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AN 
 
 ELEMENTARY TREATISE 
 
 ON 
 
 VARIABLE QUANTITIES 
 
COPYRIGHT, 1921 
 
 BY 
 ASAHBL, R. COOK 
 
AN ELEMENTARY TREATISE ON 
 
 VARIABLE QUANTITIES 
 
 IN TWO PARTS: 
 
 THE DIRECT AND INVERSE 
 By HIRAM COOK V 
 
 
 
 PRIVATELY PRINTED 
 
 LEDERER, STREET & ZEUS COMPANY 
 
 BERKELEY. CALIFORNIA 
 
Hiram Cook was born in Preston, New 
 London County, Connecticut, on Decem- 
 ber 11, 1827, and died at Norwich, 
 Connecticut, on May 26, 1917. This 
 book, published after his death, stands 
 witness to his lifelong love of mathe- 
 matics and his desire to put his knowledge 
 in this field at the service of others. 
 
 16'649 
 
PREFACE 
 
 The subject of this work is the same as that of the 
 Differential and Integral Calculus, but parts of it are treated 
 somewhat differently, especially the fundamental principles; 
 and these, it is believed, are made so clear that any ordinary 
 algebraic student can readily comprehend them. 
 
 In regard to a variable quantity, it is taken to mean as 
 qualified — that is, its value is subject to a continual change, 
 either increasing or decreasing. Now this being the case, it 
 is evident that its value must have some rate of increase or 
 decrease, uniform or variable, according to governing condi- 
 tions. Upon this theory this work is founded, and it is hoped 
 it so clears the way that it can be understandingly followed 
 by those who are so inclined. 
 
 How is it in regard to a differential, so called, and the 
 process of finding it? First an increment is added to the 
 variable, and finally, in order to obtain what is sought, this 
 increment is made equal to zero and to something at the same 
 time — the something being taken as the differential of the 
 variable. No wonder the student becomes nonplussed, for it 
 is very difficult to conceive how even an infinitesimal, or "the 
 last assignable value of a quantity" and zero can be identical. 
 Being confronted by such a dilemma, he either has to accept 
 the doctor's diagnosis or give the matter up in disgust. 
 
 Let it not be imagined that this work is claimed to be 
 perfect by its author, or that he considers himself more than 
 a tyro compared with the great mathematicians of the past or 
 present. He simply gives his theory of the subject, believing 
 it to be correct and both reasonable and comprehensible, and 
 if approved, even by a few, he will not feel he has labored 
 wholly in vain. It is a hard matter, however, to persuade a 
 man to part with his idols ; therefore, since Infinitesimal was 
 born lang syne and has done good service, possibly it is 
 unreasonable to expect that the little fellow should be 
 summarily dismissed. 
 
 Hiram Cook. 
 
 Norwich, Connecticut 
 1916 
 
CONTENTS OF PART I 
 
 Subject 
 
 Article Page 
 
 Definitions 
 
 ALGEBRAIC FUNCTIONS 
 
 Illustrations of rates ...... 
 
 Rate of u = a.t-" . . . . 
 
 Rate of the sum or difference of several variables 
 
 Rate of the sum or difference of several terms 
 
 Rate of the nth power of tw^o variables 
 
 Rate of the product of several variables 
 
 Rate of the product of several factors 
 
 Rate of a polynomial ..... 
 
 Rate of the square root of a quantity 
 Rate of a fraction . . . 
 
 Successive rates and ratal coefficients . 
 Successive rates of two or more variables 
 Special rates ...... 
 
 Classified rates ...... 
 
 Maclaurin's and Taj'lor's Theorems 
 
 LOGARITHMIC FUNCTIONS 
 
 Rate of u = a^' and x = log tt . . 
 Rate oi u=v->' . 
 Rates relative to logarithmic tables 
 Illustration of principles relative to curves 
 
 CIRCULAR FUNCTIONS 
 
 Rates of sine, cosine, etc., in terms of the arc 
 Rates of log sine, log cosine, etc.. 
 Rate of the arc in terms of sine, cosine, etc. 
 Development of sin x, cos x, etc. 
 
 1-8 
 
 9 
 
 S 
 
 10 
 
 6 
 
 . 11 
 
 8 
 
 12 
 
 8 
 
 . 13 
 
 9 
 
 14 
 
 9 
 
 15-16 
 
 10 
 
 17 
 
 11 
 
 . 18 
 
 11 
 
 19 
 
 12 
 
 . 20 
 
 13 
 
 21 
 
 13 
 
 . 22 
 
 15 
 
 23 
 
 15 
 
 24-25 
 
 16 
 
 26-27 
 
 20 
 
 . 28 
 
 22 
 
 29 
 
 23 
 
 . 30 
 
 25 
 
 . 31 
 
 26 
 
 32 
 
 29 
 
 . 33 
 
 30 
 
 . 34 
 
 32 
 
 Value of 11 = R 
 
 VANISHING FRACTIONS 
 
 {x — a)'" x» — a» tanji: 
 
 (.1- — a)' 
 
 cot 2x 
 
 -, etc. 
 
 35 33 
 
 Signification of first and second ratal coefficients 
 Curves concave and convex to the axis of x 
 Ratal equations of lines of the first and second order 
 Tangents, normals, etc., of curves .... 
 
 The cycloid ....... 
 
 The logarithmic curve ...... 
 
 Asymptotes ....... 
 
 Rates of arc, area, etc. ..... 
 
 Radius of curvature ...... 
 
 Osculatory circle ...... 
 
 Radius of curvature of lines of the second order 
 Evolutes and involutes ...... 
 
 Curves referred to polar coordinates 
 
 Tangents and normals of spirals .... 
 
 Rates of arc and area of spirals 
 
 Radius of curvature of spirals .... 
 
 Singular points of curves ..... 
 
 Maxima and minima of functions of a single variable 
 Maxima and minima of functions of two or more variables 67 
 
 . 36 
 
 38 
 
 37 
 
 39 
 
 38-39 
 
 41 
 
 40-41 
 
 43 
 
 42-43 
 
 46 
 
 44 
 
 47 
 
 . 45 
 
 49 
 
 46-49 
 
 52 
 
 50-51 
 
 54 
 
 . 52 
 
 57 
 
 S3 
 
 59 
 
 54 
 
 62 
 
 55 
 
 67 
 
 56 
 
 71 
 
 . 57 
 
 75 
 
 58 
 
 76 
 
 59-63 
 
 78 
 
 64-66 
 
 88 
 
 95 
 
CONTENTS OF PART II 
 
 Subject Article Page 
 
 Definitions and illustrations ..... 68-71 101 
 
 SIMPLE ALGEBRAIC RATES 
 
 Integral of a monomial rate ..... 72 102 
 
 Integral of the sum or difference of several rates . . 73 103 
 
 Integral of a rate of the form du = (a + bx''^)^ x^-^dx 74 104 
 
 Simple circular rates ...... 75-76 106 
 
 Integration by series ...... 77 108 
 
 BINOMIAL RATES 
 
 Any binomial rate reduced to the form 
 
 du = (o -f bxn)r x'^dx 78 111 
 
 INTEGRATION BY PARTS 
 
 Formulas .4 and 5 79 113 
 
 Formulas C and D 80 118 
 
 RATIONAL FRACTIONAL RATES 
 
 When the factors of the denominator are rational . 81 119 
 
 When the factors of the denominator are imaginary . 82 123 
 
 Irrational fractional rates .... . 83-84 126 
 
 TRANSCENDENTAL RATES 
 
 Exponential rates ...... 
 
 Logarithmic rates ..... 
 
 Circular rates ....... 
 
 Bernouilli's series ...... 
 
 Successive integration ..... 
 
 Integration of partial rates .... 
 
 Integration of total rates ..... 
 
 Integration of homogeneous rates 
 
 Length of curves ...... 
 
 Area of curves ...... 
 
 Surface of revolution ..... 
 
 Volume of revolution ..... 
 
 Curved surfaces and solids referred to three coordinate 
 axes ........ 
 
 Curve of pursuit ...... 
 
 . 85 
 
 130 
 
 86 
 
 132 
 
 . 87 
 
 134 
 
 88 
 
 136 
 
 . 89 
 
 138 
 
 90 
 
 140 
 
 . 91 
 
 141 
 
 92 
 
 143 
 
 93 
 
 144 
 
 94 
 
 149 
 
 . 95 
 
 153 
 
 96-97 
 
 157 
 
 e 
 
 . 98 
 
 162 
 
 99 
 
 164 
 
PART ONE 
 DIRECT METHOD 
 
PART ONE 
 
 DIRECT METHOD 
 
 DEFINITIONS 
 
 Art. 1. Two classes of quantities are employed: namely, 
 constants and variables. Constants are usually represented by 
 the first letters of the alphabet, a, b, c, etc., and variables by the 
 last, u, X, y, etc. 
 
 The value of a constant remains the same throughout the 
 same investigation ; while that of a variable continually in- 
 creases or decreases at either a uniform or variable rate. 
 
 2. The variable whose rate of increase or decrease is as- 
 sumed to be uniform is called the independent variable, and the 
 variable whose value depends on that of the independent vari- 
 able is called the dependent variable. Thus u is the dependent 
 and X the independent variable in 
 
 u = ax^ -{- b. 
 
 3. The dependent variable is a function of the independent 
 variable. Thus w is a function of x in 
 
 u=^x^-\- ax -{- b, 
 
 which is expressed generally thus, u = f(x), in which f is 
 simply a symbol denoting function. 
 
 4. Functions are of two general classes, algebraic and trans- 
 cendental. 
 
 A function is algebraic when the dependent variable equals 
 the expression containing the independent variable in a purely 
 algebraic form, as 
 
 u^a^ — x^. 
 
 A function is transcendental when the dependent variable 
 equals the expression containing the independent variable in 
 
4 AN ELEMENTARY TREATISE 
 
 the form of an exponent, logarithm, sine, cosine, tangent, etc., 
 as in 
 
 u = a^ ; u^ log x; u = sin x; u = cos x; u = tan x , etc. 
 
 Transcendental functions are of two classes, logarithmic 
 and circular. 
 
 5. Functions are also explicit, implicit, increasing, and 
 decreasing. 
 
 An explicit function is one in which the dependent variable 
 is directly expressed in terms of the independent variable, as in 
 
 u = ax^ -\- b OT u = log X. 
 
 An implicit function is one in which the value of the func- 
 tion is not directly expressed in terms of its variable and 
 constants. Thus in the equation 
 
 y^ -\- axy -\-bx^ -f- c = 
 
 y is an implicit function of x — that is, y is not directly ex- 
 pressed in terms of x and the constants a, b, and c. 
 
 An increasing function is one in which the dependent 
 variable will increase when the independent variable increases, 
 or will decrease when the independent variable decreases, as in 
 
 u = ax^ -\- b. 
 
 A decreasing function is one in which the dependent vari- 
 able will increase when the independent variable decreases, or 
 will decrease when the independent variable increases, as in 
 
 1 
 
 u^ — . 
 
 X 
 
 6. A function may consist of two or more independent 
 variables, as 
 
 w = ojir ± by ± C2 or u = axy 2. 
 
 7. The rate of a variable — that is, its rate of increase or 
 decrease — is designated by writing d before it, as du represents 
 the rate of u, dx of x, dy of y, etc. 
 
 8. A ratal coefficient is the rate of the dependent variable 
 . du 
 
 divided by that of the independent variable. Thus -— is the 
 
 dx 
 
 ratal coefficient of w = / (.*:). 
 
ON VARIABLE QUANTITIES 
 
 Algebraic Functions 
 9. Illustrations of the application of the rates of variables 
 
 A 
 
 B 
 
 e: 
 
 c 
 
 o 
 
 J 
 
 F 
 
 Fig. 1 
 
 represent the area 
 or du, the rate of u. 
 
 IS 
 
 Let the side AC oi the rectangle 
 ABCD (Fig. 1) be represented by a, 
 the side CD by x, and the area by 
 u = ax. Extend AB to E, CD to F, 
 and draw EF parallel to BD. Now- 
 let dx, the rate of increase of x, be 
 represented by DF, and the area of 
 ABCD by w = ax; then adx will 
 
 of BEDF, the rate of increase of ABCD, 
 
 Therefore the rate of 
 
 (1) 
 
 (2) 
 
 ax 
 
 du = adx = ax^~'^ dx. 
 
 Extend AB of the rectangle ABCD (Fig. 2) to i^ and G, 
 also CD to L and H, and draw KL, EF, and G// parallel to 
 AC. Now let AC he represented by a, FD by x, CF by y, and 
 the area of ABCD by ax -j- ay; also let dx, the rate of ;ir, be 
 represented by DH, and dy, the rate of y, by LC ; then aa?;r will 
 
 represent the area of BGDH, 
 the rate of increase of the 
 area of EBFD, and ady will 
 represent the area of KALC, 
 the rate of increase of the area 
 will represent the rate of in- 
 crease of the area of ABCD, 
 of AECF ; hence adx -{- ady 
 Therefore the rate of 
 
 u=^ax -{- ay (3) 
 
 du = adx -f- ady. (4) 
 
 Extend the side AB of the rectangle ABCD (Fig. 3) to 
 G, CD to H, and draw EF, KL, and GH parallel to BD. Now 
 let AC ht represented by a, CD by x, FD by y, CF by jr — y, 
 and the area of AECF by M = ajr — ay; also let dx, the rate 
 
 of X, be represented by DH, 
 and afy, the rate of y, by Z)L; 
 then adx will represent the 
 area of BGDH, the rate of in- 
 crease of ABCD, ady that of 
 BKDL, the rate of increase 
 of EBFD, and adx— ady, that 
 of KGLH, the rate of increase 
 of ^5CL) less that of EBFD, 
 
 is 
 
 Fig. 3 
 
6 AN ELEMENTARY TREATISE 
 
 or du, the rate of u. Therefore the rate of 
 
 u = ax — ay (5) 
 
 is du^adx — ady. (6) 
 
 Extend the side AB of the rectangle ABCD (Fig. 4) to G, 
 CD to H, AC to E, ED to F, and draw EE parallel to AB, 
 also GH to i?D. Now let CD be represented by ax, AC hy y, 
 
 and the area of ABCD 
 
 by li = a;r3r. Let adx, 
 
 the rate of a;?;, be repre- 
 
 [^ iB G sented by DH, and (Z^', 
 
 M 
 E F 
 
 the rate of y, by y4£; 
 I N then axdy will represent 
 
 I the area of EEAB, the 
 
 I rate of increase of the 
 
 C OH area of ABCD in the di- 
 
 rection of M, and aydx 
 r 1 ^^ 4- ^jl[ represent the area 
 
 of BGDH, the rate of increase of the area of ABCD in the 
 direction of A''. Hence axdy -f- adyx will represent the total 
 rate of increase of the area of ABCD, or du, the rate of u. 
 Therefore the rate of 
 
 u^axy (7) 
 
 is du = axdy -\- aydx. (8) 
 
 10. Let y^x, then dy = dx. Substituting x for 3; in (7) of 
 the last article, also x for 3; and dx for dy in (8), then 
 
 w = ax^ ( 1 ) 
 
 and c?M = axdx -\- axdx = 2axdx = 2ax'^~'^dx. (2) 
 
 Let y=^x'^, then (/y ^ 2jv(/jir. Substituting ;?;- for y in (7) 
 of the last article, also x^ for y and 2;ircf;ir for dy in (8), then 
 
 Mi^a^tr^ (3) 
 
 and du = 2a.ar-flf;r -\- ax^dx = 3ax^dx = 3ax^~^dx. (4) 
 
 Let y = :r^ then dy^Sx^dx. Substituting as before, the 
 rate of 
 
 u^:^ax^ (5) 
 
 is du = Ax^-^dx. (6) 
 
 Hence, if the exponent of x is n, n. being a positive integer, 
 
 from (1) of Art. 9 and from (2), (4), and (6) of the present 
 
 article it is evident that the rate of 
 
 u = ax'' (7) 
 
 is du^:^anx^'~'^dx. (8) 
 
ON VARIABLE QUANTITIES 7 
 
 When n is negative, as in 
 
 u^ax-'^, (9) 
 
 multiplying both sides by jr" gives 
 
 ux'^ _— ^ 
 
 Passing to the rate, 
 
 x'^du -\- nux'^''^dx = 0. 
 
 Transposing and substituting for u its value, 
 
 x^^du = — anx~'^dx, 
 
 and dividing by x'\ du = — anx~'^-^dx. (10) 
 
 When the exponent of ;ir is a positive fraction, as in 
 
 u = ax''\ (11) 
 
 raising both sides of the equation to the ^th power, 
 
 u? = a^x''. 
 
 Passing to the rate, 
 
 su^~'^du = a^rx''~^dx. (12) 
 
 Raising (11) to the (s — l)th power and multiplying by s, 
 
 j^s-i ^ a'-^sx'-''/^, (13) 
 
 and dividing (12) by (13), 
 
 r 
 du^=a—x'''^-^dx. (14) 
 
 s 
 
 When the exponent of ;ir is a negative fraction, as in 
 
 u=^ax-'-'', (15) 
 
 raising both sides of the equation to the sih power and multi- 
 plying by x'^ give 
 
 u^x^' =^ a*. 
 Passing to the rate, 
 
 su'^''^x^du -\- ru^x'"'^dx = 0. 
 Transposing and dividing by su^-'^x'' give 
 
 r 
 
 du^ — — ux'^dx, 
 s 
 
 or, smce u = ax~ 
 
 r 
 du^ — a—x-'^/^-^dx. (16) 
 
 s 
 
8 AN ELEMENTARY TREATISE 
 
 Hence, the rate of a variable affected with any constant 
 exponent, -\- or — , having also any constant coefficient, is the 
 product of the coefficient and the exponent of the variable, 
 multiplied by the variable with its exponent less unity, into 
 the rate of the variable. 
 
 EXAMPLES 
 
 1. u = ax"-'^'^ 2. M = ax^''^ 
 
 1 
 
 3. M^^7;ir(»+l)/n 4. u ^ — CX-^ 
 
 2 
 
 11. To determine the rate of a function of the sum or dif- 
 ference of several independent variables, as 
 
 u = av -\- bx -\- cy -}- es, ( 1 ) 
 
 assume u = r -\- s, r = av -\- bx, and s = cy -\- ez, the rates of 
 which [see (4) and (6) of Art. 10] are respectively 
 
 du = dr -\- ds, dr = adv + bdx, and ds ^= cdy -\- edz. (2) 
 
 Substituting the values of dr and ds in du ^ dr -\- ds gives 
 
 du = adv -j- bdx -f- cdy -f- edz. (3) 
 
 Hence the rate of the sum or difference of several inde- 
 pendent variables is the corresponding sum or difference of 
 their rates taken separately. 
 
 12. To determine the rate of 
 
 u = ax — bx^ -f- cx"^. ( 1 ) 
 
 Assume v = ax, y = bx', and z = ex"', 
 
 the rates of which are [see (2) of Art. 9, and (2) and (10) of 
 Art. 10] 
 
 dv = adx, dy = 2bxdx, and dz = cnx^^~'^dx. (2) 
 
 But, according to the assumption, 
 M = t/ — y -\- 2, 
 
 the rate of which is [see (1), Art. 11] 
 
 du^dv — dy-{-dz; (3) 
 
 therefore, substituting in (3) the values of dv, dy, and dz, then 
 
 du = adx — Zbxdx -f- cnx"-'^dx. (4) 
 
ON VARIABLE QUANTITIES 9 
 
 Hence it is evident that the rate of the sum or difference of 
 any number of terms containing the same independent variable 
 is the corresponding sum or difference of their rates taken 
 separately. 
 
 13. Required the rate of 
 
 u= {ax ± byy. (1) 
 
 Assume u = v^; (2) 
 
 then v = ax±:by. (3) 
 
 Now the rate of (2), from Art. 10 is 
 
 du = nV'-^dv, (4) 
 
 and the rate of (3) [see (3) and (5), Art. 9] is 
 
 dv == adx dz bdy. 
 
 But v'^~^ = {ax ± by)'^-'^ , therefore, by substituting in 
 (4) the values of z^""^ and dv, the result is 
 
 du = n {ax -\- by)'^~'^ {adx -\- bdy) (5) 
 
 or du = n {ax -f- by)"'^adx -\- n {ax -\- by)^^'^bdy. (6) 
 
 Hence, the rate of the nth power of the sum or difference 
 of two variables, is n times their sum or difference raised to 
 the {n — \)th power, multiplied by the sum, or difference of 
 their rates, whether n be an integer or fraction, positive or 
 negative. 
 
 EXAMPLES 
 
 1 
 
 I. u = x^ — — x4-2x^^^ 2. u = x''-\-ax-'- ^b 
 
 4 
 
 3. u = ax'^-'^ + nx'^''^ 4. u= {x + av)'*^^ 
 
 14. To determine the rate of 
 
 u = vxy. 
 Assume s^xy; ( 1 ) 
 
 then u = vs, 
 
 and the rates of these, from Art. 9, are 
 
 ds = xdy -\- ydx (2) 
 
 and du^vdz -\- zdv. (3) 
 
10 AN ELEMENTARY TREATISE 
 
 Substituting the value of z from (1); and ds from (2), in 
 (3), then 
 
 du = vxdy -\- vydx -j- xydv. (4) 
 
 Hence it is evident that the rate of the product of any num- 
 ber of variables is the sum of the products obtained by multi- 
 plying the rate of each variable by the product of the others. 
 
 15. To determine the rate of 
 
 u = x''(a-\-x) {bx^ -\- cx^). 
 Assume v = a -\- x (1) 
 
 and y==bx^-\-cx'^\ (2) 
 
 then u = x^vy. 
 
 Passing to the rate, (1) becomes 
 
 dv = dx, (3) 
 
 (2), by Art. 12, 
 
 dy = 2bxdx -f- cnx^''^dx = (2bx -{- cnx"^-^) dx, (4) 
 and u = x'^'vy, by Art. 14, 
 
 du = x^vdy -\- x^'ydv -{- rx^'^vydx. ( 5 ) 
 
 Substituting the values of v and y from (1) and (2), also 
 the values of dv and dy from (3) and (4), in (5), the result is 
 
 du = X'' (a -\- x) (2bx -\- cnx"-^) dx -\- x'' (bx^ -f cjr") dx + 
 
 rx^~^ (a -\- x) (bx' -\- ex") dx, 
 
 or du^ {x'' {a-\- x) (2bx -\- cnx''-'^) + 
 
 X'' (bx^ + ex'') -j- rx''-'^ (a^ x) (bx^ + ex"") ]dx. 
 
 Hence, the rate of the product of any number of factors 
 containing the same variable is the sum of the products ob- 
 tained by multiplying the rate of each factor by the product of 
 the others. 
 
 16. To determine the rate of 
 
 u=:x^(a — z") (b-\-y). 
 
 Assume v=a — s''^ ( 1 ) 
 
 and w^b-{-y^', (2) 
 
 then M = xH'w. ( 3 ) 
 
ON VARIABLE QUANTITIES 11 
 
 The rate of (1) is 
 
 dv = — nz''-^dz, (4) 
 
 that of (2) dw=^ry'-^dy, (5) 
 
 and that of (3) 
 
 du = x^vdw + x^wdv -{- 2xvwdx. (6) 
 
 Substituting the values of v and w from (1) and (2), also 
 the values of dv and dw from (4) and (5), in (6) gives 
 du = rx^ (a — 2"") y-^dy — nx^ (b -{- y ) z'^-'^dz -\- 
 2x (& + 3''") (a — ^") dx. 
 
 Hence the preceding rule is also applicable when each fac- 
 tor contains a different variable, or, as is evident, even when 
 each factor contains several variables. 
 
 EXAMPLES 
 
 1. u^x^yz'^ 2. u= {bx -\- c) {x" -\- ax) 
 
 3. u = x^ {y^ -\- av) 
 
 17. To determine the rate of 
 
 u= {a -{- bx -{- ex- ) ". 
 Assume y = a -\- bx -\- cx^ ; ( 1 ) 
 
 then u == y^. 
 
 Passing to the rate, 
 
 dy^^{b-\-2cx)dx (2) 
 
 and du = ny^''^dy. (3) 
 
 Substituting the value of y from (1), and dy from (2), in 
 (3), then 
 
 du = n (a -\- bx -{- cx^)"''^ (^ + 2cx) dx. 
 
 Hence, the rate of a polynomial affected with atiy constant 
 exponent is the exponent into the polynomial with its ex- 
 ponent less unity, multiplied by the rate of the polynomial. 
 
 18. To determine the rate of 
 
 u^\/ {ax -\- bx^) 
 or M ^ (ax + bx^)^''^. 
 
12 AN ELEMENTARY TREATISE 
 
 Passing to the rate, as in Art. 17, 
 
 1 
 
 du = — {ax -\- bx'^)~'^ (o + nbx'^~^) dx, 
 
 (a -}- nbx"~^) dx (a -{- nbx^^~^) dx 
 or du = ^= . 
 
 2 {ax + bx" y^ 2 V («■*■ + bx" ) 
 
 Hence, the rate of the square root of a quantity is the rate 
 of the quantity under the radical, divided by twice the radical. 
 
 19. To determine the rate of a fraction, as the function 
 
 V 
 
 z 
 Multiplying through hy z, uz=^v; 
 
 then passing to the rate, by (7) of Art. 9, 
 udz -\- zdu == dv. 
 Substituting for u its value and transposing give 
 
 vdz 
 
 zdu f^dv — 
 
 z 
 
 zdv — vdz 
 
 or zdu ■ 
 
 z 
 Therefore, dividing by z, 
 
 zdv — vdz 
 du=^ '. 
 
 Hence the rate of a fraction is the denominator into the 
 rate of the numerator, minus the numerator into the rate of 
 the denominator, divided by the square of the denominator. 
 
 If 2/ be a constant, then, since a constant has no rate, 
 
 vdz 
 
 du^^^ — ; 
 
 z'~ 
 
 that is, when t/ is a constant, w is a decreasing function of z 
 and its rate is consequently negative. 
 
 EXAMPLES 
 
 1. u^^{l -\- x^) 3. M=;tr« (.^ — a) (a — x-) 
 
 2. u = ^y(x^-\-y^) 4. u 
 
 VGr+1)— V(.r— 1) 
 
ON VARIABLE QUANTITIES 12 
 
 Successive Rates and Ratal Coefficients 
 20. In obtaining these, and at the same time to exemplify 
 the work, let 
 
 u^x'' -\- ax^. ( 1 ) 
 
 Passing to the rate, by Art. 12, 
 
 du= (nx^^~'^ -j- 2ax) dx. (2) 
 
 Passing to the rate again, regarding dx as constant, 
 
 d (du) =d^u={n (n—1) x""-^ ^2a]dx\ (3) 
 
 In like manner it will be found from (3) that 
 
 d^u = n (n — 1 ) (n — 2) x'^'^dx^. (4) 
 
 (2), (3), and (4) are successive rates of (1), and 
 (^^n-i _^ 2ax), [n (n — 1) x"-'^ + 2a} 
 
 and {n (n — 1) (n — 2) .r"~^} are respectively coefficients of 
 dx, dx^, and dx^. 
 
 Dividing (2) by dx, (3) by dx'^, and (4) by dx^, the results 
 
 are 
 
 du 
 
 = n.r""^ -\- 2ax 
 
 dx 
 
 (5) 
 
 d^u 
 
 = n (n — 1 ) x"-^ + 2a 
 
 dx'- 
 
 (6) 
 
 d'u 
 
 = n(n — 1) (n — 2)x''-\ 
 
 dx' 
 
 (7) 
 
 and 
 
 du d^u d'u 
 
 Inasmuch as , , and are respectively equal to 
 
 dx dx^ dx' 
 
 the coefficients of dx, dx^, and dx', they are called ratal co- 
 efficients ; du, d^u, and d'u are the first, second, and third rates 
 of the dependent variable u, and dx, dx^, and dx' are the first, 
 second, and third powers of the rate of the independent vari- 
 able X. 
 
 Rates of Functions 
 OF Two OR More Independent Variables 
 
 21. It has been shown in Art. 9 that the rate of u = xy is 
 
 du = ydx -\- xdy ; therefore, 
 
14 AN ELEMENTARY TREATISE 
 
 if M = x'^y, 
 
 its rate is du = nx'^-'^ydx + ji;"c?3; ; 
 
 and if u = x''y'^, 
 
 its rate is a?w ^ M;ir""^3;'^c?;ir + mx^y'^'''^dy ; 
 
 also if M ^ ;r«3;'« + ;ir'"3'^ (1) 
 
 its rate is 
 
 du = nx^'-'^y'^dx + Wjir'^3;"»-iflf3; + rx'-^y'dx + j;»r''y«-^fl?3'. (2) 
 
 See preceding rules. 
 
 Now if the rates of ( 1 ) be first taken under the supposition 
 that X varies and y remains constant, then that y varies and x 
 remains constant, the sum of the results will be the same as 
 (2) : thus 
 
 du = nx'^'^y^dx -\- rx^'^y^dx (3) 
 
 and du = mx^'y^"'~'^dy -{- sx^'y^'^dy. (4) 
 
 Adding (3) and (4), 
 
 du = nx"'^y"^dx + rx'"^y^dx -\- mx'^y^'^dy -f- sx''y^~^dy. (2) 
 
 Dividing (3) hy dx and (4) by dy, the results are 
 
 du 
 
 = nx"-'^y"' + rx''-^y^ ( 5 ) 
 
 dx 
 
 du 
 
 and = mx'^y^"'''^ -\- sx''y^~'^. (6) 
 
 dy 
 
 By taking the rate of (5) with respect to y and the rate of 
 (6) with respect to x, the following are found, 
 
 d^u 
 
 = mnx"-'^y"'-^ + rsx^'-'^y^-^ (/) 
 
 dxdy 
 
 d^u 
 
 and = mnx^'-'^y'"-'^ + rsx''-'^y^-^ (8) 
 
 dydx 
 
 in which the right-hand members are identical ; therefore 
 
 d^u d'^u 
 
 dxdy dydx 
 
 (2) is called the total rate, (3) and (4) partial rates, and 
 (5) and (6) ratal coefficients. The second, third, and higher 
 rates can be found in a similar manner as in Art. 20. 
 
ON VARIABLE QUANTITIES 15 
 
 If the function contains three independent variables, as 
 
 by proceeding in like manner the following results will be 
 obtained : 
 
 d^u d^u d^u d^u d'U d^u 
 
 dxdy dydx dxdz dzdx dydz dzdy 
 
 If the function contains four independent variables, there 
 will be six of these equalities, if five there will be ten, and so on. 
 
 Special Rates 
 
 22. Let u= {x ^yY, (1) 
 
 then, by taking the rate first with respect to x and secondly 
 with respect to y, the following are found, 
 
 du^n{x -^yY'^dx (2) 
 
 and du^=n {x -\- yy^^'^dy. (3) 
 
 Dividing the first by dx and the second by dy gives 
 
 du 
 
 — - = w(^ + y)"-i (4) 
 
 dx 
 
 du 
 
 and ——^=n{x-\-yy-'^ (5) 
 
 dy 
 
 in which it will be observed that the right-hand members are 
 identical. 
 
 The sum of (2) and (3) is 
 
 du=^n (x -\- y)^~'^(dx + dy), 
 
 virtually the same as given in Art. 13. 
 
 Classified Rates 
 
 23. When the partial rates of a function of two or more 
 independent variables are taken with respect to one variable 
 only, they are said to be of the first class; when taken with 
 respect to one variable and that rate taken with respect to an- 
 other variable, they are said to be of the second class : thus, if 
 
 u = x^y^ + x^y, ( 1 ) 
 
 by taking the rate with respect to x only, the results are 
 
 du = Sx^y^dx -\- 2xydx 
 
 d^u = 6xy^dx^ -\- 2ydx^ 
 and d^u = Gy^dx^, 
 
 which are partial rates of the first class. 
 
16 AN ELEMENTARY TREATISE 
 
 Taking the rate of (1) with respect to ;r gives 
 
 du = Sx^y'^dx + 2xydx (2) 
 
 and this rate taken with respect to y is 
 
 d^u == 6x^ydxdy -\- 2xdxdy ; (3) 
 
 (2) and (3) are partial rates of the second class, 
 
 Maclaurin's Theorem 
 This theorem explains the method of developing into a 
 series a function of a single independent variable. 
 
 24. Assume the development to be 
 
 f {x):=A-^Bx-^Cx^~-{-Dx^-^ttc., (1) 
 
 in which A, B, C, D, etc. are constants whose values depend 
 entirely upon those which enter / {x). 
 
 Now in order to determine the values of A, B, C, D, etc. 
 such as will render the assumed development true for all 
 values of x, let 
 
 u = A-\-Bx^Cx^"^Dx^^e\.c. (2) 
 
 and of this find the ratal coefificient, as in Art. 20; thus 
 
 du 
 
 = 5 + 2Cjr + 3Z).s;- + etc. (3) 
 
 dx 
 
 d^u 
 
 dx^ 
 d^u 
 
 2C + 2 • 3Dx + etc. (4) 
 
 2 ■ 3D -\- etc. (5) 
 
 dx' 
 
 Making .ar = 0, it will be found from (3), (4), (5), etc. that 
 
 du 1 d^u 1 d'u 
 
 B=—, C = , \D = ,etc. (6) 
 
 dx 2 dx^ 2-3 dx' 
 
 Since A will retain the same value, whatever the value of x, 
 substituting the values of B, C, D, etc. in (2) will give 
 
 xdu x^d^u x'd'u 
 
 u = A^ + + + etc., (7) 
 
 dx 1 ■ 2dx^ 1 • 2 • 3dx' 
 
 the theorem of Maclaurin. 
 
 If the exponent of the variable is greater than unity, as 
 M= (a -)- bx^), assume bx^^v; then substitute for v and its 
 rates their values in the development of m =(a -|- t/). 
 
ON VARIABLE QUANTITIES 17 
 
 When the function or any of its ratal coefficients becomes 
 infinite by making its variable equal to zero, it can not be 
 developed by this theorem — as, for instance, u = ax^''. 
 
 For an exemplification of this theorem, take 
 
 u^={a-^ xY. ' (1) 
 
 Determining the ratal coefficient, as in Art. 20, 
 
 = M(a + ;ir)«-i (2) 
 
 dx 
 
 n {n—l) {a^xy-^ (3) 
 
 d^u 
 
 dx- 
 
 n {n — 1) (w — 2) (a + ;^) ''-3, etc. (4) 
 
 dx 
 
 Making x = 0, then from (2), (3), and (4) are found 
 du 
 dx 
 
 d^u 
 
 = n (n — 1) a^-^ 
 
 dx' ^ 
 
 d^u 
 
 n{n — 1) (n — 2) a""^ 
 
 dx^ 
 
 and from ( 1 ) , when jr = 0, w = a" : that is, A = a". 
 
