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Full text of "Elements of the differential and integral calculus"

n ■ ■ ■ m ■ Tt,;..*» ■ I *$ si ■ ■ I .<rC OK mm Km HlfiiB ^ fflteg wmMB&BSBBmMm oo x x* x ,' W V *3> V fU. V* '^ <#? ELEMENTS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS BY CHARLES DAVIES, AUTHOR OF MENTAL AND PRACTICAL ARITHMETIC, FIRST LESSONS IN ALGEDRA, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIP- TIVE GEOMETRY, ELEMENTS OF ANALYTICAL GEOMETRY, AND SHADES SHADOWS AND PERSPECTIVE. SECOND EDITION. REVISED AND CORRECTED. PUBLISHED BY A. S. BARNES & Co, Hartford.— WILEY & PUTNAM; COLLINS, KEESE & Co, New York.— THOMAS, COWPERTIIVVAIT & Co, Philadelphia.— CUSHING & SONS, Baltimore.— TRUMAN & SMITH, Cincinnati. 183 P. 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. HENRY W. REES, STKREOTTPKK, 45 QOID STREET, HEW YORU 44 i)\) 1.-- ■X 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. It has been the intention, however, to render the sub- ject as plain as the nature of it would admit, but still, it 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 freely used, although the general method of arranging the subjects is quite different from that adopted by either of those distinguished authors. Military Academy, West Point, October, 183& CONTENTS CHAPTER I. Constants and variables, ...... 9 Functions defined, ....... 9 Increasing and decreasing functions, ... 10 Implicit and explicit functions, . . . . 11 Differential coefficient defined, .... 16 Differential coefficient independent of increment, . 20 Differential Calculus defined, ..... 23 Equal functions have equal differentials, . . 23 Reverse not true, ....... 23 CHAPTER II. Algebraic functions defined, .... 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 Taylor'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 Manner 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. 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, . . . . . . .133 Discussion of the equation y = bzizc(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, ........ 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 e volute 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. Page. 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, . . . . .194 Integration of particular binomials, . . . 195 Integration — when a logarithm, . . . . .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 V2 ax — oc 2 Integration of rational fractions when the roots of the denominator are real and equal, the roots are Integration of rational fractions when equal, ..... Integration of rational fractions when the denominator contains imaginary factors, Integration of irrational fractions, Rectification of plane curves, Quadrature of curves, " of curved surfaces. Cubature of solids, . I>ouble integrals, 216 221 226 233 243 250 261 269 274 DIFFERENTIAL CALCULUS CHAPTER I. Definitions and Introductory Remarks. 1. There are two kinds of quantities which enter into the Differential Calculus : variables and constants. The variable quantities are generally designated by the final letters of the alphabet, x, y, z, &c. ; and any values may be attributed to them which will satisfy the equations in which they enter. The constant quantities are designated by the first letters of the alphabet, a, b, c, &c. ; and these preserve the same value throughout the same investigation, what- ever values may be attributed to the variables with which they are connected. 2. If two variable quantities are so connected together that any change in the value of the one will necessarily produce a change in the value of the other, they are said to be functions of each other. Thus, in the equation of a given straight line y = ax -f b, 10 ELEMENTS OP THE if we change the value of the ordinate y, the value of x will also undergo a change : hence, y is a function of x, or a? a function of y. This general relation, which merely implies a depen- dence of value, is expressed by y = F(x), or, x = F{y); and the equations are read, y a function of x, and x a function of y. This dependence of value may also be expressed by the equation F(x, y) = 0, which is read, function of x, y, equal to 0, and merely implies, that x depends for its value on y, or y on x. 3. The letter which is placed in the first member of the equation is called the function, and the one in the second member is called the variable. In the equation y = F(x), y is the function and x the variable, and in the equation x = F{y), x is the function and y the variable. 4. In the equation of the straight line y = ax -f b, it is plain that if the value of x is increased the value of y will also increase, or if x be diminished the value of y will diminish : hence, y and x increase together, or de- crease together, and y is then said to be an increasing function of the variable x. DIFFERENTIAL CALCULUS 11 In the equation of the circle a? + y 2 = R 2 , or y 2 = R 2 -x 2 , the value of y increases when x is diminished, and de- creases when x is augmented : when this relation subsists between y and x, y is said to be a decreasing function, of the variable x. 5. If in any equation of the form y = F(x), the value of y is expressed in terms of x and con- stants, as for example, if y — ax 5 , or y = 3 x 2 + bx 3 , &c, y is then said to be an explicit function of x. But if the value of the function is not directly expressed in terms of the variable on which it depends, as in the equation y 4 — 3axy + x 5 = 0; or if the dependence is expressed by means of an inter- mediate variable, as in the equations y = F(u), u = F(x), y is then said to be an implicit or implied function of x. The roots of an equation, for example, are implicit func- tions of the coefficients. • 6 In every equation of the form y = F(x), either the function y, or the variable x, may be made to 12 ELEMENTS OF THE change its value according to any law whatever, and the corresponding change which takes place in the other, will be determined by resolving the equation. Thus, in the equation of the circle x 2 +if = R 2 , if we change the value of either x or y by a quantity ± h, the corresponding value of the other variable may be determined from the equation, and the difference between it and the primitive value, will express the change of value. The law of change is generally imposed on the variable a?, and as this law is arbitrary, x is called. -an independent variable. It simplifies the operations of the calculus, to increase or diminish the variable x uniformly; that is, to change it from one state of value to another by the addition or sub- traction of a constant quantity; and since the law of change is arbitrary, this supposition does not render the calculus less general. 