tl)e name of the curve. 
 
 *► 
 
< 
 
 THE CATENARY. 
 
 181 
 
 II X 
 
 168. Mechanics of the figare. The portion ap of the chain is 
 in equilibrium under the action of thr-e forces, viz., the hori- 
 zontal tension at a, the tension at p in the direction of the 
 tangent,, and the weight, which is vertical. Hence pbo is a 
 triangle of forces^ 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. 
 
 H 
 
 169. Geometry of the figure. Draw from a a vertical line and 
 take AO = rt ; take o as the origin, oa as axis of i/ and a hori- 
 zontal line ox as axis of x. Then osi = x, MP = i/. By this 
 choice of axes the constants of integration in the first and third 
 of the following integrations will = 0. 
 
 Since CB, BP, and cp are a, s, and ^'^iF+?, respectively, we 
 have 
 
 di/ _ s ,^. dx 
 dx~a ^ ^' di" 
 
 a 
 
 ^d' + &' 
 
 (2), 
 
 di) s 
 
 (3), 
 
182 
 
 THE CATENARY. 
 
 From (3) Ji/ = s ds/^a^ + s\ /. y = V«' + «' = CP (4) 
 
 .*. the tension at any point P is equal to the weight of a length 
 y of the chain. 
 
 From (2), dx = a ds/^/oFT?, :. x = a log (s + V«' + «') + c 
 
 where c is the constant of integration. "When x = 0, « is also 0, 
 /, = a log « + c, which determines the coastant. 
 
 :.x = a iog/« + V«' + A ^ or s + V^^HT' = «e", 
 
 whence, solving for s, we have 8 = - ( e" - e " ) , (5) 
 
 which gives the length from the lowest point to the point whose 
 abscissa is x. . 
 
 From (1) and (5), dy=-^(f-e'^ 
 
 :,y =^(e" + c'«) 
 
 (6) 
 
 the equation of the curve. 
 
 The Normal, np : mp : : cp : bc, .*. NP/y = y/a, or np = y-/a. 
 The Radius of Curvature. From (1), s/a = tan o, 
 
 .'. ds = a sec'0 d^ = a (^y^/a^) d<p, or ds = (yV«) (^o. 
 
 .'. the radius of curvature = ds/d<p = y'/a, 
 
 or, the radius of curvature is equal (and opposite) to the nor- 
 mal. 
 
 Let D be the foot of the perpendicular from M on PT. It can 
 easily be shown that md = a and dp = s. 
 Hence the locus of D is the involute of the catenary. 
 Also DM, the tangent at D to the involute, is of constant length, 
 .*. the involute is a tractrix (§ 166, Ex. 3). 
 
HI 
 
 CHAPTER XXXVI. 
 
 SOLUTION OF A DIFFERENTIAL EQUATION. 
 CHANGE OF THE INDEPENDENT VARIABLE. 
 
 170. Given ~ = - a^y, to find y in terms of x. 
 
 Multiply both sider by My and integrate, remembering that 
 dx is regarded as a constant (§ 46). 
 
 Z/'' being a constant of integration. ' 
 
 •.^^ = -=^= = i-i^^, /.xJsin-i /«^V+c 
 
 ^h y - -sin (ax -c), (1) 
 
 where c is the new constant of the integration. 
 
 We see from this that y cannot pass beyond the limits ± h/a 
 for any increase however great in x, since a sine cannot be greater 
 than 1 nor less than - 1. Also that whenever ax~c increases 
 by 27r, i.e., whenever x increases by 27r/a, the values of y repeat 
 themselves. This equation is of great use in mechanics. 
 
 The result may be expressed thus : 
 
 ^ = A cos aa; + B sin ax, (2) 
 
 and similarly if A ^ a^y we shall get 
 
 .y = A c°^ + B e 
 
 ■CU 
 
 (3) 
 
: 
 
 
 ^ I 
 
 I'l i 
 
 1S4 CHANGE OF THE INDEPENDENT VARIABLE. 
 
 Change of the Ir dependent Variable. 
 
 171. When an equation or other expression involving differen- 
 tials higher than the first has been obtained on the supposition 
 that some one quantity involved is the independent variable 
 whose differential is constant (§§46, 47), it sometimes happens 
 that we wish to know what the expression would have been if 
 some other quantity had been taken as independent variable. 
 
 The first thing to do in such a case is to find what the given 
 expression would have been if none of the variable quantities had 
 been assumed to have constant differentials. 
 
 Let us for the moment use d'^i/, D'/y, ... for the successive 
 differentials of c/y when dx is supposed constant, tPj/, dfy, ... for 
 the differentials of dy when dx is supposed variable. 
 
 d}i 
 
 Then (§ 47) 
 
 dan 
 
 = /'(«'), 
 
 5^ = /"(x) = ^Z^ = 
 
 \dx) 
 
 dx d^y - dy d'x 
 
 dx^ dx dx dx^ 
 
 Simil 1 ^^'^ ^^ ^^^ ^y ~ ^^y ^^^^ ~ ^^^^ ^^^"^ ^^y ~ ^-1 ^^) 
 dx^ dx^ 
 
 Examples. 
 
 1. Supposing X to be the independent variable in the equation 
 
 ^ '^ ^ dx^ -^ 
 
 'dy 
 
 \dx) 
 
 a^x 
 
 \dx) 
 
 (0 
 
 let it be required to find what the equation would have been '\S y 
 instead of x had been the independent variable. 
 
 Since tlie d^ij is hero taken on the supposition that dx is con- 
 stant, it i;^ what we have called above D^_y, we may .". write 
 
 which is what (1) would have been if there ha^ jeen no variable 
 with constant differential. Now if ?/ be the independent variable , 
 
 \ 
 
CHANGK OF TIIK INDEPENDENT VAllIABLE. 185 
 
 dij is constant and /. d^y = 0, and (2) becomes after reduction 
 
 2\ d'^x dx 2 n 
 dy^ du 
 
 2. What does (1) become when y = sin and o is taken as in- 
 dependent variable ? 
 
 We have y = sin «, dy = cos ^^ do, d^y = - sin « do\ Substitu- 
 ting in (2) and reducing, we get 
 
 which is solved in § 170, and thus ?/. may be found in terms of x. 
 
 Exercises. 
 
 1. Find the equivalent of 
 
 when z instead of x is the independent . .able, being given 
 
 X=l/Z. ^Py 
 
 . MS. — ^ + (1^1/ = 0. 
 
 dz' '' 
 
 2. If « = tan ^, change the independent variable from cc to ^ 
 in the equation 
 
 d?y 2x dy y 
 
 dx' rih^^'*'(l+x^) 
 
 = 0. 
 
 A d^U , n 
 
 Alls. - — + 1/ = 0. 
 ar 
 
 3. Change the independent variable from a; to ^ in 
 
 d^ 
 dx' 
 
 'dy 
 
 '-Ki)-'(:i)^ 
 
 '/ /^¥ 
 
 4. Transform the equation 
 
 0. 
 
 A d'x , 
 
 dy' 
 
 d'y 2 dy , 2 n 
 dx' X dx 
 
if ^ 
 
 I 
 
 180 
 
 CHANGE OF THE INDEPENDENT VARIABLE. 
 
 into another in which z instead of x is the independent variable, 
 
 given X = 1/2. 
 5. Transform 
 
 Ans. 2* —d + ci^ y= 0. 
 dz 
 
 2 (Pi/ d\i r. 
 
 dx^ dx -^ 
 
 into an equation in which (^ is the independent variable, beini; 
 given X = e'K . d^y r. 
 
 6. What does the equation (\ -x^) -JL-x-IL become wh 
 
 ^ ^ ^ dx^ dx 
 
 len 
 
 X = cos ^ and /^ is the independent variable ? 
 
 Ans. d^y/d^ = 0. 
 
 ^ 
 
 ( 
 
 
 
 6" 
 
 E! 
 
 i 
 
 1 
 
 
CHAPTER XXXVII. 
 
 SUCCESSIVE DIFFEBEKTIAl S OF FUNCTIONS OF MORE 
 THAN ONE VARIABLE. EXTENSION OF TAYLOR'S 
 THEOREM. MAXIMA AND MINIMA FROM TAY- 
 LOR'S THEOREM. MULTIPLE INTEGRATION. 
 
 Successive Partial Differentials. 
 
 172. Suppose u to be ax^ - xy^ + y ; we have as in § 40, 
 djii = {3ax^ - y^) dx^ d^u = Qax dx^, dj'u = Qadx\ d*n = 0, 
 
 dyH = (- 2x!J +1) dy, d^a = - 2x dy^^ d^ii = 0. 
 
 Again, d^n or {^ax^-y"^) dx contains y as well as x, and we 
 may obtain its differential on the supposition that y alone varies. 
 We then have 
 
 dydji = -2y dy dx, dj^d^u = - 2dyhlx, dy'd^u - 0. 
 
 Similarly ; d^d^u = -2y dx dy, d.dfu = - 2dx dy', dji^\ = 0. 
 
 173. In comparing these results it will be seen that 
 
 d^dyU = dyd^n, d^d,fu = d,fd^u, d/Iy% = dy^d^u ; 
 also, d^dydjt = d^dyU = dydjn ; in other words, the successive 
 operations indicated by d-, and dy may take place in any order. 
 To prove that this is true in all cases, let « =f(x, y) and change 
 <c'mtox + dx. Then 
 
 /(x + dx, y) = w + d^u. 
 
 If now y be changed into y + dy, we have 
 
 f(x + dx,y + dy) = ?/ + dyU + d^u + dyd^u. 
 
 If, on the other hand, we first change y into y + dy and after- 
 wards X into a; + £?x we pet 
 
 f(x + dx, y + dy)zzu + d^u + dyU + d^dyii, 
 
 :. dydji = d^dyU (1) 
 
 4 
 
188 
 
 SUCCESSIVE PARTIAL DIFFERENTIALS. 
 
 m 
 
 I 
 
 Henoc, d^d„^u = d^di,d„u = d/l^dyU = d/l^d^u = djdji, 
 
 and similarly for any combination. 
 
 Theso results may obviously be extended to functions of any 
 number of variables, 
 
 174. The exnressions '^'"' ^>'"' "^'^"^ 'IH'^ etc 
 
 dx* d\f dxdy dx*dy 
 
 are generally written (cf.§ 40, (3) ) 
 
 d^u d?\i d*u d*u . 
 dx^ dy'^ dx dy dx\iy 
 
 in which it must be carefully noticed that d^ii, d*u, have an en- 
 tirely different meaniTjg from tliat employed in the following 
 article. , 
 
 Successive Total Differentials. 
 
 175. Taking again the example 
 
 u = ax^ ■•'Xy^ + y, 
 
 we have, du = {Hax' - y^) dx + ^l- 2xy) dy, 
 
 dht = 6ax dx^ — 4y dx dy — 2x dy^, 
 
 d'^u = Ga dx^ - G dx dy^, 
 
 dSt^^. 
 
 These are the true values of the successive total diff«^rential3 
 of ?t when hotli a;and ?/ are independent variables simultaneously. 
 To obtain the general formulae for such successive differentials 
 substitute du for u in 
 
 d(i = dju + dyV, 
 
 Then, d (da) = dj,(dii) + dy (du) 
 
 or, d^u = d^ {dj.t -f- d,jU) + dy (d^u + dyii) 
 
 = d^u + d/lyii + d^^d^u + dy^u 
 
 ;. d^H = d^'u + 2dj.dyU + dy^a. 
 Similarly dht = dj'u + M^dyii -f- M^dyii \- d^u, 
 
SUCCESSIVE TOTAL Dll-r'?.RENTlALS. 
 
 189 
 
 ( 
 
 the law of formation of the terms being the same as in the Biuo- 
 luial Theorem. 
 
