University of California Berkeley THE THEODORE P. HILL COLLECTION of EARLY AMERICAN MATHEMATICS BOOKS . ELEMENTS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS. BY CHARLES DAVIES, LL. D., 1UTHOR OF ARITHMETIC, ELEMENTARY ALGEBRA, ELEMENTARY GEOMETRY ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE GEOME- TRY, ELEMENTS OF ANALYTICAL GEOMETRY, AND SHADES SHADOWS, AND PERSPECTIVE. IMPROVED EDITION. NEW YORK: PUBLISHED BY A. S. BARNES & CO. No. 51 JOHN STREET. 1854 Entered according to the Act of Congress, in the year one thousand eight hundred and thirty-six, by CHARLES DAVIES, in the Clerk's Office of the District Court of the United States, for the Southern District of New York. PREFACE. THE Differential and Integral Calculus is justly con- sidered the most difficult branch of the pure Mathematics. The methods of investigation are, in general, not as obvious nor the connection between the reasoning and the results so clear and striking, as in Geometry, or in the elementary branches of analysis. i It has been the intention, however, to render the sub- ject as plain as the nature of it would admit, but still, it 7 cannot be mastered without patient and severe study. This work is what its title imports, an Elementary Treatise on the Differential and Integral Calculus. It might have been much enlarged, but being intended for a text-book, it was not thought best to extend it beyond its present limits. 4 PREFACE. The works of Boucharlat and Lacroix have been t. freely used, although the general method of arranging the subjects is quite different from that adopted by either of those distinguished authors. The present is a corrected, and it is hoped an improved edition. The first chapter has been entirely re-written, and some of the other parts of the work have been considerably altered. WEST POINT, March, 1843. CONTENTS. CHAPTER I. n*e. Constants and variables, . . . .''*'. 9 Functions defined, . . v . * "'' *- 9 Increasing and decreasing functions, ' . , 10 Implicit and explicit functions, . ' .,- d .' 11 Differential coefficient defined, . . i . 16 Differential coefficient independent of increment, . 20 Differential Calculus defined, . f < . . 22 Equal functions have equal differentials, w . 23 Reverse not true, . . , 4 > . 23 CHAPTER II. Algebraic functions defined, V ". ^ ."' i 25 Differential of a function composed of several terms, . 26 " " the product of two functions, . . 27 " " " any number of functions, 28 " " a fraction, 29 Decreasing function and its differential coefficient, . 30 Differential of any power of a function, . . . . 30 " of a radical of the second degree, . . 31 " coefficient of a function of a function, . 33 Examples in the differentiation of algebraic functions, 34 Successive differentials second differential coefficient, 39 Ta-ylor's Theorem, 43 Differential coefficient of the sum of two variables, 43 Development of the function u = (a -f- a?)", . . 46 second state of any function, . 47 Sign of the limit of a series, 47 6 CONTENTS. Page, Cases to which Taylor's Theorem does not apply, . 48 Maclaurin's Theorem, . ' .... 50 Cases to which Maclaurin's Theorem does not apply, 53 Examples in the development of algebraic functions, . 54 CHAPTER III. Transcendental functions logarithmic and circular, . 55 Differential of the function u = a*, ... 55 " " logarithm of a quantity, . .58 Logarithmic series, ...... 59 Examples in the differentiation of logarithmic functions, 62 Differentials of complicated exponential functions, . 64 " " circular functions, .... 66 " " the arc in terms of its functions, . 70 Development of the functions of the arc in terms of the arc, 73 Development of the arc in terms of its functions, . 75 CHAPTER IV. Partial differentials and partial differential coefficients defined, 79 Development of any function of two variables, . . 80 Differential of a function of two or more variables, . 82 Examples in the differentiation of functions of two va- riables, ........ 85 Successive differentials of a function of two variables, 86 Differentials of implicit functions, . . .89 Differential equations of curves, .... 93 Mariner of freeing an equation of constants, . . 96 " " the terms of an equation from ex- ponents, ....... 97 Vanishing fractions, ..... .98 CHAPTER V. Maxima and minima defined, ..... 105 General rule for maxima and minima, . . . .108 Examples in maxima and minima, . . ... 109 Rule for finding second differential coefficients, . . 112 CONTENTS. 7 * CHAPTER VI. Page. Expressions for tangents and normals, . . . 116 Equations of tangents and normals, . . .118 Asymptotes of curves, ...... 122 Differential of an arc, . . . . . .125 " " the area of a segment, ; . . 127 Signification .of the differential coefficients, . . 128 Singular points defined, . . . . . . 132 Point of inflexion, ;1 . . . . . .133 Discussion of the equation y = b c(x a) m , . 134 Condition for maximum and minimum not given by Tay- lor's Theorem Cusp's, . . . . . 139 Multiple point, . . . . . . . .143 Conjugate or isolated point, ..... 144 CHAPTER VII. Conditions which determine the tendency of curves to coincide, . . . . . . . v . 147 Osculatrix defined, . . . . . . 150 Osculatrix of an even order intersected by the curve, . 152 Differential formula for the radius of curvature, . 154 Variation of the curvature at different points, . . 155 Radius of curvature for lines of the second order, . 156 Involute and evolute curves defined, . . . .158 Normal to the involute is tangent to the evolute, . 160 Difference between two radii of curvature equal to the intercepted arc of the evolute, . . . 162 Equation of the evolute, . . . . . .163 Evolute of the common parabola, . . . . 164 CHAPTER VIII. Transcendental curves defined Logarithmic curve, . 166 The cycloid, 169 Expressions for the tangent, normal, &c., to the cycloid, 171 Evolute- of the cycloid, . . . . . . 173 Spirals defined, . . . . . . .175 CONTENTS. INTEGRAL CALCULUS. Png* Integral calculus defined, . . . . 189 Integration of monomials, ..... 190 Integral of the product of a differential by a constant, 192 Arbitrary constant, . . . . . . 194 Integration when a logarithm, ... . '. . 1 94 Integration of particular binomials, . . . 195 Integration when a logarithm, . . . t . .. . 195 Integral of the differential of an arc in terms of its sine, ....... "...'.' 196 Integral of the differential of an arc in terms of its cosine, . . . . . . . 198 Integral of the differential of an arc in terms of its tangent, ..... . ... 199 Integral of the differential of an arc in terms of its versed-sine, . . . . . . 200 Integration by series, . . . ... 201 " of differential binomials, . . . . 207 Formula for diminishing the exponent of the parenthesis, 212 Formulas for diminishing exponents when negative, . 213 Particular formula for integrating the expression 214 --'+ . , /OJ.H Integration of rational fractions when the roots of the denominator are real and equal, . . . 216 Integration of rational fractions when the roots are equal, . ...... 221 Integration of rational fractions when the denominator contains, imaginary factors, .... 226 Integration of irrational fractions, .... 233 Rectification of plane curves, .... 243 Quadrature of curves, ...... 250 " of curved surfaces. . . . . 261 Cubature of solids, ....... 269 Double integrals, . . . , ; r 274 DIFFERENTIAL CALCULUS. CHAPTER I. Definitions and Introductory Remarks. 1. All the quantities which are considered in the Dif- ferential Calculus may be divided into two principal classes : constants and variables. Each constant retains the same value throughout the same investigation ; but the variable quantities are subjected to certain laws of change, in consequence of which they may assume in suc- cession, an infinite number of different values, without changing the form of the expression into which they enter. The constant quantities are generally designated by the first letters of the alphabet, a, 6, c, &c. ; and the variable quantities by the final letters, x, y, z, &c. 2. If two variable quantities are so connected together that any .change in the value of the one necessarily pro- duces a change in the value of the other, they are said to be functions of each other. Thus, in the expression y = y> *) If we make # 0, we have if we also make y==0, we have u = f(z ; and if, in addition, we make z = 0, we have u a constant, which constant may itself be equal to 0. If the function be of the form u = b + ax + yx 2 + zx 2 , in which one of the variables is a factor of several of the terms, then, if x= 0, we shall have u=l; or, if x were a factor of all the terms, we should have, for x = 0, u = 0. 14 Let us now examine the change which takes place in the function, in consequence of any change that may be made in the value of the variable on which it depends. Let us take, as a first example, u=ax 2 , (1) and then suppose x to be increased by any quantity h. 16 ELEMENTS OP THE Designate by -uf the new value which u assumes, under this supposition, and we shall have u'=a(x+h) 2 , or, by developing, u' = ace 2 - + 2axh + ah 2 . If we subtract the first equation from the last, we shall have u' u 2axh + ah 2 ; hence, if the variable oc be increased by h, the function will be increased by 2axh + ah 2 . If both members of the last equation be divided by h, we shall have u/ u o r = 2ax + ah, h which expresses the ratio of the increment of the variable to that of the function. Let us take, as a second example, u = x*, (2) and suppose x to be increased by a quantity h; desig- nating by u' the new value which u assumes under this supposition, and we shall have u' = (x + h)*, and, by developing, u f = x 3 + 3x 2 h + 3z7i 2 + h\ By transposing x 3 , substituting for it its value u, and then "dividing by h, we have DIFFERENTIAL CALCULUS. 17 From equation ( 1 ) we have u' u = 2ax + ah ; h and from equation (2), h 15. Let us now observe that the numerator in the first member of each of the above equations, is the difference between the primitive function u, and the new value u' , which arose from giving an increment h to the variable x, of which u is a function. Hence we see, that the first member of each equation is equal to the increment of the function divided by the corresponding increment of the variable. If we examine the second members of these equations, we find a term in each which does not contain the in- crement h, viz. : in the first, the term 2ax, and in the second, 3x 2 . If now, we suppose h to diminish, it is evident that the terms 2ax, and 3x 2 , which do not contain h, will remain unchanged, while all the terms which contain h will diminish. Hence, the ratio u' u ~hT in either equation, will change with h, so long as h re- mains in the second member of the equation ; but of all the ratios which can subsist between u' -u ~T~' is there one which does not depend on the value of h? We have seen that as h diminishes, the ratio in the first 2 18 ELEMENTS OP THE equation approaches to 2ax, and in the second to 3z 2 ; hence, 2ax and 3x 2 , are the limits towards which the ratios approach in proportion a h is diminished ; and hence, each expresses that particular ratio which is in- dependent of the value of h. This ratio is called the limiting ratio of the increment of the variable to the corresponding increment of the function. 16. We are now to explain the notation by means of which this limiting ratio is to be expressed. For this purpose let us resume the equation h and represent by dx the last value of 7z, that is, the vi of h, which cannot be diminished, according to the law of change to which h or x is subjected, without becoming and let us also represent by du the corresponding dif- ference between w'and u; we then have dx. The letter d is used merely as a characteristic, and the expressions du, dx, are read, differential of u, differential of a:. It may be difficult to understand why the value which h assumes in passing to the limiting ratio, is represented by dx in the first member, and made equal to in the second. We have represented by dx the last value of h, and this value forms no appreciable part of h or x. For, if it did, it might be diminished without becoming 0, and therefore would not be the last value of h. By designa- ting this last value by dx, we preserve a trace of the DIFFERENTIAL CALCULUS. 19 letter #, and express at the same time the last change which takes place in h, as it becomes equal to 0. For a like reason the last difference between u! and u is desig- nated by du. The limiting ratio in equation (2) is ^ = 3**. dx The limiting ratio of the increment of the variable to that of the function, which has been found in the pre- ceding equations, is called the differential coefficient of M regarded as a function of x. 17. Let us take, as another example, the function u = ax* : (3) if we give to x an increment h, we shall have u' = ax* + 4ax^h + 6ao: 2 /i 2 -f- 4axh 3 + ah*, and r^ = 4a* 3 + Gax*h + *ah* + ah 3 , h and, by taking the limiting ratio, we have for the differ- ential coefficient, du . = 4aar*. dx 18. If it were required to find the differential of the function u, after we had formed its differential coefficient, it could be done by simply muliplying the differential coefficient by the differential of the variable ; thus, from equation (l) we should have du Zaxdx ; from the second, 20 ELEMENTS OF THE and from the third, du = The differential of each function may also be written under the following form : Eq. 1, --dx = 2axdx ; Eq. 2, du Eq. 3, -r-dx = 4ax 3 dx; which, indeed, is nothing more than finding the differen- tial of the function by multiplying the differential coefficient expressed in the first member of the equation, by the dif- ferential of the variable. 19. Let us now examine each of the three equations which we have considered, and observe the/orwi of the expression for the difference between the two states of the function ti. From the first equation we have u' u = 2axh + ah 2 ; from the second, and from the third, u' - u = 4.ax 3 h + 6ax*h z + 4a*7i 3 + a/* 4 . We see in each of the expressions for the difference between the two states of the function u, that the first term of the difference contains the first power of the in- crement /*, and that the coefficient of this term is the differential coefficient of the function u, or the limiting DIFFERENTIAL CALCULUS. 21 ratio of the increment of the function to that of the vari- able; This differential coefficient is, in general, a func tion of x. If, now, in either of the expressions, we represent the differential coefficient, or limiting ratio, by P, and all the following terms of the difference by P'h 2 (in which P will in general be a function of h), the difference may be written under the form u' -u = Ph + P'h a ; and we shall assume that what has been proved in regard to the three forms of the function u which we have con- sidered, is equally true for all other forms. This form, for the difference between the two states of the function, is important, and should be carefully remembered. If, then, we have a function of the form =/(*), and give to x an increment h f we shall have iS-u= Ph + P'h\ If, now, we wish the ratio of the increments, we have and, passing to the limiting ratio, and, if we wish the differential of the function w, we have du = Pdx, du, or -rdx = Pdx. dx If we represent the increment of the variable by k, and 22 ELEMENTS OF THE the differential coefficient by Q, the difference would be represented by u' - u = Qk + Q'k a du and fadx = Qdx. We may conclude from the above, that if we have the difference between two states of a function, as vf - u = Ph + P'h\ that we can immediately pass to the differential ofu,by writing du for u' u, substituting dxfor h in the second member, and omitting the terms which contain h a . 20. If two functions, u and v, dependent on the same variable, are equal to each other, for all possible values which may be attributed to that variable, the differentials of those functions will also be equal. For, suppose x to be the independent variable. We shall then have (Art. 15), v'-v = Qh + Q'h 2 , in which Q is the differential coefficient of v, regarded as a function of #. But, since w'and v' are, by hypothesis, equal to each other, as well as u and v, we have Ph + P'h 2 =Qh+ Q?V, or, by dividing by h and passing to the limiting ratio, P = Q, du dv hence ' 55 = 5? DIFFERENTIAL CALCULUS. 23 du dv and -j- dx -j-dx, dx dx that is, the differential of u is equal to the differential of v. 21. The reverse of the above proposition is not gene- rally true ; that is, if two differentials are equal to each other, we are not at liberty to conclude that the functions from which they were derived are also equal. For, let in which A is a constant, and u and v both functions of x. Giving to x an increment A, we shall have u' = v'A, from which subtract the primitive equation, and we ob- tain u' u =v / v, and, by substituting for the difference between the two states of the function, we have Ph + P'h 2 =Qh+Qfh 2 Dividing by A, and passing to the limiting ratio, we obtain du dv P = Q; that is -y- = -y- ; dx dx' du . dv _ hence, -y- ax= -r-aff ; dx dx or, what is the same thing, by merely changing the form, du = dv. Here we see that although v may be greater or less than u by the constant quantity A, still its differential will always be equal to that of u. 24 ELEMENTS OF THE Hence, also, we conclude that every constant quantity, connected with the variable by the sign plus or minus, will disappear in the differentiation. The reason of this is apparent; for, as a constant ad- mits of no increase or decrease, there is no ultimate or last difference between two of its values ; and this ulti- mate or last difference is the differential of a variable function. Hence the differential of a constant quantity is equal to 0. 22. If we have a function of the form u Av, in which u and v are both functions of x, and give to x an increment h, we shall have u' u A(v' v\ or Ph + P'tf = A(Qh+ Q'h z ) ; and, by dividing by h, and passing to the limiting ratio, we have P = AQ 9 or Pdx = AQdx. But Pdx = du, and Qdx = dv 9 hence, du Adv ; that is, the differential of the product of a variable quantity by a constant, is equal to the differential of the variable Multiplied by the constant. DIFFERENTIAL CALCULUS. 25 CHAPTER II. Differentiation of Algebraic Functions Succes- sive Differentials Taylor's and Madauriris Theorems. + 23. Algebraic functions are those which involve the sum or difference, the product or quotient, the roots or powers, of the variables. They may be divided into two classes, real and imaginary. 24. Let it be required to find the differential of the function. u = ax. If we give to x an increment A, and designate the second state of the function by u f , we shall have u f = ax + ah = u + ah, u'-u -JT = a: hence, du adx, or jdx = adx. dx 25. As a second example, let us take the function u = ax*. 26 ELEMENTS OF THE If we give to x an increment h, we have u' = ax* + 2 ahx -f ah 9 , hence, du = 2 ax dx. 26. For a third example, take the function u = ax 3 : giving to x an increment h, we have h or passing to the limit - 1 = 3 ao? 2 ; hence, du = 3 ax 2 dx. dx 27. Let us now suppose the function u to be composed of several variable terms : that is, of the form u y-\-z w f(x\ in which y, z, and w, are functions of x. If we give to x an increment h, we shall have _ y) + (/_ ^ ) - (,'_ W ) : hence, (Art. 19), or, /i or by passing to the limit -^ dx DIFFERENTIAL CALCULUS. 27 and multiplying both members by dx, we have ax But since P, Q, and L, are the differential coefficients of y, z, and to, regarded as functions of x, it follows (Art. 18) that, the differential of the sum, or difference of any number of functions, dependent on the same variable, is equal to the sum or difference of their differentials taken separately. 28. Let us now determine the differential of the product of two variable functions. If we designate the functions by u and v, and suppose them to depend on a variable x, we shall have and by multiplying wV = (u + Ph + P'h 2 ) (v + Qh + = uv + vPh + uQh + PQh 2 + &c ; hence u'v' uv - - - = vP + wQ-f terms containing h, h z , & h\ If now, we pass to the limiting ratio, we have d(uv) n , > --J. = vP + uQ; dx therefore, d(uv) = vPdx + uQdx = vdu + udv. Hence, the differential of the product of two functions dependent on the same variable, is equal to the sum of the 28 ELEMENTS OF THE products which arise by multiplying each by the differ ential of the other. If we divide by uv, we have d(uv) du dv uv u v ' that is, the differential of the product of two functions, di- vided by the product, is equal to the sum of the quotients which arise , by dividing each differential by its function. 29. We can easily determine from the last formula, the differential of the product of any number of functions. For this purpose, put v = ts, then dv _ d(ts) _ _dt_ ds ~v~ ' ts ~T ' ' ~s~' and by substituting for v in the last equation, we have d(uts) du . dt ds uts u t s and in a similar manner, we should find d(utsr ) = du j dt } ds ^ dr ^ & ^ utsr .... u t s r If in the equation d(uts) _ du dt ds uts ~ u t s ' we multiply by the denominator of the first member, we shall have d(uts) = tsdu + usdt + utds ; and hence, the differential of the product of any number of functions, is equal to the sum of the products which DIFFERENTIAL CALCULUS. arise by multiplying the differential of each function by the product of all the others. 