7. Although the values of the variable quantities may be changed at pleasure without affecting the values of the constants with which they are connected, there is, never- theless, a relation between them which it is important to consider. If in the equation y = F(x\ a particular value be attributed either to x or y, the other will be expressed in terms of this value and the constant quantities which enter into the primitive equation. Thus, DIFFERENTIAL CALCULUS. 13 in the equation of the straight line y = ax + 6, if a particular value be attributed to a?, the corresponding value of y will depend on a and b ; or if a particular value be attributed to y, the corresponding value of x will likewise depend on a and b. The same will evidently be the case in the equation of the circle x 2 +y 2 = R 2 , or in any equation of the form y = F(x). Hence, we see that, although the changes which take place in the values of the variables are entirely indepen- dent of the constants witli which the variables are con- nected, yet the absolute values are dependant on the constants. 8. Since the relations between the variables and con- stants are not affected by the changes of value which the variables may experience, it follows that, if the constants be determined for particular values of the variables, they will be known for all others. Thus, in the equation of the circle x*+tf = R\ if we make x = 0, we have y=±R; or if we make y = 0, we have x= ±R, 2 14 ELEMENTS 0? THE and the value of R will be equal to the distance from the origin to the point in which the circumference cuts the co- ordinate axes, whatever be the value of x or y. 9. The function y, and the variable x, may be so re- lated to each other as to reduce to at the same time. Thus, in the equation of the parabola y 2 =2px, which may be placed under the general form F(x,y) = 0, or y = F(x) t if we make x — 0, we have y = 0, or if we make y = 0, we shall have x = 0. 10. We have thus far supposed the function to depend on a single variable ; it may however depend on several. Let us suppose for example, that u depends for its value on x, y, and z, we express this dependence by u = F(x, y, z,) If we make x — 0, we have u = F(y,z); 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. 11. Let us now examine the change which takes place DIFFERENTIAL CALCULUS- 15 in the function for any change that may be made in the value of the variable on which it depends. Let us take, as a first example, u = ao? t - • and suppose x to be increased by any quantity h. De- signate by v! the new value which u assumes, under this supposition, and we shall have v! =a(x + A) 2 , or by developing v! — ax 2 -\- 2axh -f ah 2 . If we subtract the first equation from the last, we shall have u' —u — 2 axh -f- ah 2 ; hence, if the variable x be increased by h, the function will be increased by 2 axh -{-ah 2 . If both members of the last equation be divided by h, we shall have — - — = 2ax-\-ah, (1) which expresses the ratio of the increment of the function to that of the variable. 12. The value of the ratio of the increment of the func- tion to that of the variable is composed of two parts, 2 ax and ah. If now, w r e suppose h to diminish continually, the value of the ratio will approach to that of 2 ax, to which it will become equal when h = 0. The part 2 ax, which is independent of h, is therefore the limit of the ratio of IS ELEMENTS OF THE the increment of the function to that of the variable. The term, limit of the ratio designates the ratio at the time h becomes equal to 0. This limit is called the differen- tial coefficient of u regarded as a function of x. We have now to introduce a notation by which this limit may be expressed. For this purpose we represent by dx the last value of li, that is, the value of h wliich cannot be diminished according to the law of change to which h is sidjjectcdy without becoming ; and let us also represent by du the corresponding value of u : we then have du -r- = 2ax. (2) dx v The letter d is used merely as a characteristic, and the expressions du, dx, are read, differential of u, diffe- rential of x. It may be difficult to understand why the value which k assumes in passing, from equation (1) to equation (2), is represented by dx in the first member, and made equal to in the second. We have represented b}r 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 designating this last value by dx, we preserve a trace of the letter x, and express at the same time the last change which takes place in h, as it becomes equal to 0. 13. Let us take as a second example, u —. ax 3 . If we give to x an increment h, we shall have uf = a (x + h) 3 — ax 3 -f- 3 ahx 2 + 3 ah 2 x + ah 9 . hence, u r — u = 3 ahx 2 -f 3 ah 2 x -f ah 3 , DIFFERENTIAL CALCULUS. 17 and the ratio of the increments will be — 3ax^ + Sahx + ah 2 , h and the limit of the ratio, or differential coefficient, ^ = 3aa*. ax In the function nx*, we have du dx 47207 3 . 