 Substituting these values in (7) will give 
 
 na'^~'^x n (n — 1) aP~'x' 
 
 + ^-^ + 
 
 1 1-2 
 
 n {n — 1) (w — 2) a'^'^x 
 
 + etc., 
 
 1-2-3 
 the same as found by the binomial theorem, 
 
 EXAMPLES 
 
 \.u={l-\-xY 2.u={a^bx)-' 
 
 Taylor's Theorem 
 25. This theorem explains the method of developing into 
 a series any function of the sum or difference of two inde- 
 pendent variables, according to the ascending powers of one 
 of them. 
 
18 AN ELEMENTARY TREATISE 
 
 Let u = f{x + y) (1) 
 
 and assume the development to be 
 
 u = A^By-^Cy' + Dy^-^ etc., (2) 
 
 in which A, B, C, D, etc. are functions of x. 
 
 Now in order to find the values of A, B, C, D, etc., such as 
 will render the development true for all possible values which 
 may be ascribed to x and y, determine the ratal coefficients of 
 (1), first under the supposition that x varies and y remains 
 constant, then that y varies and x remains constant. By this 
 process it will be found that 
 
 du dA dB dC dD 
 
 =^ + y -h / + / + etc. (3) 
 
 dx dx dx dx dx 
 
 and -^ = B ^ 2Cy + Wy' + etc., (4) 
 
 dy 
 
 but since these ratal coefficients are identical [see (4) and 
 (5) of Art. 22] it follows that 
 
 dA dB dC dD 
 
 B^ZCy-^ 3Df- = -- + -—y + —-y' + -—y'+ etc., (5) 
 dx dx dx dx 
 
 in which the coefficients of like powers of 3; must evidently be 
 equal — that is, 
 
 dA dB dC 
 
 B = , C = — and £> = , (6) 
 
 dx 2dx 3dx 
 
 the rates of which are, regarding dx as constant, 
 
 d-A d~B dK 
 dB= , dC = , dD = ; (7) 
 
 dx 2dx 3dx 
 
 d^A d'B 
 also d^B = , d'C = . (7) 
 
 dx 2dx 
 
 From (6) and (7) it will be readily found that 
 
 dA d-A d^B d^A 
 
 B= . C = and D 
 
 dx 2dx- 2-3dx^ 2-3dx^ 
 
 Substituting these values of B, C, and D in (2) will give 
 
 dA d^A d^A 
 
 u = A-\- 3; + y- + / + etc., (8) 
 
 dx 2dx^ 2 ■ 3dx^ 
 
 known as the theorem of Taylor. 
 
ON VARIABLE QUANTITIES 19 
 
 In like manner the development of u=^f {x — y) will be 
 found to be 
 
 dA d'A d'A ^ , 
 
 u = A— 3; + / — — :; y + etc. (9) 
 
 dx 2dx^ 2 ■ 3dx^ 
 
 Although this theorem gives the general development of 
 every function of the sum or difference of two variables cor- 
 rectly, yet in some particular cases a certain value may be 
 ascribed to the variable x which will render the development 
 impossible, as will be indicated by some of the ratal coefficients 
 of the development becoming equal to infinity : thus, if in 
 
 u^a -\- (b -\- X — y)^''^ 
 
 y be made equal to zero, then 
 
 A = a^ (b-\-xy^, 
 
 the first and second ratal coefficients of which are 
 
 dA 1 d^A 1 
 
 and 
 
 dx 2{b-\-xy^ dx- 4r{b-^xY'^ 
 
 both of which become equal to infinity when x=^ — b. 
 
 For an exemplification of the theorem, develop 
 
 u= {x -\- yy\ 
 Making 3; = gives 
 
 u^A=^ x'\ 
 
 the successive ratal coefficients of which are, from Art. 20, 
 
 dA d^A 
 
 nx^~'^, = M (m — 1) x"'~, 
 
 dx dx 
 
 d^A 
 
 ^^n (n — 1) (n — 2) x"'^, etc. 
 dx' 
 
 Substituting these values in formula (8), the result is 
 
 n (n — 1) 
 u = x'' -\- nx'^-^y -\- x"-^y- -f 
 
 n (n — 1) (w — 2) 
 
 x"-"y' -\- etc.. 
 
 2-3 
 the same as found by the binomial theorem, 
 
 EXAMPLES 
 
 I. u= (x -\- yy^ 2. u =:^ (x -\- ay)- 
 
20 AN ELEMENTARY TREATISE 
 
 Transcendental Functions 
 26. Let the function be 
 
 u = a^. (1) 
 
 Assuming a = \ -\- c, 
 
 then M^ (1 -f c)^, 
 
 the development of which, by the binomial theorem, is 
 
 X {x — 1) X {x — 1) {x — 2) 
 
 u^l -\- xc -\- c~ -\- c^ 4- 
 
 2 2-3 
 
 x{x—l) {x — 2) {x — Z) 
 
 c^ + etc. (2) 
 
 2-3-4 
 Passing to the rate, 
 
 2x — 1 Zx^ — 6;r -|- 2 
 
 2 2-3 
 
 Ax^ — l^x^ + 22x — 6 
 
 c^ + etc.) dx. (3) 
 
 2-3-4 
 
 Dividing each member of (3) by the corresponding mem- 
 bers of (2) gives 
 
 du c^ c^ c^ 
 
 = (c — — + — — — -|-etc.) a?;ir, (4) 
 
 w 2 3 4 
 
 which is the ordinary logarithmic series for log (1-|- c), that 
 
 c^ c" c* 
 is log (1 + c) =c — — + — — — + etc. (5) 
 
 2 3 4 
 
 But \ Ar c^=-a, therefore 
 
 du 
 
 = dx log a. 
 
 u 
 
 Substituting for u its value and multiplying by a^, 
 
 du^a^dx log a. (6) 
 
 Substituting in (4) for u and c their values and multiplying 
 by a^, the result is 
 
 a—\ {a — \y 
 
 du = a-'' ( — -\- 
 
 1 2 
 
 (a._l)3 {a—iy 
 
 — + etc. ) dx. 
 
 3 4 
 
ON VARIABLE QUANTITIES 21 
 
 Assuming 
 
 a— I {a—\y {a—\y {a—iy 
 
 — -\- etc. =^ 
 
 12 3 
 
 then du^a'^ edx, (J) 
 
 wherein e is dependent on a for its value, which may be de- 
 termined by Maclaurin's theorem. 
 
 Since the rate of u^a'^ is du=^a''edx, (7), it is evident, 
 e being constant, that the rate of du = a'^edx is d^u =^ a^e'^dx^ 
 and the rate of d^u ^= a-'e^dx^ is d^u = a'^e^dx^. Therefore, 
 the ratal coefficients are, from Art. 20, 
 
 du d'^u d'^U' 
 
 dx dx^ dx^ 
 
 Making x^O in (1), also in the ratal coefficients, it will 
 be found that 
 
 du d^u d^u 
 
 u=^l, = ^, ^e'^, ^^^, etc. 
 
 dx dx~ dx^ 
 
 Substituting these values in (7), Art. 24, gives 
 
 ex e'^x'^ e^x^ 
 
 li = a^ = 1 4- -\- -|- + etc. 
 
 1 2 2-3 
 
 1 
 
 If .ar = — , then 
 e 
 
 1 1 1 
 
 a*- == 1 _u — ^ — -)- -1- etc. 
 
 12 2-3 
 
 The sum of the first twelve terms of this series is 2.7182818, 
 which is the base of the Naperian system of logarithms ; hence 
 e is the Naperian logarithm of a. Therefore, substituting 
 log a for e in (7), then 
 
 du = a^dx log a, 
 the same as (6). 
 
 Hence the rate of an exponential function is the function 
 into the rate of the exponent multiplied by the Naperian log- 
 arithm of the constant of which the variable is the exponent. 
 
22 AN ELEMENTARY TREATISE 
 
 27. Resuming (1) of the last article, transposing it, and 
 taking the logarithm of both members give 
 
 ji: log a := log u, 
 
 log u 
 
 whence x = . ( 1 ) 
 
 log a 
 
 Now it has been shown in the last article, that the rate of 
 
 M = a* is 
 
 du = a^dx log a, 
 du 
 
 whence dx = 
 
 a'^ log a 
 
 or, substituting for a^ its value, from (1) of Art. 26, 
 
 du 
 
 dx = , (2) 
 
 u log a 
 the rate of (1). 
 
 If a is the base of a system of logarithms, then x is the 
 
 1 
 
 logarithm of u in that system, and is the modulus of the 
 
 log a 
 
 1 
 
 system; therefore representing by M, (2) becomes 
 
 log a 
 
 du 
 dx = M . (3) 
 
 u 
 
 Hence the rate of the logarithm of a quantity is the modulus 
 of the system into the rate of the quantity, divided by the 
 quantity itself. 
 
 The modulus of the Naperian system of logarithms is 
 unity; therefore if the logarithms are taken in the Naperian 
 system (3) becomes 
 
 du 
 dx = . 
 
 u 
 
 Hence the rate of the Naperian logarithm of a quantity is 
 the rate of the quantity divided by the quantity itself. 
 
 28. To determine the rate of 
 
 u^v^, 
 in which both v and x are variables. 
 
ON VARIABLE QUANTITIES 23 
 
 Taking the logarithm of both members, 
 log M = ^ log V, 
 and passing to the rate, by Arts. 26 and 9, 
 du xdv 
 
 -j- dx log V, 
 u V 
 
 uxdv 
 or du = -|- udx log v. 
 
 V 
 
 Substituting t"^ for u and reducing, 
 
 du = xv^~'^dv -f v^dx log v. ( 1 ) 
 
 Hence the rate of a variable quantity having a variable 
 exponent is the sum of the rates obtained, first under the sup- 
 position that the quantity varies and the exponent remains 
 constant, then that the exponent varies and the quantity remains 
 constant. 
 
 29. Take u = \og{\^x), (1) 
 
 in which u is the Naperian logarithm of 1 + -^^ ^^^ the suc- 
 cessive ratal coefficients are 
 
 du 1 d'^u 1 d^u 1 • 2 
 
 -, etc. 
 
 dx \-^x dx^ (1+^)' dx^ (1+-^)' 
 
 Making ;r ^ in ( 1 ) , also in the ratal coefficients, then 
 
 du d^u 
 
 dx dx~ 
 
 d^u d*u 
 
 =1-2, = —1 •2-3, etc. 
 
 dx" dx^ 
 
 and consequently, by substitution in (7) of Art. 24, 
 
 x'^ x" X* 
 u^los: (I 4- x) =x — — -|- — — — + etc. (2) 
 
 2 3 4 
 
 [see (5) of Art. 26]. 
 
 Developing u = log (1 — x) 
 
 in like manner, it will be found that 
 
 X^ X^' X* 
 
 M^logfl — x) <= — X — — — — — — — etc. (3) 
 
 ^ V ; 2 3 4 ^ 
 
1 + ^ 
 
 v-\- 1 
 
 1—x 
 
 V * ' 
 
 1 
 
 24 AN ELEMENTARY TREATISE 
 
 Subtracting (3) from (2) gives 
 
 log(l+^)— log(l— ;r)=2(^ + ^ + y + etc.)(4) 
 
 1 A-x v+ 1 
 Assuming 
 
 then 
 
 2v-\-l' 
 
 therefore, since 
 
 log ( ) = log (z/ + 1 ) — log V, 
 
 V 
 
 by substituting the value of x in (4), the following is obtained : 
 
 log {v^ 1)— logz/ = 
 
 1 1 1 
 
 2 [ _|- 4- ^ etc.], 
 
 (2z/+l) 2>{2v^iy 5(2z/+l)s 
 
 or log (z/ + 1) ^log^ -f 
 
 1 1 1 
 
 2 [ + + + etc.], 
 
 (2z/+l) 3(2z/+l)3 5(2z/-fl)5 
 
 by which the logarithm of ^' -|- 1 can readily be found when 
 the logarithm of v is known. Thus, if 
 
 111 
 v=\, log 2= + 2(— + + + etc. )= 0.69314718 
 
 ^ 3 3 • 33 5 • 3^ 
 
 1 1 1 
 
 v=2, log 3 = log 2 + 2(— + + + etc.)= 1.09861229 
 
 t;=3, log 4 = log 2 + log 2 = 1.38629436 
 
 I 1 1 
 
 v=^, log 5 = log 4 + 2(— + + + etc.)= 1.60943791 
 
 ^ ^ ^93- 93 5- 9^ 
 
 v=S, log 6 = log 2 + log 3 = 1.79175947 
 
 II 1 
 
 v=6, log 7 = log 6 4- 2(— + + + etc.)= 1.94591014 
 
 ^ ^ ^ ^3 3-13^ 5-13^ 
 
 t/=7, log 8 = log 2 + log 4 =2.07944154 
 
 t;=8, log 9 = log 3 + log 3 = 2.19722458 
 
 v=9, log 10=-log 2 + log 5 = 2.30258509 
 
ON VARIABLE QUANTITIES 25 
 
 The logarithm of 10 in the common system is 1, and in the 
 Naperian system is 2.30258509; 1 divided by 2.30258509 is 
 0.43429448, the modulus of the common system, usually desig- 
 nated by M. 
 
 To avoid inconvenience, the Naperian logarithms are gen- 
 erally used in this work. Whenever the common system may 
 desired, it will be necessary to multiply by the modulus of that 
 system. 
 
 EXAMPLES 
 
 To determine the rate of 
 
 X 
 
 u = log . 
 
 X 
 
 Assuming u = , ( 1 ) 
 
 (a^ + x^y^ 
 
 dv 
 then u = \ogv and du = . (2) 
 
 V 
 
 From ( 1 ) , by passing to the rate, 
 
 (a^ -\- x^y dx — x^ {a" -\- x^)-^- dx 
 
 dv 
 
 {a^ -f x^) 
 
 a^dx 
 or, reducing, dv = — — — -— . (3) 
 
 {or -\- x^Y'^ 
 
 Substituting for v and dv their values in (2) gives 
 
 a^dx 
 
 du = 
 
 X (a^ -f- x^) 
 
 a^ + b'' log X 
 
 2. 
 
 a'^ -\- b^ log y 
 
 (l+;,) + (l+3,) 
 
 6. u = log 
 
 (1+.^) — (l+y) 
 
 What is the rate of the common logarithm 4300? 
 
 Illustrations of Principles Relative to Curves 
 30. If a particle impelled from A toward B along the curve 
 
26 
 
 AN ELEMENTARY TREATISE 
 
 Y 
 
 A 
 
 E 
 
 f; 
 
 
 ^' 
 
 P 
 
 G 
 
 'B 
 
 D 
 Fi a. 3 
 
 APB (see figure) be left to itself at any point in the curve, 
 
 Q as at P, it is obvious that 
 
 it would then proceed at a 
 uniform rate toward C along 
 the straight line PC tangent 
 to the curve at P. 
 
 For the point P, let x re- 
 present the abscissa AD, y the 
 ordinate DP, and z the curve 
 AP. Extend DP to E, and 
 jx:^ draw EF and PG parallel to 
 
 AX] also draw £F and EG 
 parallel to AY. Then dx will 
 be represented by PG, dy by 
 ^P, and ds by PF ; hence 
 dz' = dx^ -\- dy~ or dz = ((/;r- -[- dy~Y^. 
 
 Circular Functions 
 To determine the rate of 
 
 u ;=: sin X, 
 let the radius AC =BC = R (see 
 figure), the arc AB = x, and, as the 
 case may be, let u represent the sine, 
 cosine, tangent, etc. of the arc x; 
 then we have, by Art. 30, BE = dx, 
 BD equal to the sine, CD the cosine, 
 EF the rate of the sine, and BF the 
 rate of the cosine ; hence 
 BC:CD::BE:EF 
 or R : cos x::dx: du, 
 
 cos xdx 
 
 du^ . 
 
 R 
 
 31. 
 
 O A 
 
 Fi a. 6 
 
 whence 
 
 Therefore the rate of the sine of an arc is equal to the 
 cosine of the arc into the rate of the arc, divided by the radius. 
 
 If u^ cos X, 
 
 then, since u, the cosine CD, is a decreasing function of the 
 arc X, its rate is negative ; therefore 
 
 sin xdx 
 
 R: sin xwdx: — du or du 
 
 R 
 
ON VARIABLE QUANTITIES 27 
 
 Hence the rate of the cosine is minus the sine into the rate 
 of the arc, divided by the radius. 
 
 If tt = vers x^^R — cos x, 
 
 sin xdx 
 
 then du =^ . 
 
 R 
 
 Hence the rate of the versed sine is the sine into the rate of 
 the arc, divided by the radius. 
 
 R sin X 
 If w = tan^^ , 
 
 cos X 
 
 then, by passing to the rate, by Art. 19, 
 
 COS" xdx -\- sin- xdx 
 
 du^ , 
 
 cos^ X 
 
 or, since cos- x -\- sin- x = R'~, 
 
 R^dx 
 
 du 
 
 cos- X 
 
 Therefore the rate of the tangent of an arc is equal to the 
 square of the radius into the rate of the arc, divided by the 
 square of the cosine. 
 
 R cos X 
 
 If w = cot X = , 
 
 sin;ir 
 
 then, passing to the rate, by Art. 19, 
 
 — (cos^ X -f- sin- x) dx 
 
 du = 
 
 sm'' X 
 or, since cos^ x -\- sin^ x =■ R~, 
 
 R-dx 
 
 du^^ — , 
 
 sm- X 
 
 Therefore the rate of the cotangent is equal to minus the 
 square of the radius into the rate of the arc, divided by the 
 square of the sine. 
 
 If u^ sec X = , 
 
 cos X 
 
 then, passing to the rate, 
 
 R^ sin xdx 
 
 du ==■ . 
 
 cos^ X 
 
28 AN ELEMENTARY TREATISE 
 
 Therefore the rate of the secant is the square of the radius 
 into the sine multiplied by the rate of the arc, divided by the 
 square of the cosine. 
 
 If w=cosec;r = , 
 
 then du = - 
 
 sin;ir 
 R^ cos xdx 
 
 sm" X 
 
 Therefore the rate of the cosecant is equal to minus the 
 square of the radius into the cosine multiplied by the rate of 
 the arc, divided by the square of the sine. 
 
 li R = l, then for 
 
 u = sin X, du = cos xdx, 
 
 u = cos X, du = — sin xdx, 
 
 u = vers X, du = sin xdx, 
 
 u = tan X, dx 
 
 du = , etc. 
 
 COS^ X 
 
 If u = s\nrx, (1) 
 
 by assuming rx = v, then will sin rx = sin v, 
 
 whence dv = rdx and du = d (sin rx) = cos vdv. (2) 
 
 Substituting these values of v and dv m (1), then 
 
 du = r cos rxdx. 
 
 If w = sin"x, (3) 
 
 by assuming sin x^^v, then will sin" x = v^, 
 
 whence dv = cos xdx and du = d (sin" x) = nv^-'^dv. (4) 
 
 Substituting these values of v and dv m (4) gives 
 
 du = n sin""^ x cos xdx. 
 
 If w = sin" ;ir sin r;r, (5) 
 
 then du = d (sin";r) sin rx -{- d (sin rx) sin" x; 
 
 but, as shown, d (sin" x) =n sin"~^ x cos xdx 
 
 and d (sin rjr) ^r cos rxdx ; 
 
 therefore 
 
 du = (n sin""^ x cos x sin rx -\- r cos rx sin" x) dx, 
 
 or du == sin""^ x (n cos x sin rx -\- r cos rx sin x) dx. 
 
 By use of these equations, the rates of like expressions of 
 the cosine, versed sine, tangent, etc., can be determined. 
 
ON VARIABLE QUANTITIES 29 
 
 32. For radius R and 
 
 u = log sin X, 
 by passing to the rate, by Arts. 27 and 31, 
 
 cos xdx 
 
 R sin X 
 
 the rate of the logarithmic sine of the arc x. 
 
 If u== log cos X, 
 
 sin xdx 
 then du = 
 
 R cos X 
 the rate of the logarithmic cosine of the arc x. 
 
 If u = log vers x 
 
 or its equivalent, u = log {R — cos x), 
 
 sin xdx 
 
 then du = 
 
 R (R — cosx) 
 smxdx 
 
 or du- 
 
 R vers x 
 
 the rate of the logarithmic versed sine of the arc x. 
 
 If u = log tan X 
 
 R sin X 
 
 or, since tan x = , 
 
 cos X 
 
 R sin.jr 
 u = \og ( ); 
 
 cos X 
 
 then, passing to the rate, by Arts. 27 and 31, and reducing, 
 
 cos^ xdx + sin^ xdx 
 
 du = 1 
 
 R sin X cos x 
 
 or, since cos^ x -\- sin^ x = R^, 
 
 Rdx 
 
 du = — ; , 
 
 sin X cos X 
 
 the rate of the logarithmic tangent of the arc x. 
 
 R cos X 
 
 If u^ log cot X = log — ; ; 
 
 sin X 
 
30 AN ELEMENTARY TREATISE 
 
 — (sin^ xdx + cos^ xdx) R cos x 
 
 then du = : ^ ~ ; 
 
 sm x^ sin X 
 
 or, since sin^ x + cos^ x = R-, by reducing 
 
 Rdx 
 
 du 
 
 sm X cos X 
 
 the rate of the logarithmic cotangent of the arc x. 
 
 R^ 
 If w ^ log sec ;i; = loj 
 
 then du ^= 
 
 cos X 
 R^ sin xdx R^ sin xdx 
 
 cos X R cos X 
 
 R sin X 
 
 or, since = tan jr, 
 
 cos X 
 
 tan ;ir<f;ir 
 
 du^= , 
 
 R^ 
 
 the rate of the logarithmic secant of the arc x. 
 
 In like manner the rate of 
 
 u = log cosec X 
 cot xdx 
 
 will be found to be du 
 
 system of loga 
 
 M cos xdx 
 
 R' 
 In using the common system of logarithms, for 
 
 u = log sin X, du = 
 
 R sin X 
 
 M sin xdx 
 u = log vers x, du = — - ; 
 
 R vers x 
 
 M sin xdx 
 u =log cos X, du = — 
 
 R cos X 
 
 MRdx 
 u = log tan X, du = — ; 
 
 sm X cos X 
 
 MRdx 
 u = log cot X, du = — - , etc. 
 
 sm X cos X 
 
 33. It is often desirable to have the rate of the arc in 
 terms of that of its sine, cosine, tangent, etc., and for this 
 
ON VARIABLE QUANTITIES 31 
 
 purpose the following expressions are employed: namely, 
 X = sin-^ u, x^ cos-^ u, x^ tan'^ u, etc. , x being the arc and 
 u its sine, cosine, etc. 
 
 Let X = sin"^ u , 
 
 its equivalent being u = sin x, 
 
 the rate of which is, by Art. 31, 
 
 cos xdx 
 
 du 
 
 whence dx^ 
 
 R 
 
 Rdu 
 
 cos X 
 or, since sin.;tr:^M, consequently cos.ar^ (7?^ — u^Y^, 
 
 Rdu 
 
 dx^ 
 
 the rate of the arc in terms of the sine and its rate. 
 
 In like manner, if x=^ cos"^ u, 
 
 Rdu 
 it will be found that dx=^ 
 
 (R-2 — U^)V. 
 the rate of the arc in terms of the cosine and its rate. 
 
 If x^ vers"^ u, 
 
 its equivalent being u = vers x = R — cos x, 
 the rate of which is, by Art. 31, 
 
 sin xdx 
 
 du 
 
 whence dx = 
 
 R 
 Rdu 
 
 sm.ar 
 
 Now sin .;r :^ (R^ — cos x^)^'^^, 
 but 
 
 cos .V =^ R — vers x = R — u ; 
 therefore 
 
 sin x={R~—(R — uyy- = {2Ru — u-y^ ; 
 
 Rdu 
 hence dx = , 
 
 (2Ru — u^y 
 
 the rate of the arc in terms of the versed sine and its rate. 
 
32 AN ELEMENTARY TREATISE 
 
 If x = tan"^ u, 
 
 the equivalent being 
 
 X = tan X, 
 
 the rate of which is, by Art. 31, 
 
 R^dx 
 
 du 
 
 cos^ X 
 
 cos^ xdu 
 
 whence dx = , (1) 
 
 R^ 
 
 But sec^ X : R- : : R^ : cos^ x, or, since sec^ x^R^ -\- u-, 
 R- + w^ :R2..]^2. (,os2 ^^ 
 
 R^ 
 
 whence cos- x ■ 
 
 R^ -{-u^ 
 
 Substituting this value of cos^ ;i; in ( 1 ) gives 
 
 R~du 
 dx = , 
 
 R^~ + u^ 
 
 the rate of the arc in terms of the tangent and its rate. 
 
 In like manner, if x = cot~^ u, 
 
 R^ du 
 it will be found that dx = • 
 
 the rate of the arc in terms of the cotangent and its rate. 
 
 34. By means of Maclaurin's theorem, sin x, cos x, etc. can 
 be developed in terms of .*" : thus, if i? ^ 1, the ratal coefficients 
 of w = sin X are 
 
 du d'u 
 = cos X = — sin X 
 
 dx dx- 
 
 d^u d*u 
 
 cos X = sm X 
 
 dx^ . dx' 
 
 d^u 
 
 =^ cos X, etc. 
 
 dx^ 
 
 Making x^O, then, from Art. 24, 
 
 du d^u d^u 
 
 A=0, -—=1, -7 = 0, -— = —1, 
 
 dx ax- ax" 
 
ON VARIABLE QUANTITIES 33 
 
 = 0, =1, etc. 
 
 dx^ dx^ 
 
 By substituting these values in Maclaurin's theorem the 
 following is obtained: 
 
 X x^ x^ 
 u = s\r\ x^ — — + , — etc. 
 
 1 2-3 2-3-4-5 
 
 Proceeding in like manner, it will be found from u = cos x 
 
 x^ X* 
 
 that M = 1 — — + — etc. 
 
 1 2-3-4 
 
 EXAMPLES 
 
 Determine the rates of the following: 
 
 w = sin ;ir^ u = tan x^ 
 
 u = sin X cos X w = cos x sin x 
 
 u = tan X log cot x m = log tan x -\- log cot x 
 
 u 
 
 .s; = sin"^ 2m ( 1 — u) .ar = cos"^ 
 
 a — u 
 
 b 
 X = tan~^ — 
 
 u 
 
 Develop x = sin"^ u. 
 
 Vanishing Fractions 
 
 35. A vanishing fraction is one which reduces to the form 
 
 . . . - 
 
 — when a particular value is given to the variable. Thus, 
 
 c {x — a) 
 
 
 reduces to — when x = a. This, it will be seen, is owing to a 
 
 factor common to both numerator and denominator which 
 reduces to zero for x^=a. 
 
 Let the equation be 
 
 (1) 
 
34 AN ELEMENTARY TREATISE 
 
 and put it under the form 
 
 {x — a)'" {jir'""^ -f- amx"^-~ -\- a- m (m — \)x"^~^ -\ a'"^-'^Y 
 
 U=:i ^ — 
 
 {x — a)*' {x"~'^ -\- anx"^~'~ -j- d-n (n — 1) x"~^ -\- a"-^}« 
 
 Now the right-hand portion of this fraction does not reduce 
 to zero for x^=^a; therefore, making it equal to R, then 
 
 (x — ay 
 
 u = R- '-. (2) 
 
 {x — a)« 
 
 Let (x — a)''^P, and {x — a)* = Q, both of which 
 reduce to zero for x = a; then 
 
 RP 
 
 (3) 
 
 
 
 u — 
 
 ' Q ' 
 
 or 
 
 
 Qu = 
 
 ■ RP. 
 
 Passing to the rate 
 
 
 
 
 Qdu -f- udQ = 
 
 ■ RdP + PdR, 
 
 butQ: 
 
 = OandP = 0; 
 
 therefore 
 
 
 
 udQ — 
 
 -.RdP 
 dP 
 
 or 
 
 
 u = R 
 
 dQ ' 
 
 hence, 
 
 RP 
 since u^ , (3) 
 
 Q 
 
 P 
 
 u = R—-- 
 Q 
 
 dP 
 = R . 
 
 dQ 
 
 If both dP and dQ reduce to zero for x^a, then by pass- 
 ing to the rate again 
 
 dP d'P 
 
 R = R ; 
 
 dQ d-Q 
 
 P dP d^P 
 
 therefore u = R — =:^ R = R . 
 
 Q dQ d'-Q 
 
 Should this also reduce to zero for x == a, by continuing 
 the process a fraction may be found which will not reduce to 
 zero for x = a, and thus the true value of the primitive fraction 
 will be obtained.* 
 
 * It will be observed that the process employed simply eliminates the 
 vanishing factor, thus giving the real value of the fraction. 
 
ON VARIABLE QUANTITIES 35 
 
 Let u={hx~ + ex) { ), (4) 
 
 X — a 
 
 in which m = 2, m=1, and bx~ -\- ex represents R, and its 
 rate is 
 
 d(x- — 0") 2xdx 
 
 u=(bx^ + ex) ={bx~ + ex) = 2bx^-\-2cx- 
 
 d {x — a) dx 
 
 or, when x^a, u = 2a^b -{-2a^e. (5) 
 
 Multiplying (bx^ -\- ex) by {x^ — a^) in (4) gives 
 
 bx'^ -|- ex^ — a~bx- — a^cx 
 
 u^ _ 
 
 X — a 
 
 Therefore 
 
 (4bx^ -\- 3ex^ — 2a^bx — a^c) dx 
 
 dx 
 4bx^ + ^^•^" — 2arbx — a^c, 
 or,when;r = a, u^2a^b-\-2a^e, 
 
 the same result as (5). 
 
 xn a"" 
 
 If u = , 
 
 X — a 
 
 d (x"- — a") nx'^-^dx 
 
 then u = = ^=nx^ 
 
 d (x — a) dx 
 
 or, when x = a, u = na^~^. 
 
 (;r2_ ^2)3/2 
 
 If 
 
 (x — a)^^^ 
 
 by squaring and then taking the cube root, 
 
 (x^ — a^) 
 
 u'/' = — 
 
 (x — a) 
 
 2xdx 
 
 Therefore u~^^ = = 2x 
 
 dx 
 
 or, when x^a, 
 
 y2iz —-2a or u= {2a)^f^. 
 
36 AN ELEMENTARY TREATISE 
 
 
 EXAMPLES 
 
 u^ 
 
 u 
 
 
 (x^~2ax'-\-a^y^ {2ax^- — 2d'y- 
 
 
 {x'- — a^y^ {x^ -\- a-x — 2a^y 
 
 
 x^ — 2ax^ + a^x a — x -\- a log x — a log a 
 
 
 X- — a- a — {2ax — x)"^ 
 
 Det( 
 
 irmine the value oi u = , when x = 0. 
 
 X {x -\- c) 
 Determine the value of 
 
 cos X — sin ;ir -|- 1 
 
 cos X -\- sin X — 1 
 
 , when x = 90°. 
 
 In some cases both P and Q become infinite for a particular 
 value of the variable : thus 
 
 tan;ir 
 
 
 u 
 
 cot 2x 
 
 00 
 
 becomes 
 
 -when ;ir = 90°. 
 
 
 00 
 
 
 
 Now 
 
 1 1 
 
 tan X — anc" - - * '^ ■ 
 
 cot X 
 
 1 L,UL i-A , 
 
 tan2;ir 
 
 
 tan X tan 2x 
 
 
 
 therefore 
 
 cot 2x cot X 
 
 2dx 
 
 o' 
 
 Hence 
 
 cos^ 2x 
 
 2 sin^ X 
 
 li 
 
 — dx 
 
 cos^ 2x 
 
 s\rr X 
 
 or, since sinjr==l and cos2;ir = — 1 when ;ir = 90°, 
 
 u^ — 2. 
 
 Sometimes in a product one factor becomes zero and the 
 other infinite for a particular value of the variable : thus, in 
 
 1 
 m:^(1 — ;i:) tan — tt x (6) 
 
ON VARIABLE QUANTITIES 37 
 
 1 1 
 
 (1 — ;r) =0 and tan — 7r:ir=co, when x=l and — 7r = 90°. 
 ^ / 2 2 
 
 1 1 . , 
 
 Since tan — tt ;ir == , (6) can be written thus : 
 
 2 1 
 
 cot — ttX 
 
 2 
 
 l—x 
 
 1 
 
 cot 77 X 
 
 2 
 
 then 
 
 1 
 
 dx 2 sin^ — irx 
 
 2 
 
 1 
 
 TT dx IT 
 
 2 
 
 ~ i 
 
 sin^ — irx 
 2 
 
 1 1 
 
 Therefore, when x=l and — tt = 90°, since sin — tt jr then 
 
 2 2 
 
 equals 1, 
 
 2 
 
 u = — . 
 
 Of the difference of two quantities, both sometimes become 
 infinite for a particular value of the variable : thus, in 
 
 ^ = -^— ^, (7) 
 
 X — 1 log ^ 
 
 ^ 1 r • ' 
 
 when x=l, both and. become mfinite. 
 
 X — 1 log X 
 
 In this case put (7) under the form 
 X log X X -\- I 
 
 (x — 1) log;i; 
 
 
 which becomes — when x = 1 ; therefore 
 
 
 d (x log X — X -}- I) X log X 
 
 u = = 
 
 d (x — 1 ) log X X log X -\- X — 1 
 
38 
 
 AN ELEMENTARY TREATISE 
 
 This also becomes — when x=l, consequently 
 
 dx log X 
 
 xdx 
 
 log X -\- \ 
 
 dx log X 
 
 xdx 
 
 + dx 
 
 log;r+l + l 
 
 or, when jir = 1, since log 1^0, 
 
 1 
 
 u = — . 
 2 
 
 EXAMPLE 
 
 Determine the value of 
 
 u^ X tan X — 
 
 z cos X 
 
 -, when x = 90^. 
 
 Curves Referred to Rectangular Coordinates 
 
 36. Signification of the first and second ratal coefficients. 
 
 Every curve or line referred to rectangular coordinates 
 may generally be represented by the equation 
 
 y = f {^) 
 
 in which x represents any abscissa, as AB, and y the cor- 
 responding ordinate BP, of the curve CPD (see figure). 
 