30. To obtain the differential of any fraction, as we make = t. and hence u = tv. v Differentiating both members, we find du vdt + tdv ; finding the value of dt, and substituting for t its value u . . , we obtain v j. du udv at = , v v 2 or by reducing to a common denominator , vdiif udv ~~s~ hence, the differential of a fraction is equal to the deno- minator into the differential 'of the numerator, minus the numerator into the differential of the denominator, divided by the square of the denominator. 31. If the numerator u is constant in the fraction t = , v its differential will be (Art. 21), and we shall have , udv dt u dt= 2~, or -y-= 5. v 2 dv v 2 When u is constant, t is a decreasing function of v (Art. 5), and the differential coefficient of t is negative. This is only a particular case of a general proposition 30 ELEMENTS OP THE For, let u be a decreasing function of x. Then, if we give to x any increment, as h, we have or vf-u = Ph-t P f h\ But by hypothesis u>u f \ hence, the second member is essentially negative for all values of h ; and, passing to the limiting ratio, ^- _P dx~ hence, the differential coefficient of a decreasing function is negative. 32. To find the differential of any power of a function, let us first take the function w n , in which n is a positive and whole number. This function may be considered as composed of n factors each equal to u. Hence, (Art. 29), d(u n ) _ d(uuuu . . . . ) _ du du du du u n ~ (uuuu . . . . ) ~ u u u u But since there are n. equal factors in the first member, there will be n equal terms in the second J hence, d(u n ) _ ndu m u n ^ u ' therefore, d (u n ) = nu n ~ l du. If n is fractional, represent it by , and make o r^ v = u', whence, u r = v'; and since r and s are supposed to represent entire num- bers, we shall have ru r ~* du = sv'~ l dv ; DIFFERENTIAL CALCULUS. 31 from which we find dv - n/r ~ 1 du - rM< " 1 du su~* ( " or by reducing dv =' u' du; which is of the same form as the function V by substituting the exponent for n. s Finally, if n is negative, we shall have 1 from which we have (Art. 31), u hence, by reducing d(u- n )=-nu- n - l du. Hence, the differential of any power of a function, is equal to the exponent multiplied by the function with its primitive exponent minus unity, into the differential of the function. 33. Having frequent occasion to differentiate radicals of the second degree, we will give a specific rule for this class of functions. Let v = ^fu^ or v = w r ; 1 4~ , 1 -4 , du then, dv ~ u* du -^u *du = 32 ELEMENTS OF THE that is, the differential of a radical of the second degree, is equal to the differential of the quantity under the sign, divided by twice the radical. 34. It has been remarked (Art. 3), that in an equation of the form u = f(x), we may regard u as the function, and x as the variable, or x as the function, and u as the variable. We will now show that, the differential coefficient which is obtained by regarding u as a function of x, is equal to the recip- rocal of that which is obtained by regarding x as a func- tion of u. If we have and give to x an increment h, we have (Art. 19), u'-u^Ph + P'h 2 . (1) But, if x be expressed in u, and we have * =/()> and then give to u an increment k, we shall nave aS-x = h=:Qk + Q'k*. (2) But k = u' u. Substituting these values for u' u, and h, in equation (l), and we have k PQk + terms containing the higher powers of k. Dividing by k, and passing to the limiting ratio, we have 1=PQ, or P = . DIFFERENTIAL CALCULUS. 33 To illustrate this by an example, let j u = a?, whence x = tyu^= u 3 Now, -- dx but regarding x as the function dx 35. If we have three variables u, y, and #, which are mutually dependant on each other, the relations between them may be expressed by the equations u=f(y\ and y=f'(x). If now we attribute to x an increment h, and designate by k, the corresponding increment of y, we shall have (Art. 19), and If we* multiply these equations together, member by member, we shall have but li y' y; hence, by dividing and passing to the limiting ratio, we have du du dv -7- = T~ x 7 > dx ay dx and hence, if three quantities are mutually dependant on 3 34 ELEMENTS OF THE each other, the differential coefficient of the first regarded as a function of the third, will be equal to the differential coefficient of the first regarded as a function of the second, multiplied by the differential coefficient of the second re- garded as a function of the third. 36. Let us take as an example v = bur, u = oar. we find dv _ , 2 du du dx dv dv du But, -= -X-T- 36w 2 x 2ax = ax du dx and by substituting for w 2 , its value a 2 x* t dx ~ EXAMPLES. 1. Find the differential of u in the expression i^ Put a 2 a? = y, then w = y 2 , and the dependence be- tween u and x, is expressed by means of y, and u is an implicit function of x. Differentiating, we find du 1 - 1/ ~ d DIFFERENTIAL CALCULUS. 35 by multiplying the coefficients together we obtain du Ixo o.-i _ x hence, f? _ r 2. Find the differential of the function u = (a + bx n ) m . Place a + bx n = y: then u = y m ; and = m"" 1 = w a + for"" -^ = my"" 1 = w (a nence, du ~ j- = mnb (a + bx") a?"" 1 . du = mnb (a -f bx n ) x*~ l dx. 3. Find the differential of the function in which the operations in the last two terms are only indicated. If we perform them, we find d( x 2 ) xdx ^V 36 ELEMENTS OF THE Substituting these values, we find or, reducing to a common denominator and cancelling die like terms, _ 4. Find the differential of the function ~ _ - a?) - (a 2 - from which we find __ 5. Find the differential of the function Make y = ~= then we shall have DIFFERENTIAL CALCULUS. 37 we therefore have (Art. 32), 3 --i du = (a _ 3dy But from the equations above, we find , ,/ b \ d,Tj~x~ bdx d = d * - X - Substituting these values of i =r > &c ' dx dx dx But the differential of p is obtained by differentiating its value , regarding the denominator dx as con- dx stant; we therefore have du\ j d 2 u , d ' or ' - = ** and by substituting for dp its value, we have DIFFERENTIAL CALCULUS. 41 The notation d 2 u, indicates that the function u has been differentiated twice, and is read, second differential of u. The denominator da? expresses the square of the differential of x, and not the differential of a?. It is read, differential square of x, or differential of x squared. If we differentiate the value of q, we have d 3 u hence, j = r, &c., and in the same manner we may find The third differential coefficient - T -^, is read, third oar differential of u divided by dx cubed; and the differ- ential coefficients which succeed it, are read in a similar manner. Hence, the successive differential coefficients are du _ d 2 u d*u _ ~ "*** ~- from which we see, that each differential coefficient is deduced from the one which precedes it, in the same way that the first is deduced from the primitive function. 39. If we take a function of the form u = ax n , 42 ELEMENTS OP THE we shall have for the first differential coefficient, du ~ nax . dx If we now consider n, a, and dx, as constant, we shall have for the second differential coefficient 2? and for the third, and for the fourth, It is plain, that when n is a positive whole number, the function u = ax n , will have n differential coefficients. For, when n dif- ferentiations shall have been made, the exponent of x in the second member will be ; hence, the nth differential coefficient will be constant, and the succeeding ones will be equal to 0. Thus, n(n-l)(n-2)(n-3) ...... a.l, ax n + l and, DIFFERENTIAL CALCULUS. 43 Taylor s Theorem. 40. TAYLOR'S THEOREM explains the method of de- veloping into a series any function of the sum or difference of two variables that are independent of each other, ac- cording to the ascending powers of one of them. 41. Before giving the demonstration of this theorem, it will be necessary to prove a principle on which it de- pends, viz : if we have a function of the sum or difference of two variables u = f(xy\ the differential coefficient will be the same if we suppose x to vary and y to remain constant, as when we suppose y to vary and x to remain constant. For, make x y = z : we shall then have u=f(z) du and -JP- dz If we suppose y to remain constant and x to vary, we have dz = dx, and if we suppose x to remain constant and y to vary, we nave dz = dy. But since the differential coefficient p is independent of dz 1 (Art. 15), it will have the same value whether, dz = dx, or, dz = dy. 44 ELEMENTS OF THE To illustrate this principle by a particular example, let us take If we suppose x to vary and y to remain constant, we find du and if we suppose y to vary and x to remain constant, we find the same as under the first supposition. 42. It is evident that the must be expressed in terms of the two variables x and y, and of the constants which enter into the function. Let us then assume f(x + y) = A + By* + Cy> + .%' +, &c, in which the terms are arranged according to- the ascend- ing powers of y, and in which A, B, C, D, &c., are inde- pendent of y, but functions of x, and dependant on all the constants which enter the primitive furiction. It is now required to find such values for the exponents 0, 6, c, &c., and the coefficients A, B, C, D, &c., as shall ren- der the development true for all possible values which may be attributed to x and y. DIFFERENTIAL CALCULUS. 45 In the first place, there can be no negative exponents For, if any term were of the form - -;'; By-, '-.-... : it may be written .B_ y" and making y 0, this term would become infinite, and we should have /(*)=. which is absurd, since function of a?, which is independent of y, does not necessarily become infinite when y 0. The first term A, of the development, is the value which the primitive function assumes when' we make y 0. If we designate this value by u, we shall have /(*)=. If we make and differentiate, under the supposition that x varies and y remains constant, we shall have du' dA dB a dC b , dD -T- = -r--t--7-y+-7-y'4--7- dx dx djc dx dx and if we differentiate, regarding y as a variable and x as constant, we shall find - 1 +, &c. : But these differential coefficients are equal to each other (Art. 41); hence, the second members of the equations 46 ELEMENTS OF THE are equal, and since the coefficients of the series are independent of y, and the equality exists whatever be the value of y, it follows that the corresponding terms in each series will contain like powers of y, and that the coef- ficients of y in these terms will be equal (Alg. Art. 244). Hence, a 1 = 0, b l = a, c 1=6, &C., and consequently a = l, 6 = 2, c = 3, &c.; and comparing the coefficients, we find B = C= D= l dC dx ' 2 dx ' 3 dx And since we have made / we shall have jj du n (Pu n d?u /7'vi 1 O /"/'Y^ 1 ^ ^ fJ f*^ and consequently, . c?w d*u v 2 . d?u v 3 ' &c - 43. This theorem gives the following development for the function U n l (U i i\ n I S u = a? n , -3 = nx n , -j-f =n(n I )x n ~ l t &c. : dx dx* % DIFFERENTIAL CALCULUS. 47 hence, *>=(* + yY = x n + nrf-'y + fifedfi^i n(n-l)(n-2) y &c 1.2.3 44. The theorem of Taylor may also be applied to the development of the second state of any function of the form when x receives an arbitrary increment h, and becomes x + h. For, if we substitute h for y, we have du, d*u h 2 . ^-+ U" + U '"3+ &c - If we designate the modulus of the system of the loga- rithms by A, we shall have S If we now make x 0, we have U=0, U f = A, U"=-A, U m = 2A, &c.; hence, This series is not sufficiently converging, except in the case when a? is a very small fraction. To render the series more converging, substitute x for x : we then have DIFFERENTIAL CALCULUS. 61 and by subtracting the last series from the first, we obtain ; (1+ *)-/(i-*) = ;(g)= 2 A(!44 + &c.) If we make 1+07 Z i Z = 1 H -- , we have x = 1 x n and by observing that I we have from which we can find the logarithm of n + z when the logarithm of n is known. This series is similar to that found in Algebra, Art. 270. If we make n = 1, and z 1, we have ll = 0, and If we make the modulus A = 1 , the logarithm will be taken in the Naperian system, and we shall have I' 2 = 0.693147180, 2 L r 2 = l f 4 = 1.386294360; and by making z = 4, and n = 1 , we have V 5 =1.60943791 3, and J'2.+ V 5 = Z'10 ='2.302585093. 62 ELEMENTS OP THE If we now suppose the first logarithms to have been taken in the common system, of which the base is 10, we shall have, by recollecting, that the logarithms of the same number taken in two different systems are to each other as their moduli (Alg. Art. 267), Z10 : no : : A : 1, or, 1 : 2.302585093 : : A : 1 ; WhenC6 ' A= 2.30258509 =- 434284488 - Remark. To avoid the inconvenience of writing the modulus at each differentiation (Art. 56), the Naperian logarithms are generally used in the calculus, and when we wish to pass to the common system, we have merely to multiply by the modulus of the common system. We may then omit the accent, and designate the Naperian logarithm by Z. 59. Let us now apply these principles in differentiating logarithmic functions. 1. Let us take the function u = l Make and we shall have but DIFFERENTIAL CALCULUS. 63 whence, 2. Take the function u = ar- Vl-x and make i/l+ x+ -\/lx=y, -\/\-\-x *\/lx=z, which gives tl== /m = Zy_fe > and y z But we have dx , dx dx dx / /- - /^ \ dy= - -- : -- 7= = 7 -- ~ ( V 1 + x ~ V 1 g)> 2Vl + a? 2V1 a; 2V1 a^ v / d ^ zdx T dx dx T+x+ */l^x), Whence, dz zdx ydx and observing that y z +z z =4 : and yz = dx we have du = - xVl- 64 ELEMENTS OF THE (fe 4 ' w=: 5. w = 7 6. u l , a7 Va ad 60. Let us suppose that we have a function of the form Make Ix z, and we have u z n , du = nz*~ l dz, and substituting for z and dz their values, a? 61. Let us suppose that we have Make /a? = z, and we shall have, hence, 7 j dz , dx u = lz 9 du = , dz : z x dx U ~ xlx DIFFERENTIAL CALCULUS. 65 62. The rules for the differentiation of logarithmic func- tions are advantageously applied in the differentiation of complicated exponential functions. 1. Let us suppose that we have a function of the form = *, in which z and y are both variables. If we take the logarithms of both members, we have lu = ylz ; du j , dz hence, = dylz + y ; or, du = ulzdy + uy , or by substituting for u its value du = dz y z y lzdy -j- yz y ~ l dz. Hence, the differential of a function which is equal to a variable root raised to a power denoted by a variable exponent, is equal to the sum of the differentials which arise, by differentiating, first under the supposition that the root remains constant, and then under the sup position that the exponent remains constant (Arts. 55, and 32). 2. Let the function be of the form u=a*. Make, b* = y, and we shall then have (Art. 55), u = a y , du a y lady ; but dy = b*lbdx, hence, du = afb'lalbdx. 5 66 ELEMENTS OP THE 3. Let us take as a last example u = z<, in which z, t, and s, are variables. Make, t' = y, we shall then have u = z y , dw = z y Izdy + yz y ~ l dz. But dy = fltds + stf- W* ; hence, cfo = z f lz(fltds + **-'cfr) + Vaf~ l dz, Differentiation of Circular Functions, 63. Let us first find the differential of the sine of an arc. For this purpose we will assume the formulas (Trig. Art XIX), sin a cos b -f sin b cos a sin (a + b) = sin (a b) = s sin a cos 6 sin b cos a R If we subtract the second equation from the first, . , , x . , , N 2 sin 6 cos a sin (a + 6) sin (a b) = . and if we make a -f b = x + h, and a b = a?, we shall have 2sin Acos (x H sin (x + h)- sino? = DIFFERENTIAL CALCULUS. 67 and dividing both members by A, 2sin h cos (x-\ -- h] 2 \ 2 / sma? hR sin h cosfoH -- h\ If we now pass to the limit, the second factor of the rr\c *Y second member of the equation will become . R sin h 2 In relation to the first factor j- - its limit will be unity. * -r, Rsina sina cosa ror, tang a = - , whence - = ; cosa tanga R ' Now, since an arc is greater than its sine and less than its tangent* sina - , sina sina and - > a a tanga * The arc DB is greater than a straight line drawn from D to B, and consequently greater than the sine DE drawn perpendicular to JIB. The area of the sector J1BD is equal to - JIB X BD, and the area of the triangle J2BC is equal to -JIB X BC. But the sector is less A E B than the triangle being contained within it : hence, consequently, BD < BC. 00 ELEMENTS OF THE hence, the ratio of the sine divided by the arc is nearer unity than that of the sine divided by the tangent. But when we pass to the limit, by making the arc equal to 0, the sine divided by the tangent being equal to the cosine divided by the radius, is equal to unity : hence the limit of the ratio of the sine and arc, is unity. When therefore we pass to the limit by making h = 0, we find d sina? _ cos a? m ~dx~ ~~R~ cosxdx , , . hence, a sm x = K 64. Having found the differential of the sine, the diffe- rentials of the other functions of the arc are readily de- duced from it. cosa? = sin(90 a?), dcosx = dsin(90 a?), and by the last article, dsin(90 - x) = -^cos(90 - x)d(QQ - a?), K = -rrcos (90 x)dx : K 7 sin xdx hence, d cos x = = ; K the differential of the cosine in terms of the arc beii , negative, as it should be, since the cosine and arc axe decreasing functions of each other (Art. 31.) DIFFERENTIAL CALCULUS. 69 65. Since ihe versed sine of an arc is equal to radius minus the cosine, we have d T er-sin x = d (R - cos o?) = . R 66. Since tang x = R Sm x , we have (Art. 30), COS X 7 f R cos x d sin x R sin x d cos x a wng A = - _ - 2 C080? (cos 2 o?4- sin 2 o? but cos 2 o? + sin 2 o? = R 2 : hence, d tango? = , cos^o? 67. Since cota- = , we have tango? tang 2 o? tang 2 ^ cos 2 o? ' but, tang 2 a; R 2 dx COS0? hence, ^ cot a? = sm 2 o? which is negative, as it should be, since the cotangent is a decreasing function of the arc. 70 ELEMENTS OF THE R 2 68. Since seca? = - , we have cos x , R 2 d cosx R s'mxdx dsecx = -- - - = - - - : cos a? cos a? R sin x R 2 but, - = tango?, and - = seca?; cos a? cos a? 7 seca? hence, a sec a? = . R 2 69. Since cosec a? = - - , we have sin a? , R 2 d sin a? R cosxdx a cosec a? = -- - = -- - : coseca? cotxdx > hence, d cosec x = 70. If we make R = l, Arts. 63, 64, 65, 66, 67, will give, d sina?= cos a? da? (i), dcosx= s'mxdx (2), d ver sina? = sina?da? (3), dx (4), CM. cos a a? dx a cot. .7? BUT* The differential values of the secant and cosecant are omitted, being of little practical use. 71. In treating the circular functions, it is found to be most convenient to regard the arc as the function, and the DIFFERENTIAL CALCULUS. 71 sine, cosine, versed-sine, tangent, or cotangent, as the variable. If we designate the variable by u, we shall have in (Art. 63) sin x = u, and Rdu Rdu cosa? If we make coso? = w, we have (Art. 64), , Rdu Rdu sin x If we make ver-sina? = u, we have (Art. 65), Rdu dx = smx But, sina?= V-R 2 cos 2 #, and coso?=.R tt, therefore, cos 2 a? = R 2 2 Ru + u 2 , hence, sin a? = -\/2Ru u 2 , '' , Rdu and consequently, ax = , m V2Ru u 2 > If we make tang x = u, we have (Art. 66) cos^r R cos 2 x R 2 R 2 but = = - , hence R ~sec*' R 2 ~sec 2 o?~fl 2 +tangV R 2 du hence, - dx = 72 ELEMENTS OF THE Now, if we make R = l, the four last formulas become du ( i~ du 7P3T' Vl - u 2 ' rfr- du h Jw and these formulas being of frequent use, should be care- fully committed to memory. 72. The following notation has recently been introduced into the differential calculus, and it enables us to designate an arc by means of its functions. sin~ x w = the arc of which u is the sine, cos" 1 ^^ the arc of which u is the cosine, tang -1 M the arc of which u is the tangent, &c. &c. &c. If, for example, we have du = sin X then, dx = 1-w 2 73. We shall now add a few examples. 1 . Let us take a function of the form Make cos x z^ and then, u = z y , and (Art. 62); du = z y Izdy -h yz y ~ l dz: DIFFERENTIAL CALCULUS. 73 also, dz=. smxdx, and dy = cosxdx hence, du = z y (lz dy + dz \ = cosa? rinr Ucosa?cosa? -- }dx. \ cosa? / 2. Differentiate the function mdu x = sm mu. ax = , Vl-mV 3. Differentiate the function x = cos" 1 (u Vl u 2 J 4. Differentiate the function w , 2du 5 Differentiate the function x = sin" 1 (%u Vl w 2 ), v / 6. Differentiate the function _! a: ., ydx xdy l , du = y 74. We are enabled by means of Maclaurin's theorem and the differentials of the circular functions, to find the 74 ELEMENTS OF THE value of the principal functions of an arc in teims of the arc itself. Let u= f(x) = sin a?: then, du tPu d?u , > dx da? dx 3 = COSX > -j-r = + COS X. x dx 5 If we now render the differential coefficients independent of x, by making x = 0, we have (Art. 49), X 0? X 5 : - + 75. To develop the cosine in terms of the arc, make u = f(x) = cos x ; then, du . d 2 u d?u = sma?, -- cosa?, -Y-^-^sma?, c^a? c^r 2 dx 3 d*u d 5 u - = cosx, -7-j = sm a:, dx* dx 5 and rendering the coefficients independent of a?, we have 17=1, ^=0, V"=-l, U"'=0, hence, cosx= 1 -- + - &c. DIFFERENTIAL CALCULUS. 75 The last two formulas are very convenient in calculating the trigonometrical tables, and when the arc is small the series will converge rapidly. Having found the sine and cosine, the other functions of the arc may readily be calculated from them. 76. In the two last series we have found the values of the functions, sine and cosine, in terms of the arc. We may, if we please, find the value of the arc in terms of any of its functions. 