14. We have seen, in the preceding examples, that the differential coefficient, or limit of the ratio of the increment of the function to that of the variable, is entirely indepen- dent of the increment attributed to the variable. Indeed, when the ratio is obtained as in the examples already given, and the increment made equal to in the second member, it is plain that the first member can no longer depend upon the increment. As this, however, is an important principle, we shall add another proof of it ; in the course of which we shall discover the value of the dif- ferential coefficient under particular suppositions, and also the form under which the new value of the function u may be expressed. Every relation between a function u and a variable x, expressed by the equation u =± F{x), will subsist between the ordinate and abscissa of a curve. For, let A be the origin of the rectangular axes, AX, A Y. In the equation y u = F(x), make x — 0, which will give u = a constant : 18 ELEMENTS OF THE lay off AB equal to this constant. Then attribute values to x, from to any limit, -as well negative as positive, and find from the equation u - F{x), the corresponding values of u. Conceive the values of a? to be laid off on the axis of abscissas, and the values of u on the corresponding ordinates. The curve described through the extremities of the ordinates will have for its equation- u = F{x). (1) 15. Let x represent any abscissa, AH for example, and u the corresponding ordinate HP. If now we give to x any arbitrary increment 7^, and make HF—h, the value of u will become equal to FC y which we will designate by v! . We shall then have u'=zF(x + h). But F(x + h) = HP+CD, and HP F(x). Now, for a given value of h, CD will vary if P be moved along the curve : hence, CD will depend for its value on x and h } and we shall have CD = F f (x } h): the notation, F f , F", &c, designating new, or different functions of x. DIFFERENTIAL CALCULUS. 19 But since CD becomes 0, when h = 0, h must be a factor of the second member of the equation, and we may therefore write, CD = F"(x, h)h. Hence, vl = F{x + h) = F(x) + F" (a?, k) h, (2) and transposing F(x) — u, we have u'-u = F // (x,h)h. (3) But since v! ' —u= CD = tang CPD. h, we have u r -u= tang CPD.li = F f, {x, h)h ; hence, ^^- = Uxi S CPD = F / (x i h). (4) If now we suppose h to diminish continually, the point C will approach the point P, the angle CPD will be- come nearer and nearer equal to the angle PTH, which the tangent line forms with the axis of abscissas. If we pass to the limit of the ratio, we shall have i£- = tang PTH=F"'{x); (5) ax and it remains to show that, this differential coefficient is independent of h. To prove this, we will observe, that whatever value may- be attributed to h, a secant line, APC, can always be drawn through P and the extremity of the corresponding ordinate. The ratio of the increments of the ordinate and abscissa may then be expressed by the tangent of the an- gle CPD ; and since any secant will become the tangent PT, when we pass to the limit, it follows that, the limit 20 ELEMENTS OF THE of the ratio which is represented by the tangent of the angle PTH, is independent of the increment h. When, therefore, we pass from equation (4) which expresses the ratio of the increment of the function to that of the variable, to equation (5) which expresses the limit of trfat ratio, the second member of equation (4) must be made independent of h, which is done by making h = ; and since the second member itself does not become 0, it follows that there is at least one term in F" (x, h) which does not contain h. If then, we divide the second member of equation (4) into two parts, one independent of h, and the other con- taining h as a factor, it may be written under the form F"(x, h) = F ff \x) -f F 1Y (x, h)h. Substituting this value of F f/ (x, h) in equation (2), we obtain u'= F(x) + F ,,f {x)h -V F 1Y (x, h)h 2 , or, u! == u + F m {x) h + F l \x, h) h\ or by omitting a part of the accents, u'=u + F(x)h + F"(x,h)W. (6) Hence, also, jt^- = F'(x) + F"(x,h)h, (7) and by passing to the limit 16. Let us now resume the discussion of equation (6), uf=u + F(x)h + F" (x, h) h\ (9) DIFFERENTIAL CALCULUS. 21 This equation expresses the relation which exists be- tween the primitive function u, and the new value v! which it assumes, when an increment h is attributed to the variable x. We see that the new value of the func- tion is composed of three parts. 1st. The primitive function u. 2d. A function of x multiplied by the first power of the increment h. 3d. A function of x and h multiplied by the second power of the increment h. We may also remark that, the coefficient of h in the second term, is the differential coefficient of the function u, and that the third term will vanish when ive pass to the limit or make h = 0. In order to render the form of the equation as simple as possible, let us make F' O) = P, and F" (x, h) = P f ; the equation will then become u^u + Ph + P'K 2 , (10) or, vf -u = Ph + P f h 2 . The coefficient P is in general a function of x, yet the relation between u and x may be such as to make P a constant quantity, in which case P' will be 0, or the relation may be such as to render P' constant. These cases will be illustrated by the examples. 17. If we take equation (8), which is 22 ELEMENTS OF THE and multiply both members by dx, we have du — Pdx: hence, the differential of a function is equal to its dif- ferential coefficient multiplied by the differential of the variable. 18. The differential of the function may also be ex- pressed under another form. For, if we multiply both members of the equation by dx, and omit to cancel dx in the first member, we shall have — - -dx = Pdx, dx in which either member expresses the differential of the function u. 19. We may conclude from the preceding remarks that the differential of a variable function, is the difference be- tween two of its consecutive values, by which term we mean to designate that difference which cannot be dimin- ished according to the law of change to ivhich the function has been subjected, without becoming 0. 20. We also see that, the Differential Calculus is that branch of mathematics, in which the properties of quan- tities are determined by means of the changes which take place when the quantities pass from one state of value to another. 21 . If two variable functions u and v, are so connected DIFFERENTIAL CALCULUS. 23 together as to be always equal to each other, whatever value may be attributed to either of them, their differ- entials will also be equal. For, suppose both of them to be functions of an inde- pendent variable x. We shall then have (Art. 16), u> _ u = Ph + P7/ 2 , v f -v=Qh+Q'h 2 . But, since u! and v' are, by hypothesis, equal to each other, as well as u and v, we have Ph + P'h 2 = Qh + Qh\ or by dividing by h and passing to the limit P=Q, hence, du dv dx dx' and, du j dv 7 — — dx = -y— dx, dx dx that is, the differential of u is equal to the differential of v (Art. 18). 22. 