 Fig. 7 
 
 Draw TT' tangent to the curve CPD at P, and with radius 
 unity draw the arc EH, also draw EK tangent to EH. Then 
 the angle T'TX will be the angle of tangency, so called, and 
 
ON VARIABLE QUANTITIES 
 
 39 
 
 EK its tangent, which is designated by t. Now let dx be re- 
 presented by PG, and dy by GF (see Art. 30) ; then will 
 
 dx:dy:: TE : EK 
 
 or, since TE^l, and EK^t, by representing the rate of 
 / (x) by /' (X) 
 
 dx: dy:: I: t, 
 
 dy 
 
 whence 
 
 ^ = 
 
 dx 
 
 Passing to the rate and representing the rate oi f (x) by 
 
 dt d^y 
 
 = — ^ = /-(x). 
 
 dx dx^ 
 
 Hence the first ratal coefficient of the equation of a curve 
 represents the tangent of the angle of tangency of any point of 
 the curve and the second ratal coefficient represents the rate of 
 variation of the tangent of the angle of tangency. 
 
 Z7. Of a curve concave to the axis of X (Fig. 8), the 
 angle of tangency and consequently its tangent t decrease as 
 the ordinate increases, as is clearly shown by the tangents TP 
 and TP' of the curve CPP'D. 
 
 This, it will be observed, is also true of the curve C'QQ'D' , 
 as indicated by the tangents TQ and T'Q'; but, lying below 
 the axis of X, the angle of tangency and its tangent, as well as 
 the ordinate, are negative, while those above the axis are 
 positive. 
 
 Now, since the rate of a positive decreasing function is 
 negative and that of a negative decreasing function is positive, 
 
40 
 
 AN ELEMENTARY TREATISE 
 
 the rate of the tangent of the angle of tangency of the curve 
 CPP'D is negative, while that of the curve C'QQ'D' is posi- 
 tive. Therefore, since t represents the tangent of the angle 
 of tangency, when a curve is concave to the axis of x and its 
 ordinate is positive, dt, and consequently the second ratal co- 
 efficient of the equation of the curve, are negative, but positive 
 when the ordinate is negative. 
 
 If the curve is convex to the axis of X (Fig. 9), the angle 
 of tangency, and consequently its tangent t, increase as the 
 ordinate increases, as is shown by the tangents TP and T'P' 
 of the curve CPP'D. 
 
 This is also true of the curve C'QQ'D' , as indicated by 
 the tangents TQ and T'Q' ; but as they lie below the axis of X, 
 the angle of tangency and its tangent, as well as the ordinate, 
 are negative. 
 
 Therefore, since the rate of a positive increasing function is 
 positive, and that of a negative increasing function is negative, 
 the rate of the tangent of the angle of tangency for any point of 
 the curve CPP'D is positive, while that of the tangency for any 
 point of the curve C'QQ'D' is negative. 
 
 Hence when a curve is convex to the axis of X, and its 
 ordinate is positive and increasing, dt, and consequently the 
 second ratal coefficient of the equation of the curve, are posi- 
 tive, hut negative when the ordinate is negative. 
 
 From what precedes, the following conclusion is evident: 
 
ON VARIABLE QUANTITIES 41 
 
 When the second ratal coefficient of the equation of a curve 
 is negatizre, the curve is either concave to and above the axis of 
 X or convex to and below it; but when the second ratal co- 
 efficient is positive, the curve is either concave to and below the 
 axis or convex to and above it. . 
 
 Sometimes a particular value of x, as ;r = a, will make 
 
 d^'y 
 
 = 0. In this case, substitute a ±v ior x; then, for a small 
 
 dx'- 
 
 d^y 
 
 value of V, if and the ordinate corresponding to x=^a 
 
 dx~ 
 
 have contrary signs, the curve is concave to the axis of X at a 
 point in the curve whose abscissa is x^a, but if like signs, 
 convex. 
 
 E. g., let the equation of the curve be 
 
 y = x^ — Sx^^ AOx- — mx + 58. 
 
 Passing to the rate twice, 
 
 d^y 
 
 -^ = 20.^^ — 60^^ + 80, (1) 
 
 dx^ 
 
 d'y 
 in which = when x = 2; therefore, substituting 2 =t ^^ 
 
 dx'-^ 
 
 for JIT in (1) gives 
 
 d'y 
 
 = 60?v2 ±: 20z/3 
 
 dx' 
 
 d^y 
 or -^ = 20v' (3±v), 
 
 dx" 
 
 which is positive for any value oi v <i 3. Hence, since 3; = 10 
 when x^^2, the curve is convex to the axis of X at the point 
 whose abscissa is ;ir = 2. 
 
 Determine whether the curve whose equation is 
 
 3; = 5 -{- 4x — x~, 
 
 is concave or convex to the axis of X. 
 
 Ratal Equations of Lines 
 
 38. A ratal equation of a line is one which shows the rela- 
 tion between the coordinates and their rates, and, being inde- 
 pendent of the values of the constants which enter the primitive 
 
42 AN ELEMENTARY TREATISE 
 
 equation, determines the general nature of the Hne without 
 regard to its magnitude. 
 
 First take the general equation of lines of the first order 
 
 dy 
 
 whence ^ a, 
 
 dx 
 
 a result which is the same for all values of b. This equation 
 represents the tangent of the angle of tangency (see Art. 36) 
 and is the first ratal equation of lines of the first order. 
 
 Passing to the rate again 
 
 dy d-y 
 
 d-^=^0 or -^ = 0, 
 dx dx- 
 
 an equation entirely independent of the values of a and b and 
 consequently equally applicable to every line of the first order 
 which can be drawn in the plane of the coordinate axes. It is 
 called the general ratal equation of lines of the first order. 
 
 d~y 
 
 The equation = shows that the tangent of the angle 
 
 dx'^ 
 
 of tangency has no variation (see Art. Z7) ; hence every line 
 of the first order must necessarily be a straight line. 
 
 39. In the general equation of lines of the second order 
 
 y^ = ax~ -{- bx -\- c, ( 1 ) 
 
 passing to the rate thrice will give 
 
 2ydy 
 -^—^ = 2ax-{-b (2) 
 
 dx 
 dy^ yd^y 
 
 -^Jr^-^ = a. (3) 
 
 dx' dx- 
 
 dyd-y yd^y 
 
 -^—^ + -^^—^ = 0; (4) 
 
 dx^ dx^ 
 
 equations (2), (3), and (4) are respectively the first, second, 
 and general ratal equations of lines of the second order. 
 
 When the origin of coordinates is at the vertex of the trans- 
 verse axis, the general equation of lines of the second order is 
 
 y^ =r= ax"^ -\- bx. (5) 
 
ON VARIABLE QUANTITIES 
 
 43 
 
 Passing to the rate twice 
 
 2ydy = 2axdx -\- bdx (6) 
 
 and dy^ -\- yd^y = adx"^. (7) 
 
 Eliminating a and & in (5) by means of (6) and (7) 
 y-dx^ + x^dy- -\- x'^yd'^y — 2xydxdy = 0, 
 a general ratal equation of lines of the second order, when the 
 origin of the coordinates is at the vertex of the transverse axis, 
 differing from (4) on account of passing to the rate but twice. 
 
 Determine the general ratal equations of the circle, para- 
 bola, and ellipse. 
 
 Tangents and Normals 
 40. li X and y represent the coordinates of any point of the 
 curve APC, and z the corresponding arc AP, Art. 30, dx will 
 
 be represented by 
 DP, dy by DE, and 
 dz by PE, a tan- 
 gent to the curve at 
 the point P ; also 
 iO TR represents the 
 subtangent, TP the 
 tangent, RN the 
 
 -p -^ p nh nZ subnormal, and PN 
 
 the normal, each of 
 which are obtained as follows : 
 
 DE:DP::PR:TR; 
 
 dy: dx 
 
 that is. 
 
 or 
 
 TR 
 
 y.TR, 
 ydx 
 
 that is, 
 
 or 
 
 that is. 
 
 or 
 
 dy 
 TP^- = PR^- + TR^- ; 
 
 TP^ = 3,2 ^ £ 
 
 dy- 
 
 TP = — (dx' -\- dy^-y\ 
 dy 
 
 DP:\DE::PR:RN; 
 
 dx:dy\\y: RN 
 
 ydy 
 
 (1) 
 
 (2) 
 
 RN=' 
 
 dx 
 
 (3) 
 
44 AN ELEMENTARY TREATISE 
 
 PN^ = PR^-\-RN^; 
 that is, PN^ = 3,2 _p 3,2 
 
 or PN = -^ {dx- ^ dy'^y-. (4) 
 
 dx 
 
 1. Hence the length of the subtangent to any point of a 
 curve is equal to the ordinate into the rate of the abscissa 
 divided by the rate of the ordinate. 
 
 2. The length of the tangent to any point of a curve is 
 equal to the ordinate divided by the rate of the abscissa, into 
 the square root of the sum of the squares of the rates of the 
 abscissa and ordinate. 
 
 3. The length of the subnormal to any point of a curve is 
 equal to the ordinate into its rate divided by the rate of the 
 abscissa. 
 
 4. The length of the normal to any point of a curve is equal 
 to the ordinate divided by the rate of the abscissa, into the 
 square root of the sum, of the squares of the rates of the 
 abscissa and ordinate. 
 
 The tangent TP may also be obtained thus : 
 
 DE:PE::PR:TP ; 
 
 that is, dy.dsr.y: TP 
 
 dz 
 or TP^y--. (5) 
 
 dy 
 
 Likewise, for the normal PN, 
 
 DP:PE::PR:PN; 
 
 that is, dx:ds::y: PN 
 
 dz 
 
 or PN = y . (6) 
 
 dx 
 
 In the application of these formulas to any particular curve, 
 
 dx dy _ 
 
 the value of or , obtained from the equation of the 
 
 dy dx 
 
 curve by passing to the rate, must be substituted in each of 
 them. The result will be true for all points of the curve ; then, 
 by substituting therein the values of x and y for any particular 
 
ON VARIABLE QUANTITIES 45 
 
 point of the curve, we can find the value of the subtangent, 
 tangent, subnormal, and normal for that point. 
 
 41. To apply the formulas of the preceding articles to lines 
 of the second order, whose general equation is 
 
 y~ = ax^ -\- bx -\- c, 
 and its rate 
 
 dy 2ax -(- ^ 2a;ir -f- h 
 
 dx 2y 2 {ax^ -\- bx -{- c)''^ 
 
 dy 
 
 Substituting the value of in formulas (1), (2), (3), 
 
 dx 
 and (4) of Art. 40, will give 
 
 dx 2{ax^ -\- bx -\- c) 
 
 TR = y 
 
 dy 2ax -\- b 
 
 TP = —{dx'--^ dy^ ) ^/^ = 
 dy 
 
 ax^ -{- bx -\- c 
 
 ^{ax^- -^bx^c +4 ( -—-)! 
 
 lax -\- b 
 
 dy 2ax 4- b 
 RN = y—^ = 
 
 dx 
 
 y 1 
 
 PN = (dx"" + dy^y''=^/{ax^ -j- bx + c -\- —{2ax + by). 
 
 dx 4 
 
 By giving proper values to a, b, and c, these formulas will 
 become applicable to any line of the second order. 
 
 In the case of the parabola, a = 0, b = 2p, and c == ; 
 therefore TR = 2x 
 
 TP = {2px ^ Ax^y^ 
 RN = p 
 
 PN = (2px -{- p-y. 
 
 EXAMPLES 
 
 The major axis of an ellipse is 40 inches and the minor 
 axis is 20 inches. What are the lengths of the tangent, sub- 
 tangent, normal, and subnormal? 
 
 The transverse axis of a hyperbola is 6 inches and the 
 conjugate axis is 4 inches. What is the length of the sub- 
 normal corresponding to an abscissa of 9 inches, the equation 
 
46 
 
 AN ELEMENTARY TREATISE 
 
 being y^ 
 
 A' 
 
 52? 
 
 The Cycloid 
 42. If a circle, NPG, be rolled along a straight line AC, 
 any point P of its circumference will describe a curve called 
 a cycloid. The circle NPG is called the generating circle and 
 the point P is called the generating point. 
 
 The line AC is equal to the circumference of the generat- 
 ing circle and is 
 called the base of 
 the cycloid, and the 
 line BD, drawn 
 perpendicular to it 
 at its middle point, 
 is called the axis 
 of the cycloid and 
 is equal to the di- 
 TA n N D C ameter of the gen- 
 
 erating circle. 
 
 In determining 
 the equation of the cycloid, take the origin of the coordinates 
 at A and suppose that the generating point has described the 
 arc AP ; then if N be the point at which the generating circle 
 touches the base, AN will be equal to the arc NP. 
 
 Draw NO, the diameter of the generating circle, PR perpen- 
 dicular to the base, and PH parallel to it ; then PR will be equal 
 to NH which is the versed sine of the arc NP. 
 
 Let NO = 2r, AR = x, and PR^HN = y\ then 
 
 RN = PH= {NH-HGy-= (y2r — yy-= (Zry — y^)^-; 
 
 also X =AR = AN — RN = arc NP — PH. 
 
 Therefore, since NP is the arc whose versed sine is NH or 
 y (that is, NP = vers ''^y), 
 
 X = vers "^3; — ( 2ry — y~)V2 ( 1 -j 
 
 which is the transcendental equation of the cycloid. 
 
 rdy 
 
 The rate of vers "^3^ is and that of {2ry — y^)"^^ 
 
 {2ry—y^)'^ 
 
 rdy — ydy 
 is ; therefore 
 
 {Iry — y^-y^ 
 
dx ^ 
 
 ON VARIABLE QUANTITIES 47 
 
 rdy rdy — ydy 
 
 {2ry — y~y^^ {2ry — y^)^ 
 
 ydy 
 or a?;r = , (2) 
 
 {^2ry — y^y- 
 
 which is the ratal equation of the cycloid. 
 
 43. Dividing both members of (2) of the preceding article 
 by dy, gives 
 
 dx y 
 
 dy {2ry — y'^y^ 
 
 dx 
 
 then, by substituting this value of in formulas (1), (2), 
 
 dy 
 
 (3), and (4) of Art. 40, the values of subtangent, tangent, 
 subnormal, and normal for any point of the cycloid are as 
 follows : 
 
 TR = 
 
 (^2ry — y^y^ 
 
 y (2ryy^ 
 TP= 
 
 (2ry — Z)'/^ 
 
 RN= {2ry — y~y 
 PN=^{2ry). 
 
 The Logarithmic Curve 
 
 44. The logarithmic curve takes its name from the pro- 
 perty that, when referred to rectangular axes, any abscissa is 
 equal to the logarithm of the corresponding ordinate; hence 
 the equation of the curve is 
 
 X = log y. 
 
 If a represents the base of one system of logarithms, and b 
 that of another, then 
 
 gx :—-y ^jj(J ^a; __ y . 
 
 from which it is evident that for every different base the same 
 value of x will give a different value for 3; ; that is, every dif- 
 ferent logarithmic base will give a different logarithmic curve. 
 
 If we make x^=0, y = 1 ; therefore, since this relation is 
 independent of the base of the system, it follows that every 
 logarithmic curve will intersect the axis of ordinates at a dis- 
 tance from the origin equal to unity. 
 
 From a'^ = y 
 
 the curve can be described by points even without the aid of 
 a table of logarithms ; thus 
 
^A^ ELEMENTARY TREATISE 
 
 etc. 
 
 x=l, y = a; 
 x = — 1, y^a-'^ 
 
 Then if the origin 
 
 is at A (see Fig. 12), 
 
 2 BCD will be the curve. 
 
 Resuming the equa- 
 tion of the curve, 
 
 2 -^ = log 3' ; 
 
 then, if M represents the modulus of the system, the rate is 
 
 Mdy 
 
 
 dx = - 
 dx 
 
 y 
 
 M 
 
 or 
 
 dy 
 
 y 
 
 dx 
 
 Substituting this value of in formula (1) of Art. 40 
 
 dy 
 gives 
 
 dx 
 
 TR = y = M. 
 
 dy 
 
 Hence the subtangent 
 of the logarithmic curve 
 is constant and equal to 
 the modulus of the sys- 
 tem, in which the loga- 
 rithms are taken (see 
 Fig. 13). 
 
 In the Naperian sys- 
 tem M = \; consequently 
 the subtangent of the 
 curve in this system is 
 equal to unity or AB. 
 
 TAR X If the value of — 
 
 dy 
 
 ^§- '^ be substituted in formu- 
 
 las (2), (3), and (4) of Art. 40, then 
 
ON VARIABLE QUANTITIES 
 
 49 
 
 RN = 
 
 f 
 
 M 
 
 M 
 
 Asymptotes 
 45. An asymptote is a right line which continually ap- 
 proaches a curve and becomes tangent to it only at an infinite 
 distance from the origin of the coordinates. 
 
 Let A be the origin of the coordinates (see Fig. 14), and 
 let TE be tangent to the curve at P ; then, since the subtangent 
 
 dx 
 
 TR is equal to y and the abscissa AR = x, 
 
 dy 
 
 AT^TR — AR 
 
 or 
 
 dx 
 
 AT = y — X. 
 
 dy 
 
 (1) 
 
 Also DP:DE::AT:AB 
 
 or, since DP is represented by dx, DE by dy, and 
 
 dx 
 
 AT = y 
 dx:dy::y 
 
 dy 
 dx 
 
 ■ X 
 
 dy 
 
 x:AB, 
 
50 AN ELEMENTARY TREATISE 
 
 dy 
 
 whence AB = y — x . (2) 
 
 dx 
 
 If, when X and y become infinite, both of the expressions 
 (1) and (2) also become infinite, it is evident the curve has 
 no asymptote; but if either one or both of the expressions 
 reduce to a finite quantity, it may be inferred that the curve 
 has an asymptote. 
 
 If both expressions are finite, the asymptote will be in- 
 clined to both tlie coordinate axes ; if one becomes finite and 
 the other infinite, the asymptote will be parallel to one of the 
 coordinate axes; if both become zero, the asymptote will pass 
 through the origin of coordinates. 
 
 The general equation of lines of the second order is 
 y^ = ax^ -\- bx -\- c. 
 
 Passing to the rate, 
 
 2ydy = (2ax -\- b) dx, 
 
 dx 2y^ 
 
 whence y = 
 
 dy 2ax -f- b 
 
 or, substituting for 2y^ its value, 
 
 dx 2ax^ -(- 2bx -\- 2c 
 
 y 
 
 dy 2ax ~\- b 
 
 it will be found that 
 
 dy 2ax^ -\- bx 
 
 dx 2 (ax^ -\- bx -\- cy^ 
 
 dx dy 
 
 Substituting these values of y and x — \ — in ( 1 ) and 
 
 dy dx 
 
 (2) gives 
 
 2ax'^ -\- 2bx -\- 2c bx -\- 2c 
 
 AT = ■ — x = , (3) 
 
 2ax + b 2ax -j- b 
 
 AB= {ax- -^ bx^cy — 
 
 2ax- -\- bx bx -\- 2c 
 
 2 (ax^ -\- bx -]- cy~ 2 (ax- -\- bx ^ cy 
 
 (4) 
 
ON VARIABLE QUANTITIES 51 
 
 The equation of the parabola is 
 y^ = 2px ; 
 hence in y"" = ax^ -\- bx -\- c, a = 0, b = 2p, and c = ; there- 
 fore (3) and (4) become 
 
 2px 
 
 AT = = x 
 
 2p 
 
 2px 1 
 
 and AB= =-^{2px). 
 
 2\/{2px) 2 
 
 Making x infinite, these equations become infinite also; 
 therefore the parabola has no asymptote. 
 
 In the equation of the circle and ellipse, a in 
 
 is negative; consequently AB becomes imaginary when x is 
 infinite; therefore neither the circle nor the ellipse has an 
 asymptote. 
 
 The equation of the hyperbola is 
 
 -y- = x^ — B- ; 
 
 A^ 
 
 hence & = in y^ ^ ax^ -\- bx -\- c ; therefore (3) and (4) 
 become 
 
 2c c 
 
 AT = 
 
 and AB 
 
 2ax ax 
 
 2c c 
 
 2{ax^-^cy- {ax'--\-cy 
 
 When X is infinite both of these equations become equal to 
 zero ; hence the hyperbola has asymptotes, one to either branch 
 of the curve, both of which pass through the origin of coor- 
 dinates. 
 
 The equation of the logarithmic curve is 
 X = log y. 
 
52 
 
 AN ELEMENTARY TREATISE 
 
 Passing to the rate 
 
 dx 
 
 Mdy 
 
 y 
 
 whence 
 
 dx 
 
 dy 
 
 y 
 
 M and x = 
 
 xy 
 dy dx M ' 
 
 Substituting these values in (1) and (2) gives 
 dx 
 
 AT = y 
 
 dy 
 
 = M — X 
 
 and 
 
 AB = y 
 
 dy 
 
 xy 
 
 = y 
 
 dx ^ M 
 
 When y = 0, X is negative and infinite ; but when x is nega- 
 tive and infinite (5) and (6) become 
 
 AT =ao and AB = 0; 
 
 hence the axis of abscissas of the logarithmic curve is an 
 asymptote to that branch of the curve which lies on the left 
 of the origin of the coordinates. (See Fig. 13 in Art. 44.) 
 
 Rates of Arc, Area, Surface, and Volume 
 OF Revolution 
 46. For the rate of an arc, see Art. 30, in which the 
 formula thereof will be found, viz., 
 
 d2=(dx^-\-dy^y'^, 
 z being the arc, x the abscissa, and 3* the ordinate. 
 
 Hence the rate of an arc is equal to the square root of the 
 sum of the square of the rates of the coordinates. 
 
 47. Let X represent 
 the abscissa AD (see 
 Fig. 15), y the ordinate 
 DP, and let PG or DH 
 be represented by dx, 
 the rate of ^; then the 
 rate of increase of the 
 area APD will be repre- 
 sented by ydx. There- 
 X fore, if the area of APD 
 be represented by A, then 
 dA ^= ydx. 
 
ON VARIABLE QUANTITIES 
 
 53 
 
 Hence the rate of the area of a segment of a curve is equal 
 to the ordinate into the rate of the abscissa. 
 
 48. Let X represent the abscissa AD, y the ordinate DP, 
 
 2 the arc AP, and S 
 the surface of revolu- 
 tion made by the arc 
 AP in revolving round 
 AD or the axis of X ; 
 then the point P of 
 
 _ the curve APC will 
 describe a circle whose 
 radius is y and conse- 
 quently its circumfer- 
 ence 2 Try. Now it is 
 J- ^ >^ / evident that if 2iry be 
 
 •■'g- '-^ P'^==^^ multiplied by dz 
 
 (which equals PT) the rate at which z, or the arc APC, is 
 increasing at P, then 
 
 dS^2 nydz, 
 
 or, substituting for dz its value from Art. 46, 
 
 dS = 2 Try idx- + dy^y^. 
 
 Hence the rate of the surface of the volume of revolution 
 of an arc of a curve is equal to the circumference of a circle 
 whose radius is the ordinate of the arc, multiplied by the rate 
 of the arc. 
 
 49. Let X represent any abscissa, as AD, y the correspond- 
 
 P G 
 
54 AN ELEMENTARY TREATISE 
 
 ing ordinate DP, and let dx be represented by DH = PG ; then 
 dx multiplied by the area of the circle described by the point P, 
 in revolving round the axis of X, is equal to the rate at which 
 the volume of revolution is increasing when the arc is AP, and 
 consequently the ordinate is DP. Therefore, since DP = y, 
 TT'f- is equal to the area of the circle described by the point P ; 
 consequently, if V represents the volume of revolution gen- 
 erated by the arc AP in revolving round the axis of X, then, 
 since PG = dx, 
 
 dV = TT y^dx. 
 
 Hence the rate of the volume of revolution of an arc of a 
 curve is equal to the area of the circle whose radius is the 
 ordinate of the arc, into the rate of the abscissa. 
 
 EXAMPLES 
 
 Determine the rate of an arc of the circle whose equation is 
 
 Passing to the rate, 
 
 x'dx^ 
 ydy = — xdx or dy~ = , 
 
 but, by Art. 46, ds = (dx^ -\- dy^ y^; 
 
 therefore 
 
 x-dx'^ dx 
 dz={dx^~ + y- or dz=: {x^^-y^y. 
 
 y- y 
 
 But, since y= (r- — x^)^'^- or {x^ -\- y^)''-^r, then 
 
 rdx 
 dz = . 
 
 (^r^- — x'-y 
 
 Determine the rate of the area of the parabola ; also the rate 
 of the surface and volume of revolution of the hyperbola. 
 
 Radius of Curvature 
 
 50. Of curves tangent to each other and having a common 
 tangent line at the point of contact, the one which departs 
 most rapidly from the tangent line is said to have the greatest 
 curvature. The curvature of a circle is measured by the angle 
 formed by the radii drawn through the extremities of an arc 
 of a given length. 
 
 Let R and R' represent the radii of two circles, a the length 
 of a given arc measured on the circumference of each, c the 
 angle formed by the radii drawn through the extremities of the 
 
ON VARIABLE QUANTITIES 55 
 
 arc of the one having radius R, and c' the angle similarly 
 formed by the radii of the one having radius R' ; then 
 27ri?:360°::o:c and 2 tt R' : ?>60° : : a : c' , 
 
 360° a 360° a 
 
 vi^hence c = and c' 
 
 ZttR 2TrR' 
 
 360° a 360° a 1 1 
 
 therefore c:c':: : or c:c':: —— : ——. 
 
 2TrR 2-kR\ R R' 
 
 Hence the curvature in two different circles varies inversely 
 as their radii. 
 
 Make TNG a tangent line to the curve ANC, touching at 
 the point A'', and NM a normal line thereto (see figure) ; then 
 
 G 
 
 the circumference of every circle having its center in NM, 
 which may be described through the point N, will touch at A^ 
 both the curve ANC and the tangent line TNG. 
 
 Now it is obvious that the circumference of any such circle 
 which has a greater curvature than the curve ANC will depart 
 more rapidly from the tangent line than ANC and consequently 
 will fall wholly within ANC ; but any circumference which has 
 a less curvature than ANC will depart less rapidly from the 
 tangent line than ANC and consequently will fall between it 
 and TNG. Hence, since there may be circumferences of both 
 less and greater curvature than ANC, it follows that, with a 
 center in the line NM, a circumference may be described 
 through the point N whose curvature will correspond with that 
 of the curve ANC at N — that is, which will depart from the 
 tangent line at N at the same rate as the curve ANC. 
 
 The circle, the curvature of whose circumference corre- 
 sponds with that of any curve at any point, is called the 
 
56 
 
 AN ELEMENTARY TREATISE 
 
 osculatory circle or circle of curvature, and its radius the radius 
 of curvature of the curve. 
 
 51. Let the curves BNC and DNE be tangent to each other 
 at N, and draw TNG a tangent line to both, touching at N (see 
 figure); also let y = f {x) represent the curve BNC and 
 y'=f{x') that of DNE. Draw the ordinate LN ; then, 
 since LN is common to both curves, y^^y' for the point N ; 
 also, since the angle LTN and consequently the tangent of the 
 angle of tangency for the point A^ are common to both curves, 
 
 dy dy' _ _ dy dy' 
 
 = . These two conditions, y=^y' and ^ , 
 
 dx dx' dx dx' 
 
 existing, the curves are said to have a contact of the first order. 
 
 If at the point N the second ratal coefficients of the equa- 
 
 d'^y d^y' 
 
 tions of the two curves are also equal (that is, if = ) 
 
 dx" dx^^ 
 
 c 
 
 there will be a so-called contact of the second order. This is 
 
 evident since either or is the same as the rate of 
 
 dx^ dx'^ 
 
 variation of the tangent of the angle LTN (the angle of 
 tangency) ; consequently both curves depart from the tangent 
 line TNG at the same rate. 
 
 If, in addition, the third ratal coefficients of the equations 
 
 d^y d^y' 
 
 of the curves are equal (that is, if == ) the curves will 
 
 dx^ dx'^ 
 
 have a contact of the third order, and so on for any order of 
 of contact. 
 
 Now if BNC be given in species, magnitude, and position, 
 
ON VARIABLE QUANTITIES 57 
 
 and DNE in species only, then the constants which enter 
 y^=f {x) will be fixed and determinate, while those which 
 enter y' =f {x') will be entirely arbitrary, and therefore their 
 values may be made to answer as many independent condi- 
 tions as there are constants. Hence for a contact of the first 
 order, y^f {x) must contain at least two constants; for a 
 contact of the second order, three constants ; for a contact of 
 the third order, four constants, and so on. 
 
 In the most general equation of the straight line, which is 
 
 y^ax -]rh, 
 
 there are two constants, a and h ; therefore the straight line 
 can have only a contact of the first order. 
 
 The most general equation of the circle is 
 
 (^y — by = R-— {x — ay, 
 
 which contains three constants, a, b, and R ; therefore the 
 circle can have a contact of the second order. 
 
 In the general equation of the parabola, 
 
 { (y — h) cos V — {x — a) sin vY = 
 ^P {(y — ^) sin V -{- (x — a) cos v], 
 
 there are four constants, a, b, v, and p ; therefore the parabola 
 can have a contact of the third order. 
 
 In the general equation of the ellipse or hyperbola there 
 are five constants ; therefore either the ellipse or hyperbola 
 can have a contact of the fourth order. 
 
 The curve which has a higher order of contact with a given 
 curve than can be found for any other curve of the same 
 species is called the osculatrix of that species. 
 
 52. The general equation of the circle is 
 
 {y — by = R''—{x — ay. (1) 
 
 Passing to the rate twice, under the supposition that neither 
 X nor y varies uniformly — that is, that neither dx nor dy is 
 constant — then 
 
 (y — b) dy = — (x — a) dx 
 
 and (y — b)d''y-\-dy'' = —(x — a)d^x — dx^. 
 
 From these two equations the following are found: 
 
 (dx^ + dy^) dx 
 
 dxd^y — dyd^x 
 
58 AN ELEMENTARY TREATISE 
 
 {dx^ -\- dy^) dy 
 
 and X — a = . 
 
 dxd^y — dyd'X 
 
 Substituting these values of y — b and x — a in (1) will 
 give 
 
 {dx^ -\- dy^Y dx~ {dx- -\- dy"^)- dy- 
 
 R- 
 
 ( dxd-y — dyd-x ) - ( dxd~y — dyd'~x)- 
 
 {dx-^dy^ydx^ {dx^ ^ dy^Y dy"" 
 R' = — + ; 
 
 (dxd^y — dyd'^xY {dxd'^y — dyd'^xY 
 {dx- -^dy^Y 
 
 therefore R^ = 
 
 {dxd^y — dyd^x)- 
 
 or R = ± , (2) 
 
 [dxd-y — dyd-x) 
 
 which is the general expression for the value of the radius of 
 the osculatory circle. 
 
 If dx be constant, d-x^O and (2) becomes 
 
 ^^^ (,.^ + ,fr^ ^ (3) 
 
 dxd-y 
 
 which is the expression for the value of the radius of the 
 osculatory circle applied to curves referred to rectangular 
 coordinates in which the abscissa is supposed to vary uni- 
 formly. 
 
 Hence, in order to find the radius of curvature for any 
 particular curve, the first and second rates of its equation 
 must be taken and the values of dx, dy, d^y obtained and substi- 
 tuted in (3). 
 
 If 2 represents the arc, then (dx'^ -\- dy-)^''- ^=^ ds ; substi- 
 tuted in (3), this gives 
 
 d2^ 
 
 R=± . (4) 
 
 dxd^y 
 
 d-y 
 
 It has been shown in Art. 37 that y and , consequently 
 
 dx- 
 
 y and d-y, have contrary signs when the curve is concave 
 toward the axis of abscissas and like signs when convex; 
 therefore, if we wish the radius of curvature and the ordinate 
 of the curve to have like signs, we must employ the minus 
 
ON VARIABLE QUANTITIES 
 
 59 
 
 sign m (2), (3), and (4) when the curve is concave toward 
 the axis of abscissas and the plus sign when convex. 
 
 If P and P' be any two points in the given curve APP'B, 
 
 r the radius of the 
 osculatory circle of 
 the point P, and r' the 
 B radius of the point 
 P' (see figure) then 
 curvature at P : curv- 
 
 1 1 
 ature at P' :: — : — : 
 r r' 
 
 that is, the curvature 
 at different points of 
 a curve varies in- 
 versely as the radii of 
 the osculatory circles. 
 53. The general equation of Hues of the second order is 
 3/2 =^ ax- -\-hx -\- c ( 1 ) 
 
 {2ax -f- &) dx 
 
 Fig. ZQ 
 
 and its rate 
 therefore 
 
 dx- -j- dy- = dx'- 
 
 dy 
 
 ly 
 
 (2) 
 
 {lax^hydx- {\f- ^ {lax ^ hy-\ dx"- 
 
 or {dx'' -\- dy-yi'' 
 
 Ay- 4y2 
 
 [43,2 _|_ {2ax ^ by- Y'-' dx^ 
 
 8y2 
 
 (3) 
 
 The rate of (2) is 
 
 2aydx^ — {2ax -\- h) dxdy 
 
 d-y ^ 
 
 2y^ 
 
 {2ax -f b) dx"" 
 whence, since dxdy ^ ;; [see (2)], by substitut- 
 
 ing it and reducing, 
 
 ^y 
 
 [Aay"" — (2ax -\- by] dx^ 
 
 43,3 
 
 but [see (1)] 
 
 4a3;2 = 4a^x^ + 4abx + 4ac = {2ax ^ b)- + Aac — b- 
 or 403;^ — {2ax -\- b)^ ^ Aac — b^ ; 
 
 (4) 
 
60 AN ELEMENTARY TREATISE 
 
 therefore, substituting this in (4) and multiplying by dx, 
 
 (4ac — b-) dx^ 
 dxd'-y^ J^;^ • (5) 
 
 Substituting (3) and (5) in (3) of Art. 52, then 
 
 [4y + {lax^hyyi- 
 
 R = ± , 
 
 2{4ac — h'') 
 
 or, substituting for y its value, 
 
 {4{ax--^hx^c)-^{2ax-^hyyi- 
 
 R = ±: , (6) 
 
 2 ^4ac — b') 
 
 which is the general expression for the radius of curvature of 
 lines of the second order for any abscissa x. 
 