77. The differential coefficient of the arc in terms of its sine, is (Art. 71), ^_ 1 2x4 ^-7T^- (1 " developing by the binomial formula, we find dx 1 l- 3 1.3.5 In passing from the function to the differential coeffi- cient, the exponent of the variable in each term which contains it, is diminished by unity ; and hence, the series which expresses the value of x in terms of u, will contain the uneven powers of u, or will be of the form Cu 5 + Du 7 + &c.; and the differential coefficient is ^ = A au 76 ELEMENTS OF THE But since the differential coefficients are equal to each other, we find, by comparing the series, ~2.3' *CO' -O^T7' hence, u I u 3 l.3u 5 1.3.5 7 ^p gin ^T/ U -\~ QC 1 2 3 2.4.5 2.4.6.7 If we take the arc of 30, of which the sine is (Trig. Art. XV), we shall have and by multiplying both members of the equation by 6, we obtain the length of the semi-circumference to the radius unity. 78. To express the arc in terms of its tangent, we have (Art. 71), which gives /Jv ^^1_ du hence the function x must be of the form ac = Au + Bit 3 and consequently = A du DIFFERENTIAL CALCULUS. 77 and by comparing the series, and substituting for A, B, C, &c., their values, we find u u? u 5 u 1 , tf^tang u = - + -j- +&c. If we make x = 45, u will be equal to 1 ; hence, arc 45 = 1 + + &c. 357 But this series is not sufficiently convergent to be used for computing the value of the arc. To find the value of the arc in a more converging series, we employ the following property of two arcs, viz. : Four times the arc whose tangent is , exceeds the 5 arc of 45 by the arc whose tangent is *. 239 * Let a represent the arc whose tangent is . Then (Trig. Art. XXVI), 2 tang a 5 2 tans; 2 a __ 120 ~ 1 tang 2 2 a = 119 ' The last number being greater than unity, shows that the arc 4 a ex- ceeds 45. Making the difference, 4 a 45 == A B = 6, will have for its tangent hence, four times the arc whose tangent is -, exceeds the arc of 45 by an arc whose tangent is - . / 239 78 But tang- hence, ELEMENTS OF THE -iJL_.l i i i ang 5 ~ 5 S.S^S.S 5 7.5 7 239 239 3(239) 3 5(239) 5 7(239) 7 arc 45 = \239 3(239) 3 5(239) 5 7(239) Multiplying by 4, we find the semi-circumference = 3.141592653. | 7 DIFFERENTIAL CALCULUS. 79 CHAPTER IV. Development of any Function of two Variables Differential of a Function of any number of Variables Implicit Functions Differential Equations of Curves Of Vanishing Fractions. 79. We have explained in Taylor's theorem the method of developing into a series any function of the sum or dif- ference of two variables. We now propose to give a general theorem of which that is a particular case, viz : To develop into a series any function of two or more variables, when each shall have received an increment) and to find the differential of the function. 80. Before making the development it will be necessary to explain a notation which has not yet been used. If we have a function of two variables, as u =/(*> y), we may suppose one to remain constant, and differentiate the function with respect to the other. Thus, if we suppose y to remain constant, and x to vary, the differential coefficient will be -/(* y); (i), 80 ELEMENTS OF THE and if we suppose x to remain constant and y to vary, the differential coefficient will be The differential coefficients which are obtained under these suppositions, are called partial differential coef- ficients. The first is the partial differential coefficient with respect to x, and the second with respect to y. 81. If we multiply both members of equation (1) by dx, and both members of equation (2) by dy, we obtain d -^dx=f'(x,y)dx, and '-dy = f"(x,y)dy. The expressions, du , du j dx, dy, dx ay " are called 'partial differentials; the first a partial diffe- rential with respect to x, and the second a partial diffe- rential with respect to y : hence, A partial differential coefficient is the differential co- efficient of a function of two or more variables, under the supposition that only one of them has changed its value : and, A partial differential is the differential of a function of two or more variables, under the supposition that only one of them has changed its value. 82. If we differentiate equation (1) under the suppo- sition that x remains constant and y varies, we shall have dy DIFFERENTIAL CALCULUS. 81 and since x and dx are constant _d(du) ~ which we designate by d*u d 2 u hence ' The first member of this equation expresses that the function u has been differentiated twice, once with respect to x, and once with respect to y. If we differentiate again, regarding x as the variable, we obtain d z u riv, N __ = />>, y ), which expresses that the function has been differentiated twice with respect to x and once with respect to y. And generally d n * m u indicates that the function u has been differentiated n -f m times, n times with respect to a?, and m times with respect to y. 83. Resuming the function if we suppose y to remain constant, and give to x an arbi- trary increment h, we shall have from the theorem of Taylor, duh , d 2 u h 2 , d 3 u h 3 /(* + **) = + _+^ + -^~+&c., 82 ELEMENTS OF THE , . , du d 2 u d 3 u mwh,ch, u, -, , are functions of x and y, and dependent on the constants which enter the f(x,y). If we now attribute to y an increment k, the function u t which depends on y, will become d 2 u k z d' 3 u k 3 and the function will become dx du ffu k d 3 u k 2 d*u k 3 dx dxdy 1 dxdy 2 l.2dxdy 3 1.2.3 and the function -y-r-, will become dor d?u k d*u k 2 d 5 u k 3 & 1.2.3 " and the function - will become d*u k d 5 u k 2 d?u A 3 & 1 " da?dy* 1.2 ^rfy 3 1.2.3 &c. &c. &c. &c. Substituting these values in the development of /(* + *, y), DIFFERENTIAL CALCULUS. 83 and arranging the terms, we have duk d?u k 2 tfu k 3 du h a?u hk (Pu htf ffu tf_ cPu tfk_ V ~ + + which is the general development of a function of two variables, when each has received an increment, in terms of the increments and differential coefficients. 84. If we transpose u =f(x, y) into the first member, and apply the result of Art. 19 to a function of two vari ables, we find y . , = . , ;il The differential of f(x, y) = du, which is obtained under the supposition that both the variables have changed their values, is called the total differential of the function. 85. If we have a function of three variables, as u = f(x, y, z), and suppose one of them, as z, to remain constant, and increments h and k to be attributed to the other two, the development of / (x + h, y + k, z) will be of the same form as the development of f(x-\-h, y + k) ; but u and all the differential coefficients will be functions of z 84 ELEMENTS OF THE If then an increment I be attributed to z, there will be four terms of the development of the form du , du , du 7 u > T* 1 , -T-/C, -j-l. ax ay dz If u were a function of four variables, as = /(* y> *> *)> there would be five terms of the form du, du, du 1 du u, T~"> T-ft* T~*> -=-i ; cfo dy dz (Art. 55), dz (Art. 32) dx hence, du &lzdy + yz y ~ l dz. Remark. In chapter II, the functions were supposed to depend on a common variable, and the differentials were obtained under this supposition. We now see that the dif ferentials are obtained in the same manner, when the func- tions are independent of each other, and unconnected with a common variable. 87. We have seen (Art. 39), that a function of a single variable has but one differential coefficient of the first order, one of the second, one of the third, &c. ; while 9 DIFFERENTIAL CALCULUS. 87 function of two variables has two differential coefficients of the first order, a function of three variables, three ; a function of four variables, four ; &c. It is now proposed to find the successive differentials of a function of two variables, and also the successive differential coefficients. We have already found , du , , du , du = - ax + ay. ax ay " du du But du du and since, -j- and -y- are functions of x and y, the dx dy differentials -j-y) = we suppose the variables x and y to change their values in succession, any change either in x or y, will produce a change in u : hence, u is a function of x and y when they vary in succession. The value, however, which u assumes, when a? or y varies, will reduce to when such a value is attributed to the other variable as will satisfy the equation X*,y) = o. ( We have from Art. 83, f(x + h, y + k) u = j-j+ terms containing A 2 , du -jyi + terms containing A 2 , plus other terms containing kh, and the higher powers of h and k. But, since y is a function of #, we have in which P is the differential coefficient of y regarded as a function of x. Substituting this value of k, and we have du (x + h, y -f k) u = j-Ph + terms containing h 2 , du -j-h + terms containing A 2 , plus other terms containing the higher powers of h. DIFFERENTIAL CALCULUS. 91 But, in consequence of the relation between y and a?, the first member of the equation will be constantly equal to 0. Hence, by the law of indeterminate coefficients (Alg, Art. 244), du du\ p +^=; v; ; ?'"',:! du hence , P = f = -^. dx du Ty Hence, the differential coefficient of y regarded as a function of x, is equal to the ratio of the partial differen- tial coefficients of u regarded as a function of x, and u regarded as a function ofy, taken with a contrary sign. Let us take, as an example, the equation du da then ' dx =2x > and Ty ==< * y: du. dx x dy hence ' du = --j- = te dy Although the differential coefficient of the first order is generally expressed in terms of x and y, yet y may be eliminated by means of the equation f(x,y) = 0, and the coefficient treated as a function of x alone. In the above equation we have 92 ELEMENTS OF THE dy x hence, = / dx VR 2 -x? .,. 92. If it be required to find the second differential coefficient, we have merely to differentiate the first diffe- rential coefficient, regarded as a function of x, and divide the result by dx. Thus, if we designate the first diffe- rential coefficient by p, the second by g, the third by r, &c., we shall have dp da ^JT_ . xy _i_ __ A* fVf* dx ' dx 93. To find the second differential coefficient in the equation of the circle, we have dy _ x dx y f , /dy \ _ ydx -f- xdy \dx)~~ v* hence ' and by substituting for -^ its value , we have dx y 1. Find the first differential coefficient of y, in the equation y 2 2mxy -f x 2 a 2 = u = 0, du du Zmy + 2x, = 2y dx dy DIFFERENTIAL CALCULUS. hence, ^ = - f oa? L 2 77107 J y mx 2. Find the first differential coefficient of y in the equation y 2 + 2xy + or 2 - a 2 = 0. ' 3. Find the first and second differential coefficients of y t in the equation 3 n, m = n, m < n, for under these suppositions, respectively, it takes the form P(x-a} m - n J> P ~~3~ Q' Q(* -a)-"' Let the numerator of the proposed fraction be desig- nated by X, and the denominator by X f , and let us sup- pose an arbitrary increment h to be given to x. The numerator and denominator will then become a function of x -f- h, and we shall have from the theorem of Taylor . dX h #X A 2 cPX h 3 . dX'h _ __ V dx T~ h "3^~1.2 + da* 1.2.3^ If the value of x a, reduces to the differential coefficients in the numerator as far as the mth order, and those of the denominator as far as the wth order, the value of the fraction will become, d m X h m dx m 1.2.3.4... . o dx n 1.2.3.4....7Z If we make h = Q, the value of the fraction will be- come 0, finite, or infinite according as m > rc, m = n, m therefore, the value of the fraction when x =1, is + n. 2. Find the value of the fraction ax 2 2 acx + bx 2 - r , when a? = = 2ax-2ac, dx dx both of which become 0, when x c. Differentiating again, we have _ x'~ ' x hence, the true value of the fraction when x = c is -7- o DIFFERENTIAL CALCULUS. 101 3. Find the value of the fraction x 3 aa? a 2 x + a 3 - 2 > when x = a. Ans. 0. 4. Find the value of the fraction -, when x = a. Ans. oo. 5. Find the value of a* b* - , when x = 0. x Ans. la Ib. 6. What is the value of the fraction 1 sin 07 + cos x -r , when x 90. sin 07 + cos a? 1 Ans. 1. 7. What is the value of the fraction a x ala + alx - 7 _, when x = a. a- V2ax~x 2 Ans. 1. 8. What is the value of the fraction x '-, when x = 1 . 1 x -f Ix Ans. 2. 9. What is the value of the fraction a n x n 7 7-, when x a. la Ix Ans. ia n . 102 ELEMENTS OF THE 104. It has been remarked (Art. 47), that the theorem of Taylor does not apply to the case in which a particular value attributed to oc, renders any differential coefficient of the iunction infinite. Such functions are of the form (x-a f ' in which m and n are fractional. In functions of this form we substitute for a?, a + h, which gives a second state of the function. We then divide the numerator and denominator by h raised to a power denoted by the smallest exponent of h, after which we make h = 0, and find the ratio of the terms of the fraction. When we place a -f h for a?, we have, in arranging according to the ascending powers of 7^, F(a + h) _ Ah" + Bh b + Ch e + &c., F(a + A) ~ A'h a> + Bh b> + Clf + &c. Now there are three cases, viz. : when a > a! , a a f j a < a 1 '. In the first case, the value of the fraction will be ; in the second, a finite quantity ; and in the third it will be infinite. 105. In substituting a-\-h for a?, in the fraction (x-a)" DIFFERENTIAL CALCULUS. 103 , (2aft + ft 2 ) 2 , . M {- we have v *F = (2a + ft) 2 , ft T and by making ft = 0, which renders x = a the value of the fraction becomes 3 2. Required the value of the fraction * (a?-3ax+2a 2 ) 3 l . L when x = (a-'-o'F Substituting a -+- ft for a?, we have ft T (3a 2 + 3aft + ft 2 ) 2 (3a 2 + 3aft+ ft 2 ) 2 which is equal to 0, when ft = 0. 106. Remark. The last method of finding the value of a vanishing fraction, may frequently be employed advan- tageously, even when the value can be found by the theorem of Taylor. 107. There are several forms of indetermination under which a function may appear, but they can all be reduced to the form . 1st. Suppose the numerator and denominator of the fraction JT to become infinite by the supposition of sc = a. The fraction can be placed under the form 104 ELEMENTS OF THE which reduces to , when X and X f are infinite. 2d. We may have the product of two factors, one of which becomes and the other infinite, when a particular value is given to the variable. In the product PQ, let us suppose that x=a, makes P = and Q = oo . We would then write the product under the form, P "Q" which becomes when x = a. 108. Let us take, as an example, the function (l-tf)tang *-o?; in which T designates 180. If we make x 1, the first factor becomes 0, and the second infinite. But 1 1 tang -**?=:: ; ^ p n f ~prp COL ^TTX hence, (1 - #)tang *x = =^- , cot *x 2 the value of which is when x = 1 . DIFFERENTIAL CALCULUS. 105 CHAPTER V. Of the Maxima and Minima of a Function of a Singh Variable. 109. If we have * = /(*), the value of the function u may be changed in two ways : first, by increasing the variable x ; and secondly, by dimin- ishing it. If we designate by u' the first value which u assumes when x is increased, and by u" the first value which u assumes when x is diminished, we shall have three con- secutive values of the function !/, u, u ff . Now, when u is greater than both u r and u rf , u is said to be a maximum : and when u is less than both u' and u", it is said to be a minimum. Hence, the maximum value of a variable function is greater than the value which immediately precedes, or the value that immediately follows : and the minimum value of a variable function is less than the value which imme- diately precedes or the value that immediately folloivs. 110. Let us now determine the analytical conditions which characterize the maximum and minimum values of a variable function. 106 ELEMENTS OF THE If in the function = /(*), the variable x be first increased by h, and then diminished by 7i, we shall have (Art. 44), du h d 2 u h 2 . d?u h 3 T __ h d 2 u h 2 d?u h 3 and consequently, , du h d*u h 2 (Pu h 3 - w++ du h ffiu h 2 tfu h 3 - --- Now, if u has a maximum value, it will be greater than u' or u" ; and hence, u' u and u" u will both be negative. If u is a minimum, it will be less than u' or u" 9 and hence, u' u and u" u will both be positive. Hence, in order that u may have a maximum or a minimum value, the signs of the two developments must be both minus or both plus. But since the terms involving the first power of h, in the two developments, liave contrary signs, and since so small a value may be assigned to h as to make the first term in each development greater than the sum of all the other terms (Art. 44), it follows that u can have neither a maximum nor a minimum, unless dx DIFFERENTIAL CALCULUS. 107 and the roots of this equation will give all the values of x which can render the function u either a maximum or a minimum. Having made the first differential coefficient equal to 0, the signs of the developments will depend on the sign of second differential coefficient. But since the signs of the first members of the equa- tions, and consequently of the developments, are both negative when u is a maximum, and both positive when u is a minimum, it follows that the second differential co- efficient will be negative when the function is a maximum, and positive when it is a minimum. Hence, the roots of the equation being substituted in the second differential coefficient, will render it negative in case of a maximum, and positive in case of a minimum; and since there may be more than one value of x which will satisfy these conditions, it follows that there may be more than one maximum or one minimum. But if the roots of the equation reduce the second differential coefficient to 0, the signs of the developments will depend on the signs of the terms which involve the third differential coefficient ; and these signs being different, there can neither be a maxi- mum nor a minimum, unless the values of x also reduce the third differential coefficient to 0. When this is the case, substitute the roots of the equation ]08 ELEMENTS OF THE in the fourth differential coefficient ; if it becomes negative there will be a maximum, if positive a minimum. If the values of x reduce the fourth differential coefficient to 0, the following differential coefficient must be examined. Hence, in order to find the values of x which will render the proposed function a maximum or a minimum. 1st. Find the roots of the equation ^ = 0. dx 2d. Substitute these roots in the succeeding differential coefficients, until one is found which does not reduce to 0. Then, if the differential coefficient so found be of an odd order, the values of x will not render the function either a maximum or a minimum. But if it be of an even order, and negative, the function will be a maximum ; if positive, a minimum. 111. Remark. Before applying the preceding rules to examples, it may be well to remark, that if a variable function is multiplied or divided by a constant quantity, the same values of the variable which render the function a maximum or a minimum, will also make the product or quotient a maximum or a minimum, and hence the con- stant will not affect the conditions of maximum or mini- mum. 2. Any value of the variable which will render the* function a maximum or a minimum, will also render any root or power a maximum or a minimum ; and hence, if a function is under a radical, the radical may be omitted. DIFFERENTIAL CALCULUS. 109 EXAMPLES. 1. To find the value of x which will render y a maxi- mum or a minimum in the equation of the circle dy x ' making -- = 0, gives x = 0. The second differential coefficient is da? and since making x = 0, gives y R, we have dx* ~ R which being negative, the value of x = renders y a maximum. 2. Find the values of x which will render y a maximum or a minimum in the equation, y = a bx + tf 3 , differentiating, we find ^=-6 + 2*, and f| = 2 , ax dor making, b + 2x = Q, gives d? = ~; 2 and since the second differential coefficient is positive, this value of x will render y a minimum. The minimum 110 ELEMENTS OF THE value of y is found by substituting the value of a?, in the primitive equation. It is 3. Find the value of x which will render the function u = a 4 + tfx - c V, a maximum or a minimum, du ,, _ , b 3 fo = b ~ 2CX ' X = -2 + * : . I 4(T 4. Let us take the function we find -=- = 9 V 6 4 , and a? = =b dx 3a The second differential coefficient is d z u 2 = 18 A Substituting the plus root of a?, we have DIFFERENTIAL CALCULUS. * 111 which gives a minimum, and substituting the negative root, we have which gives a maximum. The minimum value of the function is, 9a and the maximum value 112. Remark. It frequently happens that the value of the first differential coefficient may be decomposed into two factors, X and X ', each containing x, and one of them, X for example, reducing to for that value of a?, which renders the function a maximum or a minimum. When the differential coefficient of the first order takes this form, the general method of finding the second diffe- rential coefficient may be much simplified. For, if du we shall have dx dx But by hypothesis X reduces to for that value of x which renders the function u a maximum or a minimum : d 2 u X f dX 112 ELEMENTS OF THE from which we obtain the following rule for finding the second differential coefficient. Differentiate that factor of the first differential coef- ficient which reduces to 0, multiply it by the other factor, and divide the product by dx. 5. To divide a quantity into two such parts that the mih power of one of the parts multiplied by the Tzth power of the other shall be a maximum or a minimum. Designate the given quantity by a and one of the parts by a?, then will a x represent the other part. Let the product of their powers be designated by u ; we shall then have u = x m (a #) n , whence, -r- = mx m ~ l (a x) n nx m (a x) n ~\ (J'OC = (ma mx nx)x m ~ l (a a?)"" 1 , and by placing each of the factors equal to 0, we have ma m + The second differential coefficient corresponding to the first of these values, found by the method just explained, is and substituting for x its value, it becomes (m + n) m-f-n 3 hence, this value of x renders the product a maximum. The two other values of x satisfy the equation of the DIFFERENTIAL CALCULUS. 