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, if we have the function bu + a = F(x), the values of a and b will not be affected by attributing 24 ' ELEMENTS OF THE an increment h to x : we shall therefore have (Art. 16), bu' + a = bit + a + Ph -f- P'h 2 , 7 (u' — u) „ , „,, or, b v — - — '- = P -f- P7?, & or by passing to the limit b -j— ±= P, hence, Mw = P<f a?. Now, bdu is the differential of the function bu as well as of the function bu-\- a: and hence we may conclude 1st. That every constant quantity connected with a variable by the sign -plus or minus will disappear in the differentiation. 2d. That the differential of the product of a, variable quantity by a .constant, is equal to the differential of the variable multiplied by the constant. 3d. That the differential of a constant quantity is equal to 0. DIFFERENTIAL CALCULUS. 25 CHAPTER II. Differentiation of Algebraic Functions — Succes- sive Differentials — Taylor's and Maclaurirts 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 h, and designate the second state of the function by u f , we shall have v! — ax + ah — u -f- ah, h = a du hence, du = adx, or -j-dx = a ^» 25. As a second example, let us take the function u = ax 4 3 26 ELEMENTS OF THE If we give to x an increment h, we have u / =ax z +2ahx-\-ah' i f u f —u — r — = 2 ax -f ah : h hence, du = 2axdx. 26. For a third example, take the function u — ax 3 : giving to x an increment h, we have — - — =: 3 ax 2 + 3 axh -f ah 2 , or passing to the limit — — - =3 3 ax 2 ; hence, du = 3 ax 2 dx. ax 27. Let us now suppose the function u to be composed of several variable terms : that is, of the form u = y -\- z — iu = F (a?), in which y, z, and iv, are functions of x. If we give to x an increment h } we shall have v! —u — (y r — y) + {z f — z) — (io r — w) : hence, (Art. 16), «'_ » = (Pft + PW) + ( Q& + <W - (Lk + 2/A 2 ), or, *^ = (P + P'h) + (Q + Q>h)-(L + I/h), or by passing to the limit DIFFERENTIAL CALCULUS. 27 and multiplying both members by dx, we have —r-dx = Pdx -j- Qdx — L dx. ax But since P, Q, and L, are the differential coefficients of y, z, and to, regarded as functions of x, it follows (Art. 17) 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 v/ = u + Ph -f P'h\ v' = v + Qh + Q'h 2 , and by multiplying u'v' = ( u f Ph + Fh 2 ) (v + Qh + Q'h 2 ) ; if we perform the multiplication, and omit the terms which contain Jl 2 , which we may do, since these terms will vanish, when we pass to the limit, there will result, j = vP-\-uQ + $c. or passing to the limit, d(uv) „ therefore, d(uv) = vPdx + uQdx = vdu -f- ndv. Hence, the differential of the product of two functions dependent on the same variable, is equal to the sum of the 23 ELEMENTS OF THE products which arise by multiplying each by the differ- ential of the other. 29. If the differential of the product be divided by the product itself, we shall have d(uv) du dv uv u v that is, equal to the sum of the quotients lohich arise by dividing each differential by its function. 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 i ds v ts t s ' and by substituting for v in the last equation, we have d(uts) __ du dt ds m - — |- _ ( £ uts u t & and in a similar manner, we should find djutsr . . . .) _ du dt ds dr » utsr .... u t s r If in the equation d(uts)_du dt ds " — — H I r > uts uts 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. 29 arise by multiplying the differential of each function by the product of the others. ^, 30. To obtain the differential of a fraction of the form — we make v — = t, and hence u as tv. v Differentiating both members, we find du = vdt + tdv ; find] ing the value of dt, and substituting for t its value u > V we obtain V udv v 2 ' or by reducing to a common denominator 7 ^ vdu — udv at = ; 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. 22), and we shall have , udv dt u dt= 5-, or -?-=— -o • ' v l dv v 2 When u is constant, t is a decreasing function of v (Art. 4), and the differential coefficient of t is negative. This is only a particular case of a general proposition 80 ELEMENTS OF THE For, let u be a decreasing function of x. Then, if we give to x any increment, as + h, we have u'^u + Ph + P'h 2 , or, u'-u = Ph-\-P f h\ But by hypothesis u > u r ;. hence, the second member is essentially negative - y and passing to the limit, du _ p dx hence, a decreasing function and its differential coefficient will be affected with contrary signs. 32. To find the differential of any power of a function,, let us first take the function w* 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{uwuu .. ) _du du du du u {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 ; hence, d(u n ) _ ndu u n ~~ u * therefore, d(u n ) = nu n ~ l du. r If n is fractional, represent it by — , and make s v = u'y whence, if = v"' r and since r and s are supposed to represent entire num- bers, we shall have ru T " l du = sv*~ l dv ; - DIFFERENTIAL CALCULUS. 31 from which we find r 1 r i , ru , ru 7 av = — — T du = — dUy su" or by reducing dv = — u" du; s which is of the same form as the function d{u n ) — nu n ~ x d,u, T by substituting the exponent — for n. Finally, if n is negative, we shall have -n 1 from which we have (Art. 31), 1 \ — d(u n ) __ — nu n ~ x du hence, by reducing d(u~ n )= -nu~ n - l du. Hence, generally, the differential of any power of a function, is equal to the exponent multiplied by the func- tion 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 = -y/u^ or v — u* ; then, dv=z—u % du = —u * du 2Vu~' 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. 2), that in an equation of the form u = F(x), we may regard u as the function, and x as the variable, or a? 