 If both numerator and denominator of (6) be divided by 
 8, then 
 
 1 
 
 [ax^ -\-bx + c -{-— {2ax + byy^^ 
 
 R = ± -^ . (7) 
 
 ac — — b" 
 4 
 
 The numerator of this value of R is the cube of the normal 
 [see (6), Art. 40] ; therefore, since the denominator is con- 
 stant, it is evident from Art. 52 that the radii of curvature at 
 different points of lines of the second order are to each other as 
 the cubes of the corresponding normals. 
 
 If the origin of coordinates is at the vertex of the trans- 
 verse axis, c = 0; consequently, using the minus sign, (6) 
 becomes 
 
 (4 {ax^ + bx) + {2ax + byy^^ 
 
 R = , 
 
 2b'- 
 
 which, when jr = 0, reduces to 
 
 1 
 R = —b. 
 
 2 
 
 In this case b is the parameter of the curve; therefore the 
 radius of curvature at the vertex of the transverse axis of lines 
 of the second orde? is equal to half the parameter of the curve. 
 
 In the case of the parabola whose equation is 
 
 y2 = 2pX, 
 
ON VARIABLE QUANTITIES 61 
 
 a = 0, c = 0, b^2p; therefore, substituting these values in 
 (7) and using the minus sign, 
 
 R=z , 
 
 which is the general value of the radius of curvature for any 
 point of the parabola. If x = 0, then R^=p, the radius of 
 curvature at the vertex of the axis. 
 
 In the case of the ellipse whose equation is 
 
 B^ 
 
 , & = 0, and c = B- ; therefore, substituting these 
 
 A' 
 values in (6), reducing and using the minus sign, 
 
 (^4 _^2_^2_|_ 52^2)3/2 
 
 R 
 
 A'B 
 
 which is the general value of the radius of curvature for any 
 point of the ellipse. If x = 0, then R= , which is the 
 
 radius of curvature at the vertex of the minor axis, li x = A, 
 
 B^ 
 then R = , which is the radius of curvature at the vertex 
 
 A 
 
 of the major axis. 
 
 Taking the equation of the logarithmic curve, 
 X = log y, 
 and passing to the rate twice, 
 
 dy ydx 
 
 dx=^M or dy 
 
 and d^y = 
 
 y M 
 
 dxdy 
 
 M 
 
 dy^ 
 whence {dx^ + dy^y-'^ = (M^ + y^Y'K 
 
 Substituting the values of dxd'-y and {dx- -\- dy-y- in 
 
62 AN ELEMENTARY TREATISE 
 
 (3) of Art. 52 and using the plus sign, for any point of the 
 logarithmic curve 
 
 R ^= . 
 
 My 
 
 When 3; is equal to the modulus of the system of logarithms 
 employed, 
 
 R = 2M V2. 
 From the ratal equation of the cycloid (Art. 42) 
 
 ydy 
 dx = ^-^ . (1) 
 
 {2ry — y-y^ 
 
 Passing to the rate and reducing, 
 
 ydy'^ {r — 3;) 
 0= {yd^y + dy^) {2ry — y'^Y' 
 
 {2ry — y~y'' 
 
 {2ry — 3;- ) yd-y + ryrfy^ ; 
 whence 
 
 rdy^ rydy^ 
 
 d^y = — or dxd~y = 
 
 2ry — y~ ( 2ry — y- ) ^'^ 
 
 It will also be found that 
 
 y^dy^ 
 
 dx^- + dy' = + dy^ = 
 
 2ry — y 
 y^dy^ -\- 2rydy^ — y^dy^ 2rydy^ 
 
 therefore 
 
 2ry — y^ 2ry — y^ 
 
 2rydy^ \/(2ry) 
 
 {dx^ -\- dy^y^ 
 
 (2ry — y2)^/2 
 
 Substituting the values of dxd^y and {dx" + dy^y^ in (3) 
 of Art. 52 and using the minus sign will give 
 
 R = 2^{2ry); 
 
 but the normal is equal to \/ {2ry) by Art. 43 ; hence the radius 
 of curvature for any point of the cycloid is equal to twice the 
 normal at the point of contact. 
 
 EVOLUTES AND INVOLUTES 
 
 54. An evolute is a curve from which a thread is supposed 
 to be unwound or evolved, its extremity describing another 
 curve called an involute. 
 
ON VARIABLE QUANTITIES 
 
 63 
 
 Thus, let a thread be wrapped about the curve BCC'D (Fig. 
 
 21); then, if the thread be kept tight and unwound from 
 
 p BCC'D, its extremity, com- 
 
 mencing at A, will describe 
 
 „ ^ > 5 the curve APP'S. The curve 
 
 ^^ \ BCC'D is called the evolute 
 
 of the curve APP'S and 
 APP'S the involute of 
 BCC'D. 
 
 From the manner in which 
 
 the involute is generated it is 
 evident that any portion of 
 the thread, as CP, which is 
 disengaged from the evolute 
 is a tangent to it at C and per- 
 pendicular to the involute at 
 P ] also, that any point in the 
 evolute, as C, may be consid- 
 ered as a center, and the line 
 CP as the radius of a circle of whose circumference that por- 
 tion of the involute curve at P is an arc. 
 
 The points B, C, C are therefore centers, and the lines BA, 
 CP, CP' the radii of circles of curvature of the points A, P, 
 P' of the involute ; hence any radius of curvature, as CP, is 
 equal to AB plus the arc BC of the evolute. 
 
 The value of AB will depend upon the position of the 
 point B, from which the arc of the evolute is estimated; but 
 since AB is the radius of curvature of the involute at A, if A 
 is the origin of the involute and B the corresponding origin 
 of the evolute, B will be the center of the osculatory circle to 
 the involute at its origin. Therefore, if the radius of curva- 
 ture at the origin of the involute is equal to zero, A and B will 
 coincide, and consequently AB will be equal to zero. If the 
 involute is a curve of the second order, the radius of curvature 
 at the vertex of the transverse axis is equal to half its para- 
 meter, Yzb, by Art. 53; consequently AB will be equal to 3'^&, 
 and B, the origin of the evolute, will be in the axis of abscissas 
 AX. 
 
 Hence, since the center of any circle of curvature of the 
 curve APP'S is in the curve BCC'\D, it follows that the equa- 
 tion representing the coordinates of the center of any circle of 
 curvature of the involute will be the equation of the evolute. 
 
64 AN ELEMENTARY TREATISE 
 
 Now the general equation of the circle, consequently of any 
 circle of curvature, is 
 
 {y — br = R'—{x — ay, (1) 
 
 in which a and b are the coordinates of its center and x and y 
 the coordinates of any points of its circumference; therefore 
 a and b will represent the coordinates of any point, as C, of the 
 evolute BCC'D, and its equation will be 6 = / (a) ; also x and 
 y will represent the coordinates of any corresponding point, as 
 P, of the involute APP'S, and its equation will be 3^ = / {x). 
 
 Taking the rate of ( 1 ) twice 
 
 (3' — b) dy = — {x — a) dx 
 and ^3'" + (3' — b)d~y = — dx- ; 
 
 dx^ -\- dy'^ 
 whence ^^3' + (2) 
 
 d'^y 
 
 dy dx^ -\- dy- 
 
 and a = x — -— ( ). (3) 
 
 dx d-y 
 
 These are expressions for the values of the coordinates of the 
 evolute in terms of the rates of the coordinates of the involute. 
 
 Hence, if we take the rate of the equation of the involute 
 
 dy 
 
 twice, y = f {x), obtain the values of dx", dy-, , and d-y, 
 
 dx 
 
 and then substitute them in (2) and (3), we shall have two new 
 equations, expressing the values of a and b, the coordinates of 
 the evolute, in terms of x and y, the coordinates of the involute. 
 
 Finally, by combining the equations thus found with the 
 equation of the involute and eliminating x and y, an equation 
 will be obtained containing only a and b, which will be the 
 equation of the evolute. 
 
 Taking the equation of the common parabola 
 
 y~ = 2px, 
 
 and passing to the rate twice 
 
 ydy = pdx 
 
 and dy"^ -{- yd-y = ; 
 
 whence 
 
 p- dy p p- 
 dx^-\-dy^= ( + 1) dx-, = —, 2.ndd-y = — dx\ 
 
 y^ dx y y^ 
 
ON VARIABLE QUANTITIES 
 
 65 
 
 Substituting these values in (2) and (3) and reducing 
 
 give 
 
 that 
 and 
 
 yO yh yi 
 
 6 = — or 6^ = , and a^^^x -^ -|- p. 
 
 f P' P 
 
 Substituting 2px for y~ in the last two equations, it is found 
 
 —, (4) 
 
 6- = 
 
 1 
 
 a=^^x -\- p or x=^ — (a — p). 
 
 Finally, substituting — (a — p) for x in (4), then 
 
 ,- = —^a-py. 
 
 (5) 
 
 which is the equation of the evolute and shows it to be the 
 semi-cubical parabola. 
 
 If we make & = 0, 
 then a = p; hence, the 
 evolute meets the axis of 
 abscissas at a distance 
 AB from the origin 
 (Fig. 22) equal to half 
 the parameter of the in- 
 volute. 
 
 If the origin of the 
 coordinates of the evo- 
 lute be transferred from 
 A to B, (5) becomes 
 
 27p 
 
 Since every value of a gives two equal values of b with 
 contrary signs, the curve is symmetrical with respect to the 
 axis of abscissas ; the evolute BD^ corresponding to the part 
 AP of the involute and BD to the part AP\ 
 
 From the equations relative to the cycloid. Art. 53, it is 
 found that 
 
 2rydy^ dy (2ry — 3;^) ^^ 
 dx^ -f- dy^ 
 
 2ry 
 
 dx 
 
 y 
 
66 
 
 AN ELEMENTARY TREATISE 
 
 and 
 
 d-y 
 
 rd'f- 
 
 2ry — 3;2 
 
 Substituting these values in (2) and (3) of Art. 54 will 
 give 
 
 h^= — y and a = x -\- 2 (2ry — y^)"^^, 
 
 whence y = — b and x = a — 2 ( — 2rb — b^)^^^. 
 
 Substituting* these values of x and 3' in the transcendental 
 equation of the cycloid (Art. 42) gives 
 
 a — 2 (— 2rb — b'y^ = vers -^ (— b) — (—2rb — b-'y\ 
 
 or a = vers -' {— b) + (— 2rb — b- ) ^/% 
 
 which is the transcendental equation of the evolute of the 
 cycloid, referred to the primitive axes and origin. 
 
 From the equation of the radius of curvature for the cy- 
 cloid, R^2\/(2ry) (see Art. 53), we have R = when 
 3;:^0, and when yr:=.2r = BD,R=^Ar-^A'B; therefore the 
 origin of the evolute is at A, and A'D = BD. 
 
 By transferring the origin of the coordinates of the evolute 
 
 from A to A' and 
 Q estimating the ab- 
 
 scissas from the 
 right toward the 
 left, a new equation 
 of the evolute is 
 formed which will 
 be found to be of 
 the same form and 
 to involve the same 
 constants as the 
 equation of the cy- 
 cloid ; hence the 
 evolute of a cycloid is an equal cycloid — that is, the arc AA' 
 is a facsimile of the arc AB, and A'C of the arc CB. 
 
 Since the origin of the evolute is at A and the radius of 
 curvature for the vertex B of the cycloid is 4r, the length of 
 the evolute AA' is 4r; hence the length of the cycloid ABC is 
 equal to 8r, or four times the diameter of the generating circle. 
 
 EXAMPLES 
 
 Determine the length of the radius of curvature for a point 
 in a parabola whose abscissa is four inches and ordinate six 
 inches. 
 
ON VARIABLE QUANTITIES 
 
 67 
 
 Determine the length of the radius of curvature for a point 
 in an ellipse, whose abscissa is 16 inches, measured from the 
 center, the semi-axes being 26 and 13 inches. 
 
 Determine the equation of the evolute of the equilateral 
 hyperbola, its equation being y^=^x'^ — A^. 
 
 Determine the evolute of the spiral whose ratal equation is 
 dr 
 
 dv 
 
 ar 
 
 Fig. £4 
 
 Curves Referred to Polar Coordinates 
 55. If the right line PC (Fig. 24) revolves uniformly 
 around the point P, and if at the same time a point moves from 
 P along the line PC at such a rate that at the first revolution 
 of PC it will arrive at A, at the second at B, etc., the curve 
 described by the point will be a spiral. 
 
 The point P about which the right line revolves is called 
 the pole ; the point which moves along the line PC and de- 
 scribes the curve is called 
 the generating point; a 
 straight line drawn from 
 the point P or eye of the 
 spiral, so called, to any 
 point of the curve, as A^, is 
 called the radius vector, 
 and each portion of the 
 spiral described by the 
 generating point, as PDA, 
 AEB, is called a spire. 
 
 With the pole as a 
 center and PA (the dis- 
 tance which the generating 
 point moves from P along PC during the first revolution of 
 PC) as a radius, if the circle AFG be described, the angular 
 motion of PC about the pole, consequently the radius vector, as 
 PN, is measured by an arc of this circle, estimated from A. 
 
 Now, if r represents the radius vector and v the measuring 
 arc estimated from A, it is evident that r is a function of v and 
 may generally be represented by the equation, 
 
 r = a?7", ( 1 ) 
 
 in which a and n are constants. The value of n depends upon 
 the law which governs the motion of the generating point along 
 
68 AN ELEMENTARY TREATISE 
 
 the radius vector and the value of a upon the relation existing 
 between a given value of r and the corresponding value of v. 
 
 If n is positive, the spiral represented by (1) commences 
 at the pole, for when v^O, r^O. If w is negative, the equa- 
 tion becomes 
 
 r^av-""; (2) 
 
 consequently the spiral commences at an infinite distance from 
 the pole, for when v^O, r is infinite, or when r = 0, v is 
 infinite. 
 
 When n is equal to unity, ( 1 ) becomes 
 
 r^av. (3) 
 
 Now if a = AP, the circumference of the circle AFG will 
 be 2a TT, which is the measuring arc for the first revolution of 
 PC ; therefore, since PA or a is then the radius vector, 
 
 a=^a- 2a TT 
 
 1 
 
 whence 
 
 27r 
 Substituting this value of a in (3) gives 
 
 V 
 
 2 TT 
 
 the equation of the spiral of Archimedes. 
 
 When n is equal to one half, ( 1 ) becomes 
 
 r = av^' or r^ == a^v, 
 
 which is the equation of the parabolic spiral, being of the same 
 form as that of the parabola; for substituting y for r, \/{2p) 
 for a, and x for i' gives 
 
 y^ = 2px. (4) 
 
 With 2p as radius draw the circle ABC, divide its cir- 
 cumference into any number of equal parts, as six, and draw 
 through its center P, the divisional lines DD' , EE' , FF'. With 
 
 -(2/) + !), -(2/> + 2), -(2/>-f 3), and -(2/^ + 4), (1, 
 
 2, 3, and 4 being values given x as in the construction of the 
 parabola) as radii, draw the arcs Aa, AhC, Ac, and Ad, having 
 their centers in the line AG ; then with Pa, Ph, Pc, and Pd, as 
 radii, draw the arcs aa' , hh\ cc', and dd', and the curve drawn 
 
ON VARIABLE QUANTITIES 
 
 69 
 
 Fig. Z5 
 
 from P through a', h\ c' , d' will be the required spiral. In 
 proof, it will be seen that AP : PC is equal to Pb^ or Ph'^ (see 
 Euclid, proposition 35, Book III) ; hence, since AP = 2p and 
 PC = x,ii y be represented by Pb = Pb\ then 
 
 y^ = 2px. 
 
 Also, when x = 0, 3* ^^ ; therefore the spiral commences at 
 P, its pole. 
 
 When n is equal to — 1, (1) becomes 
 
 a 
 r = av~'^ or r = — . (5) 
 
 The curve represented by this equation is called the hyper- 
 bolic spiral on account of its analogy to that of the hyperbola 
 when referred to its center and asymptote. 
 
 With a as a radius draw the circle ABC and divide its cir- 
 C vJ cumference, 2a tt, into any 
 
 number of equal parts, as 
 six; then giving to v the 
 
 Sfl TT 4a TT 
 
 values 2a TT, , , 
 
 3 3 
 
 air, etc., the corresponding 
 
 1 
 
 values of r will be , 
 
 27r 
 
70 AN ELEMENTARY TREATISE 
 
 3 3 1 
 — — , , — , etc. Let Pa, Pb, Pc, Pd, etc. represent these 
 
 b TT 4 TT IT 
 
 values of r; then the curve drawn through a, h, c, d, etc., Mrill 
 be the hyperboHc spiral. 
 
 Take any point in the spiral, as G, and draw GH perpen- 
 dicular to PL; then PG^r and the angle GPH = v; hence 
 
 GH = r sin v, 
 
 or substituting for r its value from (5) 
 
 a sin V 
 GH^ . (6) 
 
 V 
 
 Now it is evident that the smaller the value of v, the nearer 
 will V and sin v approach equality and consequently the nearer 
 will GH become equal to a; therefore, if CJ be drawn parallel 
 to PL, CJ must be an asymptote to the spiral. 
 
 The equation of the logarithmic spiral, so called, is 
 
 alogr^v. (7) 
 
 This spiral may be constructed as follows. With unity for 
 radius draw the circle ABC; then, giving to r the values 1, 2, 
 3, etc., the corresponding values of v will be 0, a log 2, a log 3. 
 
 Fig. 17 
 
 Set off from A on the circumference of the circle these values 
 of V, A, Ab, Ac; then to A and through b, c, draw PA, PD, 
 PE, the values of r, and the curve drawn through A, D, E, 
 will be the logarithmic spiral. 
 
 Since the relation between r and v is entirely arbitrary, 
 
 P 
 r^ , (8) 
 
 1 -|- cos V 
 
 is the polar equation of the parabola, the pole being at the 
 focus. 
 
ON VARIABLE QUANTITIES 
 
 71 
 
 The polar equation of the eUipse, the pole being at one of 
 the foci, is 
 
 P 
 
 r = - . (9) 
 
 1 + ^ cos V 
 
 The polar equation of the hyperbola, the pole being at one 
 of the foci, is 
 
 r = ~ . (10) 
 
 1 -|- ^ COS V 
 
 In (8), (9), and (10) p represents half the parameter, e 
 the eccentricity, and v the angle which the radius vector makes 
 with the axis of X. 
 
 Tangents and Normals 
 
 56. The suhtangent to a spiral is a line drawn from the 
 pole perpendicular to the radius vector and limited by a tangent 
 drawn through the extremity of the radius vector ; the tangent 
 is a line extending from the point of tangency to the sub- 
 tangent; the subnormal is a line drawn from the pole to the 
 foot of the normal ; the normal is a line drawn perpendicular 
 to the tangent and extending from the point of tangency to the 
 subtangent extended. 
 
 Let r be any radius vector, as PN (see Fig. 28), v the 
 measuring arc estimated from A, and z the corresponding arc 
 
 of a spiral of which BNC is a section. Then, if dz be repre- 
 sented by NM, a tangent to BNC at A'', and PN be extended 
 to N, so that the angle NN'M will be a right angle, it is evident 
 that NN' will represent dr, the rate at which the radius vector 
 PN is increasing. Now, draw NM' parallel to N'M and MM' 
 
72 AN ELEMENTARY TREATISE 
 
 parallel to NN' ; also, with F as a center and PN as a radius, 
 describe the circular arc A'NS. Then it will be seen that NM' , 
 tangent to A'NS at N, will represent the rate at which the arc 
 A'N is increasing at A''. Hence, since NM' represents the rate 
 at which the arc A'N is increasing, it is obvious that RR' , 
 tangent to the measuring circle at R, represents c?z^, the rate at 
 which V, the measuring arc estimated from A, is increasing to 
 correspond with NM or dz. 
 
 Therefore, since the triangles PRR' and PNM' are similar, 
 PR : RR' ::PN: NM', 
 or, making the radius of the measuring circle unity, 
 
 1 : dv : : r : NM', 
 whence NM' = rdv. ( 1 ) 
 
 Again, from the similar triangles MM'N and NPT, 
 MM' : NM' ::NP: PT, or, since NP = r, MM' = NN' = dr, 
 and from ( 1 ) , NM' = rdv, 
 
 dr : rdv wr: PT, 
 
 r^dv 
 
 whence PT =^ ; (2) 
 
 dr 
 
 but PT is the subtangent of the spiral, hence : 
 
 The length of the subtangent to any point of a spiral is 
 equal to the square of the radius vector into the rate of the 
 measuring arc, divided by the rate of the radius vector. 
 
 For the tangent TN , 
 
 TN- =PN^ -f PT'-— 
 
 r^dv^ 
 
 that is, TN^ ^r~ -\- 
 
 dr^ 
 
 or TN = ^{ dr^ + r^dv^ ) ^-. ( 3 ) 
 
 dr 
 
 Hence the length of the tangent to any point of a spiral is 
 equal to the square root of the sum- of the squares of the radius 
 vector and subtangent. 
 
 For the subnormal PQ, 
 
 PT:PN::PN:FQ— 
 
 r^dv 
 
 that is, -.r-.-.r: PQ 
 
 dr 
 
 dr 
 or PQ=-—. (4) 
 
 dv 
 
ON VARIABLE QUANTITIES 73 
 
 Hence the length of the subnormal to any point of a spiral is 
 equal to the rate of the radius vector divided by the rate of the 
 measuring arc. 
 
 For the normal QN, 
 
 dr''' 
 
 that is, QN^ = r'-\- 
 
 dv'^ 
 
 dr^ 
 or QN={r^~^--yK (5) 
 
 dv^ 
 
 Hence the iength of the normal to any point of a spiral is 
 equal to the square root of the sum of the squares of the radius 
 vector and subnormal. 
 
 The tangent of the angle of tangency of a spiral, PTN, 
 
 r~dv 
 
 since PN = r and PT =^ from (2), is 
 
 dr 
 
 PN dr 
 
 = . (6) 
 
 PT rdv 
 
 Hence the tangent of the angle of tangency of a spiral is 
 equal to the rate of the radius vector divided by the radius 
 vector into the rate of the m^easuring arc. 
 
 PT 
 
 The tangent of the angle PNT is equal to : but, since 
 
 b fe \i pj^ 
 
 r'-dv 
 
 PN = r and PT 
 
 dr 
 
 PT rdv 
 
 PN dr 
 which is the tangent of the angle the tangent line makes with 
 the radius vector. 
 
 Of the general equation of spirals, 
 
 r = av^, 
 
 dr dv \ 
 
 the rate is — — = anv^~'^ or 
 
 dv dr 
 
74- AN ELEMENTARY TREATISE 
 
 dr dv 
 
 Substituting the value of or , also av^^ for r, in 
 
 dv dr 
 
 formulas (2), (3), (4), (5), (6), and (7), the result will be 
 
 
 
 
 11 — 
 
 n 
 
 
 
 
 TN = 
 
 --{a 
 
 2.^2)1 1 
 
 
 av'' 
 n 
 
 {n- 
 
 + z;^)V3 
 
 n -\- 
 
 2 ^ ~ 
 
 
 
 
 PQ = anv"-^ 
 
 
 
 
 QN = 
 
 {a- 
 
 V'" + ( 
 
 PN 
 
 
 av^' 
 n 
 
 ■1 (n 
 
 2_^^2)y. 
 
 
 PT 
 
 av 
 
 V 
 
 
 
 
 
 PT V 
 
 
 
 
 PN n 
 
 V 
 
 In the equation of the spiral of Archimedes, r = , n=\, 
 
 2 TT 
 
 1 
 
 and a = . By substituting these values in the preceding 
 
 2 TT 
 
 formulas, the following are obtained : 
 
 PT = ^, TN = ^{\^v^-y\PQ=-^, 
 
 Z TT Z TT Z TT 
 
 1 PN I PT 
 
 QN = (1 + vy\ = -, = v. 
 
 2m ^ PT V PN 
 
 If v = 2 7r — that is, if the tangent is drawn at the extremity 
 of the arc generated in the first revolution of the radius 
 vector — then 
 
 pr = 2 TT— 
 
 that is, PT is equal to the circumference of the measuring 
 circle. 
 
 At the completion of m revolutions v = 2m^ tt, and conse- 
 quently PT = 2m^ TT = m • 2m TT — 
 that is, at the completion of in revolutions the subtangent is 
 equal to in times the circumference of the circle described with 
 the radius vector of the wth revolution. 
 
 In the equation of the hyperbolic spiral, r^av, n^ — 1 ; 
 therefore PT = — a. 
 
ON VARIABLE QUANTITIES 75 
 
 Hence the subtangent of the hyperbolic spiral is constant. 
 From the equation of the logarithmic spiral, 
 
 V = log r, 
 
 rdv 
 
 it will be found that = M ; 
 
 dr 
 
 rdv 
 
 but [see (7)] represents the tangent of the angle made 
 
 dr 
 
 with the radius vector by a tangent line to the curve. Hence 
 the tangent of the angle which the tangent line makes with the 
 radius vector is constant and equal to the modulus of the 
 system of logarithms employed. In the Naperian system the 
 modulus is unity; therefore, if v is the Naperian logarithm of r, 
 the angle which the tangent line makes with the radius vector 
 is 45°. 
 
 Rate of the Arc and Area of Spirals 
 57. Let BNC in Fig. 29 be a section of a spiral, P the 
 pole and TN a tangent to the curve at N. Draw PN, and NM' 
 at right angles to PN ; also extend TN to M and draw MM' 
 so that NM'M will be a right angle ; then 
 
 NM^ = M'M' + M'N^. 
 
 But since NM repre- 
 sents ds; M'M, dr; and 
 M'N, rdv [see (1) of 
 Art. 51], 
 
 dz"= dr^-{- r^dv^ or dz= 
 (dr^ + r'~dv'')\ (1) 
 
 Hence the rate of an 
 p -T- arc of a spiral is equal 
 
 to the square root of the 
 sum of the squares of the rate of the radius vector and of the 
 product of the radius vector and the rate of the measuring arc. 
 Let PN be a radius vector of the spiral PBNC in Fig 30. 
 Draw NM' and M'P, making the angle PNM' a right angle. 
 Then, representing the area by A, 
 
 1 
 dA= — PN-NM' 
 2 
 
 or, since PN = r and, by (1) of Art. 51, NM' = rdv, 
 
 1 
 
 dA= — r-dv. (2) 
 
 9 V / 
 
76 AN ELEMENTARY TREATISE 
 
 P B 
 
 This is evident from what has- been shown in Art. 51 ; for, 
 since NM' represents the rate at which the arc AN is increas- 
 ing at N, it must also represent the rate at which the extremity 
 of the radius vector is revolving when it arrives at N. Conse- 
 quently the area of the triangle PNM' represents the rate at 
 which the area of the spiral is increasing when the radius 
 vector is PN. 
 
 Hence the rate of the area of a spiral is equal to one-half 
 the square of the radius vector into the rate of the measuring 
 arc. 
 
 EXAMPLES 
 
 Determine the rate of an arc of the parabolic spiral. 
 
 Determine the rate of the area of the hyperbolic spiral. 
 
 If the rate of the measuring circle of the Naperian log- 
 arithmic spiral is three, at what rate is the area of the spiral 
 increasing when the radius vector is four? 
 
 Radius of Curvature for Spirals 
 
 58. Of the spiral PNS in Fig. 31, the subtangent 
 
 r'^dv 
 
 PT = (see (2) of Art. 56), the tangent 
 
 dr 
 
 NT = ^^ {dr^ ^r^dv'-y^ [see (3) of Art. 56], the radius 
 
 dr 
 
 vector NP = r, and CN = R, the radius of the osculator}^ 
 circle AMN , NQ being normal to the spiral. 
 
 Join CP and draw DP parallel and BP perpendicular to 
 NQ ; then, since BN = DP, 
 
 CP^~ = CN^ + NP^~ — 2CN ■ DP 
 or CP^ = R- + r^ — 2R-DP ; 
 
 but DP = r s'mPND, or, since s'm PND is also equal to 
 
ON VARIABLE QUANTITIES 
 
 77 
 
 M 
 
 Fjo. 31 
 
 PT 
 
 rdv 
 
 NT 
 
 ^' DP 
 
 C 
 
 A. 
 
 {dr- -f- r^dv'^Y' 
 
 CP^ = i?2 _^ r^ 
 
 2Rr-dv 
 
 — : therefore 
 
 Vo ' 
 
 ( dr~ -\- r^dv- ) '''^ 
 
 Now the equation of the spiral is r- 
 
 av^. 
 
 (1) 
 whence 
 
 V =^ ; hence dz/ ■ 
 
 )/n^f. 
 
 ,i/« 
 
 7.1/« 
 
 Substituting this value of (/I/ in (1), then 
 
 {r~'" + n-a^'^'Y^ 
 
 (2) 
 
78 AN ELEMENTARY TREATISE 
 
 Passing to the rate, CP and R being constant for any point 
 of the circle AMP, and reducing, 
 
 R = , (3) 
 
 which is the general value of the radius of curvature for all 
 spirals represented by the equation r = av'\ in terms of the 
 radius vector. 
 
 In the case of the spiral of Archimedes, w=l, and (3) 
 
 (r- 4- a-) 3/2 
 
 becomes R = . 
 
 r- 4- 2a2 
 
 For the logarithmic spiral, whose equation is log r = v, 
 
 Mdr 
 
 dv = ; 
 
 r 
 
 therefore, substituting this value oi dv m (2), it will be found 
 
 2RMr 
 that CP^ = i?2 _|_ ^2 
 
 (M2 + \y- 
 
 Passing to the rate and reducing, 
 
 R == . 
 
 M 
 
 If the Naperian system be used, M = 1, and R = r \/2. 
 
 Determine the radius of curvature for a parabolic spiral; 
 also for the hyperbolic spiral. 
 
 Singular Points of Curves 
 
 59. It has been shown in Art. 36 that the first ratal co- 
 efficient of the equation of a curve represents the tangent of 
 the angle of tangency; therefore, since the tangent of this 
 angle is zero when the angle is zero, and infinite when the 
 angle is 90°, it follows that the roots of the equation 
 
 dy 
 
 dx 
 will give the abscissas of all points of the curve at which the 
 
ON VARIABLE QUANTITIES 
 
 79 
 
 tangent line is parallel to the axis of abscissas; also that the 
 roots of the equation 
 
 dy 
 
 dx 
 
 will give the abscissas of all points of the curve at which the 
 tangent line is perpendicular to the axis of abscissas (see Fig. 
 32 and Fig. Z^). 
 
 Taking the equation of the 
 circle 
 
 3;=± {R' — x'^y- 
 and passing to the rate, 
 dy X 
 -^■=-^ • (1) 
 
 If (1) =0, ;r = 0, but when 
 x = 0, y=±R; therefore the 
 circle has two tangents parallel 
 to the axis of abscissas, (see Fig. 
 32). If (1) is infinite, ^ = ± i? 
 and 3; = ; therefore the circle 
 has two tangents perpendicular 
 to its axis of abscissas (see Fig. 
 33). 
 
 Of the equation of the Y 
 parabola, 
 
 y=±yy{2px), 
 the rate is 
 
 dy p 
 
 dx 
 
 (2) 
 
 V(2M) 
 If (2)=0, both .r and 3; 
 are infinite; therefore the 
 parabola has no tangent F"'g- ^^ 
 
 parallel to its axis of ab- 
 scissas. If (2) is infinite, both x and 3; are equal to zero; 
 therefore the parabola has a tangent (the axis of ordinates), 
 perpendicular to its axis of abscissas at the origin of its 
 coordinates, as shown in Fig. 34. 
 
 Points of Inflection 
 60. Those points of a curve at which the curve changes its 
 direction — that is, from being concave to its axis of abscissas it 
 
80 AN ELEMENTARY TREATISE 
 
 becomes convex, or vice versa— are called points of inflection. 
 At such a point the angle of tangency and consequently 
 its tangent must either change from increasing to decreasing, 
 or from decreasing to increasing; therefore the rate of varia- 
 tion of the tangent of the angle of tangency at a point of 
 inflection will be zero, real, or infinite; zero when the angle of 
 tangency is zero, real between 0° and 90°, and infinite when 
 
 90°. Hence since represents the rate of variation of the 
 
 tangent of the angle of tangency, every point of inflection will 
 have for its abscissa some root of the equations : 
 
 -^^0 (1), \0 (2), and = oo (3). 
 
 dx' dx- dx^ 
 
 But it does not follow that every root of these equations will be 
 
 the abscissa of a point of inflection ; hence it is necessary to 
 
 d^y 
 
 examine whether the value of x will give contrary signs 
 
 dx'~ 
 
 (see Art. 37). 
 
 Let the equation of the curve be 
 
 y^a + b (x — c)^ (4) 
 
 Then passing to the rate twice, 
 
 dy 
 
 -^=.Zb{x — cy (5) 
 
 dx 
 
 d^y 
 
 and =^6b {x — c). (6) 
 
 dx~ 
 
 Making (6) equal to zero, then 
 
 but when x = c, the first ratal coefficient is equal to zero also ; 
 therefore y = a when x^=^ c, there is a tangent line to the 
 curve at the point whose coordinates are a and c, which is 
 parallel to the axis of abscissas. 
 
ON VARIABLE QUANTITIES 
 
 81 
 
 JJ 
 
 If h is positive, the second ratal coefficient will be zero for 
 x = c, but negative when x <^c and positive when x ^ c ; 
 
 therefore there is an inflection 
 of the curve at the point whose 
 abscissa is ;ir^c (see Fig. 35). 
 If 6 is negative, the second 
 ratal coefficient will be positive 
 when .r <;; c and negative when 
 X y. c ; therefore, at the point of 
 the curve whose abscissa is 
 x^ c, there is an inflection of 
 the curve, but opposite to the 
 first (see Fig. 36). 
 ^ ^ In the first case the curve is 
 
 first concave, then convex to the 
 axis of X; in the second case it 
 is first convex, then concave, as 
 A X shown in the figures. 
 
 Let the equation of the curve be 
 
 y=^a-\-h (x — c)^/^. 
 
 Then, passing to the rate twice. 
 
 dy 
 
 3b 
 
 dx 
 
 and 
 
 d^y 
 
 dx^ 
 
 25 (x — c)' 
 
 5 7 
 
 5 {x — cy^' 
 
 Making x^ c, both expres- 
 sions become infinite ; therefore 
 the first ratal coefficient, since 
 y^a when x^ c, gives a tan- 
 gent line to the curve at the point 
 whose coordinates are a and c, 
 which is perpendicular to the 
 axis of X. 
 