113 problem, but do not satisfy the enunciation, since they are not parts of the given quantity a. Remark. If m and n are each equal to unity, the quan- tity will be divided into equal parts. 6. To determine the conditions which will render y a maximum or a minimum in the equation y 2 2mxy + a? a 2 = Q. The first differential coefficient is dy _ my x f dx y mx ' x hence, my x = 0, or y = . m Substituting this value of y in the given equation, we find ma Vl-m 2 ' and the value of y corresponding to this value of x is a y = To determine whether y is a maximum or a minimum. let us pass to the second differential coefficient. We have ^y. A hence, d>y = \fo~ ) dx 2 y mx 8 114 ELEMENTS OF THE and since ~ = 0, we have dx 1 dot? y mx* and by substituting for y and x their values, we have hence, y is a maximum. 7. To find the maximum rectangle which can be in- scribed in a given triangle. Let b denote the base of the triangle, h the altitude, y the base of the rectangle, and x the altitude. Then, u=ixy = the area of the rectangle. But b : h : : y : h x: bh bx hence, y = r - , and consequently, bhx bx z b ,, ,,v ti = - - - = (hx,-a?). h h and omitting the constant factor, du ^ h = h-2x, or x = ; dx 2 hence, the altitude of the rectangle is equal to half the altitude of the triangle : and since dx* the area is a maximum. DIFFERENTIAL CALCULUS. 115 8. What is the altitude of a cylinder inscribed in a given cone, when the solidity of the cylinder is a maxi- mum ? Suppose the cylinder to be inscribed, as in the figure, and let AB = a, BC = b, AD = x, ED = y ; EJ[~ then, BD = a x = altitude of the cylinder, and n y 2 (a x) = solidity .;. (i) From the similar triangles and JlCB, we have bx x : y : : a : b ; whence y . Substituting this value in equation (l), and we have v = a - x). f) Omitting the constant factor -> we may write u = x 2 (a x) ; tor the conditions which will make u a maximum will also make v a maximum (Art. 111). By differentiating, we have du d z u -r- = 2ax 3x 2 , and j = 2a 6x. dx ay? Placing 2ax 3x z = 0, we have x = 0, and x = -a. . * 3 Hence the altitude of the maximum cylinder is one-third the altitude of the cone ELEMENTS OF THE 9. What are the sides of the maximum rectangle in- scribed in a given circle ? Ans. Each equal to R V%. 10. A cylindrical vessel is to contain a given quantity of water. Required the relation between the diameter of the base and the altitude in order that the interior surface may be a minimum. Ans. Altitude = radius of base. 11. To find the altitude of a cone inscribed in a given sphere, which shall render the convex surface of the cone a maximum. . Ans. Altitude = R. 12. To find the maximum right-angled triangle which can be described on a given line. Ans. When the two ides are equal. 13. What is the length of the axis of the maximum parabola that can be cut from a given right cone ? Ans. Three-fourths the side of the cone. 14. To find the least triangle which can be formed by the radii produced, and a tangent line to the quadrant of a given circle. Ans. When the point of contact is at the middle of the arc. 15. What is the altitude of the maximum cylinder which can be inscribed in a given paraboloid ? Ans. Half the axis of the paraboloid DIFFERENTIAL CALCULUS. 116* CHAPTER VI. Application of the Differential Calculus to the Theory of Curves. 113. Every relation between a function y and a van- able x, expressed by the equation y =/(*), will subsist between the ordinate and abscissa of a curve Let Ji be the origin of the rectangular axes ; then if in the equation , \C we make x = 0, we have y a constant : lay off JIB equal to this con- slant. Then attribute values to a?, from to any limit, as well T A H negative as positive, and find from the equation the corresponding values of y. Conceive the values of x to be laid off on the axis of abscissas, and the values of y on the corresponding ordinates. The curve described through the extremities of the ordinates will have for its equation y=f( x \ (0 Let x represent any abscissa, J1H for example, and y the corresponding ordinate HP. If now we give to x any arbitrary increment h, and make HF h, the value of y will become equal to FC, which we will designate by y'. Through P draw the secant CPI and the tangent TP. 116 s ELEMENTS OF THE Now, y'-y=CF-PH= CD, CD T 2 ' = Pi) ~ tan s ent of tlie an s le CPD- CD PH y>-y CD and ~ But, by similar triangles Now, the limiting ratio of the increment of the variable to that of the function, is that ratio which is independent of the value of /z, and is obtained by making h equal to in the expression for the ratio of the increments (Art. 15.) It is evident that as h diminishes, the point ' C will ap- proach the point P, the point / will approach T, and the secant 1C will approach the tangent TP ; and when h becomes equal to 0, the secant 1C will coincide with the tangent TP. For every position of C we shall, have C 1 T) PTT ~pj)~~j^f = tangent CPD = tangent CIH ; .and when C coincides with P, ~W = ~ == tangent PTH -; that is, the limiting ratio, or first differential coefficient, is equal to the tangent of the angle which the tangent line makes with the axis of abscissas. Of Tangents and Normals. 114. Having found the value of dy dx we will now proceed to find the value of the subtangent, tangent, subnormal, and normal. T A R N DIFFERENTIAL CALCULUS. 117 We have (Trig. Th. II), 1 : TR :: tangT : RP; that is, 1 : TR :: -j- : y dx hence, TR y sub-tangent. 115. The tangent TP is equal to the square root of the sum of the squares of TR and RP ; hence, TP = y V 1 + j-a = tangent. 116. From the similar triangles TPR, RPN, we have TR : PR :: PR : RN, (i T hence, y : y : : y : RN, consequently, RN = y-= sub-normal. 1 17. The normal PN is equal to the square root of the sum of the squares of PR and RN ; hence, = y\/\ +-TZ = normal. 118. Let it be now required to apply these formulas to lines of the se.cond order, of which the general equation (An. Geom. Bk. VI, Prop. XII, Sch. 3), is, y 2 mx -f nx 2 . Differentiating, we have dy _m-\- 2nx _ m 118 ELEMENTS OF THE substituting this value, we find , . D dx sub-tangent TR = y-~ = y dy . 2nx - = A/ mx + fta? 2 + 4 mx iin: ! T i r> AT dy m -\-2nx sub-normal RN y-Z = - , y dx 2 PN = y\J \ + % =y nx 2 -f- (m By attributing proper values to m and ??, the above formulas will become applicable to each of the conic sections. In the case of the parabola, n = 0, and we have W=f- PN = \/ mx + m 2 . 119. It is often necessary to represent the tangent and normal lines by their equations. To determine these, in a general manner, it will be necessary first to consider the analytical conditions which render any two curves tangent to each other. Let the two curves, PDC, PEC, intersect each other at P and C. Designate the co-ordinates of the first curve by x and y, and the co-ordinates of the second by a/, y'. Then, for the common point P y we shall have x ~ a/, y = y f . DIFFERENTIAL CALCULUS. ] 19 ff we represent BG, the increment of the abscissa, by h, we shall have, from the theorem of Taylor (Art. 44), <*--* + hence, by placing the two members equal to each other, and, dividing by h, we have cfy J 2 y h dy' d?y' h + + &c " = - If we now pass to the limit, by making ^ = 0, we shall have dy dy' fa = ~djT ; in which case the point C will become consecutive with P, and the curve PEC tangent to the curve PDC. Hence, two lines will be tangent to each other, when they have a common point, and the first differential coefficient of the one equal to the first differential coefficient of the other, for this point. 120. The equation of a straight line is of the form y = ax + b, dy hence, ^ = a. But the equation of a straight line passing through a given point, of which the co-ordinates are a? /x , y", is (An. Geom. Bk. II, Prop. IV), y-y"=a(x-x"). 120 ELEMENTS OF THE . But if the point whose co-ordinates are a?", y'', is required to be on a given curve, these co-ordinates must satisfy the equation of that curve. If the straight line is required to be tangent to the curve at this particular point, the first differential coefficient --^, found from the equation doc du" of the curve, must take the particular value -g / ; that is, . G/Ou we must have dy_dy^ dx ~ da? and the equation of the line tangent at the point whose co-ordinates are a/', y", will be 121. Let it be required, for example, to make the line tangent to the circumference of a circle at a point of which the co-ordinates are x", y". The equation of the circle is x 2 + y 2 = R 2 ; dy x and, by differentiating, we have = -- . But if the straight line is to be tangent to the circle, at the point whose co-ordinates are #", y", we must have dx" ~ dx~~ y y // and by substituting this value in the equation of the line, and recollecting that sc" 2 + y" 2 = R 2 , we have yy" + xx" = R 2 , which is the equation of a tangent line to a circle. 122. A normal line is perpendicular to the tangent at DIFFERENTIAL CALCULUS. 121 the point of contact, and since the equation of the tangent is of the form the equation of the normal, at the point whose co-ordin- ates are oc", y", will be of the form (An. Geom. Bk. II., Prop. VII., Sch. 2), y If we take the equation of any curve, and find the value of - for the particular point whose co-ordinates ay are a/', y", and then substitute thac value in the above equation, we shall have the equation of the normal pas- sing through this point. The equation of th*e normal in the circle will take the form 123. To find the equation of a tangent line to an ellipse at a point of which the co-ordinates are #", y", we have By differentiating, we have hence, we have l-rf' *-*'), or and for the normal y y" = iSuyX* ~~ x " 122 ELEMENTS OF THE 124. To find the equation of a tangent to lines of the second order, of which the equation for a particular point (An. Geom. Bk. VI, Prop. XII, Sch. 3) is By differentiating, we have d 2y' f hence, the equation of the tangent to a line of the second order is ,, m + 2na/' , /A y-y"= 8y// (*-"). and the equation of the normal y-y"=- 2y Of Asymptotes of Curves. 125. An asymptote of a curve is a line which continually approaches the curve, and becomes tangent to it at an infinite distance from the origin of co-ordinates. Let AX and AY be the co-ordinate axes, and the equation of any tan gent line, as TP. R T DIFFERENTIAL CALCULUS. 123 If in the equation of the tangent, we make in succes- sion y 0, x 0, we shall find If the curve CPB has an asymptote RE, it is plain that the tangent PT will approach the asymptote RE, when the point of contact P, is moved along the curve from the origin of co-ordinates, and T and D will also approach the points R and Y, and will coincide with them when the co-ordinates of the point of tangency are infinite. Jn order, therefore, to determine if a curve have asymp- totes, we substitute in the values of AT and AD, the co- ordinates of the point which is at an infinite distance from the origin of co-ordinates. If either of the distances AT, AD, become finite, the curve will have an asymptote. If both the values are finite, the asymptote will be in- clined to both the co-ordinate axes : if one of the distances becomes finite and the other infinite, the asymptote will be parallel to one of the co-ordinate axes ; and if they both become 0, the asymptote will pass through the origin of co-ordinates. In the last case, we shall know but one point of the asymptote, but its direction may be deter- minedly finding the value of -^, under the supposition that the co-ordinates are infinite. 126. Let us now examine the equation \f = mx + no?, 124 ELEMENTS OF THE of lines of the second order, and see if these lines have asymptotes. We find A n _ which may be put under the forms and making x oo , we have AR=- and If now we make n = 0, the curve becomes a parabola, and both the limits, AR, AE, become infinite : hence, the parabola has no rectilinear asymptote. If we make n negative, the curve becomes an ellipse, and AE becomes imaginary: hence, the ellipse has no asymptote. But if we make n positive, the equation becomes that of the hyperbola, and both the values, AR f AE, become B 2 finite. If we substitute for n its value ^ we ?hall have A. AR= -A, and AE = B. DIFFERENTIAL CALCULUS. 125 Differentials of tJie Arcs and Areas of Segments of Curves. 7. It is plain, that the chord and arc of a curve will approach each other continually as the arc is diminished, and hence, we might conclude that the limit of their ratio is unity. But as several propositions depend on this rela- tion between the arc and chord, we shall demonstrate it rigorously. 128. If we suppose the ordi- nate PR of the curve, POM to be a function of the abscissa, we shall have (Art. 19), : h. and N Ji in which = -f- dx Hence, PM= Vh 2 +(P+P / h) 2 tf=hVl+(P+P'h?. We also have NQ = Ph ; hence, PN=Vh 2 + P 2 h 2 = NM = NQ-MQ=- P'tf- hence, we have PN + MN hVl+F 2 -P'h 2 PM hVl+(P + P'h) 2 Vl + (P + J26 ELEMENTS OF THE of which tho limit, by making h 0, is IT But the arc POM can never be less than the chord PM, nor greater than the broken line PNM which contains it ; hence, the limit of the ratio POM, ~M~ and consequently, the differential of the arc is equal to the differential of the chord. If we designate the arc by z, PM will be represented by z / z, and we shall have "h PM PQ~ PM 7, and, by passing to the limiting ratio, or z that is, the differential of the arc of a curve, at any point, is equal to the square root of the sum of the squares of the differentials of the co-ordinates. } 29. To determine the differential of the arc of a circle of which the equation is xdx we have xdx + ydy = 0, or ay = -- whence, dz = DIFFERENTIAL CALCULUS. 127 Rdx _ Rdx i_ 1 7 y the same as determined in (Art 71). The plus sign is to be used when the abscissa x and the arc are increasing functions of each other, and the minus sign when they are decreasing functions (Art. 31). 130. Let BCD be any segment of a curve, and let it be required to find the differential of its area. The two rectangles DCFE, B / DOME, having the same base DE, are to each other as DC to EM; and hence, the limit of their ratio is equal to the limit of the ratio of DC to EM, which is equal to unity. But the curvelinear area DCME is less than the rect- angle DOME, and greater than the rectangle DCFE : hence, the limit of its ratio to either of them will be unity. But, DCME DCME DEFC DCME <* Tji v DE DE DEFC " DEFC' or by representing the area of the segment by s and the ordinate DC by y, and passing to the limit, we have ds 7/7 =r y, or ds = yax ; hence, the differential of the area of a segment of any curve, is equal to the ordinate into the differential of the abscissa. 128 ELEMENTS OF THE 131. To find the differential of the area of a circular segment, we have a? -{- y 1 = jR 2 , and y = VR 2 a? ; hence, ds = dx VR 2 a?. The differential of the segment of an ellipse, is and of the segment of a parabola ds dx V2px. Signification of the Differential Coefficients. 132. It has already been shown that, if the ordinate of a curve be regarded as a function of the abscissa, the first differential coefficient will be equal to the tangent of the angle which the tangent line forms with the axis of abscis- sas (Art. 1 13). We now propose to show the signification of the second differential coefficient, the ordinate being re- garded as a function of the abscissa. Let AP be the abscissa and PM the ordinate of a curve. From P lay off on the axis of abscissas PP' = h, and PP" = 2h. Draw the ordinates PM, P'M.P"M"; also the lines MMN, MM"; and lastly, MQ, M'Q', parallel to the P' P" DIFFERENTIAL CALCULUS. 129 axis of abscissas. Then will M'Q = NQ f , and we shall have PM=y, - M'Q 1.2 &c. Now, since the sign of the first member of the equation is essentially positive, the sign of the second member will also be positive (Alg. Art. 85). But by diminishing h, the sign of the second member will depend on that of the second differential coefficient (Art. 44) : hence, the second differential coefficient is positive. If the curve is below the axis of abscissas, the ordinates will be negative, and it is easily seen that we shall then have 130 ELEMENTS OF THE Now, since the first metnber is negative, the second member will be negative : hence we conclude that, if a curve is convex towards the axis of abscissas, the ordi- nate and second differential coefficient will have like signs. N 133. Let us now con- sider the curve CMM'M" , which is concave towards the axis of abscissas. We shall have, p*n=v, M M p, - P'M = - dx M' f Q!- M f Q = - NM'= dor dor 1 . 2 &c. But since the first member of the equation is negative, the essential sign of the second member will also be negative : hence, the second differential coefficient will be negative. DIFFERENTIAL CALCULUS. 131 P' If the curve is below the axis of abscissas, the ordi- nate will be negative, and it is easily seen that we should then have M"Q!- M f Q = + NM" = oar &c. ; hence we conclude that, if a curve is concave towards the axis of abscissas, the ordinate and second differential coefficient will have contrary signs. The ordinate will be considered as positive, unless the contrary is mentioned. 134. Remark 1. The co-ordinates x and y, determine a single point in a curve, as M. The differential of y is derived from the ordinate PM, and is what QM' becomes when the ordinates P'Jtf' and PM become consecutive. The second differential of y is derived from JWPQ, in the same way that dy is derived from the primitive func- tion y. ^It is, indeed, what JVf'Q' becomes, when M"Q' becomes consecutive with JV/'Q. The abscissa x being supposed to increase uniformly, the difference between PP' and P'P" is : and therefore the second differential of x is 0. The co-ordinates x and y, and the first and second differentials determine three points, M, M', M", consecutive with each other. 135. Remark 2. When the curve is convex towards 132 ELEMENTS OF THE the fcxis of abscissa, the first differential coefficient, which represents the tangent of the angle formed by the tangent line with the axis of abscissas, is an increasing function of the abscissa : hence, its differential coefficient, that is, the second differential coefficient of the function, ought to be positive (Art. 31). When the curve is concave, the first differential coeffi- cient is a decreasing function of the abscissa ; hence, the second differential coefficient should be negative (Art. 31 ). Examination of the Singular Points of Curves. 136. A singular point of a curve is one which is dis tinguished by some particular property not enjoyed by the points of the curve in general. Let us, as a first example, find the points of a curve, through which the tangent lines will be parallel or per- pendicular to the axis of abscissa* 137. Since the first differential coefficient expresses the value of the tangent of the angle which the tangent line forms with the axis of abscissas, and since the tangent is 0, when the angle is 0, and infinite when the angle is 90, it follows that the roots of the equation will give the abscissas of all the points at which the tan- gent is parallel to the axis of abscissas, and the roots of the equation dy dx -2- = oo , or = 0, dx dy DIFFERENTIAL CALCULUS. 133 will give the abscissas of all the points at which the tan- gent is perpendicular to the axis of abscissas. 138. If a curve from being convex towards the axis of abscissas becomes concave, or from being concave becomes convex, the point at which the change of curvature takes place is called a point of inflexion. Since the ordinate and differential coefficient of the second order have the same sign when the curve is convex towards the axis of abscissas, and contrary signs when it is concave, it follows that at the point of inflexion, the second differential coefficient will change its sign. There fore between the positive and negative values there will be one value of x which will reduce the second differential coefficient to or infinity (Alg. Art. 310) : hence the roots of the equations will give the abscissas of the points of inflexion. 139. Let us now apply these principles in discussing the equation of the circle We have, by differentiating, dy _ x dx~ ~y and placing /yi -- = 0, we have x = 0. y Substituting this value in the equation of the curve, we have y = R ; 134 ELEMENTS OF THE hence, the tangent is parallel to the axis of abscissas at the two points where the axis of ordinates intersects the circumference. If we make dy x y JL = -- = QD or - -i- = 0, dx y x we have y ; substituting this value in the equation, we find x= R, and hence, the tangent is perpendicular to the axis of abscissas at the points where the axis intersects the cir- cumference. The second differential coefficient is equal to which will be negative when y is positive, and posrtive when y is negative. Hence, the circumference of the circle is concave towards the axis of abscissas. If we apply a similar analysis to the equation of the ellipse, we shall find the tangents parallel to the axis of abscissas at the extremities of one axis, and perpendicular to it at the extremities of the other, and the curve concave towards its axes. 140. Let us now discuss a class of curves, which may be represented by the equation y = bc(x a) m , in which we suppose c To be positive or negative, and different values to be attributed to the exponent m. DIFFERENTIAL CALCULUS. 