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 the reciprocal of that which is obtained by regarding x as a function of u. If we consider u as the function, the ratio of the in- crements will be represented by u' — u 1 x 1 — X x 1 (1) u' — u or since xf- -x = h, we have (Art. 16), v! — u 1 1 h ~ h 1 Ph + P'h 2 P + P'h or by passing to the limit du 1 dx 1 P But when we pass to the limit, the denominator of the dx second member of equation (1) becomes -7—; hence, dx _ 1 _ 1 du P / du\ \~dx~) DIFFERENTIAL CALCULUS. To illustrate this by an example, let i u = x 3 f whence x — y/u =u 3 Now, — = 3x? = 3u 3 ; but regarding x as the function dx 1-1 1 du 3 £ Su 3 35. If we have three variables u, y, and x, which are mutually dependant on each other, the relations between them may be expressed by the equations u — F(y), and y = F f (x). If now we attribute to x an increment h, and designate by k, the change which takes place in y, we shall have (Art. 16), uf=u + Pk + P'h\ y'=y+Qh+ Q'h 2 , and ^ZJL^PJrP% ?Lz]L = Q+q] h k a If we multiply these equations together, member by member, we shall have u ^ x ^Zl = (P^Pk)(Q + Qh); k II but k = y r — y ; hence, by dividing and passing to the limit, we have du _ du dy dx dy dx aid hence, if three quantities are mutually dependant on 34 ELEMENTS OP 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 we find v = bu 3 , u = ax*, ^L = 3bu*, ^ = 2acc. du dx But, — =— — x —r- = Sbu 2 x 2ax=6abu 2 x; ax du ax and by substituting for u*, its value a 2 x\ —— = 6a?bx 5 , and dv = 6a 3 bx 5 dx. dx EXAMPLES. 1. Find the differential of u in the expression u — V# 2 — oc 2 . Put a 2 — x 2 = y, then •w = y 2 , and the dependence be- tween u and x, is expressed by means of y, and u i'a an implicit function of x. Differentiating, we find ^=+ly -* = .!(<,»_*»)-*, and ^=-S*; dy 2 J 2 v ' dx DIFFERENTIAL CALCULUS. 35 by multiplying the coefficients together we obtain -j— = (a 2 — x 2 ) 2 2x = dx 2 V Va 2 -x 2 ' hence, — xdx 2. Find the differential of the function m u — {a-\- bx") . Place a -+- bx n == y : then u — tf; and ~— = my m ~ l = 7n{a -\- bx n ) hence, ax du , m ~ l „_, — -- = mnb(a + ox ) x ax du = ?nnb(a-{- bx n ) a" ' dx. 3. Find the differential of the function u = x{a 2 + x 2 ) Va 2 — x 2 , du = {{a 2 + x 2 ) Va 2 ^?) dx + x ^Tf^x 2 d{a 2 + a*), + x(a 2 + x 2 )d^d i -x 2 , in which the operations in the last, two terms are only indicated. If we perform them, we find d(a 2 + x 2 ) = d(x 2 ) = 2xdx, d( — x 2 ) — xdx dVa 2 -x 2 2^a 2 -x 2 Va'-x 2 ' 36 ELEMENTS OF THE Substituting these values, we find = [(* du = \ (a 2 + a?)Va 2 -a? + 2a?Va 2 -a?- V a 2 —oc 2 dx; or, reducing to a common denominator and cancelling the like terms, V^—x 2 4. Find the differential of the function u = a 4 + a V + x' , _ (a 4 + a¥+ x*)d(a 2 - x 2 ) - {a 2 - x 2 ) d{a*+ aV+ x 4 ) dU ~ (a 4 +aV+* 4 ) 2 from which we find , _ — 2x(2a i + 2a 2 x 2 — x*)dx 5. Find the differential of the function Make y — — — , z = -/(c 2 — or 2 ) 2 , ya? then we shall have u = iJ/(« - y + z) 3 = (a - y -f z) A ; DIFFERENTIAL CALCULUS. 37 we therefore have (Art. 32), 3 --i du = —(a - y + z)' d(a-y + z), 3 -- _ — 3dy + 3dz 4;Va — y + z But from the equations above, we find , _ / b \ d -\fx~~ — bdx dy = d(— 7 =:)= —b = =• W^ * 2*W dz = d(c 2 - x 2 ) 1 = ^{c 2 - x 2 f~ l d{c 2 - x 2 ), '2 -- — 4xdx = -^-(c 2 — x 2 ) 3 x — 2xdx = • 3/ Substituting these values of cly and dz, in the ex- pression for du, we find 36 dw= <^ 2a?Va: ^c 2 -^ 1 i 4/ b 1 » v X J 6. w = — , 0? du 1 7. w = — =, a? n ' du y dx* dx x 2 — ndx 38 ELEMENTS OF THE 8. u == V2ax-\-x 2 , <i?/ (« + x) dx V2ax+x 2 ' 10. u = a 6 +' 3 «V + 3aV + # 6 , Jm = 6 (a 2 + tf 2 ) 2 ^. xdx 11. M = Vl-a? du 12. a? + VI e?w dx 13. w= (a-f V^), T=^(^+ yT=^) 2 3 (a-}- V^) c?a? c£w = 2V'c 14. m = [a + yl 15. u = x 2 y 2 4cf ^ 3 ' 3 <ia? dw = 2x 2 ydy + 2y 2 xdx. 16. w= va+^X Vb 2 +y\ du = - ^ % y Va 2 +* 2 V& 2 +*/ 2 — o+*r 18. 1+a 2 19. x-\-y dw d« nx n x dx 4 07^07 , _z(dx-\-dy)— (x-hy)Sdz DIFFERENTIAL CALCULUS. 39 20 . .._vr+^+yi^ ,,_ .tod+vrr?) Vi4-^_Vl-a? ^Vl-a? 2 21. Find the differential coefficient of F(x)=z8x i -3x 3 -5x Ans. 32# 3 — 9.Z 2 — 5. . 22. Find the differential coefficient of F(x) = (x 3 + a)(3x 2 + b) Ans. 15a? 4 -f 3ar^-f6ao?. 23. Find the differential coefficient of F(x) = (ax + x 2 ) 2 , Ans. 2 (ax + x 2 ) (a + 2 a?). 24. Find the differential coefficient of F(x) x+ Vl — x 2 < . 1 Ans. Vl-x 2 (l-]-2xVl-x 2 ) Of Successive Differentials. 37. It has been remarked (Art. 16), that the differ- ential coefficient is generally a function of x. It may therefore be differentiated, and x may be regarded as the independent variable, A new differential coefficient may thus be obtained, which is called the second differential coefficient. 40 ELEMENTS OF THE 38. In passing from the function u to the first differ- ential coefficient, the exponent of x in every terra in which x enters, will be changed; and hence, the rela- tion which exists between the primitive function u and the variable x, is different from that which will exist between the first differential coefficient and x. Hence, the same change in x will occasion different degrees of change in the primitive function and in the first differential coefficient. The second differential coefficient will, in general, be a function of x i hence, a new differential coefficient may be formed from it, which will also be a function of x ; and so on, for succeeding differential coefficients. If we designate the successive differential coefficients by p, q, r, s, &c, we shall have du _ dp- _ * dq _ * dx dx dx But the differential of p is obtained by differentiating du its value — , regarding the denominator dx as con- dx stant. ; we therefore have <©=*. or, d 2 u dx == dp, and by substituting for dp its value, we ] have d 2 u dtf = q. 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 dec 1 expresses the square of the differential of x, and not the differential of x 2 . It is read, differential square of x, or differential of x squared. If we differentiate the value of q, we have j/d 2 u\ , d 3 u 7 d {d?) = d *> or ' ih?= dq; , d 3 u j. hence, -— - = r, &c, doer and in the same manner we may find d 3 u The third differential coefficient -7-^, is read, third ax 6 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 3 U d*u dx =P > dx 2 ~ q ' dx*- r > dx* &c 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 , 4* 42 ELEMENTS OF THE we shall have for the first differential coefficient, du n _ l — - = nax . dx If we now consider n, «, and dx, as constant, we shall have for the second differential coefficient d 2 u and for the third, 2^ = n{?i-l)ax n ~\ d 3 u ■^ = n(n-l){n-2)ax n -*, and for the fourth, d*u dx* n(n — l)(n — 2)(n — 3)ax n ~\ 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, and, 7 w .. = 0. dx n + l 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. 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 dijference of two variables of the form u = F(x ± y), the diffe?'ential 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 = x' : we shall then have u = F(a/) , du and 17 =?