 176. The analo<,7Just mentioned suggests a convenient waj' 
 of writing the above results, viz. : 
 
 du = (d^ + d^) u, 
 
 d\ = {d^-\- dyY u, 
 
 d'u= (d, + d„yn, 
 
 d^u={d,+ d^yu. 
 
 it being understood that the expressions in brackets are to be 
 expanded by the Binomial Theorem as if d_, and d^ were symbols 
 of quantity instead of symbols of operation, and u aft(^rwards 
 placed after each term. 
 
 Similarly for a function of more than two independent 
 variables. 
 
 . ^"^^ {d^-hdy + d^+ ...yii. 
 
 Exercises. 
 
 1. If w = sin-i (x/y)^ verify that d^d^n = d^d.u, 
 
 2. If u = x«, d,dyu = »«'-i (1 + log X*) dx di/. 
 
 3. u = x^f//(a^-z^), find du. 
 
 4. If u = [(a - xy + (b - yy + (c - zy\-h show that 
 
 d}U' . d,H d'^u 
 
 - — -f- 
 
 4- 
 
 = 0. 
 
 dx^ dy'^ ' dz' 
 5. If u = /(y + ax') + F(y- ax), show that 
 
 d_,hi ., d*u 
 
 -i— = cr -£_, 
 
 dx^ dy-" 
 
 f and F indicating any functions. 
 
 Extension of Taylor's Theorem. 
 177. If in Taylor's Theorem § 157, (7), we write 
 
 \ 
 
ti' 
 
 « ■ ii 
 
 190 
 
 EXTENSION OF TAYLOR S THEOREM. 
 
 d/^ , ^V(^) ... for /'(«), /"(»)' ••• we obtain as an equi- 
 dx dx^ 
 
 valent form 
 
 Now /(oj, y) being any function of x and y, let a; become 
 aj + cfoj, y for the present remaining unchanged. Then (1) 
 gives us 
 
 /(x + h, y) =/(x, y) + ^V) A ^ fMv) |!^ ... (2) 
 If now y becomes y + /<;, (2) becomes 
 
 d^f(x,y + k) A' .3. 
 
 and each term may be expanded by Taylor's Theorem as follows : 
 /ix,y^k)=/i.,y)^^^/^k^'^iy^ ' 
 
 (2x 
 
 
 <fo; 
 
 J'A^^y ln^ ^^dJi^^y) hk + ... 
 
 dx 
 
 dx d\. 
 
 d.V(x,y + k) h'_d/[ /(x,y) + ...] h' _^ d^V(x,y) h} 
 
 dx 
 
 1.2 
 
 dx' 
 
 2! 
 
 dx' 21 
 
 whence (3) becomes 
 
 f(x + h,y + k)=:/(x,y)^-[i^/^h^'Liy^ 
 
 ^ r ^.VC^. .V) ;,. 2d,d,/(x,y) ^ ^^ d,;/(x, y) ^n 1^ 
 L dx^ dx dy du* J 2! 
 
 (-1) 
 
( 
 
 > 
 
 MAXIMA AND MINIMA. 
 
 191 
 
 r l^MT^^ ""^^ ^' ""'^"''^ ^° *^' ^°^^^^i°S convenient form, 
 (ct. ^ 176) : 
 
 A similar expression will apply to functions of three 
 
 variables. 
 
 or more 
 
 178. IfK =:/(x,y)andD=|A + i^., (4) may be written 
 
 ■0^"-^^-) 
 
 c'' ?«. 
 
 u 
 
 Maxima and Minima from Taylor's Theorem. 
 
 179 By the aid of Taylor's Theorem we may verify and ex- 
 tend the conclusions of chapter XV. for maxima and minima 
 
 If a be a value of x for which any function f(x) is a max 
 or a mm., and A any small quantity, it is plain that/(a + A) 
 -/(a) and /(a - K) -/(«) must have the same si-n viz + 
 for a mm. and - for a max. Now ° ' ' 
 
 and f{a - h) -/(a) = -/'(«) h +/"(«) §*-/-(«) ^% ... 
 
 ^1 31 
 
 and by taking h small enough the sign of the right hand side 
 will depend upon that of the first term which doe. not vanish 
 Hence there cannot be a max. or a min. unless f(a) = and 
 Uiere will then be a max. if/" («) is - and a min. if /"(«) 'is 4- 
 But .f/'Xa) also = 0, there cannot be a max. or a min. unless* 
 
T 
 
 192 
 
 MAXIMA AND MINIMA. 
 
 f"'(ci) also = 0, and there will be a max. or a min. according as 
 the fourth differential coefficient is — or + . It will thus be 
 seen that there cannot be a max. or a min. unless the first differ- 
 ential coefficient which does not vanish is of an even order and 
 that /(a) will be a max. or a min. according as this differential 
 coefficient is — or + . 
 
 180. Hence, or from § 177 (4), a function f(x, y) of two in- 
 dependent variables cannot be a max. or a min. for values a and h 
 of the variables unless a and h satisfy 
 
 ^kIihJll = and 'Lil^^hyl^O, 
 dx dy 
 
 If these equations are satisfied there will in most cases be a 
 max. or a min. but it may be shown (see Williamson, Diff, Col. 
 Ch. X.) that it is also essential that 
 
 dx' 
 
 > 
 
 / a,a,,J{x,y) \' 
 \ dx dii ) 
 
 d\f "^ V dx dy 
 
 and that there will then be a max. or a min. according as 
 d^A^, y) X 
 
 dx" 
 
 IS - or + 
 
 If the function be called u we have 
 
 du p, 3 d,u f^ 
 
 — = and — = 
 
 dx dy 
 
 which two equations must be solved as simultaneous equations 
 in order to obtain the values of x and y. 
 
 Similarly for a function of three independent variables we 
 must have 
 
 du f\ du r. J du A 
 — - = 0, -5- = and — =0 
 
 dx dy dz 
 
 to solve for x, y, and z. 
 
 Examples. 
 
 1. T* = x' + xy + y^ + x- %j + 4. 
 
 I 
 
 f.7 ). 
 
 iki ex- 
 
EXAMPLES. 
 
 193 
 
 du/dx=2x + i/+l^ 
 dii/dy -x-\-2\j-1. 
 
 Putting these = and solving for x and y we i^et « = - .* 
 // = I, which make u a min., viz., If. ° ^* 
 
 2. To find a point such that the sum of the squares of its dis- 
 tances from three given points may be a min. 
 We have 
 
 (;«*/d^ = 2(x-rtO + 2(x-«,,)-{-2(x-a3)=0 
 
 c?w/c^y = 2(^ - Z,,) + 2(^ - 6^ ) + 2(3/ - 6,) = 
 
 whencex = K«,+«.+a3),// = K^+^',+6.), the centre of 
 gravity of the triangle formed by joining the given points. 
 
 3. Given 
 
 v^ =a^x + h^y + c^ 
 r^=a^x + b^y + C:^ 
 
 show that the values of x and y which make 
 
 a min. are obtained by solving the equations 
 
 These are the "normal equations " in the method of Least 
 bquares. 
 
 4. Find the corresponding equations when 
 
 ^1 =«ice + 6j^ + Ci2f (^j, etc. 
 i.e., when there are three independent variables. 
 
 5. To make with the smallest possible amount of sheet metal 
 an open rectangular box of given volume, show that the len-th 
 and breadth must each be double of the depth. ° 
 
 13 
 
194 
 
 I 
 
 MULTIPLE INTEGllATION. 
 
 Multiple Integration. 
 
 181. Let it be required to find the function which, being 
 dittcrentiated with regard to y and then with regard to x, or 
 witii regard to x and then with regard to y. will give 
 
 6x^^ dxdy. 
 
 Calling the required result u we indicate the " double integra- 
 tion " thus : 
 
 w = Qx^y dx dy. 
 
 We first integrate on the supposition that ;/ is constant and 
 then on the supposition that x is constant. Thus 
 
 u 
 
 = f (f ^^""y <i^^ dy = f(-^'^ + «) dy 
 
 = xhf + ay + b, 
 
 u and h being constants of integration. It is to be noticed that 
 u is any quantity which is independent of x ; it may therefore 
 be any function of y. Similarly b is any function of x. The 
 result is then 
 
 x^y'^+ any function of as + any function of y. 
 
 In integrating between limits these constants or functions of 
 integration may be omitted (cf. § 68). Thus if the limits of x 
 are 1 and 2 and those of y are 2 and 3 in the above example, 
 we have 
 
 r {' 6x'y dx dy = (2^^ - P) (3^ - 2^) = 35. 
 
 But if X and y are not independent of one another, one of 
 them may be a function of the other, m which case one or both 
 of the limits in one integration will contain the other variable j 
 thus 
 
 6x^y dx dy, where z = y^ + 1, 
 
 • J 1 
 
 t 
 
MULTIPLE INTEGRATION. 
 
 195 
 
 1, being 
 to X, or 
 
 integra- 
 
 bant and 
 
 ticed that 
 therefore 
 X. The 
 
 notions of 
 
 imits of X 
 
 example, 
 
 her, one of 
 >ne or both 
 r variable j 
 
 = [2 0/ + l)^ -;/]]= 29. 
 
 In the above illustrations we have supposed the limits follow- 
 ing the first sign of integration to be those corresponding to the 
 last differential (di/) ; usually, however, it is understood that 
 the first limits are those corresponding to the first differential. 
 
 Examples. 
 
 1- Find J J X dx chj, where y = Va^'-sc^ i.e., integrate 
 
 xdx dy with regard y, the limits being and \/aF^x\ and then 
 with respect to x, the limits being and a. Ans. ^a\ 
 
 2. J J rdo dr, where r = aO. Ans. ^:rV 
 
 'o Jo 
 
 (•2 (•2i /•jr + n 
 
 /•2 ^2x fix + ji 
 
 3- J J J dxdy dz. Ans. ^. 
 
 4. What is the geometrical meaning of the following ? 
 
 (1) \\dxdy. 
 
 (2) ^{[dxdydz. 
 
 (3) [Irdddr. 
 
 5. What is the mechanical meaning of the following ? 
 
 J J^'^^^^ ^^ydxdy 
 \ \ dx dy I r dx dy 
 
 (2) |J/.xVx%. 
 
 (3) \^ti(x'^f)dxdy. 
 
if, ■ i 
 
 1^ 
 
 Iff. 
 
 4 
 
 196 MULTIPLE INTEGKATIOK. 
 
 6. To find the approximate value of 
 
 pa (•aJT 
 
 '\/h-r COS 6. rdr (III, 
 Jo Jo 
 
 when r is small compared with A. 
 
 This involves double integration ; it is also an example ot 
 
 approximation to the value of an integral when the complete 
 
 integral cannot be obtained. We have 
 
 VA — r cos = 'Jh 
 
 (l-^cos«) 
 
 } 
 
 = VA(1-2^CO««-^,COS'„_-^C08».)^ 
 
 expanding by the Binomial Theorem and neglecting the fourth 
 and higher powers of r/h. Multiplying by de and integratin 
 between the limits and 27r (regarding r as constant) we get 
 
 fe' 
 
 
 Multiplying by rdr and integrating between the limits and 
 a we find 
 
 \ 32AV 
 for the approximate value of the integral. 
 
ot 
 lete 
 
 
 rth 
 
 mg 
 
 nd 
 
 APPENDIX. 
 Note A. The Exponential Series. 
 
 It is required to expand cf into a series consisting of ascend- 
 ing powers of x with coefficients which are independent of «. 
 
 Let <t" = P-l-Aa; + BxHc.r;^+... r\\ 
 
 where p, a, b, c, ... are constant coefficients (independent of «) 
 to be determined. 
 
 When a) = 0, (1) becomes a" = p, .-. p = „o ^ j ^„j ^^ ^ .^ 
 independent of .c it must always = 1. 
 