135 1st. When c is positive, and m entire and even. By differentiating, we have dy = r _, dx m > ' - dor If we place the value -~ = 0, we find x = a, and sub- dx stituting this value in the equation of the curve, we find hence, x a, y 6, are the co-ordinates of the point at which the tangent line is parallel to the axis of abscissas. Since m is even, m 2 will also be even, and hence the second V differential coefficient will be posi- tive for all values of x. The curve will therefore be convex towards the axis of X, and there will be no point of inflexion. The value of x a renders the ordinate y a minimum, since after m differentiations a differential coefficient of an even order becomes constant and positive (Art. 110). The curve does not intersect the axis of X, but cuts the axis of Y at a distance from the origin expressed by y = b -f- ca m . 136 ELEMENTS OF THE 141. 2d. When c is negative, and m entire and even. We shall have, by differentiating, y = b e(xa) m . dx and The discussion is the same as before, excepting that the second differential coefficient being nega- tive for all values of x, the curve is concave towards the axis of abscissas, and the value of x = a, renders the ordinate y a maxi- mum (Art. 110). 142. 3d. When c is plus or minus, and m entire and uneven. We shall have, by differentiating, dy , \nT_i --._ mc(x-a) , and The first differential coefficient will be 0, when x a , hence, the tangent will be parallel to the axis of abscissas, at the point of which. the co-ordinates are x = a, y = b DIFFERENTIAL CALCULUS. 137 Since tfie exponent m 2 is uneven, the factor (x a) m ~ 2 will be negative when # < a, and positive when # > a; hence, this factor changes its sign at the point of the curve of which the _ abscissa is x = a. If c is positive, the second differential coefficient will be negative for #a: hence there will be an inflexion when x = a. If c were negative, the curve would be first convex and then concave towards the axis of abscissas, but there would still be an inflexion at the point x a. At this point the tangent line separates the two branches of the curve. There will, in this case, be neither a maximum nor a minimum, since after m differentiations a differential coef- ficient of an odd order, will become equal to a constant quantity (Art. 110). 143. 4th. When c is positive or negative, and m a o fraction having an even numerator, as m = . o By differentiating, and supposing c positive, we have dy 2 , ,4-i 2c -J = ir c(*-a)' = - - -p 3(x a) 3 d* 2c If we make x = a, the first differential coefficient will become infinite ; and the tangent will be perpendicular to 138 ELEMENTS OF THE the axis of abscissas, at the point of which the co-ordinates are x = a, y = b. In regard to the second differen- tial coefficient, it will become infi- nite for x = a, and negative for every other value of x, since the factor (x a) of the denominator is raised to a power denoted by an even exponent. Hence, the curve will be concave towards the axis of abscissas. If we take the equation of the curve y = b + c(x a) 3 , and make x = a + h, and x = a h, we shall have, m either case, and hence, y will be less for x = a, than for any other value of x, either greater or less than a. Hence, the value x = a, renders y a minimum. If c were negative, the equation would be of the form j \ / . i and we should have, by differentiating, dy _ 2c dx~~ T and 2c da? DIFFERENTIAL CALCULUS. 139 The first and second differen- tial coefficients will be infinite for x = a, and the second differential coefficient will be positive for all values of x greater or less than a; and hence, the curve will be con- vex towards the axis of abscissas. If, in the equation of the curve y = b c(x a] we make x either case, a + h, and x a A, we shall have, in y = b c Jfi ; and hence, y will be greater for x a, than for any other value of x either greater or less than a. Hence, the value x = , renders y a maximum. 144. Remark. The conditions of a maximum or a minimum deduced in Art. 110, were established by means of the theorem of Taylor. Now, the case in which the function changes its form by a particular value attri- buted to x, was excluded in the demonstration of that theorem (Art. 45). Hence, the conditions of minimum and maximum deduced in the two last cases, ought not to have appeared among the general conditions of Art. 110. We therefore see that there are two species of maxima and minima, the one ckaracterized by = 0, the .other by dx J dx 140 ELEMENTS OP THE In the first, we determine whether the function is a maximum or a minimum by examining the subsequent differential coefficient ; and in the second, by examining the value of the function before and after that value of x which renders the first differential coefficient infinite. The branches DE, ME, which are both represented by the equation. y = b =fc c(x a) 7 , are not considered as parts of a continuous curve. For, the general relations between y and x which determine each of the parts DE, ME, is entirely broken at the point My where x~a. The two parts are therefore regarded as separate branches which unite at M. The point of union is called a cusp, or a cusp point. 145. 5th. When c is positive or negative and m a 3 fraction having an even denominator, as m =-j. Under this supposition the equation of the curve will become and by differentiating, we have d 3c _ dX 4(07-G)T and ** 8 . 4.4(*-)T DIFFERENTIAL CALCULUS. 141 The curve represented by this eouation will have two branches : the one corresponding to the plus sign will be concave towards the axis of abscissas, and the one cor- responding to the minus sign will be convex. Every value of x less than a will render y imaginary. The co-ordinates of the point M, are x = a, y b. 146. 6th. When c is positive or negative and m a fraction having an uneven numerator and an uneven de- nominator, as m= . 5 Under this supposition the equation will become j \ / * and by differentiating, we have dy _ 3c dx-~7, T' 3.2c 5.5(x-a) 5 from which we see that if we use the superior sign of the first equation, the curve will be convex towards the axis of abscissas for x < a, that there will be a point of inflexion for x a, and that the curve will be concave for x > a. If the lower sign be employed, the first branch will become concave, and the other convex. 147. The cusps, which have been -considered, were formed by the union of two curves that were convex to- 142 ELEMENTS OF THE wards each other, and such are called, cusps of the first order. It frequently happens, however, that the curves which unite, embrace each other. The equation furnishes an example of this kind. By extracting the square root of both members and transposing, we have and by differentiating dx 2 We see by examining the equations, that the curve has two branches, both of which pass through the origin of co-ordinates. The upper branch, which corres- ponds to the plus sign, is constantly convex towards the axis of abscissas, while the lower branch is convex for . 64 64 \ , , and concave for and x < 1 . At s 225' 225 the last point the curve passes below the axis of abscissas and becomes convex towards it. If we make the first dif ferential coefficient equal to 0, we shall find x = 0, and substituting this value in the equation of the curve, gives y = ; and hence, the axis of abscissas is tangent to both branches of the curve at the origin of co-ordinates. Al this point the differential coefficient of the second ordei is positive for both branches of the curve, hence the\ DIFFERENTIAL CALCULUS. 143 are both convex towards the axis. When the cusp is formed by the union of two curves which, at the point of contact, lie on the same side of the common tangent, it is called a cusp of the second order. 148. Let us, as another example, discuss the curve whose equation is y = b(x a) *Jx c. By differentiating, we obtain x a We see, from the equa- tion of the curve, that y will be imaginary for all values ol x less than c. For x=c, we have y=b; and for x > c, we have two values of y and conse- quently two branches of the curve, until x = a when they unite at the point M. For x > a there will be two real values of y and conse- quently two branches of the curve. The point M, at which the branches intersect each other, is called a mul- tiple point, and differs from a cusp by being a point of intersection instead of a point of tangency. At the multiple point M there are two tangents, one to each branch of the curve. The one makes an angle with the axis of abscissas, whose tangent is 10 144 ELEMENTS OF THE the other, an angle whose tangent is 149. Besides the cusps and multiple points which have already been discussed, there are sometimes other points lying entirely without the curve, and having no connexion with it, excepting that their co-ordinates will satisfy the equation of the curve. For example, the equation ay 2 x 3 + feo: 2 = 0, will be satisfied for the values a? = 0, y=Q-, and hence, the origin of co-ordinates A, satisfies the equation of the curve, and enjoys the property of a multiple point, since it is the point of union of two values of a?, and two values of y. If we resolve the equation with respect to y, we find y = and hence, y will be imaginary for all negative values of a?, and for all positive values between the limits x = and x = b. For all positive values of x greater than b, the values of y will be real. The first differential coefficient is dy_ DIFFERENTIAL CALCULUS. 145 or by dividing by the common factor a?, dy_ 3x 2b dx 2Va(x b) and making x = 0, there results dy_ 2b ~ ~ which is imaginary, as it should be, since there is no poir. of the curve which is consecutive with the isolated or con- jugate point. The differential coefficients of the higher orders are also imaginary at the conjugate points. * '% "" 150. We may draw the following conclusion s^from the preceding discussion. 1st. The equation -j = 0, determines the points at which the tangents are parallel to the axis of abscissas. 2d. The equation -~ oo , determines the points of u>x the curve at which the tangents are perpendicular to the axis of abscissas. The two last equations also determine the cusps, if there are any, in all cases where the tangent at the cusps is parallel or perpendicular to the axis of abscissas. 3d. The equation T2 = Qy or ~J = determines the points of inflexion. 4th. The equation -j- = an imaginary constant, i dec dicates a conjugate point. 146 ELEMENTS OP THE CHAPTER VII. Of Osculatory Curves Of Evoiutes. 151. Let PT be tangent to the curve ABP at the point P, and PN a normal at the same point : then will PT be tangent to the circumference of every circle passing through P, and having its centre in the 'normal PN. It is plain that the cen- tre of a circle may be taken at some point C, so near to P, that the cir- cumference shall fall with- in the curve APB, and then every circumference described with a less ra- dius, will fall entirely within the curve. It is also apparent, that the centre may be taken at some point C', so remote from P, that the circumference shall fall between the curve APB and the tangent PT, and then every circumference described with a greater radius will fall without the curve. Hence, there are two classes of tangent circles which may be described; the one lying within the curve, and the other without it. DIFFERENTIAL CALCULUS. 147 ACE 152. Let there be three curves, APB, CPD, EPF, which have a common tan- gent TP, and a com- mon normal PN ; then will they be tangent to each other at the point P. It does not follow, however, from this cir- cumstance, that each curve will have an equal tendency to coincide with the tangent TP, nor does it follow that any two of the curves CPD, EPF, will have an equal ten dency to coincide with the first curve APB. It is now proposed to establish the analytical conditions which determine the tendency of curves to coincide with each other, or with a common tangent. Designate the co-ordinates of the first curve APB by x and y, the co-ordinates of the second CPD by a/, y f , and the co-ordinates of the third EPF by x", y" . If we designate the common ordinate PR by y, y' ', y" , we shall then have w , , dy h d 2 y h 2 d?y h 3 , dy' h d 2 y f h 2 = dx" 1 da/' 2 1. 2 da/' 3 1. 2. 3 But since the curves are tangent to each other at the point P, we have (Art. 119), 148 ELEMENTS OF THE *> d = = : h..., Now, in order that the first curve AP.Z? shall approach more nearly to the second CPD than to the third EPF, we must have and consequently, in which we have represented the coefficients in the first series by A, B, C, &c., and the coefficients in the second by A', B', a, &c. Now, the limit of the first member of the inequality will always be less than the limit of the second, when its first term involves a higher power of h than the first term of the second. For, if A = 0, the first member will involve the highest power of h, and we shall have and by dividing by h 2 . and by passing to the limit DIFFERENTIAL CALCULUS 149 ' '''* But when A = 0, we have and hence, when three curves have a common ordinate, the first will approach nearer to the second than to the third, if the number of equal differential coefficients between the first and second is greater than that between the first and third. And consequently, if the first and second curves have m + 1 differential coefficients which are equal to each other, and the first and third curves only m equal dif- rential coefficients, the first curve will approach more nearly to the second than to the third. Hence it appears, that the order of contact of two curves will depend on the number of corresponding differential coefficients which are equal to each other. The contact which results from an equality between the co-ordinates and the first differential coefficients, is called a contact of the first order, or a simple tangency (Art. 119). If the second differential coefficients are also equal to each other, it is called a contact of the second order. If the first three differential coefficients are respectively equal to each other, it is a contact of the third order; and if there are m differential coefficients respectively equal to each other, it is a contact of the mth order. 153. Let us now suppose that the second line is only given in species, and that values may be attributed at pleasure to the constants which enter its equation. We 150 ELEMENTS OF THE shall then be able to establish between the first and second lines as many conditions as there are constants in the equation of the second line. If, for example, the equation of the second line contains two constants, two conditions can be established, viz. : an equality between the co- ordinates, and an equality between the first differential coefficients ; this will give a contact of the first order. If the equation of the second curve contains three con stants, three conditions may be established, viz. : an equality between the co-ordinates, and an equality between the first and second differential coefficients. This will give a con- tact of the second order. If there are four constants, we can obtain a contact of the third order ; and if there are m -f- 1 constants, a contact of the mih order. It is plain, that in each of the foregoing cases the highest order of contact is determined. The line ivhich has a higher order of contact with a given curve than can be found for any other line of the same species, is called an osculatrix. Let it be required, for example, to find a straight line which shall be oscillatory to a curve, at a given point of which the co-ordinates are a/ f , y" . The equation of the right line is of the form y = ax + b, and it is required to find such values for the constants d and b as to cause the line to fulfil the conditions, x = af', y=y", and = - DIFFERENTIAL CALCULUS. 151 By differentiating the equation of the line, we have dy ~ = a; ax and since the line passes through the point of osculation Substituting for -~ its value -A/-, we have CLOC CLCU y ~y if, ( x * )> for the equation of the osculatrix. In the equation of the circle dy x dy" x" we find ~ = -- =-/= -- - dx y dx" y" hence, the equation of the osculatrix of the first order, to the circle, is or by reducing yy" ' -\- xx 1 ' '= R 2 . 154. If * and /3 represent the co-ordinates of the centre of a circle, its equation will be of the form 11 this equation be twice differentiated, we shall have, J 52 ELEMENTS OF v THE and by combining the three equations, we obtain, ^ dxtfy If it be now required to make this circle osculatory to a given curve, at a point of which the co-ordinates are a/', y ;/ , we have only to substitute in the three last equations, the values of )dy -(x )d*-(y- P)dp = RdR, da? + dy 2 + (y P)d?y d*dx dftdy = 0. Combining these with equations (2) and (3), we obtain _ ( y _ fidfi -(x-)d* = RdR, (4) decdx dftdy = 0. The last equation gives But equation (2) may be placed under the form dx . . which represents a normal to the involute (Art. 122), and which becomes, by substituting for its value -^-, 160 ELEMENTS OF THE dp . N t x y P = -r(x *\ (6) or jB - y = ^( - ar) (Art. 120). (Ml This last equation, which is but another form for the equation of the normal to the involute, is, in fact, the equation of a tangent line to the evolute, at the point whose co-ordinates are a and/3; hence, a normal line to the involute curve is tangent to the evolute. 168. It is now proposed to show, that the radius of cur- vature and the evolute curve have equal differentials Combining equations (2) and (5) we obtain <*$M ,h ( x -.) = ( y-t)*L, (7 ) .. .A,... , or by squaring both members, combining this last with equation (1) we have |jj!] fiWJtfVtf-*. (8) Combining equations (4) and (7), we have or DIFFERENTIAL CALCULUS. 161 or by squaring both members Dividing this last by equation (8), member by member we have or dR = Vd* + dp. But if s represents the arc of the evolute curve, of which the co-ordinates are * and , we shall have (Art. 128), ds= hence, dR = ds ; that is, the differential of the radius of curvature is equal to the differential of the arc of the evolute. 169. It does not follow, however, from the last equation, that the radius of curvature is equal to the arc of the evolute curve, but only that one of them is equal to the other plus or minus a constant (Art. 22). Hence, is the form of the equation which expresses the relation between them. 162 ELEMENTS OF THE If we determine the radii of curvature at two points of the involute, as P and P f , we shall have, for the first, and for the second hence, and hence, the difference between the radii of curvature at any two points of the involute is equal to the part of the evolute curve intercepted between them. 170. The value of the constant a will depend on the position of the point from which the arc of the evolute curve is estimated. If, for example, we take the radius of curvature for lines of the second order, and estimate the arc of the evolute curve from the point at which it meets the axis, the value of s will be when R = m (Art. 163): hence we shall have or a = i and for any other point of the curve J_ 2 DIFFERENTIAL CALCULUS. 163 Either of the evolutes, FE, FE f , F'E', or F'E, corres- ponding to one quarter of the ellipse, is equal to (Art. 169) A 2 2 B A ' 171 . The evolute curve takes its name from the connexion which it has with the corres ponding involute. Let CC 7 C /7 be an evolute curve. At C draw a tan- gent AC, and make it equal to the constant a in the equa- tion Wrap a thread ACC 7 C 7/ around the curve, and fasten it at any point, as C 7/ . Then, if we begin at A, and unwrap or evolve the thread, it will take the positions PC 7 , P 7 C 77 , &c., and the point A will describe the involute APP 7 : for PC 7 -AC=CC 7 and P 7 C /7 - AC= CC 7 C 77 , &c. . . . 172. The equation of the evolute may be readily found by combining the equations dy(da?+dif) dxcPy ~' _ with the equation of the involute curve. 164 ELEMENTS OF THE 1st. Find, from the equation of the involute, the values of ~ and c^y, dx and substitute them in the two last equations, and there will be obtained two new equations involving *, , x and y. 2d. Combine these equations with the equation of the involute, and eliminate x and y : the resulting equation will contain *, /3, and constants, and will be the equation of the e volute curve. 173. Let us take, as an example, the common parabola of which the equation is y 2 = 77107. We shall then have dy _ m m 2 da? '' "" and hence _ 4y 3 /4y 2 + ^ 2 \ _ 4y 3 + ^ 2 y _ 4y "~5?A 4y 2 )- ~rf~ ~^ and by observing that the value of x is equal to that of y /3 multiplied by ---> we nave hence we have, and a? *= -- f -- : m 2 DIFFERENTIAL CALCULUS- 165 substituting for y its value in the equation of the involute y we obtain m --; and by eliminating a?, we have 16 27m' -> which is the equation of the evolute. If we make ft = 0, we have J_ ~ 2 ' and hence, the evolute meets the axis of abscissas at a distance from the origin equal to half the param- eter. If the origin of co-ordinates be transferred from A to this point, we shall have and consequently 21m The equation of the curve shows that it is symmetrical with respect to the axis of abscissas, and that it does not extend in the direction of the negative values of <*! . The evolute CC r corresponds to the part AP of the involute, and CC lf to the part AP' . 