■ If we suppose y to remain constant and x to vary, we have da/ = dx, and if we suppose x to remain constant and y to vary, we have aW — dy. But since the differential coefficient p is independent of da/ (Art. 15), it will have the same value whether, da/ = dx } or, dx/ = dy. 44 ELEMENTS OF THE To illustrate this principle by a particular example, iet us take u = (x + y)\ If we suppose x to vary and y to remain constant, we find and if we suppose y to vary and x to remain constant, we find du f . \« i the same as under the first supposition. 42. It is evident that the F(x + y), 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 a + Cy h + Dy e +, &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 function. It is now required to find such values for the exponents a, 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 and making y = 0, this term would become infinite, and we should have F(x)=cc, which is absurd, since function of x, 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 F(x) = u. If we make F(x + y) = u', and differentiate, under the supposition that x varies and y remains constant, we shall have <M__dA_ dB_ a .d£ h ,dD lC , & dx dx dx dx ' dx and if we differentiate, regarding y as a variable and x as constant, we shall find ^i=aBif- x + bCy h ~ l + cDy e ~ l +, &c. : dy But these differential coefficients are equal to each other ( Art. 41); hence, the second members of the equations 46 ELEMENTS OP 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 = &, &c., and consequently a = 1 , b = 2, c = 3, &c. ; and comparing the coefficients, we find " dA C-J-— — D- 1 dC dx 2 dx 3 dx And since we have made F(x) = A=u, and F{x-\-y) = u' i we shall have „ du - (Pu ^ d?u A = u, B = -^~, C= _ , „ ^ D dx' '■■ 1.2 dx 2 1.2.3 dx 3 ' and consequently, , , du cPu y 2 , d 3 u y 3 , p B - M + ^ y+ -^tV + ^T±3 + ' &c - 43. This theorem gives the following development for the function u f =(x + y) n , du __, d 2 u f , \ „_i p u = x% —~=znx n \ -j^-=n(n-\)x n \ &c: DIFFERENTIAL CALCULUS. 47 hence, w' = (x + y)' 1 = x n + nx n ~ l y + ??(n ~ 1) ^~ 2 y 2 , ' n(n—l)(n — 2) n _ 3 3 , o 1.2.3 y ' 44. The theorem of Taylor may also be applied to the development of the second state of any function of the form u = F(x), when x receives an arbitrary increment h, and becomes x 4- h. For, if we substitute h for y, we have . , du 1 , dhi h 2 . d?u li 3 , „ dx dx 2 !.* dx 3 1.2.3 ' ' hence, the difference between the two states of the func- tion is , du 1 , c^w 7<? , d?u h 3 in which the difference is expressed in terms of the differential coefficients and the ascending powers of the increment. If we now suppose h to diminish continually, the sign of the limit of the series will depend on that of the first term -— h, or if h is positive, on that of the coefficient — . dx dx For, by dividing by h, we have u' — u _du d?u h d?u h 2 - 48 ELEMENTS OF THE and by passing to the limit , du 7 au — —~xax: ax hence, when a series is expressed m the powers of a variable which we suppose to be continually diminished, the sign of the limit of the series will depend on the sign of the term which contains the loioest power of the variable. 45. Remark. The theorem of Taylor has been demon- strated under the supposition, that the form of the function u'=F(x + y), is independent of the particular values which may be attributed to either of the variables x or y. Hence, when we make y = 0, and obtain F(x) — u ; this function of x ought to preserve the same form as F(x + y) ; else there would be, values of x in one of the functions, u r =F{x + y), u = F(x), which would not be found in the other, and consequently some of the values of x would be made to disappear when a particular value is assigned to y, which is entirely con- trary to the supposition. If the function be of the form u 1 =b-\- Va — x-\-y, we shall have u = b+ Va — x. DIFFERENTIAL CALCULUS. 49 If we row make is — a, we shall have v! = b + VyT and u = b, hi which we see, that v! and u are expressed under dif- ferent forms ; and hence, the particular value of y = changes the form of the function, which is contrary to the hypothesis of Taylor's theorem. When, therefore, the function v! — F(x 4" y), shall change its form by attributing particular values to x or y, the development cannot be made by Taylor's theorem. 46. The particular supposition which changes the form of the function will, in general, render the differential coefficients in the development equal to infinity. If we have then, k' = --c+^f+x- ~!A u = da 1 dx ~ d 2 u 2(/W 1 dx 2 ~ dht 2x2(/ + 1 . 3 3 x)z dx s ~ &c. 2x 2x2(/ -f &c. 5 in which all the coefficients will become equal to infinity when we make — x —f. ^ 50 ELEMENTS OF THE 47. If we have a function of the form v! — b -f Va — x + y, in which n is a whole number, all the differential coeffi- cients of u, for x— a will become infinite. For, we have hence, u - = b + y a — X - ~b-{-(a-x) n 7 da dx~~ 1 n 1 n-1 > ! — ^) " d 2 u (1- a) 1 dx 2 ?i 2 2b-1» (a — #) " &c. &c. all of which become infinite when we make x = «. MaclaurirCs Theorem. 48. Maclaurin's Theorem explains the method of developing into a series any function of a single variable. Let us suppose the function to be of the form u = F{x). It is plain that the value of F(x) must be expressed in terms of x, and of the constants which enter into F(x). Let us therefore assume u = A -f Bx a + Cx b + Dx e + , &c., in which the terms are arranged according to the ascend- ing powers of oc, and in which A, B 7 C, D, &c, are DIFFERENTIAL CALCULUS 51 independent of x, and dependent on the constants which enter into F(x). It is now required to find such values for the exponents a, b, c, &c, and the coefficients A, B, C, D, &c, as shall render the development true for all possible values which may be attributed to x. If Ave make a? = 0, u takes that value which the F(x) assumes under this supposition, and if we designate that value by U we shall have U~A. The first differential coefficient is d ^.