 ,-. a' =1-1- Ax + BX^ + cjc' + . . . C2) 
 
 Write 2x for rr., /. n'-^ = 1 + 2a.x + 4bx=^ + 8c.x» + . . . (3) 
 
 But a^^ = (a-)2 = (1 + AX -t- Ba;« 4- ca;' + ...)* 
 
 = 1 + 2a.x+(a^+2b)x^+(2ab + 2c)cb'+... (4) 
 
 :N"ow, since the right hand members of (3) and (4) both re- 
 present a'' they must be identical term by term ; 
 
 .-. 4b = a^ + 2b, .♦. B = U^ 
 
 8c = 2ab + 2c, .-. 6c = 2ab = a', .-. c = ^a'. 
 Substituting in (2) 
 
 aV 
 
 1.2 ^1.2.3"^ 
 
 (6) 
 
 ^ The value ofA is still unknown. To determine what meanin- 
 IS to be attached to it, we observe that since a is independent 
 of .T It must depend upon a for its value. Let the value of a 
 for which A IS unity be called e. Then 
 
 e'= l+x + JL.^ 
 
 x^ 
 
 1.2^1.2.3 
 
 + 
 
 (6) 
 
108 
 
 EXPONENTIAL SERIKS. 
 
 IK: 
 
 Let a; = 1, /. c = 1 + 1 + A +^ + ^\ + ... - 2'7I82818 ... 
 
 which is taken as the base of the sypcem of logarithms called 
 Napierian. 
 
 When X = 1, (6) becomes a = 1 + A + ^a^ + ... 
 
 and when « = A, (6) becomes e* = 1 + a + ^a^ + ... 
 
 .*. a = e^, i.e., a is log« a, the Napierian logarithm of a. 
 
 (5) is the general exponential series of which (6) is a parti- 
 cular case. 
 
 Corollary. If n is any positive integer, x^/e" approaches the 
 value as X approaches oo. For 
 
 x" 
 
 X" 
 
 1 
 
 x^ x^ 
 
 r , , , x" x" 1 1 
 
 1 +X 4- — + ^--+ ... — H +••• + 
 
 X 
 
 2 2.3 
 
 x" x""^ 
 
 2.3...(n+l) 
 
 - = when x = oo. 
 
 + 0+ ...+ Q0 + ... 
 
 Note B. Partial Fractions. 
 
 The algebraic sum * of certain fractions beina: siven, it is re- 
 
 quired to find the fractions. 
 
 o o' 
 
 Case 1. When the factors of the denominator arc all simple 
 and unequal. 
 
 Example. 
 
 x' + 3x + 1 
 xXx-l)(x + 2)' 
 
 The denominator indicates that this result may be obtained by 
 the addition of three fractions whose denominators are x, x - 1, 
 X + 2, respectively ; our object then is to find the numerators. 
 
 * The sum is assumed to be a proper fraction (the numerator of 
 lower dimensions than the denominator). If not, the fraction should 
 be reduced to a mixed quantity. 
 
 
 f 
 
PAKTIAL FRACTIONS. 
 
 199 
 
 Call them a, b, c* 
 
 x^ + 3x + 1 
 
 A B 
 
 % X - 1 a; + 2 
 
 (1) 
 
 '■x(x-l) (x-t-2) 
 
 Clearing of fractions, 
 
 rc^ + 3x + I = A (x - 1) (x + 2) + Bx (x f 2) + ex (x - 1). (2) 
 
 It is to be noticed that (1), and .". (2), is to be an identity 
 and therefore true for all values of x. Now, if we give any three 
 successive arbitrary values to x in (2), we shall obtain three 
 equations by solving which as simultaneous equations, the quan- 
 tities A, B, c, may be found. The arbitrary values should, 
 however, be such as to render these three equations as easy of 
 solution as possible. A little inspection will show that this will 
 be accomplished by giving x successive values which make x, 
 X — 1, and X + 2 equal to 0, i.e., by making x = 0, 1, and - 2. We 
 thus get from (2), 
 
 when X = 0, 1 = a ( - 2), 
 
 .-. A = -i 
 
 when X =s 1, 5 = b (3), 
 
 /. B=f 
 
 when x = -2, - 1 = c (6), 
 
 .'. = - i 
 
 x' + 3x+\ -^1 ^ ^ -i 
 
 
 x(x-l)(x + 2) X x-1 X4-2 
 
 
 1 
 
 5 1 
 
 2x 3(x-l) b'(x + 2) 
 
 Case 2. When the factors of the denominator are simple, but 
 two or more of them are equal. 
 
 Examples. 
 
 1. 
 
 l+3x 
 
 x(x + l)2 
 The denominator shows that the partial fractions have denora- 
 
 * The A, B, are aspunied to be independent of x. If one of them 
 contained x the sum would not be a proper fraction. 
 
>l 
 
 
 If 
 
 200 PARTIAL FUACTI0N8. 
 
 iimtorH X and (»+!)' and (probably) rc + l. We therefore 
 
 assume 
 
 1 + 3a; A II , c 
 
 a; (x + 1 )* "" a; x -f- 1 (a; -h 1 y 
 
 :. 1 + 3.7; = A (X -f- 1)* + BX (.T i 1) -I- ex. 
 If X = 0, 1 = A, .-. A = 1 . 
 
 Ifx = -1, -2 = -o, .-. = 2, 
 
 and B may be found by giving any value other than and - 1 
 to X, e.g.^ if X = 1 we have (v A = 1, and c = 2) 
 
 4 =-- 1 x2'+TJx2 + 2x 1, .•.«=-!, 
 
 . l + 3x 1 1 ._ 2_ 
 
 " x(x+l)" X XI- I (x + 1)' 
 X+2 A B C D 
 
 A B C 
 
 + _+. 
 
 (xfl)(x-l)' X4-1 x-l (x-1)^ (a^-l)' 
 
 .'. X + 2=A(x-iy+B(x4-l) (X-1)^ + C(X + 1)(X-1) 
 
 + D(X + 1). 
 Ifx=-1, 1 =A(-2)' .-. A=--^. 
 
 Ifx = ],3 = D(2), .'. i) = f 
 
 To get B and c give x any two arbitrary values other than 1 
 and —1 ; thus (remembering that a and D are foundj if x = 0, 
 
 2 = ^+B-c-f-§ orB-c-f, 
 
 and if x = 3, 2b + c = ; hence from these two equations 
 
 ^ = h c = - i. 
 
 x + 2 
 
 1 
 
 1 
 
 (x+l)(x-iy' 8(x4-l) 8 (x-l) 4(x-l)=' 
 
 + 
 
 2 (x-l) 
 
 ^' 
 
 Case 3. When the denominator contains a quadratic factor. 
 We must now assume the numerator of the fraction with a 
 quadratic denominator to be of the form A x + b. 
 
HYI'EKHOLIC FUNCTIONS. 
 
 201 
 
 Example. 
 
 l+x 
 
 AX+ ti c 
 
 ,- + 
 
 x(l^x') ~ 1 i-x' x' 
 :. 1 -I- 0! » Aa-'^ + HX + 0(1 i X^). 
 If .9: = 0, 1 = r, .-. c = 1 . 
 
 If » = 1, 2 = A f B + 2. .-. A ! B = 1 .-. A = - 1, 
 
 2, .-. A-B=-2J B=l. 
 
 If X = - 1 , = A - B -I- 
 
 1+05 
 
 - x+l 1 
 
 + 
 
 Besides the uiotl.ods explainc^d i„ the above examples others 
 may sometimes be employed with advantage. For instance, in 
 the last example 
 
 1+X =: AX- + BX ■•rC(l -i- X*), 
 
 Since the left and right hand sides are to be identical, the coeffi- 
 cients of like powers of x on the two sides must be equal ; we .'. 
 have 1 = c, 1 = b, = a + c, which give the same results as before. 
 
 Note C. Hyperbolic Functions. 
 
 V 
 
 Ihe quantities ^e' - e-), ^(e^ + .^^ are called the hyperbolic 
 sine of .-c sinh.x) and hyperbolic cosine of x (cosh-Tj), respect- 
 ively The hyperbolic tangent is defined to be sin h .r /cosh x 
 and the hyperbolic secant, cosecant, and cotangent are the reci- 
 procal of the cosine, sine and tangent, respectively. 
 
 These functions are represented graphically in figs 92, 9.3 
 In hg. 92, ABC is the curve jj = cosh..; doe is .y = sinh x, which, 
 for points beyond e coincides almost exactly with bc rthough 
 always a little below it) ; poo is the curve .y = tanh x, which, 
 beyond f and g. almost coincides with the lines y = ± 1 thou-h 
 always between these lines. In fig. 93 abc is y = sech x, be and 
 Po are .y = cosech x, hk and ml are .y = cothx. The curve 
 :y = cosh X IS a catenary. 
 
 It should be noticed that the extreme limits of variation of 
 
HYPKKHOI.lc FUNirjIONi*. 
 
 Fig. 92. 
 
 Pig. 93. 
 
 sinhxare - oo and + co ; ofcoshic, + 1 and +00; oftanhr<;, 
 - 1 and + 1. 
 
 The relations connecting the hyperbolic functions are verj 
 similar to those connecting trigonometrical ('• circular ") func- 
 tions, and are easily proved by ordinary algebra, etc. Some of 
 them are as follows : 
 
 cosh = 1, sinh = 0, tanh = 0, etc. 
 
 cosh {-x) = cosh X, sinh (-x) = - sinh x, tanh (-x) =- tanh x, etc. 
 
 cosh^a; - sinh^x = 1, sech"' = 1 - tanh^x, cosech^x = coth^x - 1. 
 
 cosh (x±y) = cosh x cosh y ± sinh x sinh y, 
 
 sinh (x±y) = sinh x cosh 3/ ± cosh x sinh y, 
 
 cosh X + cosh y = 2 cosh \(x-ty) cosh ^(x- y), 
 
 

 HYPEllUOLIC FUNCTIONS. 
 
 cosh a; - coshy = 2 sinh l(x + //) ainh ^(x - //), 
 sinl) X + sinh y = 2 sinh ^(x + y) cosh i(» - y), 
 aluhx -sinhy = 2cosh^{x^t/) 8io^(x-y), 
 
 cosh 2x = cosh^x + sinh^x. 
 
 = 2cosh»T 1-1 .-, l+28inh»x, 
 
 sinh 2x = 2 sinh x cosh x, 
 
 tanh(x±y)=^^"*V^ **»"^.'/ , 
 1 ± tanh X tanh y ' 
 
 tanh2x= 2^^"^* . 
 1 + tanh^x 
 
 203 
 
 d sinh X = cosh x tix, 
 c/ cosh X = sinh x dx, 
 d tanh x = sech'^x t^x, 
 </ coth X = ~ cosech^x dx, 
 
 I cosh x dx = sinli x. 
 sinh X rix = cosh x. 
 I secli^x </x = tanh x. 
 cosech^ X dx = - coth x. 
 
 d sech X = - sech x tanh x c?.r, Jsech x tanh x r?x . - sech x. 
 <? cosech X = - cosech x coth x c7x, fcosech x coth x c^x = ~ cosech x, 
 
 ''sinh-.?^.^, ' L^ = sinh- ? = log (.W.- + <.."). 
 
 cfcOSh-l *=:_^ 
 
 a V«* 
 
 a' •' Jx' - a- 
 
 ,x 
 
 cosh-i- B loff(x + Jx'' - a^). 
 
 d tanh'i - = 
 
 X adx 
 
 a a'~ x' 
 
 x<a. 
 