166 ELEMENTS OF THE CHAPTER VIII. Of Transcendental Curves. Of Tangent Planes and Normal Lines to Surfaces. 174. Curves may be divided into two general classes . 1st. Those whose equations are purely algebraic ; and 2dly. Those whose equations involve transcendental quantities. The first class are called algebraic curves, and the second, transcendental curves. The properties of the first class having been already examined, it only remains to discuss the properties of the transcendental curves. Of the Logarithmic Curve. 175. The logarithmic curve takes its name from the property that, when referred to rectangular axes, one of the co-ordinates is equal to the logarithm of the other. If we suppose the logarithms to be estimated in paral- lels to the axis of Y, and the corresponding numbers to be laid off on the axis of abscissas, the equation of the curve will be y =: Ix. DIFFERENTIAL CALCULUS. 176. If we designate the base of a system of loga- rithms by a, we shall have, (Alg. Art. 241) a y x\ and if we change the value of the base a to a', we shall have a fy = x. It is plain, that the same value of a?, in the two equations, will give different values of y, and hence, every system of logarithms will give a different logarithmic curve. If we make y 0, we shall have (Alg. Art. 257) x =. 1 ; and this relation being independent of the base of the system of logarithms, it follows, that every logarithmic curve will intersect the axis of numbers at a distance from the origin equal to unity. The equation a y = a?. will enable us to describe the curve by points, even with- out the aid of a table of logarithms. For, if we make we shall find, for the corresponding values of x, x = a -y/a, x = <\/a &c. a? 177. If we suppose the base of the system of logarithms to be greater than unity, the logarithms of all numbers less 168 | ELEMENTS OF THE than unity will be negative ( Alg. Art. 256) ; and therefore, the values of y corresponding to the abscissas, between the limits a? and x = AE = l, will be negative. Hence, these ordinates are laid off below the axis of abscissas. When x 0,y will be infinite and negative (Alg. Art. 264). If we make x negative, the conditions of the equa- tion cannot be fulfilled ; and hence, the curve does not extend on the side of the negative abscissas. 178. Let us resume the equation of the curve y = Ix. 11 we represent the modulus of the system of logarithms by Aj and differentiate, we obtain (Art. 56), , A dx dy = A-, dy A or /- = ' ax x But represents the tangent of the angle which the CLCC tangent line forms with the axis of abscissas : hence, the tangent will be parallel to the axis of abscissas when x oo , and perpendicular to it when x = 0. But when x = 0, y oo ; hence, the axis of ordinates is an asymptote to the curve. The tangent which is parallel to the axis of X is not an asymptote : for when x oo , we also have y oo . 179. The most remarkable property of this curve be longs to its sub-tangent FR', estimated on the axis of logarithms. We have found, for the sub-tangent, on the axis of X (Art. 114), DIFFERENTIAL CALCULUS. 169 and by simply changing the axes, we have dx hence, the sub-tangent is equal to the modulus of the system of logarithms from which the curve is constructed. In the Naperian system M = l, and hence the sub-tangent will be equal to 1 = AE. Of the Cycloid. G , B A N 180. If a circle NPG be rolled along a straight line AL, any point of the circumference will describe a curve, which is called a cycloid. The circle NPG is called the generating circle, and P the generating point. It is plain, that in each revolution of the generating circle an equal curve will be described ; and hence, it will only be necessary to examine the properties of the curve APBL, described in one revolution of the generating circle. We shall therefore refer only to this part when speaking of the cycloid. 181. If we suppose the- point P to be on the line AL at A y it will oe found at some point, as L 9 after all the 170 ELEMENTS OF THE A R N M points of the circumference shall have been brought in contact with the line AL. The line AL will be equal to the circumference of the generating circle, and is called the base of the cycloid. The line J3M, drawn perpen dicular to the base at the middle point, is equal to the diameter of the generating circle, and is called the axis of the cycloid. 182. To find the equation of the cycloid, let us assume the point A as the origin of co-ordinates, and let us sup- pose that the- generating point has described the arc A P. If N designates the point at which the generating circle touches the base, AN will be equal to the arc NP. Through N draw the diameter NG, which will be perpendicular to the base. Through P draw PR perpen- dicular to the base, and PQ parallel to it. Then, PR = NQ will be the versed-sine, and PQ the sine of the arc NP. Let us make ON = r, we shall then have = NQ=y, -y*, x = AN-RN=a.icNP-PQ: hence, the transcendental equation is x = ver-sin" 1 ^ V% ry y 2 . DIFFERENTIAL CALCULUS. 171 183. The properties of the cycloid are, however, most easily deduced from its differential equation, which is readily found by differentiating both members of the trans- scendental equation. We have (Art. 71), r/( ' - 1 ^- rd y V~2~ry-y 2 ' rdy ydy V2ry-y 2 ' hence, ^. rd !/ or y 2 which is the differential equation of the cycloid. 184. If we substitute in the general equations of (Arts. 114, 115, 116, 117), the values of dx, dy> deduced from the differential equation of the cycloid, we shall obtain the values of the normal, sub-normal, tangent, and sub-tangent. They are, normal PN = V%ry, sub-normal RN = V^ry y 2 , su b-tangent TR= These values are easily constructed, in consequence of their connexion with the parts of the generating circle. The sub-normal RN, for example, is equal to PQ of the generating circle, since each is equal to -y/2ry y 2 : hence, the normal PN and the diameter GN intersect the base of the cycloid at the same point. 172 ELEMENTS OP THE Now, since the tangent to the cycloid at the point P is perpendicular to the normal, it must coincide with the chord PG of the generating circle. If, therefore, it be required to draw a normal or a tan- gent to the cycloid, at any point as P, draw any line, as ng, perpendicular to the base AL, and make it equal to the diameter of the generating circle. On ng describe a semi-circumference, and through P draw a parallel to the base of the cycloid. Through p, where the parallel cuts the semi-circumference, draw the supplementary chords pn, pg, and then draw through P the parallels PN, PG, and PN will be a normal, and PG a, tangent to the cycloid at the point P. 185. Let us resume the differential equation of the cycloid which may be put under the form dx y y If we make y = 0, we shall have dy_ dx~ and if we make y = 2r, we shall have DIFFERENTIAL CALCULUS. 173 hence, the tangent lines drawn to the cycloid at the points where the curve meets the base, are perpendicular to the base; and the tangent drawn through the extremity of the greatest ordinate, is parallel to the base. 186. If we differentiate the equation =, V2ry y z regarding dx as qonstant, we obtain or by reducing and dividing by y, whence we obtain and hence the cycloid is concave towards the axis of abscissas (Art. 133). 187. To find the evolute of the cycloid, let us first sub- stitute in the general value of dxtfy the value of d*y found in the last article : we shall then have hence, the radius of curvature corresponding to the ex- tremity of any ordinate y, is equal to double the normal. 174 ELEMENTS OF THE The radius of curvature is when y = 0, and equal tc twice the diameter of the generating circle for y = 2r: hence, the length of the e volute curve from A to A' is equal to twice the diameter of the generating circle. Substituting the value of cPy in the values of y /3, x * (Art. 172), we obtain y P = hence we have x * 2 V 2r/3 /3 2 . Substituting these values of y and x in the transcen- dental equation of the cycloid, we have which is the transcendental equation of the evolute, re- ferred to the primitive origin and the primitive axes. Let us now trans- fer the origin of co- ordinates to the point A', and change at the same time the direction of the posi- tive abscissas : that is, instead of estima- ting them from the left to the right, we will estimate them from the rigi to the left. Let us designate the co-ordinates of tl-j evolute, referred to the new axes A! M, A f X f , by *' and f.f DIFFERENTIAL CALCULUS. 175 Since A'X' = AM = the semi-circumference of the gene- rating circle, which is equal to m, we shall have, for the abscissa A r R r of any point P x , A f R f ct! = rir *, hence, * = r* f : and for the ordinate, we shall have R f P'= p' = R'E - P'E = 2r- (- /3) = 2r + ft, hence, ft = 2r + ft', or ft = 2r /a'. Substituting these values of * and ft in the transcen dental equation of the evolute, we obtain TV a! ver-sin" 1 (2r ft f ) - or af=r* ver-sin" 1 (2 r ft') V2 rfi' ft' 2 . But the arc whose versed-sine is 2r ft f , is the supple ment of the arc whose versed-sine is ft', hence a! ver-sin -1 & -\/2rp f p' 2 , which is the equation of the evolute referred to the new origin and new axes. But this equation is of the same form, and involves the same constants as that of the involute : hence, the evolute and involute are equal curves. Of Spirals. 188. A spiral is a curve described by a point which moves along a right line, according to any law whatever, the line having at the same time a uniform angular motion. 12 176 ELEMENTS OP THE Let A B C be a straight line which is to be turned uniformly around the point A. When the motion of the line be- gins, let us suppose a point to move from A along the line in the direction ABC. When the line takes the posi- tion ADE the point will have moved along it to some point as D, and will have described the arc AaD of the spiral. When the line takes the position AD'E r the point will have described the curve AaDD f , and when the line shall have comple- ted an entire revolution the point will have described the curve AaDD'B. The point A, about which the right line moves, is called the pole ; the distances AD, AD', AB, are called radius-vectors, and if the revolutions of the radius-vector are continued, the generating point will describe an in- definite spiral. The parts AaDD'B, BFF'C, described in each revolution, are called spires. 189. If with the pole as a centre, and AB, the distance passed over by the generating point in the direction of the radius-vector during the first revolution, as a radius, we describe the circumference BEE', the angular motion of the radius-vector about the pole A, may be measured by the arcs ot this circle, estimated from B. If we designate the radius-vector by u, and the measur- ing arc, estimated from B, by t, the relation between u X DIFFERENTIAL CALCULUS. 177 and t, may in general be expressed by the equation u = at n , in which n depends on the law according to which the generating point moves along the radius-vector, and a on the relation which exists between a given value of u and the corresponding value of t. 190. When n is positive the spirals represented by the equation u = at n , will pass through the pole A. For, if we make t == 0, we shall have u 0. But if n is negative, the equation will become , or U =JT' in which we shall have u = CD for t = 0, and u = for t=co: hence, in this class of spirals, the first position of the generating point is at an infinite distance from the pole : the point will then approach the pole as the radius-vector revolves, and will only reach it after an infinite number of revolutions. 191. If we make n = 1, the equation of the spiral be- comes u = at. If we designate two different radius-vectors by u 1 and u", and the corresponding arcs by tf and if', we shall have u' = at f , and u" = at", 178 ELEMENTS OF THE and consequently . . t r . . . i . that is, the radius-vectors are proportional to the measur- ing arcs, estimated from the point B. This spiral is called, the spiral of Archimedes. 192. If we represent by unity the distance which the generating point moves along the radius-vector, during one revolution, the equation u = at, will become 1 at, or * a a = But since t is the circumference of a circle whose radius is unity, we shall have j i = 2*-, and consequently, 193. If the axis BD, of a semi-parabola BCD, be wrapped around the circum- ference of a circle of a given radius r, any abscissa, as Bb, will coincide with an equal arc Bb f , and any ordinate as ba, will take the direction of the normal Ab'a f . The curve Ba'cf, described through the extremities of the ordinates of the parabola, is called the parabolic spiral. The equation of this spiral is readily found, by observing that the squares of the lines b'a', c c f , &c., are propor- tional to the abscissas or arcs Bb f , Be . DIFFERENTIAL CALCULUS. 179 If we designate the distances, estimated from the pole y by u, we shall have Va! = u r: hence, is the equation of the parabolic spiral. If we suppose r = 0, the equation becomes u 2 = 2pt. If we make n 1 , the general equation of spirals becomes u = at~ l , or ut = a. This spiral is called the hyperbolic spiral, because of the analogy which its equation bears to that of the hyperbola, when referred to its asymptotes. 194. The relation between u and t is entirely arbitrary, and besides the relations expressed by the equation we may, if we please, make t = \ogu. The spiral described by the extremity of the radius-vec- tor when this relation subsists, is called the logarithmic spiral. 195. If in the equation of the hyperbolic spiral, we make successively, 1 1 1 t=l > = ? = 3' =4' &C " we shall have the corresponding values, u = a, u = 2a, u = 3a, u 4a, &c. 180 ELEMENTS OF THE Through the pole A draw AD perpendicular to AB y and make it equal to a : then through D draw a parallel to AB. From any point of the spiral as P draw PM perpendicular to AB, we shall then have PM = u sin MAP = u sin t. If we substitute for u its value , we shall have PM=a sin* smt Now as the arc t diminishes, the ratio of will ap- proach to unity, and the value of the ordinate PM will approach to a or CM: hence, the line DC approaches the curve and becomes tangent to it when t = 0. But when t = 0, u = oo ; hence, the line DC is an asymptote of the curve. 196. The arc which measures the angular motion of the radius-vector has been estimated from the right to the left, and the value of t regarded as positive. If we i evolve the radius-vector in a contrary direction, the measuring arc will be estimated from left to right, the sign of t will be changed to negative and a similar spiral will be de- scribed. The line DO is an asymptote to the hyperbolic spiral, corresponding to the negative value of t. DIFFERENTIAL CALCULUS. 181 197. Let us now find a general value for the subtan- gent of any curve referred to polar co-ordinates. The subtangent is the projection of the tangent on a line drown through the pole and perpendicular to the radius- vector passing through the point of contact. The equation of the curve may be written under the *brrn u=f(t\ in which we may suppose t the independent variable, and its first differential constant. Let A O = 1 be the radius of the measuring circle, P T a tan- gent to the curve at the point P, and A T drawn perpendicular to the radius-vector AP, the sub- tangent. Take any other point of the curve as P', and draw AP f . About the centre A describe the arc PQ, and draw the chord PQ. Draw also the secant PP r and prolong it until it meets AT, drawn parallel to QP, at T. From the similar triangles QPP', A T'P' y we have hence, PQ : QP' :: AT' : AP' '; QP AP 1 PQ " AT 1 ' But when we pass to the limit, by supposing the point P 1 to coincide with P, the secant TPP' will become the tangent PT, and AT will become the subtangent AT. 182 ELEMENTS OF THE But under this supposition the arc NN' will become equal to dt, the arc PQ to the chord PQ (Art. 128), AP' to u, and the line QP f to du. To find the value of the arc I PQ, w r e have 1 : NN 1 : : AP : arc PQ ; hence, 1 : dt : : u : arc PQ, and PQ = udt. Substituting these values, and passing to the limit, we have du u ~udi~~~AT : hence, we have the subtangent U 2 dt ' du ' 198. If we find the value of u 2 and du from the gen- eral equation of the spirals we shall have AT=^ DIFFERENTIAL CALCULUS. 183 In the spiral of Archimedes, we have n = I, and a= ; K t 2 hence, AT . 2-r If now we make t = %ir= circumference of the mea- suring circle, we shall have A T 27f circumference of measuring circle. After 7?i revolutions, we shall have and consequently, A T = 2 m 2 * = m . 2 m* ; that is, the subtangent, after m revolutions, is equal to m times the circumference of the circle described with the radius-vector. This property was discovered by Archimedes. 199. In the hyperbolic spiral n = 1, and the value of the subtangent becomes AT=-a; that is, the subtangent is constant in the hyperbolic spiral. 200. It may be remarked, that AT _udt AP ~ du expresses the tangent of the angle which the tangent makes with the radius-vector 184 ELEMENTS OF THE In the logarithmic spiral, of which the equation is we have dt = A ; AT udt hence, _=__= A ; that is, in the logarithmic spiral, the angle formed by the tangent and the radius-vector passing through the point of contact, is constant ; and the tangent of the angle is equal to the modulus of the system of logarithms. If t is the Naperian logarithm of u, the angle will be equal to 45. 201. The value of the tangent in a curve referred to polar co-ordinates, du*' 202. To find the differential of the arc, which we will represent by z, we have or, by substituting for QP f and PQ their values, and passing to the limit, we have DIFFERENTIAL CALCULUS. 185 203. The differential of the area ADP when referred to the Dolar co-ordinates, is not an ele- mentary rectangle as when re- ferred to rectangular axes, but is the elementary sector APP f . The limit of the ratio of the sector APP' with the arc NN', will be the same as that of either of the sectors APQ, A.P"P' between which it is contained, with the same arc NN'. Hence, if we designate the area by s, and pass to the limit, we shall have APxPQ u 2 u 2 dt ds dt 2NN f = or which is the differential of the area of any segment ol a spiral. Of Tangent Planes and Normal Lines to Surfaces. 204. Let u = F(x,y,z) = 0, be the equation of a surface. If through any point of the surface two planes be passed intersecting the surface in two curves, and two straight lines be drawn respectively tangent to each of the curves, at their common point, the plane -of these tangents will be tangent to the surface. 205. Let us designate the co-ordinates of the point at which the plane is to be tangent by a/', y", z ff . 186 ELEMENTS OF THE Through this point let a plane be passed parallel to the co-otdinate plane YZ. This plane will intersect the surface in a curve. The equations of a straight line tan- gent to this curve, at the point whose co-ordinates are a/',y",zr', are r _ rJI _ fjff .. .// ^y_(y^ y/f\ . 3 > y y ~~fo?\ z i? / the first equation represents the projection of the tangent on the co-ordinate plane ZX, and the second its projec- tion on the co-ordinate plane YZ (An. Geom. Bk. IX Art. 70). Through the same point let a plane be passed parallel to the co-ordinate plane ZX, and we shall have for the equations of a tangent to the curve The coefficient -j- represents the tangent of the angle which the projection of the first tangent on the co-ordinate plane YZ makes with the axis of Z ; and the coefficient j- represents the tangent of the angle which the projection of the second tangent on the plane ZX makes with the axis of Z (An. Geom. Bk. VIII, Prop. II). But these coefficients may be expressed in functions of the surface and the co-ordinates of its points. For, we have u = f(x,y,z) =0, and if we suppose x constant, we shall have (Art. 87) , du j . du j du = dy + dz = Q: dy * dz DIFFERENTIAL CALCULUS. 187 du hence, -r~= 7- : dz du ., and if we suppose y constant, we shall find, in a similar manner, du dx _ dz dz du ~dx hence, the equation of the projection of the first tangent on the plane of YZ becomes dy and the equation of the projection of the Second tangent on the plane of ZX is du x-a/'=:--^-(z-z"). du v dx The equation of a plane passing through the point whose co-ordinates are a/ x , y" , z" is of the form in which -will represent the tangent of the angle which the trace on the co-ordinate plane YZ makes with the f axis of Z, and ~r^ Q tangent of the angle which the A. trace on the plane of ZX. makes with the axis of Z. 188 ELEMENTS OP THE But since the tangents are respectively parallel to the co-ordinate planes YZ, ZX, their projections will be parallel to the traces of the tangent plane : therefore, du du B ~~fa' nence, > = 5*- L/ ; a* dy dz du du C A ~ dz ~ du' hence, A = r C. rfw Tx 22 Substituting these values of B and A in the equation of the plane, and reducing, we find ...du , ,/\du , ff .du (z ^ z n } _ + (x _ y/ , } _ + (y _ y : l )_^ y which is the equation of a tangent plane to a surface at a point of which the co-ordinates are a/', y v , z" . 