 If b is positive, the second 
 ratal coefficient will be positive 
 for all values of jt < c, and nega- 
 tive for all values of x ^ c; 
 hence, for all values of x less 
 than c, which makes y positive, 
 the curve will be convex to the 
 axis of X, while for all values of 
 X greater than c, it will be concave (see Fig. 37). 
 
 If b is negative, the case will be the reverse, as shown in 
 Fig. 38. 
 
82 
 
 AN ELEMENTARY TREATISE 
 
 If a = 0, P will be the point of inflection, and if o = 0, 
 also c^=0, A will be the point of inflection. 
 
 Cusp Points 
 
 61. The point at which two branches of a curve terminate 
 and have a common tangent is called a cusp point. When the 
 cusp is formed by the union of two branches, one on either 
 side of the tangent, it is called a cusp of the first order, 
 and when both branches are on the same side of the tangent, 
 it is called a cusp of the second order. 
 
 If x = c be the abscissa of a cusp point, the values of x 
 immediately preceding and following that of x^ c, when 
 substituted in the given equation, will give to 3* either two real 
 or two imaginary values ; if real, both will be greater or both 
 less than that of the cusp point; furthermore, for a cusp point 
 there will be a distinguishable term in the second ratal co- 
 efficient, either equal to zero or infinity. 
 
 Let the equation of the curve be 
 
 3; = a.ar ± h {x — c)°'^ ; 
 
 then, taking the rate twice gives 
 
 dy 5 d-y 
 
 a ± — h {x — c)"'- and 
 
 15 
 4 
 
 h {x — cy\ 
 
 dx 2 dx'- 
 
 Making the second ratal coefficient equal to zero, we have 
 x= c; hence, since for a value of x less than c, y will have two 
 imaginary values, and for a value of x greater than c, y will 
 have two real values, there is a cusp at the point of the curve 
 whose abscissa is x = c (see Fig. 39). 
 
 When x = c, the first ratal coefficient equals a ; hence the 
 tangent of the angle of tangency at the cusp point is equal to a, 
 
 and since y = ac when x^= c, 
 the tangent line to the curve at 
 the cusp passes through the 
 origin of coordinates. Also for 
 any value of x greater than c, 
 d^y 
 
 will have two values, one 
 
 dx'' 
 
 positive and the other negative ; 
 consequently one branch of the 
 curve is convex and the other 
 concave to the axis of X ; there- 
 V fore a branch must lie on either 
 side of the tangent line AN and 
 the cusp is of the first order. 
 
ON VARIABLE QUANTITIES 
 
 83 
 
 The equation of the semi-cubical parabola is 
 
 y^±ax 
 
 3/2 
 
 the rates of which are 
 dy 
 
 dx 
 
 d^y 
 
 ax' 
 
 and 
 
 dx^ 
 
 3a 
 
 ArX'^' 
 
 Making 
 
 dx^ 
 
 CO , jt; = ; then y has two imaginary values 
 
 for ;r<^0 and two real values for x^O; and, since ^ = 
 
 when x^O, there is a cusp at the origin of the coordinates; 
 
 dy 
 
 but when x^O, = 0, hence the axis of abscissas is a tan- 
 
 dx 
 
 gent to both branches of the curve at the cusp (see Fig. 40). 
 Examination of the primitive equation shows that for every 
 
 value of X greater than 0, 3; has two values, one positive and 
 the other negative ; therefore one 
 branch of the curve, AC, lies above, 
 and the other, AC , below the axis of 
 X, and the cusp is of the first order. 
 
 d^y 
 vy For any value of jt'^ 0, has two 
 
 values, one positive and the other neg- 
 ative ; consequently, since y is negative 
 for the branch AC , both branches are 
 convex to the axis of X. 
 
 Fig. 4 
 
 C 
 Of the equation 
 
 the rates are 
 
 y=^a-\- h (x — c) ^^^, 
 dy 2b 
 
 and 
 
 dx 
 d^y 
 
 3 (^x — cy^ 
 
 2b 
 
 dx^ 
 
 9 {x — cy^ 
 
 Making the second ratal coefficient equal infinity. x = c ; 
 hence, since y = a when x-^ c, and since y is greater than a 
 either for .r <^ c or x "^ c, there is a cusp at the point of the 
 curve whose coordinates are x = c, y^a. 
 
 When the first ratal coefficient is equal to infinity, jr = c ; 
 hence a tangent line to the curve at the cusp point is perpen- 
 dicular to the axis of abscissas (see Fig. 41). 
 
AN ELEMENTARY TREATISE 
 
 Of the equation 
 
 the rates are 
 
 and 
 
 For any value of x, either 
 less or greater than c, the value 
 
 of is negative; consequently 
 
 both branches of the curve are 
 concave to the axis of abscissas ; 
 also, since y has a value corre- 
 sponding to either x <^c or 
 X ■^ c, a branch lies on either 
 side of the tangent line TG. 
 
 d^y 
 
 If b is negative, then be- 
 
 dx^ 
 
 comes positive for any value of 
 
 X, either less or greater than c ; 
 
 therefore both branches of the 
 
 curve are convex to the axis of 
 
 abscissas (see Fig. 42). 
 
 4 
 
 ^ 5 
 
 dy 
 
 dx 
 
 d'y 
 
 Ax 
 
 9-V-3/2 
 
 dx' 
 
 4 ± Zx'^^ 
 
 Making the first ratal coefficient equal to zero, then ^ = 
 
 32 288 
 or 4 : but when .r ^ 0, -y ^ 0, and when .r = 4, 3/ = — - or — — ; 
 
 5 5 
 
 therefore the axis of abscissas is tangent to the curve at A, 
 
 the origin. There is another 
 
 tangent to the curve at E, 
 
 parallel to the axis of X, and 
 
 corresponding to an abscissa 
 
 32 
 of 4 and an ordinate of — — . 
 
 
 
 If the positive sign be 
 
 d^y 
 
 used (since will then be 
 
 dx- 
 
ON VARIABLE QUANTITIES 85 
 
 positive for any value of x), the left hand branch of the curve, 
 AB, is convex to the axis of abscissas. But if the negative 
 sign be used which corresponds with the right hand branch of 
 
 d-y 16 
 
 the curve AEC, since will then be positive for x <'~Z~ 
 
 dx' 9 
 
 16 
 and negative for x^ , the part AD, answering to ;ir = U to 
 
 16 
 
 , will be convex and the part DEC concave; consequently 
 
 9 
 
 there is an inflection at D. 
 
 In conclusion, since 3; has two imaginary values for ^ <^0, 
 and two real values for x '^0, and since y = when x^O, 
 the branches AB and AEC form a cusp of the second order at 
 A, their origin. 
 
 Multiple Points 
 
 62. The points at which two or more branches of a curve 
 intersect are called a multiple point. 
 
 At a multiple point it is therefore evident that there must 
 be as many tangents to the curve as there are intersecting 
 
 dy 
 branches ; hence , which represents the tangent of the angle 
 
 dx 
 of tangency, will have as many values as there are different 
 
 tangents. 
 
 Let the equation of the curve be 
 
 y = a±x {h^ — x'-y^; (1) 
 
 then, passing to the rate, 
 
 dy b^ — 2x~ 
 
 -^=± . (2) 
 
 dx {b~ — x^)'''^ 
 
 An inspection of ( 1 ) shows that values of x greater than b 
 make y imaginary, while for values of x less than b, y has two 
 values ; hence the curve has two branches which, since y=^a 
 
86 
 
 AN ELEMENTARY TREATISE 
 
 when x = 0, intersect the axis of ordinates at a distance a from 
 A, the origin (see Fig. 44). 
 
 dy 
 
 When x= b or — b, becomes infinite, consequently the 
 
 dx 
 
 tangents to the curve at B and B^ are perpendicular to the axis 
 
 dy 
 
 of abscissas. When x^O, ^± b; therefore there are 
 
 dx 
 
 two tangents, TN and T'N', to the curve passing through the 
 
 multiple point. 
 
 Let the equation of the curve be 
 
 3^=1+ (IzpV-^) (i±V^)^s 
 then, passing to the rate, 
 
 dy I zf.S\/ X 
 
 dx 
 
 4 (y/ X d= xy 
 
 (3) 
 
 (4) 
 
 From an examination of (3) we find for .*: .^0 that 3^ is 
 imaginary ; f or jr ^ 0, y ^ or 2 ; for any value between and 
 1, that y has four real values ; for x==l, 3* = 1 ; and for any 
 value of X greater than 1, that y has only two real values. 
 
ON VARIABLE QUANTITIES 
 
 87 
 
 Hence two branches of the curve must intersect each other at 
 the point whose coordinates are x=^\ and y=\ (see Fig. 45) . 
 
 dy 1 1 
 
 When x=l, is infinite, + — \/2, or — — \/2, conse- 
 
 dx 2 2 
 
 quently the curve has a multiple point corresponding to the 
 coordinates x=l, y=l, and at this point there are three 
 Y 
 
 tangent lines, TG, T'G' , T"G" : TG is perpendicular to the 
 axis of abscissas, and T'G' and VG" make angles therewith, 
 
 1 1 
 
 whose respective tangents are -f- — \/2 and — — \/2. 
 
 Isolated Points 
 63. A point which is entirely detached from a curve, but 
 whose coordinates satisfy the equation, is called an isolated or 
 conjugate point. 
 
 Since a point entirely detached from a curve can have no 
 tangent, it is evident that for an isolated point, the first ratal 
 coefficient of the equation will be imaginary. 
 
 Let the equation be 
 
 y=.±{x^a)^x; (1) 
 
 then, passing to the rate, 
 
 dy 2)X -\- a 
 
 (2) 
 
 dx 
 
 2^x 
 
AN ELEMENTARY TREATISE 
 
 By examining (1) we find that x = makes 3' = 0, and 
 for any value of jr > 0, y has two 
 real values, one positive and the other 
 negative; therefore the curve passes 
 through the origin A and has two 
 branches, AC, AC, extending to the 
 right (see Fig. 46). 
 
 Equation (1) is also satisfied by 
 the coordinates x^ — a and 3'^0; 
 
 dy 
 
 but when ;r = — a, becomes im- 
 
 dx 
 
 aginary; hence, the point P, whose 
 abscissa is x = — a, being entirely 
 detached from the curve, is an iso- 
 lated point. 
 
 The rate of (2) is 
 
 d~y 3 X — a 
 
 dx- 
 
 ^x^x 
 
 Making this equal to zero gives x=^ — a\ therefore there is 
 
 1 
 an inflection of the curve at the point whose abscissa is .r = — a. 
 
 Maxima and Minima 
 
 64. If a variable quantity increases until it attains a value 
 greater than any immediately preceding or following it, such 
 a value is called a maximum ; and if it decreases until it attains 
 
 a value less than any imme- 
 diately preceding or following 
 it, such a value is called a 
 minimum. 
 
 Illustration. Let the points 
 P and P' be so situated in the 
 curve BPP'C (see Fig. 47) 
 ^ that tangent lines, T'N and 
 T'l^' , to the curve at P and 
 P' , shall be parallel to the axis 
 of abscissas, AX; then, since it is obvious that the ordinate 
 FP is greater than any immediately preceding or following it, 
 and that the ordinate F'P' is less than any immediately pre- 
 ceding or following it, FP is a maximum and F'P' a minimum. 
 
 A.B 
 
ON VARIABLE QUANTITIES 89 
 
 Therefore, if y = / {x) is the equation of the curve, x repre- 
 senting any abscissa and y the corresponding ordinate, y is a 
 maximum when it is equal to F'P\ 
 
 Now, for that point of a curve at which a tangent Hne is 
 parallel to the axis of abscissas, since it then makes no angle 
 with this axis, and since the first ratal coefficient of its equa- 
 tion represents the tangent of the angle of tangency (see Art. 
 36), • 
 
 dy 
 
 dx 
 
 Hence when a function is either a maximum or a minimum 
 the first ratal coefficient is equal to zero. For instance, if the 
 function is of the form 
 
 y = x^ — 2ax -\- b, 
 the rate is 
 
 dy 
 
 = 2x — 2a ; 
 
 dx 
 
 therefore y is either a maximum or a minimum when 
 
 2x — 2a ^ or when x=^a. 
 
 It will be observed that the curve BPP'C is concave to its 
 axis of abscissas at the point P, and convex at the point P' ; 
 therefore, from Art. 37, since either ordinate FP or F'P' is 
 positive, the second ratal coefficient of the equation of the 
 curve will be negative for the ordinate FP and positive for the 
 ordinate F'P\ But it has been shown that FP is a maximum 
 ordinate, and F'P' a minimum ordinate; consequently, if the 
 
 d^y 
 
 equation of the curve is y^f {x), will be negative when 
 
 dx^ 
 
 3/ is a maximum and positive when 3; is a minimum. 
 
 Hence, to find the values of the variable of a function which 
 will render the function a maximum or a minimum, also to 
 distinguish the one from the other, we have the following rule. 
 
 Make the first ratal coefficient of the function equal to 
 zero and find the values of the variable in this equation; then 
 substitute these values in the second ratal coefficient of the 
 function, and each value which gives a negative result will 
 render the function a maximum, and each value which gives 
 a positive result will render it a minimum. 
 
90 AN ELEMENTARY TREATISE 
 
 It sometimes happens that a value of the variable, as jt ^ a, 
 found by making the first ratal coefficient of the function equal 
 to zero, will reduce the second ratal coefficient to zero also. In 
 
 this case, substitute a ±v for x in , and if either -\- v ov 
 
 dx'- 
 
 — V give a negative result for a small value of v, y will be a 
 maximum; but if the result is positive, 3; will be a minimum. 
 If one sign gives a negative result and the other a positive 
 result, it is clear y will be neither a maximum nor a minimum ; 
 such a result simply indicates that the curve represented by the 
 proposed equation, has an inflection at the point corresponding 
 to the abscissa x^a (see Art. 60). 
 
 For illustration, take the equation 
 
 y = x' — Ax"" + \6x + 13 ; 
 then, passing to the rate twice, 
 dy 
 
 dx 
 
 4x' — l2x' + l6, (1) 
 
 and -^=12.^2 — 24;^. (2) 
 
 dx-' 
 
 If (1)=0, then x = —l or .r = 2. 
 
 Substituting these values of x in (2), then for x^ — 1, 
 
 d^y d^y 
 
 36 and for x = 2, = 0. 
 
 dx^ dx' 
 
 Since x^2 reduces the second ratal coefficient to zero, by 
 substituting 2±v for x in (2) the result is \2v{v±2), 
 which, for a small value of v, is negative for the minus sign 
 and positive for the plus sign, which shows that there is an 
 inflection of the curve at r' , corresponding to the abscissa 
 x = 2; hence this value of x makes y neither a maximum nor 
 a minimum (see Fig. 48). There is also an inflection at r, but 
 a minimum for 3^ answering to .r^ — 1. 
 
ON VARIABLE QUANTITIES 
 
 Y 
 
 V\a. 48 
 
 91 
 
 Let the equation be 
 
 i(3^2_i2) 
 
 ay I 
 
 then = ^ 0, whence x =^ ±2. 
 
 dx {^x^ — \2xy'~ 
 
 Taking the rate of (3) and reducing, regarding 
 3jir^ — 12^0, the result is 
 
 dry Zx 
 
 dx''- (jr" — 12:1;)'''^' 
 
 3 
 
 which, for ;»; = — 2, equals — — , and for ;f = 2, equals 
 
 3 . ^ 
 
 — — V — 1 ; therefore a; is a maximum when x^= — 2 and a 
 
 2 
 
 minimum when x-^ -\-2. 
 
 If 3;2___^___^3 \2x, 
 
 dv 
 
 then 
 
 dx 
 
 3x'- — l2; 
 
 whence x ^ ±: 2, 
 
 the same values of x as before. 
 
92 AN ELEMENTARY TREATISE 
 
 Hence when the expression containing the variable is under 
 a radical, the radical may be omitted. 
 
 65. In case the equation is of the form 
 
 f{x,y)=0, (1) 
 
 in which 3; is a function of x, pass to the rate and find from 
 
 dy 
 ^ the value of x in terms of y, also of y in terms of x. 
 
 dx 
 
 Substitute the value of x in terms of 3/ in (1) and therefrom 
 determine the value of y, and with this value of y find that of x. 
 Next determine the second ratal coefficient and substitute 
 therein the values of x and y found from (1). And if the 
 result is negative, y will be a maximum ; but if positive, a mini- 
 mum. Should these values reduce the second ratal coefficient 
 to zero, proceed as in article 64. 
 
 EXAMPLE 
 
 y'^-\-2axy — x^ — 6^ = 0. (2) 
 
 Passing to the rate, 
 
 2ydy -j- 2axdy -)- 2aydx — 2xdx = 0, 
 
 dy X — ay 
 
 -j^ = —, (3) 
 
 dx ax -\- y 
 
 whence, by making it equal to zero, 
 
 X 
 
 x^ay or y^ — . 
 a 
 
 Substituting ay for x m. (2), we find 
 
 b 
 
 y = 
 
 therefore x ■ 
 
 ab 
 
 dy 
 
 Taking the rate of (3) , regarding = 0, also x — ay = 0, 
 
 dx 
 
 d^y ax -\- y 1 
 
 then 
 
 dx^ {ax -\- yY QX -\- y 
 or, substituting for x and 3; their values and reducing, 
 
 d'y 1 
 
 dx^ ~ b (1 +a2)V. ■ 
 
ON VARIABLE QUANTITIES 93 
 
 ah 
 
 Hence 3; is a minimum when x = 
 
 (l+a'-r 
 
 dy 
 
 The value of ;ir = a found from = will sometimes 
 
 dx 
 
 make the second ratal coefficient infinite. In such a case, sub- 
 
 d^y 
 stitute a ± V (v being a small quantity) for x in . Then 
 
 dx^ 
 
 if the result for both signs of v is negative, y will be a maxi- 
 mum for x = a, and if positive, a minimum ; but if the result 
 for one sign is negative and for the other sign positive, 3; will 
 be neither a maximum nor a minimum. 
 
 Let the function be 
 
 y==b — (x — a)*/^. 
 
 Passing to the rate twice, 
 
 dy 4 
 
 ~- = —-{x — ayr^, (4) 
 
 dx 6 
 
 d'~y 4 
 
 and for x = a, = — = co. (5) 
 
 dx'~ 9 (x — ay/' 
 
 If (4) ^0, x^a; but when x^=a, (5) becomes infinite; 
 therefore, by substituting a ±: v for ;ir in (5), 
 
 d^y 4 
 
 dx^~ 9(±vy/^' 
 
 which is negative for either plus or minus v; therefore y is a 
 maximum when x^a. 
 
 66. When the curve represented by a given equation forms a 
 cusp of the first order, and a tangent line to the curve at the 
 cusp point makes an angle of ninety degrees with the axis of 
 abscissas, it is evident, as may be seen in the figures 41 and 42, 
 that the ordinate of the cusp point will be a maximum or mini- 
 mum, according as both branches of the curve are concave or 
 convex to the axis of abscissas ; but when the angle of tangency 
 is 90°, the first ratal coefficient of the equation is infinite. 
 Therefore, to find the value of the variable which will render 
 the function a maximum or a minimum in such a case, a solu- 
 tion is required of 
 
 dy 
 = 00. 
 
 dx 
 
94 AN ELEMENTARY TREATISE 
 
 If the value of x thus found renders the second ratal co- 
 efficient infinite, proceed as has been previously explained. 
 
 For an example, take 
 
 y = a — 9 (x — cy/^; 
 then, passing to the rate twice, 
 
 dy 6 
 
 dx (x — c) 
 
 1/3 
 
 (1) 
 
 d^y 2 
 
 and = = 00. (2) 
 
 dx^ (x — c)*/^ 
 
 ( 1 ) is satisfied when x^ c, and by substituting c -]- v m 
 (2), it becomes 
 
 d'y _ 2 
 
 dx- ~ z/*/^ ' 
 
 therefore 3; is a minimum when x=^ c. 
 
 EXAMPLES 
 
 Determine the values of the variable that will make the 
 following maxima or minima. 
 
 1. y = x^ — 8x''^22x'' — 24x. 
 
 2. y = b—(x — a)'/\ 
 
 .3. y = 4 ± (3x'- — Ux -{- 9y^^ 
 
 4. Divide a quantity, a, into two such parts, that the wth 
 power of one part multipHed by the nih. power of the other 
 part shall be a maximum. 
 
 5. Determine the minimum hypotenuse of a right-angled 
 triangle containing an inscribed rectangle whose sides are as 
 a to b. 
 
 6. Determine the length of the axis of the largest parabola 
 that can be cut from a right cone, the length of whose side is s. 
 
 7. The perpendiculars of two right-angled triangles are 
 a and b, the sum of their bases c, and the sum of their hypote- 
 nuses a minimum. What is the base of each? 
 
 8. If the solidity of a cylinder is 2 tt and its surface a mini- 
 mum, what is its diameter? 
 
 9. What is the height of the largest cylinder which can be 
 inscribed in a cone whose altitude is a? 
 
ON VARIABLE QUANTITIES 95 
 
 10. What is the altitude of a maximum rectangle inscribed 
 in a triangle whose base is h and perpendicular height a? 
 
 11. What is the altitude of the largest cylinder that can be 
 cut from a paraboloid whose axis is a? 
 
 67. It has been shown that the value of a single variable 
 which will render its function a maximum or a minimum, is 
 found by making the first ratal coefficient of the function equal 
 to zero ; hence it is evident that the value of each variable of a 
 function of two or more variables is also to be found by mak- 
 ing the first partial ratal coefficient of the function, relative 
 to that variable, equal to zero: that is, if u = f {x,y), the 
 value of X which will render the function a maximum or a 
 minimum, is found from 
 
 du 
 
 dx 
 
 du 
 
 and of 3; from = 0, 
 
 dy 
 
 whence all the values of x and 3; can be found, which will 
 render u a maximum or minimum. 
 
 It has also been shown that the second ratal coefficient of a 
 function of a single variable is negative when the function is a 
 maximum and positive when it is a minimum ; for like reasons, 
 the values of x and y, found from the first partial rates of 
 u = f (x, y), when substituted in the second partial rates, must 
 give each a negative value when m is a maximum and a posi- 
 tive value when w is a minimum. 
 
 The second partial rates of m = / (x, y) are 
 
 d^u d^u d~u d^u 
 
 -, and 
 
 dx^ dy^ dxdy dydx 
 
 or, since the last two expressions are equal (see Art. 21), only 
 the following need be used, namely 
 
 d^u d^u d^u 
 
 , , and , 
 
 dx^ dy- dxdy 
 
 each of which must be negative when m is a maximum and 
 positive when m is a minimum. 
 
 The process is similar when there are three or more inde- 
 pendent variables. 
 
96 AN ELEMENTARY TREATISE 
 
 Let u = ax^y- — x^y- — x^y^, 
 
 the partial rates of which are 
 
 du 
 
 = Zax-y- — ^x'^y- — Zx-y^ 
 
 dx 
 
 du 
 
 and = 2ax^y — 2x^y — 3x^y^. 
 
 dy 
 
 Making these equal to zero, it will be found that 
 
 1 1 
 
 x = — a and -y = — a. 
 
 2 3 
 
 The second partial rates are 
 d^u 
 
 = 6axy^ — 12x^y^ — 6xy^, 
 dx^ 
 
 d^u 
 
 = 2ax^ — 2x'^ : — 6:1:^^ 
 
 dy^ 
 
 ■ d^u 
 
 and = 6ax~y — 8x^y — 9x^y^. 
 
 dxdy 
 
 Substituting in these the values of x and y, the results are 
 
 d^u a* d^u a* d^u a* 
 
 — and 
 
 dx^ 9 dy^ 8 dxdy 6 
 
 a a _ _ 
 
 therefore, when ;ir = — and -y = — , w is a maximum and equal 
 2 3 
 
 a« 
 
 to . 
 
 432 
 
 EXAMPLES 
 
 The volume of a rectangular solid is s. What is the length 
 of each side when its surface is a minimum? 
 
 Let X, y, and represent the lengths of the sides, and u 
 
 xy 
 
 its surface. 
 
 
 
 2s 2s 
 
 Then 
 
 u =^ 2xy 4- + • 
 
 
 X y 
 
ON VARIABLE QUANTITIES 97 
 
 The first partial rates of this are 
 
 du 2s du 2s 
 
 dx X- dy y"^ 
 
 whence, by making them equal to zero, it will be found that 
 
 x^s^'^ and y = s^^^. 
 
 The second partial rates are 
 
 d-u As d^u 4s d^u 
 
 and 
 
 dx^ x^ dy^ y^ dxdy 
 
 Since each of these are positive, m is a minimum when each 
 side is equal to s^^^. 
 
 The semi-diameter of a sphere is r. What are the lengths 
 of the sides of the greatest rectangular parallelopipedon that 
 can be cut from it? 
 
PART TWO 
 THE INVERSE METHOD 
 
PART TWO 
 
 THE INVERSE METHOD 
 
 DEFINITIONS AND ILLUSTRATIONS 
 
 68. In Part One the function is given to find the rate ; 
 herein the rate of the function is given to find the function. 
 
 69. The method of passing from the rate to the function — 
 that is, the process of restoring the function of which the rate is 
 given — is called integration, and the restored function is called 
 the integral of that rate. 
 
 The integral of a given ratal expression is indicated by the 
 character J placed before it, as 
 
 f (2axdx -f- hdx), 
 
 showing that the integral is required. 
 
 70. There can be only one rate of a given function, but 
 there may be more than one function answering to a given 
 rate. This is obvious, since x'^ and x'^ -\- a have the same rate, 
 viz., 2xdx. Therefore, in integrating, a constant term must be 
 added to the integral. This term is usually represented by C ; 
 thus the integral of 
 
 du = 2axdx is u^^ax -\- C. 
 
 C is called an arbitrary constant, and the integral before 
 the value of C is known is called an incomplete integral. In 
 the solution of a real problem, however, the value of C may be 
 determined from the known conditions of the problem and 
 consequently a complete integral obtained. 
 
 For illustration take the ratal equation of the straight line, 
 dy = adx, 
 whence y=^ax-\-C. 
 
 If the straight line passes through the origin of the coordi- 
 nates, then y = when x = 0; hence C ^ 0, and the complete 
 or true integral is 
 
 y = ax. 
 
102 
 
 AN ELEMENTARY TREATISE 
 
 But if the straight line cuts the axis of ordinates at a dis- 
 tance from the origin equal to b, then for x = 0, y = b, conse- 
 quently C = b, and the true integral is 
 
 y^ ax -{- b. 
 
 71. Of the triangle ABC, let AC be represented by x, BC 
 by 2ax, CD by dx, and the area of ABC by A ; then 
 
 dA = 2axdx, 
 whence A = ax~ -\- C. 
 
 But when x^O, ^ = 0, conse- 
 quently C = 0; therefore the true 
 integral is A^^ax~. (1) 
 
 It will be observed that the same 
 is true for the triangle AEF when 
 the area of AEF — that is 
 
 A' ^ax^. 
 
 (2) 
 
 Now if ^^n in (1) and ;r = w in (2), representing the 
 area of EBFC by A'', then 
 
 A' 
 
 A — A^ = anr — am^. 
 
 This process is termed integrating between limits. In the 
 present case, the integral of 2xdx is taken between the limits 
 of ^ = w and x^n, m being called the inferior limit of x and 
 n the superior limit. The sign of this method is placed before 
 the given rate; thus (''Xdx, X being a function of x. If 
 
 = 0, then the sign becomes (^. 
 
 Simple Algebraic Rates 
 72. According to the rules under Art. 10, if 
 
 m 
 
 u = , du = ax^dx : 
 
 therefore, it is seen that the function corresponding to the rate 
 
 ax^^dx is, by Art. 70, 
 
 ax" 
 
 n+ 1 
 
 + C: 
 
 ax" 
 
 that is Cax'^dx^ + C. 
 
 •^ n + 1 
 
 Hence the following rule : 
 
ON VARIABLE QUANTITIES 103 
 
 The integral of a monomial rate is equal to the constant 
 factor into the variable with its exponent increased by unity, 
 divided by the exponent thus increased, plus a constant term. 
 
 This rule is applicable whether n is positive or negative, a 
 whole number or a fraction, except when n = — 1, for then 
 
 ax''*'^ ax^-'^ ax^ a 
 
 n-\-l 1 — 1 
 
 But when n^ — 1, 
 
 adx 
 ax"dx = ax'^dx = , 
 
 X 
 
 which is the rate of log x, by Art. 27, 
 
 adx 
 therefore f = a log ;tr + C. 
 
 '' X 
 
 Hence the integral of a fractional rate whose numerator is 
 the rate of the denominator multiplied by a constant, is equal 
 to the constant into the Naperian logarithm of the denominator, 
 plus a constant term. 
 
 Since a constant quantity retains the same value through- 
 out the same investigation, it can be placed outside the sign 
 of integration, as a( x"dx. 
 
 73. Since the rate of a function composed of the sum or 
 difference of any number of terms containing the same inde- 
 pendent variable is the corresponding sum or difference of 
 their rates taken separately (see Art. 11), it follows that the 
 integral of a ratal expression composed of the sum or difference 
 of several terms is equal to the corresponding sum or difference 
 of their respective integrals ; thus 
 
 J {ax'^dx-{-bdx — nx^~'^dx)= aJx~dx-\-bJdx — n J x^-^dx= 
 
 1 
 
 - ax^ -\- bx — x^ + C. 
 3 
 
 From this it is evident that a polynomial of the form 
 
 du = (a ± bx zt cx^ ± etc.)" dx, 
 
 in which m is a positive whole number, can be integrated by 
 raising the quantity within the parenthesis to the nth power, 
 multiplying through by dx, then integrating each term sepa- 
 rately. 
 
104 AN ELEMENTARY TREATISE 
 
 Let du^ {a -\- bxY dx; 
 
 then du = a^dx -\- Za^bxdx -j- Zah'^x^dx + b^x^dx, 
 whence u ^^J (a^dx + Sa^bxdx -j- Sab^x^dx -\- b^x^dx) = 
 
 3 1 
 
 a^x 4- — a-bx~ 4- ab~x^ + — b^x^ + C. 
 2 4 
 
 When the rate is of the form 
 
 du^ (x^ -\- ax -\- b)^ {2xdx -\- adx), 
 
 in which n is an integer or fraction, positive or negative, and 
 when the quantity within the last parenthesis is the rate of that 
 within the first; then 
 
 M = f {x^ -\- ax -[- &)" {2xdx -\- adx) = 
 
 1 
 
 (x"" -\-ax-{- by^^ + C. 
 
 n -\- I 
 
 This case is substantially the same as that of a monomial 
 rate (see Art. 72) and is similarly inapplicable under the same 
 condition: viz., when the exponent w = — 1, for then 
 
 1 1 
 
 {x^ -\-ax-\- bY^^= {x^ + a;ir + by-^ = 
 
 n-\-l 1 — 1 
 
 {x^-\-ax^by 1 
 
 ^ — ^ 00 ; 
 
 1 — 1 
 
 but when n = — 1, 
 
 {x"^ -\- ax -\- by {2xdx + adx) = 
 
 2xdx -f- odx 
 
 (x- -y ax -\- b)~^ (2xdx -j- adx) 
 
 x^ -\- ax -\- b 
 
 in which the numerator is the rate of the denominator; there- 
 fore 
 
 2xdx -f- adx 
 
 u= C (x^ -\- ax -\- b)-^ {2xdx -\- adx)^ C = 
 
 x^ -\- ax -\-h 
 
 log {x^ ^ax^b) -\-C. 
 
 74. To determine the integral of a binomial rate of the 
 form 
 
 (/m = (a -|- bx"")"^ x'^-'^dx : 
 
 that is, one in which the exponent of the variable without the 
 parenthesis is less by unity than that of the variable within. 
 
ON VARIABLE QUANTITIES 105 
 
 Assume a -\- bx" = y, 
 
 and taking the rate nbx"~^dx = dy 
 
 or x" dx ■■ 
 
 hence du = 3;'' 
 
 nb 
 
 dy y"^dy 
 nb nb 
 
 /iiJH+l 
 
 therefore by Art. 72, u = 1- C ; 
 
 nb (ni -\- I) 
 
 or, substituting for 3; its value, 
 
 u = 1- C. 
 
 nb (m -\- 1) 
 
 Hence the integral of a binomial rate in which the ex- 
 ponent of the variable without the parenthesis is one less than 
 that within, is equal to the binomial factor with its exponent 
 increased by unity, divided by the exponent thus increased 
 into the product of the exponent and coefficient of the variable 
 within, with a constant term added to the result. 
 
 If the rate is 
 
 (a + bnx''-'^) dx 
 
 du = , 
 
 2 {ax -\- bx^^y^ 
 
 1 
 
 or du=^ — {ax -\- bx")~''^^ (« + bnx^~^)dx, 
 
 it will be seen that the quantity within the last parenthesis is 
 the rate of that within the first ; therefore, by Art. 73, 
 
 1 
 
 u^J — {ax -{- bx"")-^^" {a + bnx'^-^) dx= {ax -\- bx'^y^ 
 
 If the rate 
 
 is 
 
 
 adx 
 
 du — , 
 b ± ex 
 
 by making 
 
 
 
 b zt cx = y, 
 
 then 
 
 ± 
 
 cdx 
 
 ± dy 
 
 = dy, or dx = 
 
 c 
 
 therefore 
 
 
 
 ± ady 
 du — : 
 
 cy 
 
106 AN ELEMENTARY TREATISE 
 
 consequently, by Art. 72, 
 
 a 
 
 z* = dz — log y -\- C, 
 c 
 
 or, substituting for y its value, 
 
 a 
 
 Jt= ± — log {h ±i ex) -j- C. 
 c 
 
 EXAMPLES 
 
 ax^dx 
 
 1. f/w = 
 
 2 
 
 2. du = (:r- 4" ^^Y i^^dx -\- adx) 
 
 3. du= {1 -{- ax)~^ 2xdx. 
 axdx 
 
 4. du 
 
 {x^ H- a^) 
 5. du= {a-\- bx^y^ mxdx -\- 
 
 c -\- X 
 
 Simple Circular Rates 
 75. Referring to Art. 31, it will be seen that 
 M =j"cos ;rfl?;r = sin ;r 
 
 dx 
 
 u ^ r = tan X 
 
 cos^ X 
 u = f — sin xdx = cos x 
 
 dx 
 
 u = r — = cot X 
 
 sin^ X 
 u=^ C sin xdx = vers x 
 
 tan xdx 
 
 u = J = sec X, etc. 
 
 cos^ir 
 
 Also in (3) of Art. 31, it is shown that the rate of 
 sin x'^ = n cos x^'^dx — 
 that is, Jn cos x^'^dx = sin x"'; ( 1 ) 
 
 hence it is clear that 
 
 J — n sin x^''^dx = cos x": ( 2 ) 
 
 If n= 1, (1) and 2 become 
 
 w = sin .r -f- C" and u = cos x -\- C. 
 