 \~^. (x<«) = I tanh-i ?! log(^^•^^ 
 Ja*-x*' ^ a a 2a *'\a-x/ 
 
204 
 
 HYPERBOLIC FUNCTIONS. 
 
 
 d coth-i = ^ xyn, 
 
 a x' — fr 
 
 J^;^(..<.)= Itanh^? 1 log(^. 
 Jx -a a a Za \a + x/ 
 
 X 
 
 d sech-i^ - = _ 
 
 (K 
 
 Ix 
 
 '' X ^Jft* — X* 
 
 
 c/cosech-i^ = -__^ 
 
 " X V<«* + a*'' 
 
 J: 
 
 f/x 
 
 = - cosech-^ 
 
 X 1 
 
 X Va"" + x^ '« 
 
 
 X* x^ 
 
 sinhx = x-j — H f. ... 
 
 3! 5! ^ 
 
 co8hx=l+*' + ^%. ... 
 tanh-ix = x + |- + ^+... (see § 154) 
 
 2 3 2.4 5 2.4.6 f^ "'* 
 
 If i E \/ - 1, COS ix = cosh X, sin ix = i siuli x, cosh /x = cos x, 
 sinhix = i sinx. 
 
 If X = log tan Q -H _^ , then sinh x = tan n, tanh x = sin fl, 
 cosh X = sec ^, etc. 
 
 At any point of any ellipse ^, + '^ = 1 (fig. 94^ we may put 
 
 P 
 
p 
 
 ELLIPTIC INTEGRALS. 
 
 205 
 
 /A X 
 
 ^'g- **• Fig. 05. 
 
 X = a cosw, y = ^> sinr/, sioce cos''^ -}- sin'-'^ =.1. Jq t^jg ^ase 
 '> = 2 area oav / ab (see Ex. 12, p. 131) ; it also = the eccentric 
 Angle AOQ. 
 
 At any point of any hyperbola ^ . .^; = 1 (fi.. 95) ^^ may put 
 a; = a cosh 6, ij = b sinh 6, since cosh^6 - sinh*^y =1. In this case 
 = 2area AOV/ah=\o^ (^^+|^, (see Ex. 10, p. 114). 
 
 The equation of the catenary (§169) is y = .e cosho^j also 
 « = a sinh X. ' 
 
 jr. tPy , 
 
 ^ "^ ~ '* ^' y = A ^'os «x + B sin «ar, (§ I7O;, 
 
 ,:n^;' t" rr^'J f '•" "'' °' ''^'^^ ^'""^^^«"« ««^ »'«en- 
 hill s Differential and Integral Calculus. 
 
 NoteD. Elliptic Integrals. 
 
 J 
 
 •'(l + a sin^^ 
 
 de 
 
 and 
 
 ' (1 + a 8in=*^) Vr^ w'sin'^ 
 
 arc called elliptic integrals of the first, second, and third class 
 respectively. The constant m, which is assumed to be not 
 
*' ' i 
 
 I' 
 
 206 
 
 ELLIPTIC INTEGRALS. 
 
 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 0, H is called the amplitude of the integral. 
 The integrals are represented by the symbols p(«i, n)., E(m, 6*), 
 and n (a, m, (f), respectively ; or by F,„ (ji), &c. When the 
 limits are and |t (i.e., when the amplitude is ^ir) the integrals 
 are said to be complete^ those of the first and second class being 
 denoted by k and e, respectively. 
 If sin e = X, the integrals become 
 
 
 dx 
 
 and 
 
 (fx 
 
 
 dx. 
 
 f 
 
 J (1 4- ax/' 
 
 (1 -\- ax^) V(l - x'O (1 - mV) 
 
 and they are complete when the limits are and 1. 
 
 The elliptic integrals are frequently met with, but their values 
 cannot be expressed in finite terms. We may, however, by 
 infinite series approximate to their values. 
 
 Thus by the Binomial Theorem 
 
 dd 
 
 1.3 
 
 = (1 - m^sin^^) ^ dd 
 
 V 2 2 
 
 4-,;»^4/i , ■'■•'-'• "^ «, 6 ,,;„«£ 
 
 4 2.4.6 
 
 m''8m"ff + 
 
 "'\ dii, 
 
 and each term may be integrated by § 90 (see Ex. 4 at the end 
 of this Note). 
 
 Taking the limits as and I- we have (§ 97) for k, the com- 
 plete elliptic integral of the first class, 
 
 2L V2 J V2.4 / V2.4.(J /J 
 
 >2.4.(J 
 Similarly for the integral of the second class we have 
 
 VI - m^nm^t) do = (1 - mhin^a)^ dfj 
 
 = f 1 — - m'sin*^^ 
 
 2 2.4 
 
 wi^sin* - 
 
 1.3 
 2.4. G 
 
 w^sin'^t; - ••• 
 
 )*, 
 
 1 
 
 
1 
 
 •" ^-(^-)-l(^M- 
 
 ELLIPTIC INTEGRALS. 
 
 1/1.3.5 
 
 207 
 
 5V2.4.6 
 
 m 
 
 ■] 
 
 for E, thp complete integral of the second class 
 
 In this way the values of the integrals may be obtained but 
 more convenient formulae have been devised by which the la'beur 
 of calculation .s lightened. Tables giving the values of the 
 .ntegrals of the hrst and second class to 10 decimal places were 
 construeted by Legendre. Threcfigure tables are gL attl 
 end of this volume. 
 
 It may be noticed that 
 
 E(0, ^) = P(0, ft) = ^^ (in radians) ; 
 also, E(l, (,)= Lose dd = sin 0, 
 
 F(l, ,i) . f-^^Iogtan (l,') (E^ 5 "o.; 
 
 i. IZV^'] "bove expansions and the integral of sin"« ch a 90) 
 It may be shown that ^^ ^ 
 
 E(to, ?i-+ft) ^27iE±E(w, ft), 
 
 F(m, Wrrift) = 2nK±F(wi, ft), * 
 
 E and K being the values of ,he integrals for the amplitude A. 
 and . being any integer. Hence a table of the elliptic inte^rah 
 in which the amplitude varies from to ^. may be^sed for a 
 higher values of the amplitude. 
 
 Examples. 
 
 1. To find the length of an arc of the ellipse ^^-ZI^ i. 
 
 The complement of the eccentric angle being denlted by ft we 
 havex = asmft, and^ = 6cosft. " uy '^ we 
 
 ■''<^^ = (ioosOdt),f/^ = -bsm6do, 
 whence ds' . dx^ + df = («^cos*ft + J^sin'^ft) dl^, 
 
 = [a^ - (a^ - /.^) sinv,] dtf' = a* (1 - n^,i^,>, ,^ft-' 
 
208 
 
 i 
 I* 
 
 u 
 
 ELLIPTIC INTEGliALS. 
 
 
 h 
 
 where m is the eccentricity of the ellipse. Hence the length of 
 the elliptic arc measured from the end of tiie minor axis is 
 
 a { Vl - m'sin^^y dfi = a e(w, flj ) 
 Jo 
 
 an elliptic integral of the second class. The length of the qua- 
 drant of the ellipse - 
 
 a E(m. in-). ' 
 
 2 Find the length of a quadrant of the ellipse 
 
 16a:'' -25/ = 400. Ans. 7-335. 
 
 Also find the co-ordinates of the middle point of this arc. 
 
 Ans. x = 3-41, ^ = 2-93. 
 3. Am arc of the lemniscatc r* = a-'oos 2^. 
 From ds'^ = r^db' + dr^ we have 
 
 Let 2 sin^^ = sin> Then 
 
 V2 Jo Vl-f^V V2 W2'^V' 
 
 an elliptic integral of the first class. The length of a quadrant 
 of the lemniscate is therefore 
 
 ^f(1 i.) = 1.311„. 
 
 V2 VV2' 
 4. Show that 
 
 F(w, (f) =e-\- Im^ {6 - sin 6 cos ii) 
 + -iurn^ ( 3fl - 3 sin ti cos 0-2, sin^e cos ^0 + • • . 
 
 E(m, a) = 6- ^ni* (^~sin coaO) 
 - ^^m^ (36 - 3 sin 6 cos (? - 2 sin^'d cos n)- ... 
 
INDEX. 
 
 ( 7Vie references are to pages.) 
 
 Arc, difterential of, 14, 117. 
 Areas, 71, 107, 128. 
 Asymptote, 23, 122, 148. 
 Asymptotic circle, 124. 
 
 Cardioid, 119,128. 
 
 Catenary, 30, 180. 
 
 Centres of gravity (centroids), 
 
 150. 
 Change of independent variable, 
 
 184. 
 Circle, 113, 153, 160, 162. 
 Cissoid, 27, 113, 131, 154. 
 Companion to the cycloid, 34, 52, 
 
 114. 
 Concavity and convexity, 51, 125. 
 Conchoid, 133. 
 Conjugate point, 19. 
 Curvature, 53, 126; direction 
 
 of, 49. 
 Curve tracing, 12. 
 Cusp, 18. 
 
 Cycloid, 33, 58, 59, 114, 153. 
 Cylinder, 154, 161. 
 
 Definite integrals, 10.3. 
 Demoivre's Theorem, 170. 
 Differences, small, 42. 
 Differential coefficient, 8, 46. 
 Differentials, definition, 6. 
 
 binomial, 91. 
 
 of algebraic functions, 9. 
 
 of angles, 35. 
 
 of exponentials, 28. 
 
 Differentials {continued). 
 
 of logarithms, 28. 
 
 of the trigonometric ratios, 31. 
 
 of triangles, 43. 
 
 partial, 38, 187. 
 
 successive, 47, 187. 
 
 total, 39, 188. 
 Double integration, 194. 
 Double point, 18, 125, 
 
 Element of integral, 70. 
 Ellipse, 5, 57, 113, 144, 153, 161, 
 
 207. 
 Elliptic integrals, 205. 
 Elliptic sector, 131. 
 Envelopes, 140. 
 Epicycloid, 138. 
 Equations, approximate solution 
 
 of, 44. 
 Euler's exponential formulae, 
 
 170. 
 Evolutes, 54, 144. 
 Expansion of functions, 165. 
 
 Folium, 19, 60, 68, 132, 148. 
 Functions, definition, 6. 
 
 derived, 46. 
 
 expansion of, 165. 
 
 hyperbolic, 201. 
 
 implicit, 39. 
 
 limits of, 172. 
 
 Guldin's Theorems, 152. 
 Gyration, radius of, 158. 
 
INDEX. 
 
 Helix, curvature of, 61. 
 Hyperbola, 25, 114, 131, 144. 
 
 rectangular, 16,115, 118, 131. 
 Hyperbolic functions, 201. 
 Hypocycloiil, 138. 
 
 four-cusped, 15, 41, 6d, 101), 
 
 153, 164. 
 
 Increments, 6. 
 
 Independent variable, 6, 46, 184. 
 Infinitesintals, 2. 
 Inllexion, points of, 50, 124. 
 Iiitegrals, definite, 103. 
 fundamental, 75, 
 elliptic, 205. 
 Integration, by sub.stitution, 89. 
 by parts, 96. 
 
 by successive reduction, 99. 
 double, 194. 
 multiple, 194. 
 Inverse curves, 134. 
 Involute, 54. 
 of circle, 127. 
 of catenary, 182. 
 Kinetic energy of rotation, 156. 
 Lemniscate, 19, 119, 130, 208. 
 Lengths of curves, 107, 129, 
 Limayon, 119, 120, 131. 
 Limits, 1. 
 
 of functions, 172. 
 of integrals, 70. 
 Lituus, 119, 123, 
 Logarithmic, series, 166. 
 spiral. See Spiral. 
 
 Maclaurin's Theorem, 168. 
 Maxima and minima, 62, 191. 
 Moments of inertia, 156. 
 Multiple points, 18, 125. 
 
 Napierian loirarithms, 28, 166 
 
 198. 
 Node, 18. 
 Norm«ls, 14, 40, 118. 
 
 Pappus's Theorems, 152. 
 Parabola, common, 15, 58, 71 
 
 K53, 112, 118, 152. 
 Parabola, cubical, 2, 71. 
 Parabola, semi-cubical, 19, 108. 
 Paraboloid of revolution, 73, 152, 
 
 163. 
 