206. A normal line to the surface being perpendicular to the tangent plane at the point of contact, its equations will be of the form du du dz dz ELEMENTS OF THE INTEGRAL CALCULUS, Integration of Differential Monomials. 207. The Differential Calculus explains the method of finding the differential of a given function. The Integral Calculus is the reverse of this. It explains the method of finding the function which corresponds to a given differential. The rules for the differentiation of functions are explicit and direct. Those for determining the integral, or func- tion, from the differential expression, are less direct and are deduced by reversing the process by which we pass from the function to the differential. 208. Let it be required, as a first example, to integrate *.he expression. x m dx. We have found (Art. 32), that whence, 190 ELEMENTS OF THE x m+l and consequently -- , * the function of which the differential is x m dx. The integration is indicated by placing the character / Before the differential which is to be integrated. Thus, KG write from which we deduce the following rule. To integrate a monomial of the form x m dx, augment the exponent of the variable by unity, and divide by the exponent so increased and by the differential of the variable. 209. The characteristic / signifies integral or sum. The word sum, was employed by those who first used the differential and integral calculus, and who regarded the integral of x n dx as the sum of all the products which arise by multiplying the mth power of a?, for all values of x, by the con stant dx. 210. Let it be required to integrate the expression g. We have, from the last rule, /(A/UU / i Q X X JL _ =/ ^. = ___ = _ = In a similar manner, we find INTEGRAL CALCULUS. 191 211. It has been shown (Art. 22), that the differential of the product of a variable multiplied by a constant, is equal to the constant multiplied by the differential of the variable. Hence, we may conclude that, the integral of the product of a differential by a constant, is equal to the constant multiplied by the integral of the differential : that is, / ax m dx = a f x m dx = a J J Hence, if the expression to be integrated have one or more constant factors, they may be placed as factors with- out the sign of the integral. 212. It has also been shown (Art. 22), lhat every con- stant quantity connected with the variable by the sign t)lus or minus, will disappear in the differentiation ; and hence, the differential of a + x m , is the same as that of x ; viz. mx m ~ l dx. Consequently, the same differential may answer to several integral functions differing from each other in the value of the constant term. In passing, therefore, from the differential to the integral or function, we must annex to the first integral obtained, a constant term, and then find such a value for this term as will characterize the particular integral sought. For example (Art. 94), ~ = a, or dy = adx y CLCC is the differential equation of every straight line which makes with the axis of abscissas an angle whose tangent is a. Integrating this expression, we have 13 192 ELEMENTS OF THE = afdx, or y aX) . or finally, y = ax + C. If now, the required line is to pass through the origin of co-ordinates, we shall have, for x = 0, y = 0, and consequently, C = 0. But if it be required that the line shall intersect the axis of Y at a distance from the origin equal to + 6, we shall have, for x 0, y = -f 6, and consequently, C = + b ; and the true integral will be y = av-\-b. If, on the contrary, it were required that the right line should intersect the axis of ordinates below the origin, we should have, for x = 0, y s=a b, and consequently, C = b ; and the true integral would be y = ax b. 213. It has been shown (Art. 95), that xdx + ydy = is the differential equation of the circumference of a circle By taking the integral, we have , or or finally, :r* + ^ + C = 0. INTEGRAL CALCULUS. 193 If it be required that this integral shall represent a given circumference, of which the radius is R, we shall have, by making a? = 0, y*=-C = R 2 , and hence, C= R 2 ; and consequently the true integral is R 2 = 0, or The constant C, which is annexed to the first integral that is obtained, is called an arbitrary constant, because such a value is to be attributed to it as will cause the required integral to fulfil given conditions, which may be imposed on it at pleasure. The value of the constant must be such, as to render the equation true for every value which can be attributed to the variables. 214. There is one case to which the formula of Art. 208 does not apply. It is that in which m = 1. Under this supposition, * W " M -- ar * +1 - X = 1 - + l ~ 1 + 1~~ ~~ " But when m = 1, dx fx m dx = fx~ l dx = / , x and C = log a? + C. (Art. 57). / x 215. Since the differential of a function composed of several terms, is equal to the sum or difference of the diffe- rentials (Art. 27), it follows that the integral of a differen- 194 ELEMENTS OF THE tial expression, composed of several terms, is equal to the sum or difference of the integrals taken separately. For example, if du = adx -- s- -|- x -y/ x dx, we have or fdu =f(adx -- 3- + x v/tfcfo), and 216. Every polynomial of the form (n ~ l dx[(ax~ n L IS = x m ~ l dx(ax~ n Now, if we add unity to the exponent of x without the parenthesis, and divide by n, the quotient will be ( --- h X and the expression will have an exact integral when this quotient is a whole number (Art. 235). Hence, every differential binomial has an exact integral, when the exponent of the variable without the parenthesis augmented by unity and divided by the exponent of the variable within the parenthesis, ^plus the exponent of the parenthesis, is an entire number. 237. The integration of differential binomials is effected by resolving them into two parts, of which one at least has a known integral. We have seen (Art. 28) that d(uv} = udv -f vdu, 210 ELEMENTS OF THE whence, by integrating, uv =fudv -}-fvdu, and, consequently, fudv = uv fvdu. Hence, if we have a differential df the form Xdx, in which the function X may be decomposed into two factors P and Q, of which one of them, Qdx, can be integrated, we shall have, by making / Qdx = v and P = u t fPQdx = Pv-fvdP, in which it is only required to integrate the term fvdP. 238. To abridge the results, let us write p for , in which case p will represent a fraction, and the differential binomial will take the form If now, we multiply by the two factors x n and a?~ n , the value will not be affected, and we obtain Now, the factor x n ~ l dx(a + bx n ) p is integrable, whatever be the value of p (Art. 217) ; and representing this factor by dv, we have and, consequently, ni-n INTEGRAL CALCULUS. 211 But, fx m ~ n - l dx(a + bx n ) p+l = fx m - n ~ l dx(a + bx n ) p (a + bx n ) = afx m - n - l dx(a + bx n Y + bfx m ~ l dx(a + bx n ) p ', substituting this last value in the preceding equation, and collecting the terms containing the integral fx m - l dx(a+bx n ), we have x m ~ n (a + bx n ) p + l - a(m - n)fx m - n ~ l dx(a + bx) p . (p+l)nb whence, formula (A.) .......... fx m ~ l dx(a + bx n ) p = x m - n (a + bo?)'* l -g(m- n)fx m ~ n - l dx(a + bx n }*> b(pn-\- m) This formula reduces the differential binomial fx m - l dx(a + bx n ) p to that of fx m - n ~ l dx(a + bx n ) f ; and by a similar process we should find fx m - n - l dx(a + bx n ) p to depend on fx m ~' 2n - l dx(a-\-bx n y > ; and consequently, each process diminishes the exponent of the variable without the parenthesis by the exponent of the variable within. After the second integration, the factor m n, of the second term, will become m 2n ; and after the third, m 3n, &c. If m is a multiple of TI, the factor m n, m 2n, m 3?z, &c., will finally become equal to 0, and then the differential into which it is multiplied will disap- 212 ELEMENTS OF THE pear, and the given differential will have an exact integral which corresponds with the result of Art. 235. 239. Let us now determine a formula for diminishing the exponent of the parenthesis. We have fx m ~ l dx(a + bx n Y = fx m - l dx(a + bx n ) p ~ l (a + bx n ) = afx m ~ l dx(a Applying formula (A) to the second term, by placing m 4- n for m, and p 1 for p, we have x m (a 4- bx n ) p - amfx m - l dx(a + bx n ) p ~ l b(pn 4- m) Substituting this value in the last equation, we have formula (B) ................ fx m ~ l dx(a + bx n ) f = x m (a + bx n Y + pnafx m ~ l dx(a + bx n ) p ~ l pn + m which diminishes the exponent of the parenthesis by unity for each integration. 240. By means of formulas (A) and (B), we reduce to rn being the greatest multiple of n which can be taken from m 1, and s the greatest whole number which can be subtracted from p. ^ For example, fx 7 dx(a + &r 3 ) 2 is reduced, by formula (A), to i. fx*dx(a + bx*Y> and then to f xdx(a INTEGRAL CALCULUS. 213 and by formula (B) Jxdx(a -f for 3 ) 2 , reduces to 1 1 fasdx (a + bx*) 2 , and finally to fxdx (a + bx 3 ) 2 . 241. It is evident that formulas (A) and (B) will only diminish the exponents m 1 and p, when m and p are positive. We will now determine two formulas for dimin- ishing these exponents when they are negative. We find from formula (A) x m ~ n (a + bx n ) p+l - b(m + np)fv m - l dx(a a(jri ri) and placing for m, m + n, we have formula (C) ............. fx~ m - l dx( x~ m (a + bx*) p+l + b(m n np\f x~ m+n - l dx(a + bo?)* am in which formula, it should be remembered that the nega- tive sign has been attributed to the exponent m. 242. To find the formula for diminishing the exponent of the parenthesis when it is negative. We find, from formula (B), x m (a + bx n ) p (m-\- np)fx m ~ l dx(a + la*)* pna writing for p, p + 1, we have formula (D) ............. fx m - 1 dx(a + bx n )~ p = x m (a + bx n )~ p+l (m + n np)fx m ~ l dx(a + bx n )~ p+l (p l)na 214 ELEMENTS OF THE This formula does not apply to the case in which p = l. Under this supposition, the second member becomes infi- nite, and the differential becomes that of a transcendental function. 243. It is sometimes convenient to leave the variable in ooth terms of the binomial. We shall therefore determine a particular formula for integrating the binomial This binomial may be placed under the form JL _L fa? ~*dx(2a-x)~t, and if we apply formula (A), after making , n=l, p , a = 2a, b = - 1, we shall have 9 ? and if we observe that 9-1 7-1 1 ?-- 9-1 -1 x 2 =x x 2 x *=x x % and pass the fractional powers of x within the parentheses we shall have formula (E) /2ax-x* t (2q-l)a f V2ax-x?' INTEGRAL CALCULUS. 215 which diminishes the exponent of the variable without the parenthesis by unity. If q is a positive and entire num- ber, we shall have, after q reductions Integration of Rational Fractions. 244. Every rational fraction may be written under the form Pa?-'+Qa?- g .... +RX + S. P'x n +QV- 1 ____ +R!x+S' ' in which the exponent of the highest power of the varia- ble in the numerator, is less by unity than in the denomi- nator. For, if the greatest exponent in the numerator was equal to or exceeded the greatest exponent in the denomi- nator, the division might be made, giving one or more entire terms for a quotient and a remainder, in which the exponent of the leading letter would be less by at least unity, than the exponent of the leading letter in the divisor. The entire terms could then be integrated, and there would remain the fraction under the above form. Place the denominator of the fraction equal to : that is, make PV+ QV- 1 ...... R'x+ S' = 0, and let us also suppose that we have found the n binomial factors into which it may be resolved (Alg. Art. 264). These factors will be of the form x a, x b, x c, x -d, &c. Now there are three cases : 216 ELEMENTS OF THE 1st. When the roots of the equation are real and unequal. 2d. When they are real and equal. 3d. When there are imaginary factors. We will consider these cases in succession. 1st. When the roots are real and unequal. 245. Let us take, as a first example, -. By decomposing the denominator into its factors, we have adx _ adx a? a 2 ~ (x a) (x -f a) ' and we may make adx /A B \ , L I _^I_ ^^^___ 1 rj /y (x a) (x + a) \x a x-{-aJ in which A and B a.re constants, whose values are yet to be determined. In order to determine these constants, let us reduce the terms of 'the second member of the equation to a common denominator ; we shall then have adx (Ax + Aa -f- Bx Ba) dx (x a)(x + a) (x a)( In comparing the two members of the equation, we find a = Ax + Aa + Bx Ba ; or by arranging with reference to a?, (A + B)x + (A-B-l)a = 0. But, since this equation is true for all values of x } the INTEGRAL CALCULUS. 217 coefficients must be separately equal to (Alg. Art. 208) : hence A + B = 0, and (A-B-l)a = 0, which gives Substituting these values for A and B, we obtain adx -ndx fdx a? a 2 x a a? -f a ' and integrating, we find (Art. 218) /cidx 1 1 rftf = Y lo s(* ~ a ) - y lo s(* + fl ) + c > and, consequently, /adx 1 . fx a ?^ == 1 sfe 246. Let us take, as a second example, ^- dx. (TX 3? The factors of the denominator are x and a 2 x 2 ; but hence, the given fraction becomes dx. x(a-x)(a-}-x') Let us now make =^+-*-+ x(a x}(a + x) x a x a 218 ELEMENTS OP THE reducing the terms of the second member to a common denominator, we have a? + bx 2 Act 2 - Aoc?+ Bax + Bx 2 + Cax -Cx 2 x(a x}(a-i-x) x(a x)(a and, comparing the like powers of x (Alg. Art. 208), From these equations, we find = a, = , =--. and substituting these values, we obtain a 3 + bx 2 i dx a -4-b 7 a 4-b , - -dx a -- 1 -- - I - -dx -- - - -dx : a 2 x x* x 2a cc 2a x and integrating (Art. 218), - - [log(a a?) + log(a + 0:)] + C = alog,r --- lg( a #) (a + x) + C = aloga? (a + 6) log -y/a 2 x 2 + C. Q X _ ^ 247. Let us take, for a third example, -- dx. cc? _ 6 x -f 8 INTEGRAL CALCULUS. 219 Resolving the denominator into the two binomial factors (Alg. Art. 142), (x 2), (a? 4), we have 3 a; 5 A B - = hence x 2 6x + 8 x 2 x 3x 5 Ax 4A+Bx 2B "~ a? 607 + 8 and by comparing the coefficients of x, we have -5=-4A-2, 3 = A + B, which gives ,.2, ,__.,,_ and substituting these values, we have r 3 J a?- 337-5 , j_ r dx 7 r dx ~ } ~~ 248. Let us take, as a last example, xdx x 2 -\-kax W Resolving the equation we find x 2a + V4a 2 + tf, x=2a and consequently, for the product of the factors, r )(a?+2a- 220 ELEMENTS OF THE To simplify the work, represent the roots by K and L y and the factors will then be K, x and we shall have x A B ^2 r"^ H rr- : hence b 2 x + K x + L x Ax + AL + 5o? f # _ _ _ x 2 -f 4 ax b 2 " x 2 + 4 ax whence, and, consequently, K E _ L ~' K-L' K-L' hence, .- lo g ( X +L)+C. b z KL K L 249. In general, to integrate a rational fraction of the form -' ____ +Rx+S d 1st. Resolve the fraction into m partial fractions, of which the numerators shall be constants, and the denomi- nators factors of the denominator of the given fraction. 2d. Find the values of the numerators of the partial fractions, and multiply each by dx. INTEGRAL CALGLLUS. 221 3d. Integrate each partial fraction separately, and the sum of the integrals thus found will lie the integral sought. 250. The method which has just been explained, will require some modification when any of the roots of the denominator are equal to each other. When the roots are unequal, the fraction may be placed under the form (x a)(x b) (x c)(x d) (x e) A + * + - C -+-A+ E x a x b x c x d x e y if several of these roots are equal, as for example, a = b = c, the last equation will become Px* + Q* 3 4- &c. A + B + C D E H (x a) 3 (x d) (x e) x a x d x e' in which A + B + C may be represented by a single con- stant A! . t Now, in reducing the second member of the equation to a common denominator with the first, and comparing the coefficients of the like powers of a?, we shall have five equations of condition between three arbitrary constants, A', D, and E : hence, these equations will be incompati- ble with each other (Alg. Art. 103). If, however, instead of adding the three partial fractions ABC a? x a' x d' which have the same denominator, we go through the 222 ELEMENTS OF THE process of reducing them to one, their sum may be placed under the form A' + B'x+C'a* (x-of > or, by omitting the accents, A + Bx + Cx 2 (x-a)* 251. Let us now make x a = z, and consequently, x = z-\-a] we shall then have A + Bx + Cx 2 _ A + Ba + Co 2 + Bz + 2Caz + C# (x-a) 3 ~^ r ~ A + Ba+Ca 2 B -\-2Ca , C ! _ -i _ ; z* z 2 z substituting for z its value, and representing the numera- tors by single constants, we have A + Bx+Ca? = A 1 B' | C r the form under which the fraction may be written. Since the same reasoning will apply to the case iu which there are m equal factors, we conclude that PaT- 1 4- Qx m ~ z . . + Rx 4- S A A! A!' A! r . . ! x-a m -x-~ x-a 252. In order, therefore, to integrate the fraction p^*_|_ QV+ &c. , (^-a) 3 "^^^)^-^ *' INTEGRAL CALCULUS. 223 place it equal to (x-a) 3 ^(x-a) 2 ^ x-a^ x-d l x-e' then, reducing to a common denominator, and comparing the coefficients of the like powers of x, we find the values of the numerators of the partial fractions. Multiplying each by dx, and the given fraction may be written under the form . (xa) 2 (xa) xd x e The first two fractions may be integrated by the method of Art. 217, and the three last by logarithms. Hence, finally, r J rf (x-a) 3 (x-d)(x-e) 2(x-a) 2 x-a + A"log(ar a) + jDlog(a? d) + E\og(x e) -f C. 253. Let it be required to integrate the fraction 2 ax ' , We have 2 ax (x 4- a) (a7-far #-fa reducing the fractions of the second member to a common denominator, and comparing the coefficients of x in the two members, we have 2a = A f and A-\-A f a 0: hence, A= 2a 2 , and A / = 2a; 15 224 ELEMENTS OP THE and, consequently, 2axdx 2a 2 dx 2adx } 2 ~ ~(# + a) 2+ (x + a)' hence, (Arts. 217 & 218), /2axdx 2 a? T = ^ 254. Let us find the integral of x 2 dx x 3 ax 2 a 2 x -\- a? * By placing the denominator equal to 0, we see that, by making x = a, the terms will destroy each other : hence, a is a root of the equation, and x a a factor. Dividing by x a, the quotient is x 2 a 2 : hence, the fraction may be placed under the form (a?-a 2 )(x-a) ~ (x + a)(x- a)(x-a) Let us now make x 2 A A' B (*-) Reducing the terms of the second member to a common denominator, we have x 2 = A(x-\-a)-\-A r (x 2 -a 2 )-\-B(x-d)\ (x-a) 2 (x + a)~ (x- a) 2 (x + a) and developing, and comparing the coefficients of the like INTEGRAL CALCULUS. ^ 225 powers of a?, we obtain the equations A' + B=1, A-2Ba = 0, Aa - A'a 2 + Be? = 0. Multiplying the first equation by a 2 , and adding it to the third, we have then multiplying the second by a, and adding it to the last, we have a 2 = 2 Aa, and consequently, A = a ; substituting this value of A, we find T> 1 At 3 B = and A' = . 4 4 Substituting these values of A, A', and B, we have a. "l" ^ / (x-a) 2 (x and consequently, /x*dx a 3 , , . xt-axt-cfx + a^ -^T^ (x 255. We can integrate, in a similar manner, when the denominator contains sets of equal roots. Let us take, as an example, adx adx 226 ELEMENTS OF THE Make a A A f B B 1 #- 1)2(^+1)2 (ay. 1)8 a;-! ^ (tf + 1 )* ^ 0? + 1 ' reducing the second member to a common denominator, we find the numerator equal to and comparing the coefficients with those of the numera- tor of the first member, we have the following equations : A' + B' = 0, A + A'+ B-B' = 0, %A -A f -2B-B f = 0, A - A! + B + B' = a. Combining the first and third equations, we find A = B; and combining the second and fourth, gives 2 A -f 2B = a: hence, we have A T> a At a R/ a . = T' "T = T J consequently, thp given differential becomes dx dx dx dx and by integrating, 256. If an equation of the second degree has imaginary roots, the quantity under the radical sign will be essentially INTEGRAL CALCULUS. 227 negative (Alg. Art. 144), and the roots will be of the form x = =F a + b V 1, x = + b -y/ 1, and the two binomial factors corresponding to the roots will be ( x a - b V^Hf ) (a? =fc a + 6 V^^T) = a? 2ax + a 2 + b 2 . Hence, for each set of imaginary roots which arise from placing the denominator of the fraction equal to 0, there will be a factor of the second degree of the form x 2 2ax + a 2 + b 2 . 257. If the imaginary roots are equal, we shall have, = 0, x = + b V 1 , x bi/^~I, and the factor will become a? -f b 2 . In the equation, a? -6c#+ lOc^O, the roots are, x = 3c + c V 1, # = 3c c V 1 ; comparing these values of x with the general form, we have a = 3c 6 c, and the given equation takes the form a? 6cx-\-9c 2 -\-c 2 = Q. Comparing the roots of the equation, 0^-1-4^+12 = 0, with the values of x in the general form, we have 228 ELEMENTS OF THE and the equation may be written under the form 258. Let us first consider the case in which the deno- minator of the fraction to be integrated contains but one set of imaginary roots. The fraction will then be of the form, _ P+Qx + Ra?-\-Sa?-\- &c. __ , (x-a)(x-b) ---- (x - h) (a?+2ax + a 2 + P) ' which may be placed under the form Adx Bdx Hdx Mx + N ' x a x b' x h The first three fractions may be integrated by the methods already explained : it therefore only remains to integrate the last, which may be written under the form Mx + N If we make x + a = z, the expression becomes Mz + N-Ma , z 2 + b 2 dz > and making N Ma P, it reduces to Mz+P which may be divided into the parts, Mzdz , Pdz Z 2+tf + z2 + which may be integrated separately. INTEGRAL CALCULUS. 