^aBx a ~ l + bCx b ~ l + cDx~ x + &c, ax and since this does not necessarily become when we make x = 0, it follows that there must be one term in the second member of the form x° : hence, a — 1 = 0, or a = 1 ; and making x = 0, we have du __ rt dx The second differential coefficient is C ^ = b(b- l)Cx b ~ 2 + c(c- l)Dx- 2 + &c. ; but since the second differential coefficient does not neces- sarily become 0, when x = 0, we have b - 2 = 0, or b = 2: 52 ELEMENTS OF THE hence, by making x = 0, we have 2C, or £*4*X dx 2 ' da? 2 We may prove in a similar manner that dhi 1 c = 3 and D = -^-^ , &c. dx 3 1.2.3 If then we designate by U what the function becomes when we make x = 0, and by U, V" , V 1 ", &c, what the successive differential coefficients become under the same supposition, we shall have Fix) =U+irx+U /f — + V" jt~ -f &c. k ■ 1.2 1.2.3 49. The theorem of Maclaurin may be deduced imme- diately from that of Taylor. In the development , , du d 2 u y 2 , d 3 u if , e the coefficients u, -r—, -~-p &c, are functions of x, and also dependent on the constants which enter into F(x + y)» If we make x = 0, the F(x -+- y) becomes F(y), and each of the differential coefficients being thus made inde- pendent of x, will depend only on the constants which enter into F(x-\-y\ and which also enter into F(y}* Hence, if we designate by U, U f , V", U f \ V""\ &c. f DIFFERENTIAL CALCULUS. 53 the values which the coefficients assume under this hypothesis, we shall have 50. If we take a function of the form u = (a + x) n , we shall have Tx = n{a + ocy-\ &c. == &c. which become, when we make x = 0, U = a", V = na n ~\ V" = n(n-l) a n ~\ &c. ; hence, {a + x) n = <f + na n ~ l x + n<<n ~ l \ n - 2 x 2 + &c. 1 • 10 51. Remark 1. The theorem of Maclaurin has been demonstrated under the supposition that the F(x) reduces to a finite quantity when we make x = 0. The case, therefore, is excluded in which x = renders the function infinite. Thus, if we have u = cot x, u = cosec x, or u = log x, and make x = 0, we find u = oo ; hence, neither of these functions can be developed by the theorem of Maclaurin. 5* 54 ELEMENTS OF THE Remark 2. We have already seen (Art. 45.), that the theorem of Taylor does not apply to those cases in which the form of the function is changed by attributing a particular value to one of the variables : the theorem therefore fails for -particular values, but is true for all others, and hence, the general development never fails. In the theorem of Maclaurin the failure arises from the form of the function : hence, it is the general development which fails, and with it, all the particular cases. EXAMPLES. 1. Develop into a series the function u= Va 2 ~T^ = a(l+^y. 2. Develop into a series the function 3. Develop into a series the function a + oc \ as 4. Develop into a series the function u= 4y - — a (l-H-i) . DIFFERENTIAL CALCULUS. 55 CHAPTER III. Of Transcendental Functions. 52. If we have an equation of the form u = a x , in which a is constant, it is plain that u will be a function of x ; and if a be made the base of a system of logarithms, x will be the logarithm of the number u (Alg. Art, 257). When the variable and function are thus related to each other, u is said to be an exponential or logarithmic func- tion of x. 53. The functions expressed by the equations u = sin x, u— cos x, u — tang x, u = cot x, &c., are called circular functions. The logarithmic and circular functions are generally called transcendental functions, because the relation be- tween the function and variable is not determined by the ordinary operations of Algebra. Differentiation of Logarithmic Functions. 54. Let us resume the function u = a*. 56 ELEMENTS OF THE If we give to x an increment h, we have and v!- u = a** h - a x = a x (a h -l). In order to develop a h , let us make <2 = l+6, we shall then have rf = (i + 6y = i + Aft + ^- 1 )y + ^" 1 X^-gy +&c . V ; 1 1.2 1.2.3 hence, 1 1.3 1.3.3 \ X 1 2 1.2 3 / from which we see, that the coefficients of the first power of h will be Vl 2 3 / ' replacing b by its value a — 1, and passing to the limit, we obtain du_da*__ ,/a-l (a-lf j («-l) 3 C?07 da? or if we make VI 3 3 /* «-l (a-iy («-l) 3 . k-— — + 8 - «c, — — = ka*, or da* == ka'dx ; ax in which k is dependent on a. DIFFERENTIAL CALCULUS. 57 The successive differential coefficients are readily found. For we have da x a x k> d (^j-) = da*k = a x k 2 dx ; dx da 1 hence, -———(fl?, d " a * -a x k 3 &c. &c. d n a x , „ dx n 55. It is now proposed to find the relation which exists between a and k. For this purpose, let us employ the formula of Maclaurin, u = F{x) = U+ U— + V"— + V'"— — + &c. v ; 1 1.2 1.2.3 If in the function and the successive differential coefficients before found, we make x = 0, we have 17 = 1, ir==k t U"=k 2 , U"=k\ &c; hence, _ * , kx t k 2 x 2 | /cV , B a = 1 H h &c. If we now make a? = — , Ave shall have k a "= 1+ T + T<r + Tih- t - &c - : 58 ELEMENTS OF THE designating by e the second member of the equation, and employing twelve terms of the series, we shall find e = 2.7182818; hence, a k —e, therefore a = e k . But, 2.7182818 is the base of the Naperian system of logarithms (Alg. Art. 272) ; hence, the constant quantity k is the Naperian logarithm of a. By resuming the result obtained in Art. 54, da x — a x Jc dx, we see that the differential of a quantity obtained by raising a constant to a power denoted by a variable ex- ponent, is equal to the quantity itself into the Naperian logarithm of the constant, into the differential of the exponent. 56. If now we take the logarithms, in any system, of both members of the equation e* = a. we shall have kle = : la, or k = la : 7? whence, da x = = ka T dx = la ■Te a x dx ; or by recollecting that u - =v, we have du dx la , -Te a; DIFFERENTIAL CALCULUS. 59 or, if we regard x as the function, and u as the variable, we have (Art. 34), dx _ Ze 1 du la a'' Let us now suppose a to be the base of a system of logarithms. We shall then have x= the logarithm of w, Za=l, and le = the modulus of the system (Alg. Art. 272); and the equation will become d{lu) = le — , that is, the differential of the logarithm of a quantity is equal to the modulus of the system into the differential of the quantity divided by the quantity itself. 57. If we suppose a = e the base of the Naperian system, and employ the usual characteristic V to desig- nate the Naperian logarithm, we shall have d(l' u ) = — ; u that is, the differential oj the Naperian logarithm of a quantity is equal to the differential of the quantity divided by the quantity itself. The last property might have been deduced from the preceding article by observing that the modulus of the Naperian system is equal to unity. 