 Partial diflerentials, 38, 187, 
 Partial fractions, 198. 
 Pedal curves, 135. 
 Polar leciprocals, 137, 
 Prismoidal formula, 116. 
 Probability curve, 115. 
 
 Roulettes, 137. 
 
 Series, 165. 
 Sign of an area, 110. 
 Simpson's formula, 116, 
 Sine curve, 51, 72, 
 Singular forms, 172. 
 Solid of revolution, 72, 107, 
 Sphere, 113, 153, 161, 163. 
 Spheroids, 113, 153, 163. 
 Spiral, of Archimedes, 119, 127 
 129. 
 
 hyperbolic, 119, 123, 127. 
 
 logarithmic, 120, 121, 127, 130. 
 Subnormal, 14, 118. 
 Subtangent, 14, 118. 
 Surface of revolution, 107, 129. 
 Symmetry, 145. 
 
 Tangent curve, 51. 
 Tangents, 14, 40, 118, 
 Taylor's Theorem, 167, 189. 
 Tractrix, 180. 
 Triple point, 18, 125. 
 Trochoid, 137. 
 
 Value of e, 198; of tt, 171. 
 Variables, 6, 46, 184. 
 Volumes, 72, 107. 
 
 Witch, 51, ILS, 154. 
 
 i 
 
 s 
 
 ; 
 
I 
 
 TABLES. 
 
 I 
 
 ; 
 
 POWERS, NAPIERIAI^ LOGARITHMS, ETC. 
 CIRCULAR (TRIGONOMETRIC) FUJ^CTIONS. 
 HYPERBOLIC FUNCTIONS. 
 ELLIPTIC INTEGRALS. 
 
POWERS, NAPIERIAN LOGARITHMS, ETC. 
 
 X 
 
 x-^ 
 
 X"' 
 
 X"' 
 
 I 
 
 (fo)^ 
 
 1-000 
 
 x^ 
 
 '-(fo) 
 
 logcJT 
 
 1 
 
 I 
 
 I 
 
 0-316 
 
 I -000 
 
 3-697 
 
 0-000 
 
 2 
 
 •500 
 
 4 
 
 8 
 
 0-447 
 
 1-414 
 
 I -260 
 
 2-391 
 
 0-693 
 
 3 
 
 •333 
 
 9 
 
 27 
 
 0-548 
 
 1-732 
 
 1-442 
 
 2-79S 
 
 1-099 
 
 4 
 
 •250 
 
 16 
 
 64 
 
 0-632 
 
 2-000 
 
 1-587 
 
 1-084 
 
 1-386 
 
 6 
 
 •200 
 •167 
 
 25 
 
 36 
 
 125 
 
 0-707 
 
 2-236 
 2-449 
 
 1-710 
 
 1-307 
 
 i-6o9j 
 
 6 
 
 216 
 
 0-775 
 
 1-817 
 
 1-489 
 
 1-792 
 
 7 
 
 •143 
 
 49 
 
 343 
 
 0-837 
 
 2-646 
 
 1-913 
 
 1-643 
 
 1-946 
 
 8 
 
 •125 
 
 64 
 
 512 
 
 0-894 
 
 2-828 
 
 2- 000 
 
 1-777 
 
 :-079j 
 
 9 
 
 •in 
 
 81 
 
 729 
 
 0-949 
 
 3-000 
 
 2 -080 
 
 J -895 
 
 2-197 
 
 10 
 
 •100 
 
 100 
 
 1 000 
 133' 
 
 I 000 
 
 1-049 
 
 3-162 
 
 2-154 
 
 2 224 
 
 o-ooo 
 
 2-303 
 
 11 
 
 •091 
 
 121 
 
 3'3'7 
 
 095 
 
 2-398 
 
 12 
 
 •083 
 
 144 
 
 1728 
 
 I -095 
 
 3-464 
 
 2-289 
 
 0-182 
 
 2-485 
 
 x3 
 
 ■077 
 
 i6q 
 
 2iq7 
 
 I- 140 
 
 3-606 
 
 2-351 
 
 0-262 
 
 2-561; 
 
 14 
 
 •071 
 
 196 
 
 2744 
 
 1-183 
 
 3-742 
 
 2-410 
 
 0-336 
 
 2-639 
 
 16 
 
 •067 
 
 225 
 256 
 
 3.>75 
 
 1-225 
 1-265 
 
 3-«73 
 
 2.-\(iii 
 
 0-405 
 
 2 - 708' 
 ~2^773| 
 
 16 
 
 •063 
 
 4096 
 
 4-000 
 
 2-520 
 
 0-470 
 
 17 
 
 '059 
 
 289 
 
 4913 
 
 1-304 
 
 4-123 
 
 2-571 
 
 0-531 
 
 2-»33 
 
 18 
 
 •056 
 
 324 
 
 5832 
 
 1-342 
 
 4-243 
 
 2-621 
 
 0-588 
 
 2-890 
 
 19 
 
 •053 
 
 ;,6l 
 
 6859 
 
 1-378 
 
 4-35^) 
 
 2-668 
 
 0-642 
 
 2-944 
 
 20 
 
 •050 
 
 400 
 441 
 
 8000 
 
 1-414 
 
 4-472 
 
 2-714 
 
 0-693 
 
 2-996 
 
 21 
 
 •048 
 
 9261 
 
 1-449 
 
 4-583 
 
 2.759 
 
 0-742 
 
 3-045 
 
 22 
 
 •04s 
 
 484 
 
 10648 
 
 1-483 
 
 4 -690 
 
 2 -802 
 
 0-788 
 
 3-091 
 
 23 
 
 •043 
 
 529 
 
 12167 
 
 ••517 
 
 4-796 
 
 2-844 
 
 0-833 
 
 3*135 
 
 24 
 
 •042 
 
 ^1^ 
 
 13824 
 
 1-549 
 
 4-899 
 
 2 884 
 
 0-875 
 
 3-178 
 
 25 
 
 •040 
 •038 
 
 625 
 676 
 
 15625 
 
 1-581 
 
 5-000 
 
 2-924 
 
 0-916 
 
 3*219 
 
 26 
 
 17576 
 
 1-612 
 
 5-099 
 
 2-962 
 
 956 
 
 3*258 
 
 27 
 
 •037 
 
 729 
 
 19683 
 
 1-643 
 
 5-196 
 
 3.000 
 
 0-993 
 
 3*296 
 
 28 
 
 •036 
 
 784 
 
 21952 
 
 1-673 
 
 5-292 
 
 3-037 
 
 1-030 
 
 3*332 
 
 29 
 
 •034 
 
 841 
 
 24389 
 
 1-703 
 
 5'3«5 
 
 3-072 
 
 1-065 
 
 3*367 
 
 30 
 
 •033 
 
 900 
 
 27000 
 
 1-732 
 I -761 
 
 5-477 
 
 3-107 
 3-141 
 
 1-099 
 
 3.401 
 
 31 
 
 •032 
 
 961 
 
 29791 
 
 5-568 
 
 1131 
 
 3*434 
 
 32 
 
 •031 
 
 1024 
 
 32768 
 
 1-789 
 
 5-657 
 
 3-175 
 
 I -163 
 
 3*466 
 
 33 
 
 •030 
 
 1089 
 
 35937 
 
 1-817 
 
 5-745 
 
 3-208 
 
 1-194 
 
 3*497 
 
 34 
 
 •029 
 
 1 156 
 
 39304 
 
 1-844 
 
 5-«3i 
 
 3-240 
 
 1-224 
 
 3*526 
 
 35 
 
 •029 
 •028 
 
 1225 
 
 1296 
 
 42875 
 
 1-871 
 
 5-916 
 
 3-271 
 
 1*253 
 
 3*555 
 
 36 
 
 46656 
 
 1-897 
 
 6-000 
 
 3-302 
 
 1-281 
 
 3*584 
 
 37 
 
 •027 
 
 1369 
 
 50653 
 
 1-924 
 
 6-083 
 
 3-332 
 
 1-308 
 
 3*6" 
 
 38 
 
 •026 
 
 1444 
 
 54872 
 
 1-949 
 
 :!6-i64 
 
 3-362 
 
 1-335 
 
 3*638 
 
 39 
 
 •026 
 
 1 52 1 
 
 59319 
 
 1-975 
 
 6-245 
 
 3*391 
 
 1-361 
 
 3-664 
 
 40 
 
 •025 
 
 1600 
 
 64000 
 
 2-000 
 
 6-325 
 6.403 
 
 3-420 
 
 1-386 
 
 3*689 
 
 41 
 
 •024 
 
 1681 
 
 68921 
 
 2-025 
 
 3*448 
 
 1-411 
 
 3*714 
 
 42 
 
 •024 
 
 1764 
 
 74088 
 
 2-049 
 
 6-481 
 
 3*476 
 
 1-435 
 
 3*738 
 
 43 
 
 •023 
 
 1849 
 
 79507 
 
 2074 
 
 6-557 
 
 3*503 
 
 1-459 
 
 3*761 
 
 44 
 
 •023 
 
 1936 
 
 85184 
 
 2-098 
 
 6-633 
 
 3-530 
 
 1-482 
 
 3*784 
 
 45 
 
 •022 
 
 2025 
 
 91125 
 
 2-121 
 
 2-145 
 
 6-708 
 
 3-557 
 
 1-504 
 
 3-807 
 3-829 
 
 46 
 
 •022 
 
 2116 
 
 97336 
 
 6-782 
 
 3-583 
 
 1-526 
 
 47 
 
 •021 
 
 2209 
 
 103823 
 
 2-168 
 
 6-856 
 
 3*609 
 
 1-548 
 
 3*850 
 
 48 
 
 •021 
 
 2304 
 
 1 10592 
 
 2- 191 
 
 6-928 
 
 3*634 
 
 1-569 
 
 3*871 
 
 49 
 
 •020 
 
 2401 
 
 1 1 7649 
 
 2-214 
 
 7.000 
 
 3*659 
 
 1-589 
 
 3-892 
 
 50 
 
 •020 
 
 2500 
 
 125000 
 
 2-236 
 
 7-071 
 
 3-684 
 
 1-609 
 
 3-912 
 
 ' 
 
 
 
 l.i 
 
POWERS, NAPIERIAN LOGARITHMS, ETC. 
 