ON VARIABLE QUANTITIES 107 
 
 EXAMPLES 
 
 1 dx 
 1- /sin 7 
 
 2/- 
 
 X X- 
 2xdx 
 
 cos^ (1 — x^) 
 
 76. It is shown in Art. 33, making R = l and omitting the 
 constant C, that 
 
 du 
 
 1. x = C — 
 
 ^ (I- 
 
 = sin"^ u 
 
 du 
 2. x^=^ — = cos"^ w 
 
 du 
 Z. x = r = vers"^ u 
 
 J {^Zu — u-'Y- 
 
 du 
 4. x^ C = tan~^ u 
 
 •^ 1+u^ 
 
 du 
 
 Let dx = , ( 1 ) 
 
 (a2_w2p 
 
 and assume u^av; 
 
 then du = a4v and (a- — u^Y^=za{\ — v'^Y^. 
 
 Substituting these values in ( 1 ) gives 
 
 dv 
 dx^ : 
 
 dv 
 hence [see (1)1, .jr= C 
 
 dv du u 
 
 or, since = and v== — 
 
 du , u 
 
 X == r = sin~^ — . 
 
 (a^ — M-)'- a 
 
 du 
 Let dx = , 
 
 {2au — u^)'^^ 
 
 and assume u = av; 
 
 — = sin"^ V 
 
 V2. 
 
108 AN ELEMENTARY TREATISE 
 
 then 
 
 du adv dv 
 
 du = adv and 
 
 {2au — u^) '''2 a ( 2v — v- ) ^^ ( 2v — v~ ) '^^ 
 
 du 
 
 therefore [see (3)], jir^ f = vers"^^' = 
 
 {Zv — z/-) 
 
 du u 
 
 r __ vers"^ — . 
 
 {2au — u~Y^ a 
 
 du 
 
 Let dx = 
 
 a'-^-u^ 
 and assume u^av; then du = adv 
 du adv dv 
 
 and 
 
 a^ -\- u~ a-{l-{-v^) a (1 -\- V-) 
 
 therefore 
 
 1 dv 1 du 1 u 
 
 X = — r = — tan"^ V =( = — tan"^ — , 
 
 a 1 -\- v^ a a^ -\- u^ a a 
 
 EXAMPLES 
 
 du 
 1. dx^ 
 
 2. dx 
 
 3. dx 
 
 (^c — u^y- 
 
 du 
 
 {Au — 2u-y^ 
 du 
 
 5 + M^ 
 
 4. dx = — + 
 
 du 
 
 (l — u^y^ (2u — u^y 
 
 Integration by Series 
 
 77. Any expression of the form 
 
 du = Xdx, 
 
 in which X is such a function of x that it can be developed into 
 a series of the powers of x, may be integrated in the follow- 
 ing manner. Supposing the development to be 
 
 X = Ax'' + Bx^ -f Cx'' -f etc., 
 
ON VARIABLE QUANTITIES 109 
 
 then multiplying by dx and integrating each term separately 
 give 
 
 ABC 
 
 u = CXdx = x"^^ + x^^^ -f x'^^ + etc. 
 
 •^ a + 1 ^+1 c + 1 
 
 This method is often the best, if not the only course to 
 pursue, for when the series are rapidly converging, an approxi- 
 mate value of the integral may be readily determined. 
 
 dx 
 Let du = . 
 
 a -{- X 
 
 Then, developing by the binomial theorem, 
 
 1 I X x~ x^ 
 
 —==(o + ;r)-^ = - — - + - — - + etc.; 
 
 a -\- X a a- a-' a* 
 
 multiplying by dx, 
 
 dx dx xdx x~dx x^dx 
 
 = — + — + etc., 
 
 a -\- X a a- a^ a* 
 
 and integrating, the result is 
 
 (Xa^ Jv jC % Jv 
 
 u = C =(- — -f — + etc.) +C. 
 
 -^ a + x a 2d' Za^ 4a* 
 
 It has been shown in Art. 72 that 
 
 dx 
 
 u = ( = log (a 4- ;i;) + C ; 
 
 a -\- X 
 therefore 
 
 X X^ x^ x^ 
 
 u^los (a4- x) ^ — — -f — + etc. — 
 
 a 2a' Za^ 4a* 
 
 a result, when a= 1, the same as found in Art. 29. 
 
 dx 
 
 Let du = =^(\-irX^)-^dx. 
 
 Developing, 
 
 (1 -^ x^)-^ ==\ — x^ + x^ — x"" -^ ^iz. 
 
 Multiplying by dx and integrating, 
 
 x^ x^ x'' 
 u=C (I -\- x^) dx^x — — + — — — + etc. 
 
no AN ELEMENTARY TREATISE 
 
 It has been shown in Art. 76 that 
 
 dx 
 
 r = tan"^ X : 
 
 ^ l^x- 
 
 therefore w = J ( 1 -\- x-y^ dx = tan"^ x ^= 
 
 X x^ x^ x'^ 
 
 13 5/ 
 
 When x = 0, the arc, and consequently C, equals ; there- 
 fore 
 
 X x^ x^ x'^ 
 w-=tan"^;r^ — — — + — — — + etc. 
 1 3 5 7 
 
 dx 
 
 Let du = =(1 — x^)-'^'dx. 
 
 (l-x^-r^ 
 
 Developing and integrating, the result is 
 
 dx 
 w = C = 
 
 X x^ 3x^ 3 • 5x'^ 
 
 (— + + + + etc. ) + C. 
 
 1 2-3 2-4-5 2-4^6-7 
 
 Referring to Art. 76, it is found that 
 
 dx 
 C ^= sin~^ X ; 
 
 therefore u = sin"^ x = 
 
 X x^ Zx^ ?>-Sx' 
 
 (— + + + + etc. ) + C. 
 
 1 2-3 2-4-5 2-4-6-7 
 
 Let du = . ( 1 ) 
 
 (x — x~y^ 
 
 Assuming x = v^, then dx = 2vdv 
 
 dx 2dv 
 and = = 2(1 — v')-^''^ dv. 
 
 {x — x^y^ (i_z-2)% 
 
 Developing 2 (1 — v^y^, multiplying by dv, and integrat- 
 ing give 
 
ON VARIABLE QUANTITIES 111 
 
 J2 (l_z;2)-% dv = 
 
 V v^ 3f ^ 3 • 5z;^ 
 
 2 (— + + + + etc.) +C; 
 
 12-3 2-4-5 2-4-6-7 
 
 but J"2 (1 — t;^)-^^ a?z/ = 2 sin"^ z^; therefore, substituting for 
 V its value, x^'^, the result is 
 
 dx 
 
 u = r = 2 sin"^;l^'''^ 
 
 •^ (;ir — ;ir^)y^ 
 
 2dx 
 Putting (1) under the form and assum- 
 
 (2-2;r — 4x^)'''^ 
 
 ing 2x = V, then 
 
 2dx dv 
 
 du^^ 
 
 {2-2x — 4x^y- (2v — v^y^ 
 
 dv 
 
 but u=C = vers"^ v ; 
 
 2dx 
 
 therefore u = f = vers"^ 2x. 
 
 ^ {2-2x — Ax^y 
 
 EXAMPLES 
 
 1. du={l +x^ydx 
 
 2. du=^ (2ax — x^y^ dx 
 
 3. du= (a -\- x)^ dx 
 
 4. du= (x^ — l)-'^^dx 
 
 Binomial Rates 
 78. If the rate is of the form 
 
 du=(a-\- hx-'^y x'^'dx, 
 assume x = v-'^, then afjr = — v^dv and x^ = z^""* ; therefore 
 
 c?M = — {a-\-hv''yv-'^-^dv, 
 in which the exponent of v within the parenthesis is positive. 
 If the rate is of the form 
 
 du ={ax^ -\- hx'^y x'^dx, 
 it can be written thus, .y being less than n : 
 
 flfw = (a + bx'^-'y x'^^^'dx, 
 
 in which only one term within the parenthesis contains the 
 variable. 
 
112 AN ELEMENTARY TREATISE 
 
 Finally, if the rate is of the form 
 
 du= {a-\- bx'^y x'^dx, 
 
 in which ni and n are fractional, by substituting for x another 
 variable having an exponent equal to the least common 
 multiple of the denominators of m, and n, a new binomial rate 
 can be found in which the exponents of the variable will be 
 whole numbers. Thus, if in the rate 
 
 du^ {a^ bx"^' ) »" x^/^dx, 
 
 v^ be substituted for x, then, since dx = 6v^dv, 
 
 du = 6 {a -\- bv^Y v^dv. 
 
 Hence any binomial rate can be reduced to one of the form 
 
 du^={a^bx''y x'^dx, ( 1 ) 
 
 in which the exponents m and n are whole numbers and n is 
 positive. 
 
 When r, the exponent of the parenthesis, is a positive whole 
 number, ( 1 ) can be integrated as shown in Art. 72> ; also, when 
 m^n — 1, as shown in Art. 74. 
 
 Assuming a + bx" = v 
 
 in (1), then (a -\- bx")'' ^^v^', (2) 
 
 V — a 
 whence x'"- = 
 
 b 
 
 V — a 
 
 and x"^^^ = ( ) ('»+i>/™ ; 
 
 b 
 
 hence, by passing to the rate and dividing by m -\- 1, 
 
 1 V — a 
 
 Multiplying this by 2 gives 
 
 {a -\- bx'^y x'^dx = ( )^^^^y^-^v'-dv; (3) 
 
 bn b 
 
 1 V — a 
 hence Jm = ( ym^D/n-iyr^y (4) 
 
 bn b 
 
 w + 1 . 
 which can be integrated when is a positive whole num- 
 
ON VARIABLE QUANTITIES 113 
 
 m -{- 1 
 
 ber, or when w + 1 = w^ (see Art. 72). If is negative, 
 
 n 
 
 see formula D, Art. 80. 
 
 Let du= {a -\- bx^Y^ x^dx 
 
 fn -\- 1 1 
 
 in which m = 5, w = 2, = 3, andr = — ; then by sub- 
 
 n 2 
 
 stituting these values in (4), the following is obtained: 
 
 du = ( ^ " y v'^'dv = (z/^/2 — 2av^^' + a^z/^^) dv. 
 
 2b b 2b^ 
 
 Then, by integrating and reducing, 
 
 1 v^ 2av' a^v 
 
 b- / :) 3 
 
 therefore, since v^ (a-\- bx-), 
 
 1 {a + bx'Y 2a{a-{-bx^y 
 
 u^— { — + 
 
 b' 7 5 
 
 a^ (a-^bx^) 
 
 -^ '-}(a^bx^Y^C. 
 
 m> -\- 1 
 
 If is not a whole number, ( 1 ) may be written thus : 
 
 n 
 
 du = [x"" {ax-'' -\-h)Y x'^dx = {ax-"" -\- bY x'^^^'^'dx. 
 
 By substituting in the right hand member of (3) m -{- nr 
 for m, — n for n, a for b, and b for a, then 
 
 du= (b -{- ax-'')'' x"'^'"'dx = 
 
 1 v—b 
 
 ( ym+nr+l)/-n-l^rdz;^ (5) 
 
 — an a 
 
 m -\- nr -{- 1 
 
 which can be integrated by Art. 72 when is a 
 
 — n 
 
 positive whole number; if negative, by formula D of Art. 80. 
 
 79. Referring to Art. 9, it is found that 
 
 d {vz) •= vdz -\- zdv, 
 
114 AN ELEMENTARY TREATISE 
 
 whence, by integrating, 
 
 V2^ f vdz -\- ^zdv ; 
 
 hence ijvdz = vz — ^zdv, ( 1 ) 
 
 in which it is seen that the integral of vdz depends upon that 
 of zdv. 
 
 Resuming (1) of the last article, 
 
 c?M = (a + hx^'Y x'>'dx, (2) 
 
 and assuming z= {a -\- bx")^, 
 
 in which the exponent j may have such a value assigned to it as 
 may be found most convenient; then, by passing to the rate, 
 
 dz^bns {a-{- bx" ) «-^ x'^-^dx. (3 ) 
 
 Again, assuming vdz ^ (a -\- bx"Y x^'^dx, 
 and dividing it by (3), 
 
 zi^ 
 
 bns 
 and, passing to the rate, 
 
 r { (m — n-\- 1) (a + bx'')'-'*^ x'^-'' -\- 1 
 
 bn (r — s -^ I) (a -{-bx'^y-' x"^} dx 
 
 du = = 
 
 bns 
 
 r a {m — w+l);ir'»-"+ ] 
 b (m -\- nr — ns -\- 1 ) x"^ 
 
 { } (a + bx'')'-^ dx. 
 
 bns 
 
 Now let the value of ^ be such that 
 
 fii -\- 7ir -\- I 
 
 m -\- nr — ns -\- I ^0 or ^^ 
 then dv 
 
 n 
 a {m — n-\-l) (a -{- bx") (-»»-iVn x'>'-''dx 
 
 b (m -\- nr -\- 1) 
 
 Substituting the values of v, z, dv, and c?^ in ( 1 ) and inte 
 grating, 
 
 f (a + bx'^Y x'^'dx ^= 
 
 (a + bx'')''^^x»'-"^^ — a (m — n -\-l) f (a -\- bx")''x"'-"dx 
 
 b {m -\- nr -\- 1) 
 
ON VARIABLE QUANTITIES 115 
 
 in which the integral of (2) is made to depend upon that of 
 
 {a ^ bx'^y x'^-^'dx. 
 
 In a similar manner it will be found that 
 
 ^ {a-\- hx^'Y x'^'-'^dx 
 
 depends upon ^ {a -\- hx'^Y x'^-^'^dx; 
 
 and by continuing the process, the exponent of x without the 
 parenthesis can be diminished until it is less than n. 
 
 Hence the integral of a binomial rate may be m>ade to 
 depend upon the integral of another rate of the same form, but 
 in •which the exponent of the variable without the parenthesis 
 is diminished by the exponent of the variable within. 
 
 If the rate is of the form 
 
 du^ (a- — x^) "^^ x^^''dx, 
 
 1 
 substituting a^ for a, — 1 for b, 2 for n, and — — for r in 
 
 2 
 formula (A) gives 
 
 1 
 u^f(a^ — x^Y'^x'^dx^ — — (a- — X'Y''' ^'"'^ + 
 
 m- 
 
 a^ (m — 1) 
 
 — ^ C (a^ — x^Y"^- x'^-^dx. (a) 
 
 m 
 
 In this f (a^ — x^)-'''' x^'^dx 
 
 depends on f (a^ — x^) ~^= x^'^dx, 
 
 and this, by a similar process, will be found to depend upon 
 
 ^{a'^ — x^Y'^'^x'^-^dx, 
 
 1 
 
 and so on ; so that after — m operations, since m is an even 
 
 number, the integral will depend upon 
 
 §{d' — x^Y'/''dx, (4) 
 
 X 
 
 which is, by Art. 76, sin"'^ — . 
 
 a 
 
 If c/m = ( a- + ;ir2 ) -% x'^dx, 
 
 by substituting in formula {A), a- for a, 1 for b, 2 for n, and 
 
 1 
 — — for r, the result will be 
 
 9 
 
116 AN ELEMENTARY TREATISE 
 
 1 
 
 u 
 
 = J ( a^ + ;ir- ) -^^ x'»dx = — ( a^ ^x^ ) ^^ x^-^ 
 
 m 
 
 a^ (m — 1) 
 
 — ^ ({a^-^x^) -% x'^-^'dx, ( h ) 
 
 m 
 
 in which J (a~ -\- ;r^ )"'''= x'^dx 
 
 depends on J (a- -f- •*'^)'^^ x^^'^dx, 
 
 and by continuing the process, when m is even, the integral 
 
 will depend on 
 
 J {a- -\- x^y^^ dx. (5) 
 
 Assuming v = x -\- (a- -\- x^)"^^, 
 
 x-^ (a^^x^y- 
 
 then dv = dx -[- (a~-\-x^) "'/= xdx = dx, 
 
 (a'-{-x'y 
 
 dv dx 
 
 and = = {a^ -\- x-)-"^^ dx ; 
 
 V i^c? -\- x-y 
 
 therefore ^ {a" -\- x~y^- dx = \og {x ^ (a^ + jir^)^} + C. 
 
 If Jm = ( 2a;tr — x'^ y^-'X'^dx, ( 6 ) 
 
 assume z/= (2a;ir — ;r- ) '^^ ;r'"-^ = (2o;ir2'«-i — ^ir^'w)^^; 
 
 a {2m — 1 ) x'^'^'-'^dx — mx'^'^-^dx 
 then dv = = 
 
 a (2m — 1 );ir"*"^ dx mx'^dx 
 
 (2ax — x'-y ~ (2ax — x^y 
 
 But the last term is equal to mdu ; therefore 
 
 a {2m — 1 ) x^^~'^dx 
 dv =■ — mdu 
 
 {2ax — x^y 
 
 dv a {2m> — 1) x"^''^dx 
 or du=^ — -\- . 
 
 m m { 2ax — x-) ^^- 
 
 Hence, by integrating and substituting for v its value, it 
 will be found that 
 
 x'^dx 1 
 
 / = — — {2ax — x^-) y= x^-^ + 
 
 ( 2ax — x~y^ m 
 
 a (2m — 1 ) x"^~'^dx 
 
 — -§ , (0 
 
 m {2ax — x^y 
 
ON VARIABLE QUANTITIES 117 
 
 x^'^dx . x"^-'^dx 
 
 in which f depends on (' , 
 
 ^ {2ax — x^y^ ^ {2ax — x^y- 
 
 and this can be found to depend on 
 
 x'^-^dx 
 
 ^ (2ax — x^y' 
 
 and so on; so that after m operations, when m is a positive 
 whole number, the integral will depend on 
 
 dx 
 
 ^ (2ax — x^y' 
 
 which is, by Art. 76, 
 
 vers"^ — . 
 a 
 
 In order to obtain formulas when m is negative, multiply 
 formula (A) by b (m -\- nr -{- I) ; then 
 
 Z? (w + wr -f 1 ) J (a + bx'^)"- x^dx = 
 
 (a + bx'^Y'"'^ ^m-n+i — ^(^^ — ^_j_l)j(flr4- bx'')'' x'^-^'dx. 
 
 Transposing the terms containing the sign of integration 
 and dividing by a {m — n -\- 1) give 
 
 J ( a + bx"" ) ^ A-™-'* dx = 
 (a + bx'')''^^ ^m-n^i — Ij (^^ _^ nr ^ 1) f {a -{- bx^^y x'^dx 
 
 a {m. — w-j- 1) 
 in which f (^^ + bx'^y x'^-'^dx 
 
 depends on ( {a -{- bx^^)^ x^^^dx. 
 
 In a similar manner it will be found that 
 
 ^ {a -\- bx'^y x'^dx 
 
 depends on / (^ + bx'^y x'^'-^'dx. 
 
 and, finally, the exponent of x without the parenthesis of the 
 last term of {B) can be increased until it is less or greater than 
 n and positive. 
 
 Substituting a^ for a, ± 1 for b, and 2 for w in (5), it will 
 be seen that 
 
 {B) 
 
118 AN ELEMENTARY TREATISE 
 
 j {a" -\- x^^y x'^-^dx = 
 {a^± x^y*^ X"'-'' ±. (w + 2r+ 1) J(a2 ± x^y x'^dx 
 
 a^ (m — 1 ) 
 
 80. Again resuming 
 
 f/w = (a + bx")'' x'^dx, 
 and assuming 2 = x^, 
 
 in which such a value may be assigned to the exponent j as 
 may be desired; then, passing to the rate, 
 
 d2 = sx'-^dx. (1) 
 
 Assume vds = (a -\- bx^ ) '" x'^dx ; 
 
 then, dividing it by (1), 
 
 1 
 
 z/ = — (a4- bx'^y x^^^*'^, 
 s 
 
 the rate of which is 
 
 1 
 dv=^— {m — s-\-l) {a-\- bx'^y x'^-^dx -{- 
 s 
 
 bnr 
 
 (a + bx'^y-^ x'^-'^^'dx. 
 
 s 
 
 But (a -\- bx'^y = (a -\- bx"") (a -\- bx'')'-'' ; 
 
 1 
 hence dv = — {a (m — ^+1) -\- b (m — s -\- nr -\- I) x""} 
 s 
 
 • (a -^ bx'^y-'^ x'^-'dx. 
 
 Now let the value of ^ be such that 
 
 m — s -\- nr -\- 1 ^0 or s = m -}- nr -\- I ; 
 
 — anr (a-\- bx'')"-^ x'^-'dx 
 then dv = . 
 
 m -{- nr -\- I 
 
 Substituting the values v, 2, dv, and dz in (1) of the last 
 article, the result is 
 
 u = ^ {a -\- bx^y x^dx = 
 
 (a + bx'^y x"^^^ + anr f (a + bx'^y-'' x'^dx 
 
 i— , (C) 
 
 m -\- nr -\- \ 
 
ON VARIABLE QUANTITIES 119 
 
 in which J (^ + hx'^Y x^dx 
 
 depends on j {a -\- bx'^y-'^ x'^dx, 
 
 and this, by a Hke process, will be found to depend on 
 
 J (a 4- bx'^y-'' x'^dx, 
 
 and so on, till r, the exponent of the parenthesis of the term 
 containing the sign of integration, will be reduced to less 
 than unity when positive. 
 
 To obtain a formula when r is negative, multiply (C) by 
 m -{- nr -{- 1 , transpose the terms containing the sign of inte- 
 gration, and divide by anr; then 
 
 u ^J (a -f- bx'^y-^ x^'dx = 
 
 — (o + bx'')'' x"^-"^ + (m + wr + 1) r (a -f &jr")'" x'^dx 
 
 . (D) 
 
 anr 
 
 In (D) ^ {a -\- bx'^y-'' x'^dx 
 
 depends on ^ {a -\- bx'^y x'^dx, 
 
 and, by repeating the process, can be made to depend upon a 
 rate in which the exponent of (o + bx^) will be positive. 
 
 EXAMPLES 
 
 Determine the integrals of the following : 
 1, du={a — x'^Y x^dx 
 x^dx 
 
 2. du — 
 
 3. du = —(a-^bx^y'>^x^dx 
 
 4. du=(l -^xY^'^x-^dx 
 
 Rational Fractional Rates 
 
 81. Every rational fractional rate can be reduced to the form 
 
 {px"^ -\- qx'^-'^ -\- rx -\- s) dx -\- 
 
 A'x^'-^dx + B'x^-^'dx .... -\- R'xdx + 6"'^^ 
 
 Ax'' + 5;ir"-i -\-Rx ^s 
 
 in which the exponents of the variable are all positive whole 
 numbers, and the greatest in the numerator of the fraction is 
 at least one less than in the denominator. Hence, since that 
 part of the expression which is not fractional can readily be 
 
120 AN ELEMENTARY TREATISE 
 
 integrated, it only remains to integrate the fractional part, or 
 
 A'x'^-^dx + B'x^'-^dx . . . . + R'xdx + S'dx 
 
 du = . (1) 
 
 Ax'' + Bx'>-^ .... -^Rx-\-S 
 
 By resolving the denominator of this fraction into factors 
 of the first degree, and assuming them to be 
 
 X — a, X — b, X — c, etc., 
 
 the equation may be written under the form 
 
 Edx Fdx Gdx Kdx 
 
 du = + - + .... + -, (2) 
 
 X — a X — b X — c X — k 
 
 in which E, F, G, etc. are arbitrary constants whose values can 
 be determined in terms of a, b, c, etc. and A^, B' , C , etc. by 
 reducing (2) to a common denominator, and comparing the 
 coefficients of the like powers of x in the numerator of the 
 resulting fraction with those in the numerator of (1). 
 
 Hence, when no two or more factors of the denominator of 
 (1) are alike, the integral of (2) is, by Art. 72, 
 
 M = £ log {x — a) -\- F log {x — b) -\- 
 
 G\oz{^x — o)....^K\og{^x — k)^C. (3) 
 
 When, however, two or more factors are equal, as 
 a=b=^ c, (2) becomes 
 
 Edx Fdx Gdx Kdx 
 
 du = + + ■ .... + -, (4) 
 
 X — a X — a X — a x — k 
 
 in which E, F, and G have the same denominator ; consequently 
 these can be represented by a single constant, as in 
 
 Hdx Kdx 
 
 du = . . . . -|- . 
 
 (x — a) X — k 
 
 Here it will be seen that there are two more equations to 
 satisfy than there are arbitrary constants to be determined; 
 this condition, however, can be obviated by writing the equa- 
 tion thus : 
 
 Edx Fdx Gdx 
 
 du = -\- -|- . . . 
 
 (x — c)^ (x — a)- X — a 
 
 which retains the common denominator of (4). 
 
 In like manner, if there are two or more factors, as 
 (x — a)'^, (x — b)'', the equation can be written thus: 
 
ON VARIABLE QUANTITIES 121 
 
 Edx Fdx Gdx 
 
 du = j- . . . . -f- 
 
 i^x — ay {x — a)"'-^ {x — by 
 
 Hdx Kdx 
 
 (x — &)'-^ X — k 
 
 Edx Fdx Edx 
 
 The terms , in (5), also 
 
 (x — a)^ (x — ay {x — by 
 
 Fdx 
 
 etc. in this equation are equivalent to 
 
 {x — b)"'-^ ' 
 
 E (x — a)-^dx, F{x — ay^dx; and E {x—by^dx, 
 
 F (x — b)-"'^^dx, etc. can be integrated by Art. 74; and the 
 terms having denominators of the first power, by logarithms. 
 
 ax^dx — c^dx 
 If du = , (6) 
 
 x^ — c^x 
 
 the factors of the denominator are 
 
 X, X — e, and x -\- c ; 
 ax^dx — c^dx 
 
 therefore du ■ 
 
 X (x — c) (x -{- c) 
 
 Edx Fdx Gdx 
 Making du = + + , (7) 
 
 X X C X -]- c 
 
 and reducing it to a common denominator give 
 
 Ex^dx — Ec^dx -\- Fx^dx -\- Fcxdx-{- Gx^dx — Gcxdx 
 du^ 
 
 Comparing the numerator of this with that of (6), it will 
 be found that 
 
 E^F -}-G = a,Fc — Gc = 0,SindEc' = c\ 
 
 1 1 
 
 whence £ = c, F = — (o — c),and G= — (a — c). 
 
 Substituting these values in (7) gives 
 
 1 1 
 
 — (a — c) dx — (a — c) dx 
 edx 2 ^ 2 
 
 du = -f- -\- - 
 
 X -\- c 
 
122 AN ELEMENTARY TREATISE 
 
 and integrating, 
 
 1 
 
 u=^c\ogx-\-—{a — c) log {x — c) + 
 
 1 
 
 — (a — c) log {x-\-c) ^C 
 
 1 
 
 = c log ;r + — (c — c) log (^ — c) (;r + c) + C 
 
 1 
 
 = c log X -\- — (a — c) log {x'^ — C-) -\- C 
 
 = c \ogx -)- (a — c) log (;r^ — c^)'/- -)- C. 
 If du = , (8) 
 
 {x-\y-{x — 2) 
 
 then [see (4) and (5)] 
 
 Edx Fdx Gdx 
 
 du = + + . (9) 
 
 {x — 1)- X — 1 X — 2 
 
 Reducing to a common denominator, 
 
 E (x — 2) -\-F {x^ — 3x-\-2) -\rG {x' — 2x-^ 1) 
 du = dx. 
 
 ix—ir(x—2) 
 
 Comparing the numerator of this with that of (8), the 
 following are obtained : 
 
 E = S, F = 3, and G = — 3. 
 
 Substituting these values in (9) gives 
 
 5dx 3dx 3dx 
 
 du^=^ -|- 
 
 {x — 1)- X — 1 X — 2 
 
 3dx 3dx 
 
 5 {x — \)~- dx -\- 
 
 X — 1 X — 2 
 
 and integrating by Arts. 74 and 72, 
 u = — S (.r— l)-^ + 31og (;r— 1)— 31og(^ — 2) + C. 
 
 To verify the principle set forth in (5), let 
 
 {ax~-\- hx -\- c) dx 
 
 du = . 
 
 ( X — ry 
 
ON VARIABLE QUANTITIES 123 
 
 Assume x — r = v; then x=-v -[- r, dx=^dv, and 
 a {v^ + 2rv -\- r"^) dv -{- b {v -[- r) dv ^ cdv 
 
 du 
 
 or, collecting like powers of v, 
 
 (ar^ -{- br -\- c) dv -\- (2ar + b) vdv -\- av^dv 
 
 and, reducing, 
 
 (ar'^ -[- br -\- c) dv ( 2ar A- b) dv adv 
 du = - + + 
 
 Substituting for v and dv their values, x — r and dx, then 
 
 {ar~ -\- hr 4- c) dx (2ar -\- b) dx adx 
 
 du = - ■ — -{-- + , (10) 
 
 (x — r)^ (x — r)- X — r 
 
 in which ar^ -\- br -\- c is represented in (5) by E, 2ar -\- b 
 by F, and a by G. 
 
 Integrating (10) by Arts. 74 and 72 gives 
 
 1 
 
 u = — — {ar~ -\- br -\- c) {x — r)~^ — 
 
 {2ar -\- b) {x — r)~'^ -\- a\og {x — r). 
 
 82. When the denominator contains a single pair of imagi- 
 nary factors, as x -\- r -\- s \/ — 1 and x -\- r — s \/ — 1 
 (whose product is x~ -\- 2rx -]- r^ -\- s^) , the fraction becomes 
 
 A'x''-^dx + B'x^-^dx + S'dx 
 
 du = , (1) 
 
 (Ax^-- + Bx^-^ +5") (x"" -^ 2rx ^ r- -\- s'~) 
 
 which, assuming the factors of the denominator, other than 
 the imaginary pair, to be x — o, x — b, etc., may be written 
 thus : 
 
 Edx Fdx Kdx 
 
 du= + r+---- + r- + 
 
 X — a X — X — k 
 
 Pxdx 4- Qdx 
 ^^ . (2) 
 
 x'^ -\- 2rx -f- ^^ + -^^ 
 
 By reducing this to a common denominator and compar- 
 ing the numerator with that of (1), the values of E, F, etc., 
 also of P and Q, may be determined. 
 
124 AN ELEMENTARY TREATISE 
 
 All but the last term of the second member of (2) can be 
 integrated by the methods in Art. 81 ; therefore it will only 
 be necessary to integrate the last term, which may be put under 
 the form 
 
 (Px^Q) dx 
 dv^— . (3) 
 
 {x -\- r)'^ -\- s'^ 
 
 Assume x-\-r^z; then since dx = d2, (3) becomes 
 
 {Pz — Pr+Q)dz Pzdz (Pr—Q)dz 
 dv = = — , 
 
 Z~ -\- S" z'^ -\- s- z^ -\- s^ 
 
 the integral of which is by Arts. 72 and 76 
 
 1 Pr—Qz 
 
 v = — Plog (z^-{-s^)— tan-i — + C. 
 
 2 .y .y 
 
 Therefore, substituting for z its value x -\- r, the integral 
 of (3) is found to be 
 
 1 Pr—Qx + r 
 v^^ — P log {x"^ + 2rx -\- r"^ -\- s^') — tan"^ -|" ^ 
 
 9 
 
 .f 
 
 or 
 
 Pr — Q X -\- r 
 
 v = P\og (x^ + 2rx + r' + s^Y^ — tan-^ + C. 
 
 s s 
 
 (2 — x) dx (2 — x) dx 
 If du=^ = : (4) 
 
 x^ +1 (^+1) {x^ — x+l) 
 
 Edx (Px -\- Q) dx Edx 
 then du = -\- = -\- 
 
 X -[- I X- X -\- I X -\-\ 
 
 Pxdx Qdx 
 
 ~ — rr+ ^ .1-1 ' (=> 
 
 X' — X -^\ x^ — .ar-j-l 
 
 whence it is found that £ = 1, P = — 1, and Q = 1. 
 
 1 
 Substituting z for x — — , ( 5 ) becomes 
 
 1 
 
 {z-^ — )dz 
 dz 2 dz 
 
 du = — -I- = 
 
 3 3 3 
 
 z^- z'-}-- ^2 + - 
 
 2 4 4 
 
ON VARIABLE QUANTITIES 125 
 
 1 
 
 — dz 
 
 dz zdz 2 
 
 3 3 3 
 
 z + — ^2 + — ^2 + — 
 2 4 4 
 
 the integral of which is 
 
 3 1 3 V3 2^V3 
 
 w = log(^ + -)— -log(^2+-)-h — tan-^-y— 
 
 1 
 
 M = log(;r + 1)— --log {x^ — x^ 1) + 
 
 V3 (2;r— 1)V3 
 
 tan"' 
 
 3 3 
 
 or 
 
 ;ir+l V3 (2;r— 1)V3 
 u = log + tan-^ + C. 
 
 When the denominator contains several sets of imaginary 
 factors, respectively equal to each other, the factor 
 X -\- 2rx -\-r -\- s will enter the denominator several times ; 
 hence, for that part of the fraction containing only sets of 
 equal imaginary factors, may be put under the following form, 
 thus 
 
 {Ex-^F)dx {Gx + H)dx 
 du= + . • • • + 
 
 (x^ -f 2rx + r=^ + ^-)'» (x^ + 2rx + r' + ^')'"-' 
 
 (Px4-Q) dx 
 
 ^ ^^ (6) 
 
 (x^ -\- 2rx -\- r~ -\- S-) 
 
 The values of the constants E, F, G, etc., may be determined 
 as heretofore explained; then the integral of each term taken 
 separately. 
 
 Since the terms of the second member of (6) are all of 
 the same general form, it will only be necessary to integrate 
 the first term, which may be placed under the form 
 
 Exdx 4- Fdx 
 dv = ■ . (7) 
 
 {x^- ^ 2rx -{- r^- ^ s-'Y' 
 
 Assuming x = z — r, this expression becomes 
 
 Ezdz — {Er — F) dz 
 dv = = 
 
126 AN ELEMENTARY TREATISE 
 
 Ez {z^ + j^)-'« ds—{Er — F) {z^ + s'-)-''^ dz, 
 and, by Art. 74, 
 
 E (^- + J2)i-m 
 (Ez (z^ 4-3-) -"> dz — . 
 