229 To integrate the first term, we have Mzdz , C zdz M r 2zdz in which the numerator, 2zdz, is equal to the differential of the denominator: hence (Art. 218), CMzd*__M_ lQ (z* + b 2 )' or by substituting for z its value, x + a, *Mzdz M = M log yV + 2ax + a 2 + i 2 . Integrating the second term by Art. 224, gives or by substituting for z its value, x -\- a, and for P, N Ma, we have . = ___ tang . and finally, Af log VV + 2ax + a 2 + b 2 -\ ^ tang" 1 (~ ^~)- 259. Let us take, as an example, the fraction 230 ELEMENTS OF THE in which, if +1 be substituted for a?, the denominator will reduce to : hence, x 1 is a factor of the denomi- nator. Dividing by this factor, the fraction may be put under the form .; : ; .__^+___ (ir , - - iv in which x 2 -f x -+- I is the product of the imaginary factors. Placing this product equal to 0, finding the roots of the equation, and comparing them with the general values in the form INTEGRAL CALCULUS. 245 and integrating the second term by the formula of Art. 266, we have, after maKing h=p\ c 2 =l, and consequently, If we estimate the arc from the vertex of the parabola, we shall have y = for z = : hence C or C = and consequently, and hence, the value of the arc, for a given ordinate y, can only be found approximative^. 276. The curves represented by the equation are called parabolas This equation may be placed under the form or by placing p n =p f , and = n f , we have 246 ELEMENTS OF THE or finally, by omitting the accents, the form becomes y=px n . By differentiating, we have dy = npx n ~ l dx, and by substituting this value of dy in the differential of the arc, we have The integral of this expression will be expressed in a finite number of algebraic terms when - - - is a whole number and positive (Art. 235), If we designate such whole and positive number by i, we have for the condition of an exact integral in algebraic terms, 1 2i+l and substituting for n, we have which expresses the relation between x and y when the length of the arc can be found in finite algebraic terms. There is yet another case in which the integral will be ex- pressed in finite and algebraic terms, viz. when - ^+TT liTi - is a positive whole number (Art. 236 and 235.) o 277. If we make i 1, we have n = , and which is the equation of the cubic parabola. INTEGRAL CALCULUS. 247 Under this supposition, the arc becomes (Art. 217) i + )* f c ; and hence, the cubic parabola is rectifiable (Art. 273). If we estimate the arc from the vertex of the curve, we have x = 0, for z = : hence or =- and consequently, z = 278. If the origin of co-ordinates is at the centre of tne circle, the equation of the circumference is and the value of the arc, If the origin be placed on the curve y 2 = 2Rx a?, dx and both of which expressions may be integrated by series, and the length of the arc found approximatively. 279. It remains to rectify the transcendental curves. The differential equation of the cycloid is (Art. 182) 7 dx = , V 2ry y 2 ELEMENTS OF THE 248 which gives Substituting this value of da? in the differential of the arc, we obtain 2ry-y* _ = (2rY(2r-y) But (Art. 217) and hence, If now, we estimate the arc z from B, the point at which y = 2r, we shall have, for z = 0, \ y = 2 r ; hence -A F = + C, or C = 0, and consequently, the true integral will be the second member being negative, since the arc is a decreasing function of the ordinate y (Art. 31). If now, we suppose y to decrease until it becomes equal to any ordinate, as DF = ME, DB will be repre- sented by z, or by 2 -y/2r(2r y\ and BE = 2r y. But 5G 2 = BM x BE : hence INTEGRAL CALCULUS. and consequently or the arc of the cycloid, estimated from the vertex of the axis, is equal to twice the corresponding chord of the generating circle : hence, the arc BDA is equal to twice the diameter BM ; and the curve ADBL is equal to four times the diameter of the generating circle. 280. The differential of the arc of a spiral, referred to polar co-ordinates, is (Art. 202) Taking the general equation of the spirals u - at", we have du 2 = nW n -W', and substituting for du 2 and u 2 their values, we obtain If we make n = l, we have the spiral of Archimedes, (Art. 191), and the equation becomes . dz = adtVl -I-* 2 ; which is of the same form as that of the arc of the com- mon parabola (Art. 275). 281. In the logarithmic spiral, we have i = logw, and the differential of the arc becomes dz = duV2+C; and if we estimate the arc from the pole, 250 ELEMENTS OF THE Consequently, the length of the arc estimated from the pole to any point of the curve, is equal to the diagonal of a square described on the radius-vector, although the number of revolutions of the radius-vector between these two points is infinite. Of the Quadrature, of Curves. 282. The quadrature of a curve is the expression of its area. When this expression can be found in finite alge- braic terms, the curve is said to be quadrable, and may be represented by an equivalent square. 283. If 5 represents the area of the segment of a curve, and x and y the co-ordinates of any point, we have seen (Art. 130), -that ds = ydx. To apply this formula to a given curve : 1st. Find from the equation of the curve the value of y in terms of x, or the value of dx in terms of y, which values will be expressed under the forms y = f(x\ or dx=f(y)dy. 2d. Substitute the value of y, or the value of dx, in the differential of the area : we shall have ds = / (x) dx, or ds = f (y} dy : the integral of the first form will give the area of the curve in terms of the abscissa, and the integral of the second will give the area in terms of the ordinate. INTEGRAL CALCULUS 251 284. Let us take, as a first example, the family of para- bolas of which the equation is we shall then have and - no" **'*'* n fF(x)dx=fp n x n dx=^-x n = -xy+C; 1. by substituting y for its value, p n x n . If, instead of substituting the value of y in the differential of the area ydx, we find the value of dx from the equation we have and consequently, by substituting x for its value, 7, which is the same re- pm suit as before found. Hence, the area of any portion of a parabola is equal to the rectangle described on the abscissa and ordinate 252 ELEMENTS OF THE multiplied by the ratio - . The parabolas are there- m + n fore quadrable. In the common parabola, n = 2, m=l, and we have = xy, that is, the area of a segment is equal to two thirds of the area of the rectangle described on the abscissa and ordinate. 285. If, in the equation f-jf, we make n 1, and m 1, it will represent a straight line passing through the origin of co-ordinates, and we shall have Jf(x)dx = xy, which proves that the area of a triangle is equal to half the product of the base and perpendicular. 286. It is frequently necessary to find the integral or 1 unction, between certain limits of the variable on which it depends. A particular notation has been adopted to express such integrals. Resuming the equation of the common parabola and substituting in the equation ydx the value of dx ~ -, we have INTEGRAL CALCULUS. 253 or, ' the area be estimated from the vr/tex A, we have C = 0, and 3P If now, we wish the area to terminate J at any ordinate PM 6, we shall then take the integral between the limits of y = and y = b; and, to express that in the differential equation, we write which is read, integral of y*dy between the limits y = and y = b. If we wish the area between the ordinates MP = b, MP' = c, we must integrate between the limits y = 6, y c. We first integrate between and each limit, viz. : 1 rb 2 , 6 3 ' pj ; y 3p' we then have PMM'P = AMM'P' - AMP = C y z dy -- 3p 3p 3p v 287. Let us now determine the area of any portion of the space included between the asymptotes and curve of an hyperbola. 254 ELEMENTS OF THE The equation of the hyperbola referred to its asymp- totes (An. Geom. Bk. VI, Prop. IX,) is xy M. In the differential of the area of a curve ydx, x and y are estimated in parallels to co-ordinate axes, at right an- gles to each other. The differential of the area BCMP, referred to the oblique axes AX, A Y, is the parallelogram PMMP', of which PM=y and PP' = dx. If we designate the angle YAX = MPX by /3, we shall have area PMM'P = ydxsmp ; and substituting for y its value , and representing x the area BCMP by s, we have , ,- . dx ds = Msir\& , x /dx = M sin /slog a? + C. If AC is the semi-transverse axis of the hyperbola, and we make AB=l, and estimate the area s from J5C, we shall have, for x 1 , 5 = 0, and consequently C = ; and the true integral will be s = INTEGRAL CALCULUS. 255 But, since ABCD is a rhombus, and M = AB x BC (An. Geom. Bk. VI, Prop. IX, Sch. 2), and since AB = 1, we have M=l, and consequently, s = sin /slog x. Now, since s, which represents the space BCMP for. any abscissa a?, is equal to the Naperian logarithm of x multi- plied by the constant sin/3, s may be regarded as the loga- rithm of x taken in a system of which sin(S is the modu- lus (Alg. Art. 268). Therefore, any hyperbolic space BCMP is the logarithm of the corresponding abscissa AP, taken in the system whose modulus is the sine of the angle included between the asymptotes. If we would make the spaces the Naperian logarithms of the corresponding abscissas, we make sin/3 = 1, which corresponds to the equilateral hyperbola. If we would make the spaces the common logarithms of the abscissas, make sin/3 =, 0.43429945, (Alg. Art. 272). 288. The equation of the circle, when the origin of co- ordinates is placed on the circumference, is t/ 2 = 2rx x 2 , or y and hence, the differential of the area is dx v*2rx x 2 ; and this will become, by making x = r u, If we integrate this expression bv formula (B, Art. 239, 17 I 256 ELEMENTS OF THE we have " 2 But we have (Art. 253) -du /u .-* s = yr* u and placing for u its value 2 and taking this integral between the limits a? = and # = 2r, we shall have the area of a semicircle. For 07 = 0, the area which is expressed in the first member becomes 0, the first term in the second member becomes 0, and the second term also becomes 0, since the arc whose cosine is 1, is 0: hence the constant C = 0. If we now make x = 2r, the term (r x) -/ 2rx of 2 reduces to 0, and the second term to -i-r'cos-X- 1) = r 3 * 1 (Trig. Art. XIV), and consequently, the entire area is equal to r 2 ^, which INTEGRAL CALCULUS. 257 corresponds with a known result (Geom. Bk. V, Prop. XII, Cor. 2). The equation of the ellipse, the origin of co-ordi- nates being at the vertex of the transverse axis (An. Geom. Bk. IV, Prop. I. Sch. 8), gives B y =A and consequently, the area of the semi-ellipse will, be equal to T> / I Integrating, as in the last example, between the limits x = 0, and x 2A, and multiplying by 2, we find AB* for the entire area. This corresponds with a known result (An. Geom. Bk. IV, Prop. XIII). 289. The differential equation of the cycloid (Art. 183) is whence and applying formula E, (Art. 243) twice, it will reduce to -; and (Art. 226) But we may determine the area of the cycloid in a more simple manner by introducing the exterior segment AFKH, 258 ELEMENTS OP THE Regarding FB as a F K line of abscissas, and de- signating any ordinate as KH, by z = 2r y, we shall have B But whence d(AFKH) = zdx = , V2ry y 2 But this integral expresses the area of the segment of a circle, of which the abscissa is y and radius r (Art. 288): that is, of the segment MIGE. If now, we estimate the area of the segment from M, where y = 0, and the area AFKH from AF, in which case the area AFKH= for y ;= 0, we shall have AFKH = MIGE; and taking the integral between the limits y = and y = 2r, we have AFB = semicircle MIGB, and consequently, area AHBM = A FBM - MIGB. But the base of the rectangle AFBM is equal to the semi- circumference of the generating circle, and the altitude is equal to the diameter, hence its area is equal to four times the area of the semicircle MIGB ; therefore, area AHBM =3 MIGB, INTEGRAL CALCULUS. 259 and consequently, the area AHBL is equal to three times the area of the generating circle. 290. It now remains to determine the area of the spirals. If we represent by s the area described by the radius-vec- tor, we have (Art. 203) u 2 dt ' dS = ; and placing for u its value at n (Art. 189) ~2,2n 7, A/ 2 / 2n + l -. a i (11 , u L I /-i ds = - and s = - - - + C, 2 4n + 2 and if n is positive C = 0, since the area is when t = 0. After one revolution of the radius-vector, t 2 T, and we have = 471 + 2 which is the area included within the first spire. 291. In the spiral of Archimedes (Art. 192) a =. and n = 1 ; 2?r hence, for this spiral we have * 3 = which becomes ~, after one revolution of the radius- 7T vector ; the unit of the number being a square whose 3 side is unity. Hence, the area included by the first spire, is equal to one third the area of the circle whose radius is equal to the radius-vector after the first revolution. In the second revolution, the radius-vector describes a 260 ELEMENTS OF THE second time the area described in the first revolution ; and m any revolution, it will pass over, or redescribe, all the area before generated. Hence, to find the area at the end of the mth revolution, we must integrate between the limits t = (m 1)2^ and t = m.2vr, which gives 3 If it be required to find the area between any two spires, as between the mth and the (m -f 1 )th, we have for the whole area to the (m + l)th spire equal to (m-fl) 3 -m 3 ~3~ and subtracting the area to the mth spire, gives o for the area between the mth and (m -f l)th spires. If we make m 1, we shall have the area between the first and second spires equal to 2?r: hence, the area be- tween the mth and (m+ l)th spires, is equal to m times the area between the first and second. 292. In the hyperbolic spiral n = 1, and we have ds = - dt and s = -- . 2 2t The area s will be infinite when t = 0, but we can find the area included between any two radius-vectors b and c by integrating between the limits t = b, t = c, which will give _ 1 S ~ INTEGRAL CALCULUS. 261 293. In the logarithmic spiral t = logw : hence, dt = , u?dt udu hence, 6' = 2 2 udu _u z ~ = ~ L; and by considering the area s = Q when u = 0, we have C = and Determination of the Area of Surfaces of Revolution. 294. If any curve EMM, be re- volved about an axis AX, it will de- scribe a surface of revolution, and every plane passing through the axis AX will intersect the surface in a me- j ridian curve. It is required to find the differential of this surface. For this A P purpose, make AP x, PM = y, and PP f h : we shall then have / B PM = f(x) = y, P'M' = f(x + h) = y + ^-, - + &c. 262 ELEMENTS OF THE In the revolution of the curve BMM f , the extremities M and M of the ordi- nates MP, M'P', will describe the cir- cumferences of two circles, and the chord MM' will describe the curved surface of the frustum of a cone. The surface of this frustum is equal to (Geom : Bk. VIII, Prop. IV.) (circ.MP + circ.M'P') I M B M t>\ f P P f X X chord MM' : that is, to X chord MM' = v (MP +MP') X chord MM ; and by substituting for MP, M'P' their values, the expres- sion for the area becomes ' & t I 7i +$2+ &c -) chord MM - If now we pass to the limit, by making h = 0, the chord MM will become equal to the arc MM 1 (Art. 128), and the surface of the frustum of the cone will coincide with that of the surface described by the curve at the point M. If we represent the surface by s and the arc of the curve by z, we have, after passing to the limit, ds = Zvydz, and by substituting for dz its value (Art. 128), we have ds 2?ry V doc? + dy z : whence, the differential of a surface of revolution is equal to the circumference of a circle perpendicular to the axis, into the differential of the arc of the meridian curve. INTEGRAL CALCULUS. 263 Remark. It should be observed that X is the axis about which the curve is revolved. If it were revolved about the axis Y, it would be necessary to change x into y and y into 07. 295. If a right angled triangle CAB be revolved about the perpendicular CA, the hypothenuse CB will describe the surface of a right cone. If we represent the base BA of the triangle by 6, the altitude CA by h, and suppose the origin of co-ordinates at the vertex of the angle C, we shall have x : y : : h : b : hence b b , y = x and dy = dx. h h Substituting these values of y and dy, in the general for- mula, we have ,. bx , / T0 . 10 bx 2 and integrating between the limits x = and x = h, we obtain CB surface of the cone = nb AV = circ.AB x 296. If a rectangle ABCD be revolved around the side AD, we can readily find the surface of the right cylinder which will be. described by the- side BC. Let us suppose the axis AD = h, and AB b : the equation of the iine DC will be y = b: hence, dy = Q. Substituting these values in the general expression of the differential of the surface, we have 264 ELEMENTS OF THE and taking the integral between the limits x = 0, x Ji t we have surface =2^bh = circ. AB x AD. 297. To find the surface of a sphere, let us take the equation of the meridian curve, referred to the centre as an origin : it is and by differentiating, we have xdx + ydy = ; hence , xdx , j 2 dy = -- and ay 2 = y Substituting for dy its value, in the differential of the surface ds = 2 Try yda? -f dy 2 , we obtain s = 2^ da? + dx 2 =f2irRdx = 2*Rx+ C. If we estimate the surface from the plane passing through the centre, and perpendicular to the axis of X, we shall have s = for x = 0, and consequently C = 0. Now, to find the entire surface of the sphere, we must integrate between the limits x + j^ and x R, and then take the sum of the integrals without reference to their algebraic signs, for these signs only indicate the po- sition of the parts of the surface with respect to the plane passing through the centre of the sphere. INTEGRAL CALCULUS. 265 Integrating between the limits x = and x -f R, we find and integrating between the limits x = and x = R, there results s=-2*R 2 ; hence, surface = 4w.R 2 =:27rjR x 2R ; that is, equal to four great circles, or equal to the curved surface of the circumscribing cylinder. 298. The two equal integrals indicate that the surface is symmetrical with respect to the plane passing through the centre. 299. To find the surface of the paraboloid of revolution, take the equation of the meridian curve y 2 = 2px, which being differentiated, gives = yy and = p p 2 Substituting this value of dx in the differential of the sur face it reduces to 266 ELEMENTS OF THE But we have found (Art. 217) hence, and if we estimate the surface from the vertex at which point y = 0, we shall have, = -- + C, whence, C= -~- t o 3 and integrating between the limits y=0, y = b, we have 300. To find the surface of an ellipsoid described by revolving an ellipse about the transverse axis. The equation of the meridian curve is whence B 2 xdx B xdx Ay A -y/^ 2 a? 2 substituting the square of this value in the differential of the surface and for y its value --x/ A* r 2 A VA -or we have = 2 Tr- dx INTEGRAL CALCULUS. 267 and if we represent the part without the sign of the inte- gral by D, and make A ^ B , = R\ we shall have s Dfdx^K* x 2 . But the integral of dx ^/R 2 x 2 is a circular segment of which the abscissa is x, the radius of the circle being jR. If, then, we estimate the surface of the ellipsoid from the plane passing through the centre, and also estimate the area of the circular segment from the same point, any portion of the surface of the ellipsoid will be equal to the corresponding portion of the circle multiplied by the con- stant D. Hence, if we integrate the expression between the limits x = and x = A, and designate by D' the corresponding portion of the circle whose radius is R t we shall have surface ellipsoid = D x IX; hence, surface ellipsoid = 2JD x I? . 301. To find the surface described by the revolution of the cycloid about its base. The differential equation of the cycloid is , _ ydy 268 ELEMENTS OF THE Substituting this value of dx in the differential equation of the surface, it becomes Applying formula (E), Art. 243, we have hence, s = If we estimate the surface from the plane passing through the centre, we have C = 0, since at this point s = Q and y 2r. If we then integrate between the limits y = 2r and y 0, we have 1 t 32 ^ i. 6' = surface = r" 71 " 7 ? hence, du d*u h the limit of which, when h =. 0, is du dx whence and finally du = ny 2 dx ; the differential of the solidity ny^dx being a cylinder whose base is ny 2 and altitude dx. 304. Remark. The differential of a solid, generated by revolving a curve around the axis of Y, is 305. Let it be required to find the solidity of a right cylinder with a circular base, the radius of the base V a in Inte- 282 ELEMENTS OF THE grating between these limits we have since, sm"^!)^ hence, and taking this last integral between the limits y = and y = A C = R, we obtain irR* ' which represents that part of tne sphere that is contained in the first angle of the co-ordinate planes, or one-eighth of the entire solidity. Hence, 4 1 solidity of the sphere = R*K = D 3 7r. o u We might at once find the solidity of the hemisphere which is above the horizontal plane YX, by integrating between the limits a? = _ V.R 2 - y 2 and x = -f VR 2 -y 2 . Taking the integral between the limits x = and x = V R 2 y 2 , we have fzdx -- (R 2 y 2 ) ; and between the limits x = and x = + INTEGRAL CALCULUS. 283 we ha ve Jzdx =(R 2 y 2 )', hence, between the extreme limits, we have Then taking the integral between the limits y = R and y = + R, we find the solidity to be or the solidity of the entire sphere is, THE END 303