 * 
 
 ;r-i 
 
 -v- 
 
 -r-' 
 
 2-236 
 
 J 
 
 J 
 
 -G^) 
 
 log, X 
 
 60 
 
 •020 
 
 2300 
 
 125000 
 
 7*071 
 
 3.684 
 
 I • 609 
 
 I *629 
 
 3*912 
 
 61 
 
 •020 
 
 2601 
 
 1 3265 1 
 
 2-258 
 
 7*141 
 
 3-708 
 
 3-932 
 
 62 
 
 •019 
 
 2704 
 
 1 40608 
 
 2-280 
 
 7-211 
 
 3 733 
 
 1*649 
 
 3-95' 
 
 63 
 
 •019 
 
 2809 
 
 148877 
 
 2 j02 
 
 7-280 
 
 3-756 
 
 1*668 
 
 3-970 
 
 64 
 
 •019 
 
 29 1 6 
 
 '57464 
 
 2-324 
 
 7-348 
 
 3*780 
 
 1-686 
 
 3-989 
 
 66 
 
 •018 
 
 3025 
 
 16637s 
 175616 
 
 2-345 
 2-366 
 
 7-416 
 7-483 
 
 3-803 
 3*826 
 
 1*705 
 1*723 " 
 
 4-007 
 
 o6 
 
 •018 
 
 3 '36 
 
 4-025 
 
 67 
 
 •018 
 
 3249 
 
 '85193 
 
 2-387 
 
 7-550 
 
 3-849 
 
 1-740 
 
 4-043 
 
 68 
 
 •017 
 
 3364 
 
 195112 
 
 2-408 
 
 7-616 
 
 3-87' 
 
 1-758 
 
 4-060 
 
 69 
 
 •017 
 
 34S1 
 
 205379 
 
 2-429 
 
 7-681 
 
 3-893 
 
 '•775 
 
 4*078 
 
 60 
 
 •017 
 
 3600 
 "3721 
 
 2 1 6000 
 
 2-449 
 ^•470 
 
 7-746 
 7-810 
 
 3-9«5 
 
 1*792 
 1*808 
 
 4*094 
 4*in 
 
 61 
 
 •016 
 
 226981 
 
 3-936 
 
 62 
 
 •016 
 
 35^44 
 
 238328 
 
 2-490 
 
 7 874 
 
 3 958 
 
 1*825 
 
 4-127 
 
 63 
 
 •016 
 
 3969 
 
 250047 
 
 2*510 
 
 7-937 
 
 3-979 
 
 1*841 
 
 4-143 
 
 64 
 
 •016 
 
 4096 
 
 262144 
 
 2-53" 
 
 8- 000 
 
 4-000 
 
 1*856 
 
 4-'59 
 
 65 
 
 •ois 
 
 4225 
 4356 
 
 274625 
 
 2-569 
 
 8-062 
 
 4-021 
 
 1*872 
 
 4-174 
 4-190 
 
 66 
 
 •015 
 
 287496 
 
 8*124 
 
 4-041 
 
 1-887 
 
 67 
 
 •015 
 
 44^9 
 
 300763 
 
 2-588 
 
 8*185 
 
 4-062 
 
 I -902 
 
 4-205 
 
 68 
 
 •015 
 
 4624 
 
 3 '4432 
 
 2^608 
 
 8-246 
 
 4-082 
 
 1-917 
 
 4-220 
 
 69 
 
 •014 
 
 4761 
 
 32S509 
 
 2-627 
 
 8-307 
 
 4- 102 
 
 1-932 
 
 4-234 
 
 70 
 71 
 
 •014 
 •014 
 
 4900 
 5041 
 
 343000 
 3579" 
 
 2-646 
 2-665 
 
 8*367 
 
 4*12! 
 
 1-946 
 
 4J 248 
 4-263 
 
 8-42b 
 
 4*141 
 
 1 -960 
 
 72 
 
 •014 
 
 5i«4 
 
 373?48 
 
 2-683 
 
 8-485 
 
 4* ItO 
 
 1-974 
 
 4-277 
 
 73 
 
 •014 
 
 5329 
 
 389017 
 
 2-702 
 
 8-544 
 
 4-170 
 
 1-988 
 
 4 -29c 
 
 74 
 
 •014 
 
 5476 
 
 405224 
 
 2-720 8 -602 
 
 4-198 
 
 2 -001 
 
 4-304 
 
 75 
 
 •013 
 
 5625 
 5776 
 
 421875 
 
 2^739 i 8*660 
 
 4-217 
 4*236 
 
 2-015 
 
 4-317 
 
 76 
 
 •013 
 
 438976 
 
 2-757 
 
 8*718 
 
 2-02S 
 
 4-33' 
 
 77 
 
 •013 
 
 5929 
 
 ■156533 
 
 2*775 
 
 8-775 
 
 4*254 
 
 2-041 
 
 4-344 
 
 78 
 
 ! -013 
 
 6084 
 
 ^74552 
 
 2-793 
 
 8*832 
 
 4-273 
 
 2-054 
 
 4-357 
 
 79 
 
 ! -013 
 
 62^1 
 
 493039 
 
 2*8ll 
 
 8-888 
 
 4*291 
 
 2-067 
 
 4-369 
 
 80 
 
 •013 
 •012 
 
 6400 
 "6561' 
 
 5 1 2000 
 
 2*828 
 
 8*944 
 
 4-309 
 4-327 
 
 2*079 
 
 4-382 
 
 81 
 
 53 144 1 
 
 2-846 
 
 9 * OOd 
 
 2*092 
 
 4-394 
 
 82 
 
 1 -012 
 
 6724 
 
 55 '368 
 
 2-864 
 
 9-055 
 
 4-344 
 
 2* 104 
 
 4-407 
 
 83 
 
 •012 
 
 6889 
 
 57'787 
 
 2-881 
 
 g*no 
 
 4-362 
 
 2- n6 
 
 4*419 
 
 84 
 
 i '012 
 
 7056 
 
 592704 
 
 2-898 
 
 9*165 
 
 4-380 
 
 2-128 
 
 4-431 
 
 85 
 
 •012 
 
 7225 
 7396 
 
 614125 
 
 2-915 
 2-933 
 
 1 9*220 
 "9^74 
 
 4-397 
 
 2- 140 
 2- 152 
 
 4-443 
 
 86 
 
 •012 
 
 636056 
 
 4-4'4 
 
 4-454 
 
 87 
 
 •oti 
 
 7569 
 
 658503 
 
 2-950 
 
 9-327 
 
 4-431 
 
 2*163 
 
 4*466 
 
 88 
 
 •Oil 
 
 7744 
 
 681472 
 
 2*966 
 
 9*381 
 
 4-448 
 
 2-175 
 
 4-477 
 
 89 
 
 •Oil 
 
 7921 
 
 704969 
 
 2-983 
 
 9-434 
 
 4-465 
 
 2*186 
 
 4*489 
 
 90 
 
 •Oil 
 
 8100 
 
 729000 
 
 753571 
 
 3.000 
 
 3-017 
 
 9*487 
 
 4*481 
 
 4*498 
 
 2*197 
 2*208 
 
 4-500 
 
 91 
 
 •Oil 
 
 8281 
 
 9-539 
 
 4*511 
 
 92 
 
 •on 
 
 8464 
 
 778688 
 
 3-033 
 
 1 9-592 
 
 4-5'4 
 
 2*219 
 
 4*522 
 
 93 
 
 •on 
 
 8649 
 
 804357 
 
 3 0,0 
 
 1 9-644 
 
 4-531 
 
 2*230 
 
 4-533 
 
 94 
 
 •on 
 
 8836 
 
 830584 
 
 3-066 
 
 9-695 
 
 4-547 
 
 2*241 
 
 4-543 
 
 95 
 
 •on 
 •oio 
 
 9(;25 
 
 857375 
 884736 
 
 3*082 
 
 i 9-747 
 
 4-563 
 4-579 
 
 1 2*251 
 ; 2*262 
 
 4' 554 
 4-564 
 
 96 
 
 9216 
 
 3*098 
 
 i 9-798 
 
 97 
 
 •010 
 
 94C9 
 
 912673 
 
 3 •"4 
 
 : 9*849 
 
 4-595 
 
 ' 2*272 
 
 4-575 
 
 98 
 
 •010 
 
 9604 
 
 941192 
 
 3-130 
 
 9*899 
 
 4-bio 
 
 2*282 
 
 4-585 
 
 99 
 
 010 
 
 9801 
 
 970299 
 
 3-146 
 
 9-950 
 
 4*626 
 
 , 2*293 
 
 4-595 
 
CIRCULAR 
 
 (TRIGONOMETRIC) FUNCTIONS. 
 
 
 
 H 
 
 sin 6 
 
 conec f 
 
 tan H 
 
 cottf 
 
 aecf) 
 
 eoBd 
 
 
 1 
 
 
 
 Kadi inn 
 
 
 
 
 
 
 
 
 
 OOOO 
 
 o-ooo 
 
 00 
 
 0-000 
 
 00 
 
 i-oco 
 
 i-ooo 
 
 '•571 
 
 90 
 
 1 
 
 0017 
 
 0*017 
 
 57-30 
 
 0-017 
 
 57-29 
 
 I -00c 
 
 1 -000 
 
 J-553 
 
 89 
 
 2 
 
 0-035 
 
 0035 
 
 28-65 
 
 0*035 
 
 28-64 
 
 I 001 
 
 0-999 
 
 1-536 
 
 88 
 
 , 3 
 
 0052 
 
 0-052 
 
 19*1 1 
 
 0052 
 
 it)*o8 
 
 i-ooi 
 
 0-999 
 
 i-5'8 
 
 87 
 
 4 
 
 0070 
 0-087 
 
 0-070 
 0*087 
 
 "4 '34 
 11-47 
 
 0070 
 0087 
 
 14-30 
 
 1-002 
 I 004 
 
 0-998 
 0-996 
 
 1-501 
 
 86 
 
 6 
 
 "-43 
 
 1-484 
 
 86 
 
 6 
 
 0*105 
 
 0*105 
 
 9-567 
 
 0* 105 
 
 9-5H 
 
 1*006 
 
 0-995 
 
 1-466 
 
 84 
 
 7 
 
 0-I22 
 
 0- 122 
 
 8 206 
 
 o-i'J3 
 
 8-144 
 
 i-oo8 
 
 0-993 
 
 1-449 
 
 83 
 
 8 
 
 o- 140 
 
 0*139 
 
 7-185 
 
 0*141 
 
 7-115 
 
 i-oio 
 
 0-990 
 
 1-43' 
 
 82 
 
 9 
 
 o;i57 
 0-175 
 
 o* 156 
 
 ^•392 
 
 0-158 
 0-176 
 
 6-314 
 
 1-012 
 
 0-988 
 
 1*414 
 
 81 
 
 10 
 
 0-174 
 
 5-759 
 
 5-671 
 
 1-015 
 
 0*985 
 
 1-396 
 
 80 
 
 11 
 
 o- 192 
 
 0- 191 
 
 5-241 
 
 0-194 
 
 5-145 
 
 1019 
 
 0*982 
 
 1-379 
 
 79 
 
 12 
 
 0*209 
 
 0*208 
 
 4-810 
 
 0-213 
 
 4-705 
 
 1022 
 
 0-978 
 
 i-;^6i 
 
 78 
 
 13 
 
 0-227 
 
 0225 
 
 4-445 
 
 0-231 
 
 4-33' 
 
 1-026 
 
 0-974 
 
 «-344 
 
 77 
 
 14 
 
 0-244 
 
 0262 
 
 0-242 
 
 ■i-i34 
 
 0-249 
 
 4011 
 
 3-73^- 
 
 1031 
 
 0-970 
 0*966 
 
 1*326 
 I • 309 
 
 76 
 
 16 
 
 0-259 
 
 3-864 
 
 0-268 
 
 1-035 
 
 7& 
 
 16 
 
 0-279 
 
 0-276 
 
 3*628 
 
 0-287 
 
 3 -487 
 
 1-040 
 
 0-961 
 
 1 *292 
 
 74 
 
 17 
 
 0-297 
 
 0-292 
 
 3-420 
 
 0306 
 
 3-271 
 
 I -046 
 
 0-956 
 
 1-274 
 
 73 
 
 18 
 
 0.314 
 
 0-309 
 
 3-236 
 
 0-325 
 
 3-078 
 
 1*051 
 
 0-951 
 
 1-257 
 
 72 
 
 19 
 20 
 
 o*33i! 
 0-349 
 
 0-326 
 
 3-072 
 
 0-344 
 
 2-904 
 
 2-747 
 
 1-058 
 
 0-946 
 0-940 
 
 JL3S> 
 
 1-222 
 
 71 
 
 0-342 
 
 2 924 
 
 0-364 
 
 I -064 
 
 70 
 
 21 
 
 0*367 
 
 o-35« 
 
 2-790 
 
 0-384 
 
 2-605 
 
 1-071 
 
 0-934 
 
 I *204 
 
 69 
 
 22 
 
 0-384 
 
 0-37S 
 
 2-669 
 
 0-404 
 
 2-475 
 
 1*079 
 
 0-927 
 
 1*187 
 
 68 
 
 23 
 
 401 
 
 0-391 
 
 2-559 
 
 0-424 
 
 2-356 
 
 1-086 
 
 o-92i| 
 
 1-169 
 
 67 
 
 24 
 
 0*419 
 0*436 
 
 0*407 
 
 2-459 
 
 0-445 
 
 2*246 
 
 1-095 
 
 0-914! 
 