 2(1 — w) 
 
 By formula (D) of Art. 80, f—(Er — F) {z^ -\- s^Y'^'dz 
 can be made to depend upon J — (Er — F) (z^ -{- s^)~^ dz, 
 
 (Er — F) z 
 
 which is — tan'^ — , thus completing the integration 
 
 of (7). 
 
 From the preceding, it is evident that the fraction can be 
 integrated, even when the denominator contains several dif- 
 ferent imaginary factors ; providing, however, said factors 
 can be determined, and this condition applies to all fractional 
 rates. 
 
 EXAMPLES 
 
 adx 
 1. du^:=^ 
 
 2. du 
 
 3. du^ 
 
 x~ — a" 
 2axdx -\- adx 
 
 x^—\ 
 2(1 — x^ dx 
 
 {\^x-y 
 
 Irrational Fractional Rates 
 83. Any irrational fractional rate will admit of integration 
 when it can be changed to a rational form. Thus, let 
 
 {x""^ + ax'^"' ^h)dx 
 du^==^ , 
 
 X -\- cx'^^ -\- e 
 
 and assume x = z'^ ; then 
 
 (z^ -\- az -\- b) Gz'dz ( 6z^ + 6az^ -^ 6bz^) dz 
 du = = , 
 
 z'^ -\- cz^ -\- e z^ -{- cz^ + ^ 
 
 which is a rational form and consequently can be integrated 
 by the methods explained in Arts. 81 and 82. 
 
 When the quantity under the radical sign is a polynomial, 
 the rate can not in general be changed to one of a rational 
 form. If, however, the rate is of the form 
 
 du = {a" -^hx-\- c^x-)"^ Xdx, ( 1 ) 
 
ON VARIABLE QUANTITIES 127 
 
 in which X is a rational function of x, it can be changed to a 
 rate which will be rational; thus, assuming 
 
 {a^ -^bx-^ c^x'y- = 2 — cx, (2) 
 
 then a^ -\~ bx -\- c^x^ = s' — 2czx -\- c^x"^ ; 
 
 z^ — d^ 
 
 whence x^= . (jj 
 
 2cz+b 
 
 This value of x substituted in the second member of (2), 
 
 by reducing, gives 
 
 cz~ -{- bz -\- a^c 
 
 (a^ -\- bx ^ c^x^y^ = . (4) 
 
 ^ ^ 2cz+b 
 
 The rate of (3) is 
 
 2 (cz^-\-bz-\- a^c) dz 
 dx = — . (5)' 
 
 {2cz^by 
 
 (5) divided by (4) gives 
 
 2dz 
 
 (a'-^bx-\-c^x^y^^dx = ; (6) 
 
 ^ 2cz-\-b 
 
 2Xdz 
 
 hence du= (a^ -\- bx -{- c'^x'^y'^ Xdx = , (7) 
 
 ^ 2cz^b 
 
 which is a rational form ; for, since X is a rational function of 
 X, it must also be a rational function of s ; that is, if the value 
 of X be substituted in X, it will give the value of X in rational 
 terms of z. 
 
 1 
 
 If X = — , then substituting this value of X in (7), since 
 
 X 
 
 z^ — a^ 
 
 x^ [see (3)1, (7) becomes 
 
 2cz+b ^ ^J ^ ^ 
 
 2dz 2cz -\- b 2dz 
 
 du={ ) ( ^—) = . 
 
 2cs '\- b z^ — d^ z^ — d^ 
 
 The integral of this is, by Art. 80, 
 
 1 z — a 
 % = — log ; 
 
 a z ^ a 
 
 but from (2), z=^ (a^ -{- bx -j- c^a~y^ -\- ex, 
 
 1 (a^ 4- bx + c^x^y-{- ex — a 
 therefore m =^ — log -[" ^• 
 
 a {d^ -\- bx -\- e^x'^y^-\- ex -{- a 
 
128 AN ELEMENTARY TREATISE 
 
 If Z= 1, the integral of (7) will be 
 
 1 
 
 u = — log {2c2 -\- b) ; 
 c 
 but [see (2)], 
 
 2c2+b=^ 2c {a" -\-bx-\- c^x^)"^ + 2c''x + b ; 
 
 therefore 
 
 1 
 
 u = — \og{2c (a^ -\- bx -\- c^x^y^ -\- 2c''x -\- b} + C. 
 c 
 
 Ub = 0, then 
 
 1 
 
 u = — \og2c {(a^ + c'^x^y^ 4- ex} + C. 
 c 
 
 If X^;!: then (7) becomes 
 
 2 (2^ — a^) dz 
 
 du = , 
 
 {2cz^bY 
 
 which can be integrated by Art. 81. 
 
 {a" -\- bx -\- x^y dx 
 
 If du 
 
 X 
 
 assume {a^ -\-bx -\- x^Y^ = x -\- z; 
 
 then [see (3) and (4)] 
 
 x=^ (8) 
 
 b — 2z 
 
 and {a'^bx^x^y = — . (9) 
 
 b — 2z 
 
 Taking the rate of (8) and reducing [see (5)] 
 
 2 (2^- — hz^a^) dz 
 du = — — ^^—- . (10) 
 
 (b — 2zy 
 
 Multiplying (10) by (9) gives 
 
 2 (z^ — bz -^ a'y dz 
 
 (a^ J^ bx -\- x^y dx 
 
 therefore du 
 
 (b — 2zy 
 
 2 (z^ — bz -\- a^y dz 
 
 (b — 2zyX 
 which is rational in terms of z, as previously explained. 
 
ON VARIABLE QUANTITIES 129 
 
 84. When the rate is of the form 
 
 Xdx 
 
 {c -]- dx — x^y^ 
 assume c = ab and d^a — b; then 
 
 Xdx 
 
 du=^ ~~. 
 
 {ab-\- (a—b) x — x^y- 
 
 Now, since ab -\- {a — b) x — x^ =^ {a — x)(^b-^x), 
 
 assume V [(« — x){b-^x)] = {a — x) z; (1) 
 
 then, squaring both members, 
 
 {a — x) (b + x) = {a — xy2^ 
 
 or b-\-x^(a — x) z^, 
 
 0-^^ — ^ 
 
 whence x^ ; (2) 
 
 ^2 + 1 
 
 and therefore, 
 
 az^ — b a -\- b 
 a — x = a — ^= . (3) 
 
 ^2 + 1 5^ + 1 
 
 Substituting this value of a — x in the second member of 
 
 (1), the resuh is 
 
 (a-{-b)z 
 ^'[(a-x){b-^x)]= (4) 
 
 z~ -\- I 
 
 The rate of (2) is 
 
 2(a-\-b)zdz 
 dx = . (. J ) 
 
 Dividing this by (4) and reducing give 
 
 dx 2dz 
 
 yj[{a — x) {b-^x)-\ z' + l 
 
 Therefore, muhiplying both members by X, it is found that 
 
 Xdx 2Xdz 
 
 du = = , 
 
 ^[(a — x) (b-^x)] ^^ + 1 
 
 which is rational in terms of z, as shown in Art. 83. When 
 X =1, this becomes 
 
 2dz 
 
 du^ 
 
 z^-\-l 
 
130 AN ELEMENTARY TREATISE 
 
 Hence u^2 tan"^^: -|- C. 
 
 V [(a — x) (b^x)] dx 
 
 If du = , (6) 
 
 Ji. 
 
 then, proceeding as before, it will be found that 
 
 du = ^— I— ^ , (7) 
 
 X (^2 + 1)^ 
 
 which is also a rational fraction. 
 
 (x — x^)'''^ dx 
 Let du = . (8) 
 
 (i — xy 
 
 1 
 
 Here a=l, b = 0, (l — xY = , (x — x^y^ = 
 
 (z^ + iy 
 
 -YTV, and dx= \ \ [see (2), (3), (4), and (5)]; 
 
 therefore, substituting these values in (8) and reducing, it will 
 be found that 
 
 2z'^dz 2dz 
 
 du = ^2 di 
 
 + 1 ^^ + 1 
 
 the integral of which is 
 
 u^2z — 2 tan"^^ -|- C. 
 
 X 
 
 Substituting for z its value, ( )'/% 
 
 1 — X 
 
 u = 2{ )%_2tan-i ( ^)v. _|_ C. 
 
 1 — X 1 — X 
 
 EXAMPLES 
 
 1. du= (x^ -\- a)'/^ dx 
 dx 
 
 2. du 
 
 3. du 
 
 3 (x — x~)^^^ dx 
 
 x"" 
 
 Transcendental Rates 
 85. Simple rates of this class, which admit of direct inte- 
 gration, have been previously treated ; a few of those whose 
 integrals are less readily obtained will now be considered, 
 omitting the constant C. 
 
ON VARIABLE QUANTITIES 131 
 
 Let du = Xa^dx, 
 
 in which X is an algebraic function of x, and its wth ratal co- 
 efficient is constant, represented by A in the formula. 
 
 dX dX' 
 
 Assume v^^ X, dz = a^^'dx, 2ind— — =^X^, =^X'', etc., 
 
 dx dx 
 
 then dv = dX and s = 
 
 log a 
 
 These values of v, z, dv, and a?^; in ( 1 ) , Art. 79, give 
 
 Za^ 1 
 
 (Xa'^dx^ — j'dXa'', 
 
 log a log a 
 
 1 X'a*- 1 
 
 fdXa-'^ — ~-\- -j'dX'a^, 
 
 log a (loga)2 (logo)- 
 
 1 X''a^ 1 
 
 and CdX'a^^ — CdX^'a^, etc.; 
 
 (loga)^ -^ (loga)^ (loga)« -^ 
 
 from which the following is obtained : 
 
 X X' X" A 
 
 u=^a'={ — + ± }. (1) 
 
 log a (logo)- (loga)^ (logo)" 
 
 EXAMPLE 
 
 d%^ {hx"^ + cx^)aF. 
 
 Here X = hx"^ -\- cx'^, the ratal coefficients of which are 
 
 2bx + 4cx'', 2x + Ucx^, 24cx, and 24c = A. 
 
 Substituting these values in ( 1 ) gives 
 
 bx^ -\- ex* 2bx -j- 4cx^ 
 
 u^^a"" { — -j- 
 
 log a (loga)^ 
 
 26 + Ucx^ 24cx 24c 
 
 + - —}' 
 
 (loga)3 (log a)* (loga) = 
 
 from which it will be seen that when the greatest exponent of 
 X is even, the sign of the last term of ( 1 ) will be positive, and 
 when it is odd, the sign of the last term will be negative. 
 
 If du = x"^a''dx, 
 
 then X = x"^, whose ratal coefficients are {m being a positive 
 number) : 
 
 fyix"^~'^, m (m — 1) x^~^, m (m — 1) (m — 2) x"^~^, etc. 
 
132 AN ELEMENTARY TREATISE 
 
 Substituting these values in ( 1 ) gives 
 
 mx 
 
 m-i 
 
 u^a^ { — + 
 
 log a (loga)- 
 
 m (m — 1) x"^~^ m (m — 1) (m — 2) jr"*"^ 
 
 i^ogay (logc)^ 
 
 m (m — 1) .... 1 
 
 )• (2) 
 
 (loga)'«+i 
 
 If m be negative or fractional, then develop a^ by Mac- 
 laurin's theorem, Art. 24, multiply both members by x"^ and 
 integrate. 
 
 86. When the rate is in the form of a logarithm, as 
 
 du = x^ log xdx, 
 
 assume v = log x and dz = x'^dx ; 
 
 dx .jr"+^ 
 
 then dv = and z 
 
 X n -\-\ 
 
 These values of v, z, dv, and dz'va ( 1 ) , Art. 79, give 
 
 x^^^ ^«+i dx 
 
 du =Cx'" log xdx = log X — J ( ) ; 
 
 n-j-l n -\- \ X 
 
 x"--"^ dx x'^dx .ar""-^ 
 but /( ) = f == ; 
 
 jr"+^ 1 
 therefore u = ( log x — ) . 
 
 n -\- 1 w -j- 1 
 
 Let du^ (logx)''dx, (1) 
 
 in which w is a positive integer. 
 
 Assume v= (log.*")^ and dz = dx; 
 
 dx 
 then dv^=n (log x)^~'^ and z = x; 
 
 X 
 
 and, by substitution, 
 
 du^^J (log x)" dx = x (log.*:)" — wJ(log.^)""^ dx; 
 but — n f (log x)"-'^ dx = 
 
 — nx (log^ir)"^"^ -}- n (n — 1) r(log;r)"~^ dx, 
 and n (n — I) f (log x)^~^ dx = 
 
ON VARIABLE QUANTITIES 133 
 
 n (n — 1) X (logx)"--^ — n (n — 1) (n — 2) J(log;i;)"-^ dx; 
 
 whence u^^^x {(log x)^ — n {log x)^'^ -{- 
 
 n (n— 1) {logx)^-^ .... -\-n (w— 1) (.... 1)}, (2) 
 
 in which the last sign will be plus when n is even, and minus 
 when n is odd. If n be negative; that is, if du = (log x)'^ dx, 
 assume dv^dx and z={logx)~^, and proceed as before; 
 
 dx 
 
 u, however, will be found to depend on the integral of , 
 
 log^ 
 
 sometimes called Soldner's integral, which can be obtained by 
 series. 
 
 If du^^x""- {logxY dx, 
 
 assume y =^ x'"^*'^ ; 
 
 then log 3/ = {m -\- 1 ) log x 
 
 1 
 or (logJir)»= ( )» (log3;)*». 
 
 w -f- 1 
 
 Therefore, since ^3;= (w -)- 1) x^dx, 
 
 1 
 
 or x^dx = dy, 
 
 m, -\- \ 
 
 1 
 
 d%=^x'^ {logxYdx= ( )«+i {'^ogyYdy, 
 
 m -\- 1 
 
 the integral of which is the same as that of (1) multiplied by 
 
 m -\- I 
 
 If du = (log x)"" Xdx, 
 
 X-^dx Xr,dx 
 assume jXdx = X-^, J = X„, j'-^ = X^, etc. 
 
 X X 
 
 and z^^(log:ir)" and dz^Xdx; 
 
 dx 
 then dv^:^n { log x)'^~'^ and 2 = (Xdx = X^. 
 
 These values of v, z, dv, and dz substituted in (1), Art. 79, 
 give u^ {logxY X^ — w (log;i;)"-^ Xo + 
 
 n{n—l) {logxY'^X _ .... w (w— 1) (.... 1)X . (3) 
 
134 AN ELEMENTARY TREATISE 
 
 If the integrals of 
 
 X^dx X^dx Xndx 
 Xdx, , , .... , 
 
 can be found in finite terms, the proposed rate will have an 
 exact integral. 
 
 Let du= (log x)^ (1 -\- x^) dx. 
 
 1 
 Here w = 3, J Xdx = x -\- — .r^ = X^, 
 
 X^dx 1 
 
 / = x^-x' = X„ 
 
 X 9 
 
 ^X,dx 1 ^X.dx 1 
 
 r = x -r- .«■■' = X„ and (" ^ x 4- x^ = X,. 
 
 ^ X 27 ^ X 81 
 
 Substituting these values in (3), the result is 
 
 1 1 
 
 u={\ogxY (^ + — ^3)— 3 (logji;)^ {x^ — x^) + 
 
 1 1 
 
 6(log;r) {x^-—x^-)—6{x^—-x^-). 
 
 Complex Circular Rates 
 
 87. Let du^ s'm"^ X cos" xdx, (1) 
 
 and assume sin ;ir ^ 7/ ; 
 
 then cos x=(l — z/^)'''^ and cos xdx=^dv, 
 
 dv dv 
 
 whence dx ■ 
 
 cos .^r (1 — v^Y^ 
 Substituting these values of v and dv m ( 1 ) , gives 
 
 du = v''' (l—v^-)^"-^''-dv. (2) 
 
 Assuming cos x^z, 
 
 then will du = — ^"^ (1 — s-)^"'-^^^- ds. (3) 
 
 These can be integrated by Art. 78: (2), when m is a posi- 
 tive odd integer and m positive or negative, integral or frac- 
 tional; (3), when m is a positive odd integer, and n positive 
 or negative, integral or fractional. When these conditions do 
 not exist, they can be integrated in many cases by one of the 
 formulas A, B, C, or D. 
 
ON VARIABLE QUANTITIES 135 
 
 In du = sin^ x cos^ x dx, 
 
 m = 2 and n^^3; therefore (2) becomes 
 
 du = v^ (1 — V') dv, 
 
 1 1 ^ v^ 
 
 hence m = — v^ — — v^ = (S — 3z/-) . 
 
 3 5 15 
 
 or, substituting the value of v, 
 
 sin' X 
 
 u-= (5 — 3 sin- x) . 
 
 ^ ^ 15 
 
 (3) also becomes 
 
 du = —{l—z^y-z^^dz, 
 
 which by formula A can be made to depend on 
 
 ^{\—z^y-zdz, 
 
 which is integrable by Art. 74. 
 
 In du = sin x"^ dx, 
 
 assume sin ;i; = z/ ; then, since cos x = {\ — v-)"'' and 
 
 dv 
 dx^ , 
 
 cos X 
 
 du^ (1 — v^)-'^' V'dv, 
 
 which, when w is a whole number, either positive or negative, 
 by the application of formula A or B may be made to depend 
 on (1 — v^Y^^^ dv or (1 — v'^Y^'^vdv. 
 
 The first of these can be integrated by Art. 75 and the 
 second by Art. 74. 
 
 If du =^ tan^ xdx, 
 
 let tan x^v; then 
 
 dv 
 dx^ 
 
 1 +^2 
 V^dv 
 
 and du - 
 
 l+v- 
 
 a rational fraction. 
 
 tan xdx 
 If du^ 
 
 sm^ X 
 
 sinx 
 
 then, since tan x = , by substitution the result is 
 
 cos X 
 
136 AN ELEMENTARY TREATISE 
 
 dx 
 
 du- 
 
 xdx 
 
 sin X cos X 
 therefore u = log tan x ; 
 
 (see Art. Z2). 
 
 Let du = tan"^ xdx. 
 
 Now d (x tan"^ x) = tan"^ xdx -\- 
 
 l-{-x'' 
 
 xdx 1 
 
 and r = — loff(l+^"); 
 
 •^ 1 + ;^2 2 
 
 1 
 
 therefore . u^xtain~^x — — log(l+;r^). 
 
 Let du = X sin"^ xdx, 
 
 and assume 
 
 Xdx = dz and sin"^ x = v, also f X(/;i: = X^ ; 
 
 then c/z/ ^ (1 — x^) '^^ dx and z = fXdx = X^. 
 
 Therefore u = X^ sin~^ ;ir — ("X^ (1 — ^r^)"'/^ dx, 
 in which the integral of the proposed rate is made to depend 
 upon that of another, whose form is algebraic. 
 
 A similar process will apply to any of the following forms : 
 
 X cos~^ X, X tan"^ x, X cot~^ x, etc., 
 
 since the rates of cos "^ x, tan~^ x, cot~^ x, etc., all depend upon 
 the integral of an algebraic expression. 
 
 Examples 
 
 1. du^ a^x'^dx 
 
 x^dx 
 
 2. du 
 
 log^ X 
 
 3. du = sin^ x cos^ xdx 
 
 4. du^ X cos~^ xdx 
 
 Bernouilli's Series 
 88. Bernouilli's series expresses the integral of any rate of 
 the form 
 
 du = Xdx, 
 
 in which X is a function of x, in terms of X, its ratal co- 
 efficients, and X. 
 
ON VARIABLE QUANTITIES 127 
 
 To obtain this series, assume 
 
 X^v and dx^==^dz; 
 
 then dv = dX and ^ = ^. 
 
 Substituting these values in (2) of Art. 79, the result is 
 
 j'Xdx = xX — jxdX, 
 
 or, since dx is included in dX, 
 
 dX 
 
 i'Xdx = xX — j xdx ( ) . 
 
 •^ -^ dx 
 
 dX 
 And assuming = v and jra(;ir ^= dz ; 
 
 (/^Z x^ 
 
 then, since ofz/ = and z = , 
 
 dx"^ 1 • 2 
 
 by substituting these values as before, 
 
 dX X' dX x^dx d^X 
 
 — (xdx ( ) = — ( ) + f ( ) . 
 
 ^ dx 1-2 dx ^ 1-2 dx' 
 
 In a similar manner, it will be found that 
 
 x^dx d^X x^ d'X ^ x^dx d^X 
 
 ^ 1-2 ~d^ ~ 1-2-3 dx'- '^ 1 • 2 ■ 3 dx' 
 
 x^ dX 
 
 therefore, by the substitution of xX — ( ) + 
 
 •^ 1-2 dx 
 
 x' d'-X 
 
 ( ) — etc., the integral of (1) is found to be 
 
 1-2-3 ^dx' ^ ^ ^ ^ 
 
 x^ dX x' d'X 
 
 u = xX — ( ) + ( ) — etc. 
 
 1-2 dx 1 • 2 • 3 dx- 
 
 This series was obtained by John Bemouilli in 1694 and is 
 probably the first general development discovered ; it is, how- 
 ever, but a particular case of Taylor's theorem, discovered in 
 1715. Such expressions as log (1 + x), sin .ir, and others can 
 be readily developed into a series by Bernouilli's theorem, as 
 shown by him. 
 
 Let du = (1 -\- 2x -{- 3x^ ) dx, 
 
 in which (1 + 2 + 3x^) represents X ; then 
 xX = x -{- 2x^ -j- 3x', 
 
138 AN ELEMENTARY TREATISE 
 
 X' dX x^- d^X 
 
 — ( ) = — X- — Zx^, and ( ) = x^ : 
 
 1-2 dx 1-2Z dx' 
 
 therefore 
 
 u = X -\- 2x~ -\- Zx^ — X" — Zx^ -{- x^ = X -\- X- -{- x^ . 
 
 EXAMPLES 
 
 1. du={\ -j-x^y^dx 
 
 dx 
 
 2. du^ 
 
 Successive Integration 
 89. In the expression 
 
 d^u = (^^ -j~ ^^^) dx'^, 
 
 two integrations are required to determine the primitive func- 
 tion, or u in terms of x. Placing the expression under the form 
 
 d'^u 
 
 = x^dx -f- ax^dx, 
 
 dx 
 
 and integrating, 
 
 dw X- ax^ 
 
 — = — H- — + C. 
 
 dx A ?) 
 
 Multiplying through by dx and integrating again, 
 
 x^ x'^ 
 
 u = + + C-,x 4- C. 
 
 4-5 3-4 
 
 The foregoing may be written thus : 
 
 d^u 
 
 dx- 
 
 du 
 
 = /, (x) + Ci and m = /g (x) -\- C-^x -f C^. 
 
 dx 
 
 From the preceding it will be seen that, if 
 
 dnu = f (x) dx", 
 
 by taking n successive integration the following ' will be 
 obtained, 
 
 w=/ ^ (x) + 
 
 C,x''-^ C.x"-- 
 1 I -. U....C x + C . 
 
 1-2 .... (w—1) 1-2 .... (7z — 2) "-^ 
 
ON VARIABLE QUANTITIES 139 
 
 The nth integral of 
 
 d^'u^^f {x) dx"" 
 
 may be represented thus : 
 
 u=ff (x)dx''. 
 
 d^'u 
 Developing ^/ {x) 
 
 dx"" 
 
 by Maclaurin's theorem (Art. 24), the result is 
 
 d^u df (x) X dH (x) x"' dH ix) x^ 
 
 = A^ + 4- + etc. 
 
 dx"" dx 1 • 2dx'^ 1 • 2 • Zdx"^ 
 
 df {x) d'-f {x) 
 
 Now, by substituting for A, , , etc., their 
 
 dx dx^ 
 
 values as shown in Art. 24, then multiplying by dx and integrat- 
 ing n successive times, plus a constant each integration, the 
 result will be a series expressing the value of u in terms of x. 
 
 d^u 1 
 Let = , 
 
 dx'^ I ^ x 
 
 the development of which is 
 
 d'^u 
 
 ^1 — X -\- x~ — x^ -\- etc. 
 
 dx^ 
 
 Integrating this as explained, gives 
 
 x^ x^ x^ 
 
 + T-— ^— — etc. + 
 
 2-3-4 2-3-4-5 3-4-5-6 
 
 C^x^ C.x^ 
 -Z-r + -T- + C,x + C, 
 
 EXAMPLE 
 
 d'^u = 6dx^ -{- 36xdx^ -\- 30x'^dx~. 
 
 Note : In successive integration, it sometimes becomes ex- 
 pedient to integrate between limits, especially when there are 
 two or more independent variables. For instance, in the 
 equation of the circle, y^ = 1 — x~, x can never be greater 
 than 1 nor less than zero. 
 
140 AN ELEMENTARY TREATISE 
 
 Integration of Partial Rates 
 
 90. Partial rates are obtained with reference to one variable 
 only, or with reference first to one variable, then to another, 
 etc., (see Art. 22). 
 
 Of the first class, as 
 
 du = Zx^ydx, 
 
 the integral is u = x^y, 
 
 or, since the primitive function may contain terms in y alone, 
 an arbitrary quantity must be added, as Y , a function of y, as 
 such terms will disappear in passing to the rate ; thus, 
 
 u = x^y -\-Y-\-C. 
 
 This class of partial rates can be expressed generally thus : 
 
 f/"w = / {x,y, z, etc.) dx'^. 
 
 Taking one of the second class, as 
 
 d~u = 9x^y^dxdy -\- 2xdxdy, 
 
 and integrating first with respect to x, then with respect to y, 
 
 du = Zx^y~dy -\- x^dy-\-Y and u = x^y^-\- x^y-\-J Ydy-\-X-\- C. 
 
 This class of partial rates can be expressed generally thus : 
 d^u^f {x, y, z, etc.) dx'^dydz^, etc., 
 
 in which m is equal to the sum of the exponents of the rates of 
 the independent variables ; that is, 
 
 m = n-\-r-\-s-\- etc. 
 
 Let d^u == 6xy"dxdy. ( 1 ) 
 
 The integral of this with respect to x is 
 
 du = 3x^y^dy (2) 
 
 or, since there may have been a term containing y alone in (2) 
 which would disappear in (1) by passing to the rate, 
 
 du = Sx^y^dy -\-Y. 
 
 Integrating again with respect to y, it will be found that 
 
 u = x~y^-\-fYdy-^X. 
 
 EXAMPLES 
 
 1 . d^u = arx^dy- + ydy^ 
 
 2. d^u = x^ydxdy^ 
 
ON VARIABLE QUANTITIES 141 
 
 Integration of Total Rates 
 91. Let du = f^{x,y) dx-\-f2ix,y) dy, (1) 
 
 of which the partial rates are 
 
 du=f^ {x,y) dx 
 and du^f2(x,y)dy. 
 
 Dividing the first by dx and the second by dy give 
 du 
 
 -r^h{^,y) (2) 
 
 dx 
 
 du 
 
 and -— = f^_(x,y) (3) 
 
 dy 
 
 Taking the rate of (2) with respect to y and dividing by 
 dy; then the rate of (3) with respect to x and dividing by dx, 
 the results are 
 
 d^u 
 
 = h(^,y) 
 
 dxdy 
 
 d^u 
 
 and ——— = f^(x,y). 
 
 dydx 
 
 Now, as is shown in Art. 23, in order that ( 1 ) be integrable, 
 
 d~u d^u 
 
 must equal , 
 
 dxdy dydx 
 
 that is /s (x,y) =f^ (^,y). 
 
 This is termed the test of integration. 
 
 If du = f (x, y, z) dx -\-f {x, y, s) dy -\- f (x, y, s) dz, 
 in order that this expression be integrable the following condi- 
 tions must be fulfilled : viz., 
 
 d^u d^u d^u d^u d~u d^u 
 
 dxdy dydx dxdz dzdx dydz dzdy 
 
 and similarly if there are four or more independent variables. 
 
 Let du = {Zx^y"" ^y^l)dx-\-{2x^y^x^a)dy, (4) 
 the partial rates of which are 
 
 du= {Zx'y"^ -\- y -\- \) dx (5) 
 
 and du = {2x^y -\- x -\- a) dy; (6) 
 
 whence are obtained the following, 
 
 d^u 
 
 = 6x^y -\- 1 
 
 dxdy 
 
142 AN ELEMENTARY TREATISE 
 
 d~u 
 
 and = 6x^y -\- 1, 
 
 dydx 
 
 which fulfill the conditions stated above; therefore (4) is 
 integrable. 
 
 It will be seen that the original function must have con- 
 tained all the terms in x indicated in (5), also all the terms in 
 3; indicated in (6). Now the integral of (5) is 
 
 u-^x^y- -\- xy -{- X (7) 
 
 and of (6) u = x^y^ -\- xy -\- ay, (8) 
 
 but it will be observed that the terms in (8) containing x are 
 also included in (7), and therefore should be omitted in 
 integrating; consequently the integral of (4) is 
 
 u = x^y^ -|- ;r3; -|- ;ir -|- 03;. 
 
 Let du = ay-dx -\- 2xdy, (9) 
 
 of which the partial rates are 
 
 du = ay~dx and du = 2xdy, 
 
 from which are obtained 
 
 d^u d^u 
 
 = 2ay and = 2, 
 
 dxdy dydx 
 
 which are not equal ; therefore (9) is not integrable. 
 
 Let du = (2xy -{- 2^ -\- l)dx -\- (x- -\- 3y^z -f- ^) ^3* + 
 
 (2x2 + y^ + 4z^ ^ b) dz. (10) 
 
 It is obvious here, as in rates of two independent variables, 
 that the integral of the coefficient of dx must have all the terms 
 containing x in the original function ; therefore, in integrating 
 the coefficient of dy, the terms containing x must be omitted, 
 and in integrating the coefficient of ds, the terms containing 
 both X and y must also be omitted. 
 
 Proceeding thus, it is found that 
 
 u = x-y -|- ^^^ -\- ^ -\- y^^ + QJ + ■s'* + b2. 
 
 EXAMPLES 
 
 du= {2xy -\- 3x^n) dx -j- {2xy + ^) dy 
 
 ydx xdy xydz 
 
 du = -j" + 
 
 a — z a — z (a — z) 
 
ON VARIABLE QUANTITIES 143 
 
 3 2^1:3; 3xy^ — x"^ 
 
 du==^— {x^ — y-)dx — dy + as 
 
 2 z z^ 
 
 du^ (sin 3; — 3;sinjir) dx -{- (cos;ir + x cosy) dy 
 
 Homogeneous Rates 
 
 92. A homogeneous rate is one in which the sum of the 
 exponents of the variables is the same in each term ; this sum 
 is called the degree of the rate, and is here designated by n. 
 
 When such a rate fulfills the conditions given in the last 
 article, the integral can be obtained by substituting, for instance, 
 X, y, z for dx, dy, dz, etc., in their respective factors of the 
 functional rate, thus increasing by unity the exponent each of 
 X, y, z, etc. in its said factor; then collecting like terms and 
 dividing hy n -\- \. 
 
 To prove this, let 
 
 du = Pdx-\-Qdy^Rdz-\-^i(i. (1) 
 
 be a homogeneous rate, in which P, Q, R, etc. are algebraic 
 functions of x, y, z, etc. of the nth degree. 
 
 Now it is evident that this must have been deduced from a 
 homogeneous algebraic function of the form 
 
 u = P'x-{-Q'y-{-R'z-\- etc., (2) 
 
 of the degree n -\- I, since taking the rate diminished by unity 
 the exponent of the variable so treated in each term of (1). 
 
 Substituting xy' for y, xz' for z, etc. in (2) , each term in the 
 value of u will contain jr"""^, consequently 
 
 w = A^.a^«+^ (3) 
 
 in which iV" is a function of y' , z' , etc., but does not contain x; 
 hence, the rate of (3) with respect to x, is 
 
 du 
 
 =(n+l)A/';r^ (4) 
 
 dx 
 
 The rates of xy' , xz' , etc. with respect to x, are y'dx, z'dx, 
 etc., and these rates substituted in (1) and divided by dx, give 
 
 du 
 
 —- = P^Qy'^Rz'^eXc. (5) 
 
 dx 
 
 du 
 
 but = (w + 1) Nx"", (4), therefore 
 
 dx 
 
 (w 4- 1 ) Nx"" = P ^Qy' ^ Rz' + etc. 
 
144 AN ELEMENTARY TREATISE 
 
 or, by multiplying by x, 
 
 (n + 1 ) iV^«^i = Px-{- Qxy' + Rxz' + etc. 
 Therefore, substituting y for xy' , z for xz' , etc., and divid- 
 ing by (w + 1), 
 
 ,, , P^ + Q3' + ^^ + etc. 
 
 W+ 1 
 
 or, since Nx^^'^'^ = u [see (3)], 
 
 Px -\- Qy -i- Rz -{- etc. 
 2* = — . (6) 
 
 n -\- 1 
 
 EXAMPLES 
 
 Integrate du= (Sx -\- mxy) dx -\- {x -\- mxy) dy and 
 {nx'^-'^y + y) J.r -f (.r" + xy''-^) dy -\- {n -\- \) z"". 
 
 Length of Curves 
 93. In case of curves referred to rectangular coordinates, it 
 has been shown in Art. 46 that 
 
 dz={dx~ + dy^y^, 
 whence z =J {dx'^ -\- dy'^Y^, 
 
 which is a general expression for the length of a curve, or the 
 length of any arc thereof, estimated from the origin of the 
 coordinates or some special point. When the radical is ex- 
 pressed in terms of x and dx, or y and dy, obtained from the 
 equation of the curve, its integral may be determined. 
 
 In case of polar curves, the rate of an arc is [see Art. 
 57, (1)]: 
 
 dz= (dr~ -\- r-dv-y^, 
 whence z = ^ { dr- -f r^dv^ ) ^% 
 
 which is the general expression for the length of an arc of a 
 curve referred to polar coordinates, estimated from the pole or 
 some special point. When the radical is expressed in terms of 
 r and dr, or v and dv, its integral may be determined. 
 