 I 152 
 
 66 
 
 25 
 
 0-423 
 
 2 '3^6 
 
 0-466 
 
 2-145 
 
 1103 
 
 0-906 
 
 I -134 
 
 65 
 
 26 
 
 0-454 
 
 0-438 
 
 2-281 
 
 0-488 
 
 2 050 
 
 i*m 
 
 0-899 
 
 IH7 
 
 64 
 
 27 
 
 0-471 
 
 0-454 
 
 2-203 
 
 0-510 
 
 1-963 
 
 1122 
 
 0-891 
 
 1 - 100 
 
 63 
 
 28 
 
 0-489 
 
 0-469 
 
 2-130 
 
 0532 
 
 I -881 
 
 1-133 
 
 0-883 
 
 1-082 
 
 62 
 
 29 
 
 506 
 0-524 
 
 0*485 
 
 2-063 
 
 0-554 
 
 1-804 
 1-732 
 
 1-J43 
 
 0-875 
 
 1*065 
 
 61 
 
 30 
 
 0-500 
 
 2-000 
 
 0-577 
 
 I-J55 
 
 0-866 
 
 1-047 
 
 60 
 
 31 
 
 0-541 
 
 o-5'5 
 
 1-942 
 
 0*601 
 
 1-664 
 
 1-167 
 
 0-857 
 
 1-030 
 
 69 
 
 32 
 
 0-559 
 
 0-530 
 
 1-887 
 
 0-625 
 
 1-600 
 
 1-179 
 
 0-848 
 
 1012 
 
 58 
 
 33 
 
 0576 
 
 0-545 
 
 I -836 
 
 0-649 
 
 1-540 
 
 1 • 192 
 
 0*839 
 
 0-995 
 
 57 
 
 34 
 
 0593 
 
 0559 
 
 1*788 
 
 0-675 
 0-700 
 
 1-483 
 
 i*2o6 
 
 0*829 
 
 _°1977 
 0*960 
 
 56 
 
 36 
 
 0-611 
 
 0-574 
 
 »-743 
 
 1*428 
 
 I-22I 
 
 0-819 
 
 06 
 
 36 
 
 0*628 
 
 0-588 
 
 1-701 
 
 0*727 
 
 1-376 
 
 1-236 
 
 0-809 
 
 0*942 
 
 54 
 
 37 
 
 0*646 
 
 0-602 
 
 1-662 
 
 0-754 
 
 1*327 
 
 1-252 
 
 0-799 
 
 0-925 
 
 53 
 
 38 
 
 0-663 
 
 0-616 
 
 1-624 
 
 0-781 
 
 1*280 
 
 1*269 
 
 0-788! 
 
 0-908 
 
 52 
 
 39 
 
 o-68i 
 
 0*629 
 
 '•589 
 
 0-810 
 
 1-235 
 
 1*287 
 
 0-777 
 
 0-890 
 
 51 
 
 40 
 
 0698 
 
 0-643 
 
 '-556 
 
 0839 
 
 1*192 
 
 ^•305 
 
 0*766 
 
 0*873 
 
 60 
 
 41 
 
 0-716 
 
 0-656 
 
 1-524 
 
 0-869 
 
 I- 150 
 
 "•325 
 
 0-755 
 
 0*855 
 
 49 
 
 42 
 
 0733 
 
 0669 
 
 I 494 
 
 0-900 
 
 I- III 
 
 1*346 
 
 o'743 
 
 0-838 
 
 48 
 
 43 
 
 0-750 
 
 0-682 
 
 1*466 
 
 0933 
 
 1*072 
 
 1-367 
 
 0*731 
 
 0*820 
 
 47 
 
 44 
 
 0*768 
 
 o-69<; 
 
 1.440 
 
 0-966 
 
 1-036 
 
 1-390 
 
 0-719 
 
 0*803 
 0*785 
 
 46 
 
 45 
 
 0-785 
 
 0*707 
 
 1-414 
 
 I *ooo 
 
 I -ooo 
 
 1-414 
 
 0-707 
 
 46 
 
 e 
 
 
 COS 6 
 
 Heed 
 
 cote 
 
 tand 
 
 cosec (1 
 
 sin 
 
 6 
 
 
 
 
 
 
 
 
 
 Radinnfl 
 
 Deltrcos 
 
 t 
 
 1 
 
 I rdn. = 57°'29578, 1° = 0*01745 rdn., i rdn. = 206265*, i* = -ooooo484Srdn. 
 
HYPERBOLIC FUNCTIONS. 
 
 ^ 
 
 X 
 
 f 
 
 f ' 
 
 binh X 
 
 cosh X 
 
 tanh JT 
 
 coth j: 
 
 00 
 
 secli X 
 
 cosech X 
 
 0-0 
 
 I-OOC) 
 
 I- 000 
 
 o-ooo 
 
 I -000 
 
 o-oco 
 
 I -occ 
 
 00 
 
 0-1 
 
 I- 105 
 
 0-905 
 
 o-ioc 
 
 I 005 
 
 0- 100 
 
 0-000 
 
 0-996 
 
 lo-ooo 
 
 0-2 
 
 1-221 
 
 0-819 
 
 0-201 
 
 I -020 
 
 0-198 
 
 15 067 
 
 0-980 
 
 4-975 
 
 3 
 
 ••350 
 
 0741 
 
 0-304 
 
 I C45 
 
 0-291 
 
 3-43- 
 
 0-957 
 
 3-289 
 
 0-4 i 
 
 1-492 
 
 0-670 
 
 0-411 
 
 I -081 
 
 0-380 
 
 2-632 
 2-164 
 
 0-925 
 
 2-433 
 
 0-5 
 
 I 629 
 
 0-607 
 
 0-521 
 
 1-128 
 
 0-462 
 
 0-887 
 
 1-919 
 
 6 
 
 1-822 
 
 0549 
 
 0-637 
 
 1-185 
 
 0-537 
 
 1-862 
 
 0-844 
 
 1-570 
 
 0-7 ' 
 
 2-014 
 
 0-497 
 
 0-759 
 
 '-255 
 
 0604 
 
 1-655 
 
 0-806 
 
 1-318 
 
 0-8 
 
 2-226 
 
 0-449 
 
 0-8SS 
 
 '•337 
 
 0-664 
 
 I - 506 
 
 0-748 
 
 1-126 
 
 0-9 
 
 2 • 460 
 
 0407 
 
 I -027 
 
 '•433 
 
 0-717 
 
 1-396 
 
 - 698 
 
 0-974 
 
 10 
 
 2-718 
 
 0-368 
 
 >-i75 
 
 '•543 
 
 0-762 
 
 '-313 
 
 0-648 
 
 0-851 
 
 11 
 
 3-004 
 
 0-333 
 
 1-33^' 
 
 1 • 669 
 
 0-800 
 
 1-249 
 
 0-599 
 
 0-749 
 
 1-2 
 
 3-320 
 
 0-301 
 
 1-509 
 
 1-811 
 
 0-833 
 
 I -200 
 
 0552 
 
 0-663 
 
 13 
 
 3-669 
 
 0-273 
 
 1-698 
 
 1-971 
 
 0-861 
 
 1161 
 
 0-507 
 
 0-589 
 
 1-4 
 
 4'o55 
 
 0-247 
 
 1-904 
 
 2- 151 
 
 0-885 
 
 I- 129 
 
 0-465 
 
 0-525 
 
 1-5 
 
 4-482 
 
 0-223 
 
 2-129 
 
 2-352 
 
 0-905 
 
 I- >o5 
 
 0-425 
 
 0-470 
 
 1-6 
 
 ; 4-953 
 
 0-202 
 
 2-376 
 
 2-577 
 
 0-922 
 
 1-085 
 
 0-388 
 
 0-421 
 
 1-7 
 
 - 5"474 
 
 0-183 
 
 2-646 
 
 2-828 
 
 0-936 
 
 I -069 
 
 0-354 
 
 0-378 
 
 1-8 
 
 6-050 
 
 0- l6t; 
 
 2-942 
 
 3-107 
 
 0-947 
 
 1 -056 
 
 0-322 
 
 0340 
 
 19 
 
 6-6»6 
 
 0-150 
 
 3-268 
 
 3-4«8 
 
 0-956 
 
 1-046 
 
 0-293 
 
 0-306 
 
 2 ' 
 
 7-389 
 
 0-135 
 
 3-627 
 
 3-762 
 
 0-964 
 
 1-037 
 
 0-266 
 
 0-276 
 
 2 1 i 
 
 8' 166 
 
 0- 122 
 
 4.022 
 
 4- '45 
 
 0-970 
 
 1-031 
 
 0-241 
 
 0-249 
 
 2-2 
 
 9-027 
 
 0- I 11 
 
 4-458 
 
 ■3-568 
 
 0-976 
 
 1 -025 
 
 0-219 
 
 0-224 
 
 2-3 
 
 9-975 
 
 o- loo 
 
 4-938 
 
 5-037 
 
 0-980 
 
 1 020 
 
 0-199 
 
 0-203 
 
 2 4 
 
 11-02 
 
 0091 
 
 5-467 
 
 5-556 
 
 0-984 
 
 1-016 
 
 0- 180 
 
 0-183 
 
 2 5 
 
 12-18 
 
 0-082 
 
 6-050 
 
 6-132 
 
 0-987 
 
 1014 
 
 0-163 
 
 0-165 
 
 2-6 
 
 13-47 
 
 0-074 
 
 6 • 694 
 
 6-769 
 
 0-989 
 
 l-OII 
 
 0-148 
 
 0-149 
 
 1 2-7 
 
 14-88 
 
 0-067 
 
 7-406 
 
 7-473 
 
 0-991 
 
 1-009 
 
 0-134 
 
 o-'35 
 
 2-8 1 
 
 16-44 
 
 o-o6i 
 
 8-193 
 
 8-252 
 
 0-993 
 
 1-007 
 
 0-121 
 
 0- 122 
 
 2-9 
 
 i8-i8 
 
 0-0S5 
 
 9-059 
 
 9-114 
 
 0-994 
 
 1-006 
 
 o- no 
 
 o-iio 
 
 3 
 
 20-09 
 
 0-050 
 
 10-02 
 
 10-07 
 
 0-995 
 
 1-005 
 
 0-099 
 
 0-099 
 
 3 1 
 
 22-20 
 
 0-045 
 
 II -08 
 
 ii-ii 
 
 0-996 
 
 I -004 
 
 0-090 
 
 0-090 
 
 3-2 
 
 24-53 
 
 0-041 
 
 12-25 
 
 12-28 
 
 0-997 
 
 1-003 
 
 0-081 
 
 0-082 
 
 3 3 
 
 J27-II 
 
 0-037 
 
 13-54 
 
 13-57 
 
 0-997 
 
 1-003 
 
 0-074 
 
 0-074 
 
 3-4 
 
 29-96 
 
 0-033 
 
 14-96 
 
 15-00 
 
 0-998 
 
 I 002 
 
 0-067 
 
 0-067 
 
 1 3-5 
 
 33-11 
 
 0030 
 
 16-54 
 
 16-57 
 
 998 
 
 1-002 
 
 o-o6o 
 
 0-060 
 
 i 22 
 
 '36 ■ 60 
 
 0-027 
 
 18-29 
 
 18-31 
 
 0-999 
 
 i-ooi 
 
 0-055 
 
 0-055 
 
 3-7 
 
 40 45 
 
 0-025 
 
 20-21 
 
 20 • 23 
 
 0999 
 
 I -001 
 
 0-049 
 
 0-049 
 
 b-8 
 
 44-70 
 
 0-022 
 
 22-34 
 
 22-36 
 
 0-999 
 
 I -ooi 
 
 0-045 
 
 0-045 
 
 3-9 
 
 49-40 
 
 0-020 
 
 24-69 
 27-29 
 
 24 . 7 1 
 
 0-999 
 
 i-ooi 
 
 0-041 
 
 0041 
 
 4-0 
 
 54-60 
 
 o-oi8 
 
 27-3' 
 
 0999 
 
 1001 
 
 0-037 
 
 0-037 
 
 ■ 
 
 If*'>4, then ^approximately) sinh jc - cosh -v - ^ic = %, the rumber 
 V hose Napierian log. is x ; also, tanh ;c = colh ~\ - i. 
 