 Taking the circle whose radius is unity, its sine x and 
 cosine (1 — x-)^''-, then 
 
 X 
 
 t = tan z = , 
 
 (l—xn'' 
 whence 
 
 (I — x-y^ dx + x'- (I — x'-y^^ dx dx 
 dt = - ~ ^ = . (1) 
 
ON VARIABLE QUANTITIES 145 
 
 x^ 1 
 
 Now 1 + ^2^1+ = . (2) 
 
 1 X' 1 — x"^ 
 
 Dividing (1) by (2), the result is 
 
 dt dx 
 = = dz, 
 
 or dz^{\-\-f")-^dt, 
 
 the rate of an arc of a circle in terms of the tangent and its rate. 
 
 Developing, dz= {\ — f- ^ t^ — t'' ^ etc. ) dt 
 
 f t" f 
 and integrating, z=t — — + — — — -[-etc., (3) 
 
 which needs no correction, since z^O when ^ = 0. 
 
 Now, by means of the trigonometrical formula 
 
 2 tan a 
 
 tan 2a = ; 
 
 1 — tan- a 
 
 1 
 
 when tan a = — , we find 
 5 
 
 120 
 
 tan 4a = . 
 
 119 
 
 Also, by means of the formula 
 
 tan A — tan 5 
 
 tan (^ — 5)= , 
 
 1 + tan ^ tan 5 
 
 120 
 
 when tan A = and tan B = tan 45° = 1, we find 
 
 119 
 
 1 
 tan (A — B)=- 
 
 239 
 
 1 
 Hence, four times the arc whose tangent is — exceeds the 
 
 1 
 
 arc of 45° by an arc whose tangent is . In a similar manner, 
 
 239 
 
 1 
 
 we shall find that twice the arc whose tangent is exceeds 
 
 10 
 
 1 . 1 
 
 the arc whose tangent is — by an arc whose tangent is . 
 
 5 515 
 
146 AN ELEMENTARY TREATISE 
 
 Therefore, if ^ = arc of 45°, since (as has been shown in 
 the paragraph immediately preceding) 
 
 1 1 1 
 
 arc 45° = 8 tan — 4 tan — tan , 
 
 10 515 239 
 
 by applying these values to (3) and multiplying by 4, since arc 
 180° ^TT, the following are obtained: 
 
 1111 \ 
 
 ( — + — + etc.) 
 
 3(10)3 5(10)^ 7(10)^ ^ 
 
 1 1 1 
 
 + — + etc.) 
 
 515 3(515)2 5(515)=^ 7(515)' 
 
 11 11 
 
 i— ( — + — -fete.) 
 
 ' 239 3(239)3 5(239)^ 7(239)^ 
 
 Six terms of the first line and three each of the second and 
 third will give 
 
 7r = 3.141592653589793. 
 The transcendental equation of the cycloid is (see Art. 42) 
 
 ydy 
 
 du^- 
 
 (2ry — y^y 
 
 Squaring this equation and substituting the value of dx^ in 
 the rate of the arc give 
 
 y-dy'^ 
 
 d^=(dy^ + - -)\ 
 
 Iry — y 
 
 2r 
 
 or, reducing, d2^^=^dy{ Y^. 
 
 2r — y 
 
 Putting this under the form 
 
 ds={2ry- (2r — y)-'^dy, 
 and integrating by Art. 74, 
 
 ^ = — 2(2r)^ (2r — 3;)% + C, 
 or s = —2yy{2r(2r — y)}-\-C. 
 
ON VARIABLE QUANTITIES 
 
 147 
 
 A F 
 
 
 
 C 
 
 
 
 F 
 
 g 
 
 If 
 
 
 
 
 then 
 
 
 
 
 and since 
 
 APD 
 
 — 
 
 AI 
 
 50 
 
 Estimating the arc 
 from A, 2^ = for 
 y = 0, consequently 
 = — 4r + C or 
 C = 4r, hence ^ = 
 4r— 2V{2r(2r— y)}, 
 which represents the 
 B length of an arc of 
 the cycloid, estimated 
 from A to any point, 
 as P. 
 y=CD = 2r 
 
 z = APD = Ar 
 
 .AP = DP=4r~ [4r — 2v'{2r(2r— y)}], 
 
 DP = 2^/{2r{2r — y)} (4) 
 
 which represents the length of the arc estimated from D to any 
 point P. 
 
 By similar triangles, 
 
 CD:DL::DL:DE 
 
 or DL={CDDEy-; 
 
 but, since CP = 2r and y = PF = CE, DE :^2r — y, hence 
 
 DL = ^y{2r(2r — y)} 
 therefore, comparing this with (4), it will be found that 
 arc DP = 2DL ; 
 
 that is, the arc of the cycloid, estimated from the vertex of the 
 axis CD, is equal to twice the corresponding chord DL of the 
 
 Y 
 
 generating circle. 
 
 The equation of the 
 logarithmic curve is 
 
 X = log y. 
 
 Passing to the rate and 
 squaring, 
 
 dy^ 
 
 dx' 
 
 r 
 
 Ft a. ^1 
 
 Substituting this value 
 of dx^ in the rate of the arc 
 and reducing give 
 
148 AN ELEMENTARY TREATISE 
 
 y 
 
 Integrating by formula C of Art. 80, 
 
 dy 
 
 ^=(1+/)^'^+/ 
 
 3; (1 ^y-y^ 
 Integrating again by Art. 83, 
 
 ^=(i+r)^- — log ~ + C; 
 
 (1 +r)'^— (1 — 3') 
 
 or, multiplying both numerator and denominator of the fraction 
 in the second member by ( 1 + 3'" ) '"^^ + (1 — 3') and reducing, 
 
 y 
 
 With C the origin of coordinates, when x^O, 3; = 1 and 
 2 = 0; consequently 
 
 0-=V2 — log(l+V2)+C, 
 
 or C = — V2 + log(l+V2), 
 
 therefore 
 
 1 + (1+3'')^^ 
 ^=(l-|-3;2)V3_iog ^ ^_V2 + log(l + V2), 
 
 y 
 
 which represents an arc of the logarithmic curve AB, esti- 
 mated toward B from the point where it cuts the axis of 
 coordinates. 
 
 The equation of the spiral of Archimedes is 
 
 V 
 
 r 
 
 7 . 
 
 ^ 77 
 
 Taking the rate and squaring. 
 
 dr^ = - 
 
 Substituting this value of dr in the rate of the curve and 
 reducing, 
 
 dv 
 
 d2 = --(i+v^~)y^. 
 
 First integrating by formula C of Art. 80, then by Art. 
 83, (6), 
 
ON VARIABLE QUANTITIES 149 
 
 1 
 
 Z = — [V (1 -f t;2)%__log{(l J^v^y- — V)] 
 4'nr 
 
 which represents the length of any arc of the spiral of Archi- 
 medes, estimated from the pole; no correction is needed, since 
 ^ = when v==0. 
 
 EXAMPLES 
 
 1. Determine the length of an arc of the common parabola. 
 
 2. Determine the length of an elliptic quadrant in terms of 
 its eccentricity, the semi-major axis being unity and the semi- 
 minor axis a. 
 
 3. Determine the length of an arc of the logarithmic spiral, 
 estimated from the point where r= 1. 
 
 Area of Curves 
 
 94. The rate of the area of a curve referred to rectangular 
 coordinates is, by Art. 47, 
 
 dA = ydx, 
 
 which can be integrated when the second member is given in 
 terms of y and dy, or x and dx. 
 
 The rate of the area of a polar curve is, by Art. 57, 
 
 1 
 
 dA = — r-dv, 
 
 2 
 
 which can be integrated when the second member is expressed 
 in terms of r and dr, or v and dv. 
 
 Multiplying both members of the equation of the circle 
 by dx gives 
 
 ydx= (R^ — x^y^dx, 
 
 hence dA={R~ — x^- ) ^'^- dx. 
 
 Integrating, first by formula C of Art. 80, then by Art. 76, 
 
 1 i?2 ^ 
 
 A= — xiR'- — x-Y-^ sin-i— , (a) 
 
 2 2 R 
 
 which requires no correction, since A^O when x==0. 
 
 1 
 Making x = R, since the arc of sine unity is — tt, 
 
 1 
 A= — i?2 
 2 
 
150 AN ELEMENTARY TREATISE 
 
 which gives the area of a quadrant of a circle whose radius is 
 R ; therefore the area of the entire circle is R^tt. 
 
 The equation of the ellipse, referred to its center and axis, 
 when both members are multiplied by dx, is 
 
 b 
 
 ydx = — (a- — x-)"^^ dx ; 
 a 
 
 b 
 hence dA^— (a^ — x^y^ dx. 
 
 a 
 
 Integrating, first by formula C of Art. 80, then by Art. 76, 
 
 A = -{-x{d^ — x-y^ + ^sm-^-}, (b) 
 
 a Z Z a 
 
 which requires no correction, since A=^0 when x=^0. 
 
 1 
 
 If jir = a, since the arc of sine unity is equal to — tt, then 
 
 M 2 
 
 ^ \ E^ 1 
 
 r G FMGor A= — ab-jT, 
 
 \^^~-^-j;l -^y^ which represents the area of a quarter of 
 
 \v^^___l__,.„^^' an ellipse whose semi-major axis is a and 
 
 n' semi-minor axis is b; therefore the area of 
 
 j_. the entire ellipse is equal to abTr. 
 
 '5"^ Comparing (a) with (b), it will be seen 
 
 that the area of a segment of the ellipse, as 
 
 CDEF, is equal to the area of the corresponding segment of 
 
 . b 
 
 the circumscribing circle, CMNF, multiplied by — ; hence 
 
 a 
 
 b 
 area DEE'D' = — ( area MNN'M' ) . 
 a 
 
 Taking the general equation of the parabola 
 
 yn = ax or y = a^^^x'^^", 
 and multiplying both members by dx, the result is 
 
 ydx == a^/"x'^^"dx ; 
 
 hence dA = a^^'^x^/^'dx. 
 
 n 
 Integrating, A = Qi/n^cn+D/n _|_ c. 
 
 M -)- 1 
 
ON VARIABLE QUANTITIES 151 
 
 Estimating the area from the vertex of the parabola, A=0 
 when x=^0, and consequently C = 0; therefore 
 
 n n 
 
 n-{- 1 n -\- I 
 
 or, substituting 3; for a^^"x'^^"', 
 
 n 
 
 A= -^3^, (1) 
 
 n -\- I 
 
 which represents the area of a segment of any parabola, and is 
 equal to the rectangle described by the abscissa and ordinate, 
 
 n 
 multiplied by the constant term . 
 
 n -\- \ 
 
 li n = 2, (1) becomes 
 
 2 
 A^ — xy: 
 3 
 
 that is, the area of a segment of the common parabola is equal 
 to two-thirds of the area of the rectangle described by the 
 abscissa and ordinate. 
 
 If w=l, (1) becomes 
 
 1 
 
 A=jxy; 
 
 that is, the area of a triangle is equal to half the product of its 
 base and perpendicular. 
 
 Multiplying both members of the equation of the hyperbola, 
 referred to its center and axes, by dx gives 
 
 b b 
 
 ydx^— {x"- — a^y^ dx or dA=^— {x"- — a~y^dx. 
 a a 
 
 Integrating first by formula C of Art. 80, then by Art. 83, 
 
 bx (x^ — a^)^^ ab 
 
 A^ ^ ^_ log{^+ (x^---a'y^}^C. 
 
 2a 2 
 
 When A^O, x = a; consequently 
 
 ab log a 
 
 C 
 
 2 
 therefore 
 
 bx (x^ — a^y^' ab x -[- (x^ — a^y^ 
 
 A=-^ ^— — log( ^ '—}, 
 
 2a 2 a 
 
152 
 
 or, since 
 
 AN ELEMENTARY TREATISE 
 b 
 
 a 
 
 {x' — d'Y 
 
 y, 
 
 1 
 
 A=^ — xy ■ 
 
 2 
 
 ah bx + ay 
 
 log( ^-^). 
 
 2 ab 
 
 Squaring both members of the equation of the spiral of 
 
 ^^ . . 1 . 
 
 Archimedes (r = — ) and muUiplying by — dv give 
 27r 2 
 
 1 v-dv 
 — r~dv = , 
 
 2 Stt^ 
 
 or 
 
 dA=^ 
 
 v^dv 
 
 whence, integrating. 
 
 24 TT^ 
 
 + c:. 
 
 Estimating the area from the pole, y^ = when ^' = 0, and 
 consequently C = 0; therefore 
 
 i7;3 
 
 If v^2iT, then 
 
 ^=- 
 
 24- 
 
ON VARIABLE QUANTITIES 152 
 
 which represents the area of PBA, described by one revolution 
 of the radius vector : that is, the area of the first spire is equal 
 to one-third of the area of a circle, whose radius is equal to the 
 radius vector of the spiral at the end of the first revolution. 
 
 If v = Att, then 
 
 8 
 
 3 
 
 which represents the area described by the radius vector in 
 two revolutions; but it will be seen that the radius vector 
 describes the portion PBA a second time ; therefore, to obtain 
 the area of PB'A', the area described by the first revolution 
 must be deducted : that is, 
 
 8 1 7 
 
 area PBA = — tt — — ■7r = — tt. 
 3 3 3 
 
 EXAMPLES 
 
 1. Determine the area of the cycloid. 
 
 2. Determine the area of a segment of the logarithmic curve, 
 lying between the curve and axis of ordinates, estimated from 
 the point where the curve cuts the axis of ordinates. 
 
 Surface Areas of Revolution 
 95. For a curve referred to rectangular coordinates, revolv- 
 ing about the axis of X, the rate of the surface area of rotation 
 is (see Art. 48) 
 
 dS^^Zir ydz. 
 
 In case the curve is revolved about the axis of Y, it is evi- 
 dent that the rate of the surface area will then be 
 dS = 2Tr xdz. 
 
 When the second member of either of these equations is 
 expressed in terms of x and dx or y and dy, the integral thereof 
 may be determined. 
 
 From the equation of the common parabola it will be 
 found that 
 
 ydy 
 
 dx = . 
 
 p 
 
 Substituting this value of dx in the rate of the area of the 
 surface of revolution, 
 
 y'^dy- 
 dS = 27ry{- + df'y\ 
 
154 AN ELEMENTARY TREATISE 
 
 or dS = -^^(y-^p^y^dy. 
 
 P 
 
 Integrating by Art. 78, 
 
 3p 
 Estimating the arc from the origin of coordinates, 6' = 
 
 2p-7r 
 
 when 7 = 0: hence C=: — and 
 
 3 
 
 ^=-7^{(f' + p'r'—p'}, 
 
 3p 
 
 which represents the surface area of revolution of the common 
 parabola for any ordinate 3^. 
 
 The equation of the ellipse is 
 
 a-y^ = a^b^ — b^x^, 
 
 b a^ — b^ 
 
 ydz^ — (a- — x'Y^ dx 
 
 a a~ 
 
 or, representing the eccentricity of the ellipse by e, by substi- 
 tuting a^e'^ for a- — b^, since a^ — b~ = d-e^, 
 
 b 
 
 ydz = — (a- — e-x-y^dx; 
 a 
 
 2be TT a^ 
 
 therefore dS = ( — — x'^Y^ dx. 
 
 a e- 
 
 Integrating, first by formula C of Art. 80, then by Art. 76, 
 gives 
 
 be IT a^ ab-jT ex 
 
 6" = (— — x-y^ X ^ sin-^ , 
 
 a e^ e a 
 
 which needs no correction, since S ^0 when x = 0; hence 
 the expression represents the surface area of that part of an 
 ellipsoid estimated from the vertex of the minor axis and cor- 
 responding to the abscissa x, the arc being revolved about the 
 major axis. By making x^a and reducing, 
 
 abiT 
 
 S == b~iT -{- sin"^ e, 
 
 e 
 
 which gives one-half the area of the surface of the ellipsoid; 
 
ON VARIABLE QUANTITIES 155 
 
 therefore if S' represents the area of the entire surface, then 
 
 S' = Zb^TT + sin-i e. 
 
 e 
 
 When a^b, ^ = 0, and the equation becomes 
 
 S' = Zb^TT + 2&-7r = 4^ V, 
 
 the area of the surface of a sphere whose semi-diameter is b. 
 
 If the elHpse be revolved about its minor axis, then will 
 
 a 
 
 xds = — {b* -\- a-e-y^ ) ^^^ dy ; 
 
 2a TT 
 
 hence dS ^^ ( ^* + d^^'y^ ) ^^^ dy. 
 
 b~ 
 
 Integrating, first by formula C of Art. 80, then by Art. 83, 
 gives 
 
 S = -^ {¥ + a^e^-y^y^y^ 
 b'~ 
 
 ^^\og{{b' + a'e'~y^y^ + aey) + C. 
 e 
 
 Estimating the surface from the vertex of the major axis, 
 5 = when 3' = 0, in which case 
 
 C = — log b- ; 
 
 e 
 
 a-TT 
 
 therefore 5* = ( ^* + a^e-y"^ Y^ y -\- 
 
 b^ 
 
 7 9 L2 
 
 ^^log {(^^ + a^e'y'^y -f 0^3;} —^^ log b\ 
 e e 
 
 air b'Tz (b"^ -\- a^e^y-Y^ 4- aey 
 
 or 5- = -(&* + aVy)V3 3, + log{^ ^ '-}. 
 
 b- e b- 
 
 Now, since b^a{\ — e^Y^ and (&- + a^^^)"''^ = a, when 
 y^^b, this becomes 
 
 feV ab(l-\-eY' 
 
 S = a^ir + log 
 
 e ^ab(l — eY' 
 
 b^TT (l-\-eY' 
 or 5^ = a^TT -\- log 
 
 e (1—^)% 
 
156 
 
 AN ELEMENTARY TREATISE 
 
 which represents half the area of the surface of a spheroid. 
 If S' represents the entire surface, then 
 
 If = 
 
 ■ 2aV -|- 2&^7r{ 
 b, then 
 
 log (14-^)y^_log(l — ^)y^ 
 
 }• 
 
 :2&-7r{l + 
 
 log(l + 0'''^ — log(l — ^)' 
 
 Now, when a=b, ^ = ; but by Art. 35, 
 log(l + ^)% — log {l — ey- 
 
 1; 
 
 therefore the surface of a sphere whose semi-diameter is b, is 
 
 S' = Ab-7r. 
 
 From the equation of the logarithmic curve, it will be 
 found that 
 
 ydz ^ ( 1 -f- y'^y^dy, hence dS = 2 7r (1 -}- y^y^ dy. 
 
 Integrating by Arts. 80 and 83, 
 
 log[(l+3'^)^/^ + 3']}+C. 
 
 Estimating the surface from P, 
 the point where the axis of ordi- 
 nates cuts the curve, S^O when 
 y^l; consequently 
 
 C = vr{— V2 — log(V2 + l)}; 
 therefore 
 
 s = ^{y(i+y'y-h 
 log [{l^f-y + y]}— 
 
 -{V2 + log(V2+l)}.. 
 This represents the area of the surface of revolution of any 
 arc of the logarithmic curve, estimated from P, as PC, and cor- 
 responding to the ordinate y = DC, the curve being revolved 
 about the axis of abscissas AB. 
 
 EXAMPLES 
 
 1. Determine the area of the convex surface of a right conoid 
 whose perpendicular height is a and diameter of base is b. 
 
 2. Determine the area of surface of revolution of the 
 cycloid when revolved about its base. 
 
ON VARIABLE QUANTITIES 157 
 
 3. Determine the area of the convex surface of a cubical 
 paraboloid, when the axis of ordinates is the axis of revolution. 
 The equation is 3'^ = ax. 
 
 Volume of Revolution 
 
 96. The rate of the volume of revolution generated by an 
 
 arc of a curve revolved about its axis of abscissas is, by Art. 49, 
 
 dV = 7ry^dx. (1) 
 
 When the second member of this equation is expressed in 
 
 terms of either x and dx, or y and dy, its integral can be 
 
 determined. 
 
 When the axis of Y is the axis of revolution, the rate of the 
 volume of revolution thus generated is 
 
 dV =^ IT x^dy. 
 
 From the general equation of the parabola, 3;" = ax, it will 
 be found that 
 
 n 
 
 dx^ — y'^-^dy. 
 
 a 
 
 Substituting this value oi dx in (1), 
 
 n 
 dV = — 7r y^^dy, 
 a 
 
 and integrating, the result is 
 
 n n y" 
 
 a (n -\- 2) n -\-2 a 
 
 or since — ^x, V^ny^ ( — x) + C, 
 
 a n -\-2 
 
 which needs no correction, since v^O when x = Q. If m= 1, 
 
 1 
 
 then V ^ — TT y^x, 
 
 which represents the volume of a right cone whose altitude is x 
 and y one-half of the diameter of its base. 
 
 1 
 
 If w = 2, then V = — Try'-x, 
 
 which represents the volume of the common parabola. 
 
158 AN ELEMENTARY TREATISE 
 
 The equation of the elHpse, when the origin is at the vertex 
 of its minor axis, is 
 
 a- 
 
 x- = —{2by — y-); 
 b^ 
 hence [see (2)], 
 
 dV = —^ (2by — /) dy. 
 b" 
 
 aV 1 
 
 Integrating, V = {by'^ — — 3;^) -\- C, 
 
 b~ 3 
 
 in which C = when y = 0; therefore 
 
 a-TT 1 
 
 V = --{bf- — -y^). 
 b- 3 
 
 li y=^b, then 
 
 2 
 
 V ^ — a-biT, 
 
 3 
 
 which represents the volume of one-half of a spheroid; hence 
 the entire volume is 
 
 4 2 
 
 V' = — a'bTr = — (Za^bir). 
 3 3 
 
 But 2a~bTr represents the volume of a cylinder, whose altitude 
 is 2b and the radius of whose base is a; therefore the volume 
 of a spheroid is equal to two-thirds of the volume of a circum- 
 scribed cylinder. 
 
 The equation of the hyperbola, when the origin of the 
 coordinates is at the vertex of the transverse axis, is 
 
 b^ 
 
 3;^ = — {x^ -\- 2ax) . 
 d~ 
 
 Substituting this value of 3)- in (1), 
 
 dV = {x^ -f 2ax) dx. 
 
 a- 
 
 feV 1 
 
 Integrating, V = ( — x^ -)- ax-) -\- C. 
 
 a^ 3 
 
 Estimating the volume of revolution from the origin of 
 coordinates, we have V = when jr = 0, and consequently 
 C ^ ; therefore 
 
ON VARIABLE QUANTITIES 159 
 
 b'-TT 1 
 
 a~ 3 
 which represents the volume of revolution of the hyperbola 
 for any abscissa. 
 
 The ratal equation of the cycloid is 
 
 ydy 
 
 dx^ . 
 
 {2ry — 3'")'''^ 
 
 Substituting this value of af;r in ( 1 ) , 
 
 TTj^dy 
 
 dV 
 
 {2ry — y'^Y^ 
 the integral of which is 
 
 1 5 y 
 
 V =^ tt{ — {2y^ 4~ 5ry -\- 15r') {2ry — y^)''^ -j- — r" vers"^ — }. 
 6 2 r 
 
 If3; = 2f, then V^ — r^ vers"^ 2, 
 
 or, since vers"^ 2 ^ tt, 
 
 5 
 
 9 
 
 which represents one-half the volume of revolution generated 
 by the cycloid revolved about its base. The entire volume is 
 
 F' = 5rV^ 
 
 EXAMPLES 
 
 1. Determine the volume of rotation of the ellipse when the 
 origin of the coordinates is at the vertex of the major axis. 
 
 2. Determine the volume of revolution of the logarithmic 
 curve when revolved about its axis of abscissas. 
 
 3. Determine the volume of revolution about its axis of 
 abscissas of the curve whose equation is 
 
 y = x (x -\- a). 
 
 97. Let BDEF be a plane moving from A toward X along 
 the axis of X and at right-angles thereto, and let AC be repre- 
 
160 
 
 AN ELEMENTARY TREATISE 
 
 sented hy x, BC by y, and FC 
 by v; then the rate of the 
 volume of the solid thus gen- 
 erated will be 
 
 dV = f {v, y) dx, 
 or, since y^f{x) and 
 
 dV = f{x)dx (1) 
 
 This formula is applicable 
 to the volume of any solid, 
 when the area of the plane 
 BDEF can be expressed in terms of x and dx. 
 
 Determine the volume of a right pyramid whose base is a 
 rectangle. 
 
 Let the perpendicular Aa be represented by x, the side BC 
 by y, and the side C\D by v ; also let y = ax and v = hy. Then 
 the area of BCDE will be 
 vy = ahx^ ; hence 
 
 dV = abx^dx, 
 and integrating, 
 
 1 
 
 F = — abx^. 
 
 3 
 
 But abx^ is the area of BCDE; 
 therefore the volume of the pyra- 
 mid is equal to the area of its 
 base multiplied by one-third of 
 its perpendicular height. 
 
 Required the volume of a parabolic paraboloid, the fixed 
 parabola being the semi-cubical and the generatrix the common 
 
 parabola. 
 
 Let x = AC, y = BC, and 
 v=^CE = CD; then for ABX, 
 
 3;3/2 := ax, 
 
 and for BCE, v~ = by. 
 
 From these equations it 
 will be found that the area of 
 
 4 
 BDE is — ab'/'x, 
 3 
 
 Fi 
 
 g- 
 
ON VARIABLE QUANTITIES 
 
 161 
 
 also 
 
 and integrating, 
 
 dV = — ab^''^xdx, 
 3 
 
 2 
 3 
 
 Required the volume of an elliptical ellipsoid, the equations 
 being for the fixed ellipse a^y^ = a^b^ — b^x^ and for the gen- 
 eratrix a^v^ = a^c^ — c^x'^ and the origin of coordinates being 
 at the center. 
 
 Let A'C = a,BC = b, CF = c, 
 CC'=x,B'C'=y, and C'F'=v; 
 then from the equations, 
 
 y = -{a' — x^y% 
 
 (a2_;ir2)v^, 
 
 and vy 
 
 be 
 
 (^a'—^x^). 
 
 therefore 
 
 Fin. JS 
 
 be 
 
 But 7rz;3; = area of B'D'E'F', 
 
 Trbc 
 
 dV = (a^ — x'') dx={7rbe~ x^) dx, 
 
 IT be 
 and integrating, V=^iTbex — ;r^ 
 
 which requires no correction since F = when x=^Q; hence, 
 making ;ir = a, 
 
 1 2 
 
 V^Trabe — — irabe^ — Trabe. 
 3 3 
 
 Since V is one-half the volume of the ellipsoid, F', the volume 
 
 4 
 of the entire solid, is — tt abe. 
 
 3 
 
 EXAMPLES 
 
 1. Determine the volume of an elliptical conoid whose 
 altitude is a' and the radius of whose base is b\ 
 
 2. Determine the volume of a groin formed by the inter- 
 section of two equal semi-cylinders at right-angles to each 
 other, the equation being y = 2rx — x. 
 
162 
 
 AN ELEMENTARY TREATISE 
 
 Curved Surfaces Referred to Three 
 Rectangular Coordinates 
 
 98. To obtain a formula for the volume of a solid bounded 
 by a curved surface and referred to three rectangular coordi- 
 nates, X, y, and s, of which z^f {x,y). 
 
 Let the plane C'CPD be para- 
 llel to the plane A'ZX and the 
 plane EPBB' parallel to the plane 
 YZA' ■ also let A'B' = C'P'= x, 
 A'C' = B'P' = y and P'P = z. 
 Represent dx by P'a' = c'h' and 
 dy by P'c' = a'b' ; then zdxdy 
 will represent the rate of the 
 volume of the solid; that is, 
 
 dW = zdxdy. {A) 
 
 To obtain a formula for the 
 surface area, let Pa^cb repre- 
 sent dx, Pc = ah represent dy, 
 and Nc = Ma represent dz ; also 
 let PM be a tangent to the curve 
 
 CPD Sit P (Fig. 60), PA^ a tangent 
 
 to the curve BPE at P, and PQ a 
 
 perpendicular to NM. Then will 
 
 PN= (dx^- + dz^y^ 
 PM= (dy^ -\-dz^y^ 
 and NM = (dx^ -\- dy^ ) ^/^ 
 
 From these three equations the 
 following is obtained : 
 
 (dx^dy^4-dx'-dz^4-dy^dz-y' 
 
 PQ=- ^-^= ^— ^^ —; 
 
 (dx^^dy^y^ 
 
 but PQ ■ NM = area of NPML = 
 
 (dx^dy"" + dx^dz^- + dy^dz-y^ ; 
 
 therefore d'^S = 
 
 (dx^dy^ -^ dx'-dz^- ^ dy''dz^-y\ (B) 
 
 Required the volume, also the surface area of a sphere, the 
 equation being 
 
 ^2 = r- — (x- -{- y-) . _ (1) 
 
 For the volume [see formula (A)], it will be seen that 
 d^V= {r^ — x'^ — 3;^ ) '/^ dxdy = zdxdy. 
 
ON VARIABLE QUANTITIES 162 
 
 The integral of this with respect to y, between the limits 
 y^^O and y^ (r^ — x-)^''', is 
 
 1 3' 1 
 
 2 ^ (,■■' — x-'y^ 4 
 
 y 1 
 
 since {r- — x^)'^^=^y and sin ^ — = — tt. 
 
 3' 2 
 
 Integrating this expression with respect to x gives 
 
 1 1 
 
 F = — TT (r^x — — x^) 4- C, 
 4 3 
 
 or between the limits x = and x^r, 
 
 1 1 1 
 
 V = — TT (r^ — — r^) = — r^ TT, 
 4 3 6 
 
 which represents one-eighth of the volume of a sphere; there- 
 fore the volume of the entire sphere is 
 
 4 
 
 V' = — 7rr\ 
 3 
 
 Resuming (1), 
 
 ^2 ^ ,^2 ^2 y2^ 
 
 and taking the rate, first with respect to x, then with respect 
 to y, the results are 
 
 xdx ydy 
 dz = — and ds^ — : 
 
 z 
 
 y^dy^ x^dx^ 
 hence dz~ = and dz^ = 
 
 z^ z^ 
 
 Substituting these values of dz^ in formula B, so that 
 dx^dz^ dy^dz^ 
 
 x^dx'^dy'^ y^dy^dx^ 
 shall read -|- 
 
 z^ z~ 
 
 and reducing (since dz will be eliminated), the result is 
 
 dxdy 
 d-S = — (x^ -^y^-\- z^y^ 
 
164 
 
 AN ELEMENTARY TREATISE 
 
 or, since 
 
 {x- ^y^- ^s^-y- = r, 
 
 and 
 
 z^ (r- — X- — y'^y^, 
 
 
 rdxdy 
 
 J9.C -^ 
 
 {r^ — x- — y^y 
 
 The integral of this with respect to y, between the Hmits 
 of y = and 3'^ (r — x), is 
 
 ydx 
 
 dS ^r sin~^ 
 
 y={r- — x-y% 
 
 1 
 
 therefore, since sin~^ 1 = — tt, 
 
 (^r- — x-y^ 
 
 y 
 
 1; 
 
 1 
 
 dS = — rrr dx, 
 
 2 
 
 the integral of which, between the limits of ;r :^ and jr = r, 
 
 1 
 
 is S=^ — r^TT, 
 
 which represents one-eighth of the surface area of a sphere, 
 therefore the entire area is 
 
 99. A body T, with a uniform velocity, proceeds from C , 
 toward A, along the straight line CA, 
 B^ and a body P, with a velocity which 
 is to that of T as 1 to n, proceeds 
 from B in pursuit of T and always in 
 the direction of T. Required the 
 equation of the curve APB, called the 
 curve of pursuit, which is described 
 by P. 
 
 Let A be the origin, BC = a, 
 AS^x, PS^=y, and the arc AP=z; 
 then AT = nz and (see Art. 40) the 
 
 subtangent ST ^ 
 
 ydx 
 
 dy 
 
ON VARIABLE QUANTITIES 165 
 
 Now, since 
 
 ydx 
 AT = AS — ST = x 
 
 dy 
 
 ydx 
 
 X — = nz. 
 
 dy 
 
 Taking the rate of this, regarding dy as constant, and re- 
 ducing give 
 
 — ^= ndz ; ( 1 ) 
 
 dy 
 
 dx^ 
 
 but dz={dx' ^dy-y^ = dy { + l)'''^ 
 
 dy" 
 
 From this and ( 1 ) , the following is found : 
 
 ndy dx^ d^x 
 
 y dy~ dy 
 
 dx 
 
 Integrating by Art. 84, regarding as the variable, the 
 
 dy 
 
 result is 
 
 — nlog3' = log {-—+(-— +l)'/^}+C. 
 dy dy- 
 
 dx dx 
 
 Since = tan SPT, when = 0, y=^a] therefore 
 
 dy dy 
 
 — n log a^C, 
 
 hence, transposing the value of C, it is found that 
 
 dx dx^ 
 
 n\oga — w log 3^ = log (^ + (tT + 1)'''')' 
 
 dy dy- 
 
 a" dx dx- 
 
 or log-=r.log{— -+ (-— -f 1)''^}; 
 
 3;" dy dy- 
 
 a" dx dx^ 
 
 hence — = + ( + 1 ) '^'■ 
 
 y"- dy dy~ 
 
 By resolving this, it will be found that 
 
 1 1 
 
 dx^=^ — a'^y'dy — — a-"y''dy, ( 2 ) 
 
166 AN ELEMENTARY TREATISE 
 
 the integral of which is 
 
 a" 1 
 
 X = 3;^-" — 3;^+", (3 ) 
 
 2(1 — n) 2a» (1 +w) 
 
 which needs no correction, since 3' = when x^O, and 
 therefore is the required equation. 
 
 Dividing (3) by n, then, when y = a and x = AC, 
 X a a a 
 
 n 2w (1 — n) 2n {I -\- n) 1 — n^ 
 
 AC 
 or, since x = AC and =^APB, 
 
 APB = . (4) 
 
 19 
 — n- 
 
 1 
 
 When n = — , (3) and (4) become 
 
 -y% 1 
 
 ^ = — -(a — — 3;), 
 
 4 
 and APB= — a. 
 
 3 
 
 From (1), (2), and (3) it will be found that 
 
 1 (fjr a" 1 
 
 — {x — y ) = 3'''" + 3''"" ; 
 
 n dy 2(1— w) 2a"(l + >t) 
 
 and s = 3'"'+ 3''""' (5) 
 
 2 (1— w) 2a" (1 +w) 
 
 1 
 which represents the length of any arc, as AP. When w = — , 
 
 (5) becomes 
 
 1 3f^^ 
 
 ^ = av^yv^ + -— //^ = -— ■ (3a + y ) . 
 3a'^ 3a^^ 
 

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