 
 FIRST 
 
 KLMPTIC 
 
 INIKGRAL, vim 
 
 t, W). 
 
 
 
 u 
 0° 
 
 /;/ = o 
 
 m = -I 
 
 m = -2 
 
 ;// » -3 
 
 w = -4 
 
 HI - 'i\m c -6 
 
 w = -7 
 
 /// « -8 
 
 /;/ a '9 
 
 ;// s 1 
 
 O'OOO 
 
 O'OOO 
 
 0000 
 
 O'OOO 
 
 0-000 
 
 0-000 o-ooo 
 
 0-000 
 
 0-000 
 
 0000 
 
 0000 
 
 5" 
 
 0-087 
 
 0-087 
 
 0-087 
 
 0-087 
 
 0087 
 
 00871 0-087 
 
 0-087 
 
 0*087 
 
 0-087 
 
 0-087 
 
 10" 
 
 0-175 
 
 o'75 
 
 0-175 
 
 0-175 
 
 0-175 
 
 0-175 0-175 
 
 0-175 
 
 0-175 
 
 0-I75 
 
 0-175 
 
 15° 
 
 0-262 
 
 0-262 
 
 0-262 
 
 0262 
 
 0-262 
 
 0-2631 0-263 
 
 0-263 
 
 0-264 
 
 0*264 
 
 0-265 
 
 20° 
 
 0-349 
 0-436 
 
 •J "349 
 
 0-349 
 
 0-350 
 
 0-350 
 
 o-35'i 0-352 
 
 0-353 
 
 0-354 
 
 0-355 
 
 0356 
 
 25° 
 
 0-436 
 
 0-437 
 
 0-438 
 
 0-439 
 
 0-440 
 
 ! 0-441 
 
 0-443 
 
 0445 
 
 0*448 
 
 0-45' 
 
 30° 
 
 0-524 
 
 0-524 
 
 0-525 
 
 0-526 
 
 0-527 
 
 0-529 
 
 0-532 
 
 0-536 
 
 0-539 
 
 o*544 
 
 o*S49 
 
 35° 
 
 0-611 
 
 0-611 
 
 0-612 
 
 0-614 
 
 0-617 
 
 0-620 
 
 0-624 
 
 0-630 
 
 0-636 
 
 0-644 
 
 0-653 
 
 40° 
 
 0-698 
 
 0-699 
 
 0-700 
 
 0-703 
 
 0-707 
 
 0-71^ 
 
 0-718 
 
 0-727 
 
 0-736 
 
 0*748 
 
 0-763 
 
 45° 
 
 0.785 
 0-873 
 
 0-786 
 
 0789 
 
 0792 
 
 0-798 
 
 0-804 
 
 0-814 
 
 0-826 
 
 0-839 
 
 0-858 
 
 0*881 
 
 50° 
 
 0-874 
 
 0-877 
 
 0-882 
 
 0-889 
 
 0-898 
 
 0911 
 
 0-928 
 
 0-947 
 
 0-974 
 
 1*011 
 
 55° 
 
 0-960 
 
 0-961 
 
 0-965 
 
 o-97i 
 
 0-981 
 
 0-993 
 
 1-010 
 
 1-034 
 
 1-060 
 
 1099 
 
 1-154 
 
 60° 
 
 1-047 
 
 1-049 
 
 J-054 
 
 1 -062 
 
 1-074 
 
 1090 
 
 1- 112 
 
 I -142 
 
 1-178 
 
 1-233 
 
 '-3»7 
 
 65° 
 
 I -134 
 
 I -137 
 
 ••'43 
 
 •■'53 
 
 1-168 
 
 I 187 
 
 1-215 
 
 1-254 
 
 I -302 
 
 '•377 
 
 1*506 
 
 70° 
 
 1-222 
 
 1-224 
 
 1-232 
 
 1-244 
 
 • - 262 
 
 1-285 
 
 1-320 
 
 1-370 
 
 »-43' 
 
 J-S34 
 
 '-735 / 
 
 75' 
 
 1-309 
 
 1-312 
 
 1-321 
 
 1-336 
 
 1-357 
 
 '-385I 
 
 1 -426 
 
 1-488 
 
 1-566 
 
 1-703 
 
 2*028 \ 
 
 80° 
 
 1-396 
 
 1-400 
 
 1-410 
 
 1-427 
 
 •-452 
 
 1-485; 
 
 '-534 
 
 i-6o8 
 
 1-705 
 
 1-885 
 
 2-436 
 
 85° 
 
 1-484 
 
 1-487 
 
 1-499 
 
 '•519 
 
 1*547 
 
 i-5«5 
 
 1-643 
 
 '-731 
 
 1-848 
 
 2-077 
 
 3-131 
 
 1 90° 1-571 
 
 '•575 
 
 1-588 
 
 1-610 
 
 1-643 
 
 1-686 
 
 1-752 
 
 1-854 
 
 •-993 
 
 2-275 
 
 00 
 
 
 
 SECOND ELLIPTIC INTEGRAL, e(/,/, tt). 
 
 d 
 
 m = 
 
 m = - 1 
 
 m = -2 
 
 VI = -3 
 
 m= -4 
 
 ;;/ = -5 
 
 /// = -6 
 
 w = - 7 w/ = -8 
 
 m = "9 
 
 m = I 
 
 0° 
 
 0-000 
 
 0-000 
 
 0*000 
 
 0-000 
 
 0-000 
 
 O'OOO 
 
 0-000 
 
 - 000 * 000 
 
 0-000 
 
 0-000 
 
 5° 
 
 0-087 
 
 0-087 
 
 0*087 
 
 0-087 
 
 0*087 
 
 0*087 
 
 0-087 
 
 0*087 0*087 
 
 0-087 
 
 0-087 
 
 10° 
 
 0-175 
 
 0-175 
 
 0*174 
 
 0*174 
 
 0*174 
 
 0*174 
 
 0-174 
 
 0-174 0-174 
 
 0-174 
 
 0*174 
 
 15° 
 
 0-262 
 
 0-262 
 
 0*262 
 
 0*262 
 
 0-261 
 
 0*261 
 
 0-261 
 
 0-260 0-260 
 
 0-259 
 
 0*259 
 
 20° 
 
 0-349 
 
 0-349 
 
 0-349 
 
 0*348 
 
 0-348 
 
 0-347 
 
 0-347 
 
 0*346 0-345 
 
 0-343 
 
 0-342 
 
 25° 
 
 0-436 
 
 0-436 
 
 0*436 
 
 0-435 
 
 0-434 
 
 0*433 
 
 0-431 
 
 0-430 0-428 
 
 0-425 
 
 0*423 
 
 30° 
 
 0-524 
 
 0-523 
 
 0-523 
 
 0*521 
 
 0*520 
 
 0-518 
 
 o-5'5 
 
 0*512 0-509 
 
 0-505 
 
 0500 
 
 35" 
 
 0-611 
 
 0-610 
 
 0*609 
 
 0*607 
 
 0-605 
 
 0-602 
 
 0-598 
 
 0-593 0-588 
 
 0*581 
 
 0574 
 
 40" 
 
 0-698 
 
 0-698 
 
 0*696 
 
 0*693 
 
 0*690 
 
 0*685 
 
 0-679 
 
 0*672 0-664 
 
 0-654 
 
 0*643 
 
 45" 
 
 0-785 
 
 0-785 
 
 0*782 
 
 0-779 
 
 0*773 
 
 0-767 
 
 0-759 
 
 0-748: 0-737 
 
 0723 
 
 0-707 
 
 50° 
 
 0*873 
 
 0-872 
 
 0*869 
 
 0*864 
 
 0-857 
 
 0-848 
 
 0-837 
 
 0-823: 0-808 
 
 0-789 
 
 0-766 
 
 55" 
 
 0*960 
 
 o*9S9 
 
 0-955 
 
 0-948 
 
 0*939 
 
 0928 
 
 0-914 
 
 0-895 
 
 0-875 
 
 0*850 
 
 0-819 
 
 60° 
 
 1*047 
 
 1-046 
 
 I 041 
 
 1*032 
 
 1021 
 
 1-008 
 
 0-989 
 
 0965 
 
 0-940 
 
 0*907 
 
 0-866 
 
 65" 
 
 I -134 
 
 1*132 
 
 1*126 
 
 1*116 
 
 1*103 
 
 I -086 
 
 1-063 
 
 ••033 
 
 o-ooi 
 
 0*960 
 
 0*906 
 
 70" 
 
 1*222 
 
 1*219 
 
 1*212 
 
 1-200 
 
 1*184 
 
 1-163 
 
 »-i35 
 
 1-099 
 
 1-060 
 
 1-008 
 
 0*940 
 
 75° 
 
 1-309 
 
 1*306 
 
 1-297 
 
 1*283 
 
 1*264 
 
 1-240 
 
 1-207 
 
 1-163 
 
 r-117 
 
 I -053 
 
 0-966 
 
 80" 
 
 1-396 
 
 1-393 
 
 i-3«3 
 
 1-367 
 
 1-344 
 
 1-316 
 
 1-277 
 
 1-227 
 
 1-172 
 
 1-095 
 
 0*985 
 
 85" 
 
 1-484 
 
 1*480 
 
 1-468 
 
 1-450 
 
 1*424 
 
 1*392 
 
 »-347 
 
 1-289 
 
 1-225 
 
 I -'35 
 
 0996 
 
 90" 
 
 i-S7» 
 
 1*566 
 
 1-554 
 
 J-533 
 
 i-504 
 
 1-467 
 
 1-417 
 
 I-35I 
 
 1-278 
 
 »-i73 
 
 1*000 
 
m 3 •( 
 
 ) m s I 
 
 O-CKX 
 
 ) 0000 
 
 o-o8; 
 
 1 0087 
 
 o-i7( 
 
 i 0-175 
 
 0'26i) 
 
 ^ 0-265 
 
 o-35i 
 
 0356 
 
 0-448 
 
 0-45' 
 
 0-544 
 
 0-549 
 
 0-644 
 
 0-653 
 
 0-748 
 
 0-763 
 
 0-858 
 
 0-881 
 
 0-974 
 
 i-oii 
 
 I 099 
 
 I -154 
 
 ••233 
 
 '•3«7 
 
 '•377 
 
 1-506 
 
 1-534 
 
 »-735 / 
 
 I - 703 
 
 2-028 \ 
 
 1-885 
 
 2-436 
 
 2-077 
 
 3-'3i 
 
 2-275 
 
 00 
 
 I = -9 
 
 1 
 m = I \ 
 
 o-ooo 
 
 ooooj 
 
 0-087 
 
 0-087 
 
 0-174 
 
 0-174 
 
 0-259 
 
 0-259 
 
 0-343 
 
 0-342 
 
 3-425 
 
 0-423 
 
 3-505 
 
 0-500 
 
 3-581 
 
 0574 
 
 3-654 
 
 0-643 
 
 3-723 
 
 0-707 
 
 )-789 
 
 0-766 
 
 )-85o 
 
 0-819 
 
 )-907 
 
 0-866 
 
 )-96o 
 
 0-906 
 
 •008 
 
 0-940 
 
 •053 
 
 0-966 
 
 -095 
 
 0-985 
 
 •135 
 
 0-996 
 
 -173 I -000 1