58. The theorem of Maclaurin affords an easy method of finding a logarithmic series from which a table of logarithms may be computed. If we have a function of the form, u — F(x) = Ix, 60 ELEMENTS OF THE we have already seen that the development cannot be made, since F(x) becomes infinite when x = (Art. 51.) But if we make u = F(x) = l{l-\-x\ the function will not become infinite when x = ; and hence the development may be made. The theorem of Maclaurin gives v ' 1 1.2 1.2.3 If we designate the modulus of the system of the loga- rithms by A, we shall have du dx A- 1 1 + - = A(l X + x)~\ dhi - A i 1 = -a(i + x)-\ 1 -f xf d 3 U dx*~ -ZA (i 1 + oof ~ 2A{l+x)- -3 If we now make t v = 0, we have U = 0, U f = A, &'* = 'Za, V" = 24, &c. : hence, 1(1+sd) = A(x- — + - — + - &c.^ v; V 2 3 45 ) This series is not sufficiently converging, except in the case when x is a very small fraction. To render the series more converging, substitute — x for x : we then have z(i-,)=^(_,-4-4-4.4_ & c.) DIFFERENTIAL CALCULUS. 61 and by subtracting the last series from the first, we obtain ^+*)-^-J=KS)= 84 (t + t + t + &c If we make 1 -fa? t , z , z 1 -\ , we have x — 1 — a? n* 2n-\- z y and by observing that we have l{n + z)-ln = 2A[-?—+ -V-^- V+— (-^— Y+ &c,l 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 = l, we have 71 = 0, and If we make the modulus A = 1, the logarithm will be taken in the Naperian system, and we shall have Fa = 0.693147180, 2l f 2 = Z'4= 1.386294360; and by making z = 4, and n = 1, we have Z'5 = 1.609437913, and l f 2 + Z'5 = HO = 2.302585093. 6 62 ELEMENTS OF 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), HO : ^10 : : A : 1, or, 1 : 2.302585093 : : A. : 1 ; whence, A = = 0.434284482. 2.30258509 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 I. 59. Let us now apply these principles in differentiating logarithmic functions. 1. Let us take the function u = l( . — ). \Va 2 +a?J Make and we shall have a?dx dx ya 2 + oc 2 Va 2 +x 2 aHx but dz= a 2 -f x 2 (a 2 +a?)%' whence, du — DIFFERENTIAL CALCULUS 63 a 2 dx x(a 2 +a?) 2. Take the function u — l\ , — , ; Lvi-h^— VI — x J and make yT+#+ -y/l— x—y, -y/T+x— ^\/l—x=z i which gives u = I (—) — ly — lz, and du = -± . \ z J J v z z/< y z But we have dx dx — dx T zdx , dx dx — dx / /— /- \ dy — t - = -. ( -y/l+X— Vl— X), 2-Vl + x 2^1-* 2Vl-^ V V /' 2Vl — a*' 2V1+J? 2V1-07 2Vl-a^ / Whence, y *" 2yVl-ar 2*Vl-# 2 ' — (y 2 -}-* 2 )^ 2y2:Vl — a? 2 and observing that ?/ 2 -}-2; 2 =:4 and yz = 2x, •L 7 da? we have dw = . ^Vl-^ 2 64 ELEMENTS OF THE 3. U = l(c 0+ -y/ 1 + a ^, 5. u TV a + x + <\/a — x 0. u = / - / / • L V a + a? — Va — #_ dx Vl + x 2 c?w dx Vl-x 2 f/Wr dx Vl + x 2 ^M = adx xVa 2 -*? 60. Let us suppose that we have a function of the form u == (lx) n . Make Za? = z 9 and we have w == z n , du == rcz* ~ ! cfe, and substituting for z and dfe their values, d(lxf= n V x)n ~ l dx. x 61. Let us suppose that we have u = l(lx). Make Ix = z, and we shall have, 7 j dz t dx u = lz, du = — , dz — — ; z x hence, du dx 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 hi = ylz ; iience, — — dylz+y — ■ ; u z or, du = ulzdy + uy — t or by substituting for u its value du = dz y = zHzdy + yz y ~~ 1 dz. Hence, the differential of a function which is equal to a variable root raised to .a poiv-er denoted by a variable ■exponent, is equal to the sum of the differentials ivhich •arise, by differentiating, first under the supposition that the root remains constant, and then under the sup- position that the exponent reiqmns constant (Arts. 55, and 32). 2. Let the function be of the form U F= of. Make, b* = y, and we shall then have (Art. 55), w = o^, du = a v lady; but dy~b*lbdx 7 X tience, du = a h b*lalbdx.. 6* 66 ELEMENTS OF THE 3. Let us take as a last example u = z*, in which z, t, and s, are variables. Make, f = y, we shall then have u = z y , du = z y Izdy + yz y " 1 dz. But dy = fltds + sf ~ l dt ; hence, du = z f lz(t s ltds + sf~ l dt) + Fz^dz, du^/fQtlzds + ^+^y 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) = R sin a cos b — sin b cos a If we subtract the second equation from the first, 2 sin b cos a sin (a + b) — sin (a — b) R and if we make a-\-b = x + h, and a — b = x, we shall have 2 sin — A cos (ff + -^-^) sin (a? + A) — sma? = 5 — , " DIFFERENTIAL CALCULUS. 67 and dividing both members by A, 2sin — hcos(x-\ h) x + h)-smx 2 \ 2 / sin (x + A) h hR sin — A cosfaH h) 2 \ 2 / 1a * ' 2 If we now pass to the limit, the second factor of the COS X second member of the equation will become R • 1 i sin — a 2 In relation to the first factor — z its limit will be unity, -n Rs'ma , sin« cos« ror, tanffa = , whence == ; cos a tang a R Now, since an arc is greater than its sine and less than its tangent* sin a ■ , , sin a * sin a <1, and > a tang a * 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 ABD is equal to ~ABX BD, and the area of the triangle ABC 2 is equal to —AB XBC. But the sector is less A E B & than the triangle being contained within it : hence, ^ABXBD^ABXBC, consequently, BD < BC. DO ELEMENTS OF THE hence, the ratio of the sine divided by the arc is nearer unity than thcLt 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 since __ cos a? > dx R cosxdx hence, dsinx = R 64. Having found the differential of the sine, the diffe- rentials of the other functions of the arc are readily de- duced from it. cos# = sin (90° —a?), dcosx = cZsin(90° — x\ and by the last article, dsin(90° - x) = -^-cos(90° - x)d{§0° - x), R = ——cos (90° — w)dx: R , 7 smxdx hence, dcosx = - — ; R the differential of the cosine in terms of the arc being negative, as it should be, since the cosine and arc are decreasing functions of each other (Art. 31.) DIFFERENTIAL CALCULUS. 65. Since the versed sine of an arc is equal to radius minus the cosine, we have 7 . 1/T . s sin xdx a ver-sin x = a(R — cos x) = R R sin x 66. Since tang x = , we have (Art. 30), cos X 7 R cosxd sina? — R s'mxd cosa? a tang x = (cos 2 j?+ sin 2 x)dx ._ 2 ~~ ' COS^*? but cos 2 a? + sin 2 07 = R 2 : RHx hence, d tango? = R 2 67. Since cot# = , we have tang a? 7 R 2 d tang x R^dx a cot# = ^— 5= 5 5—; tang^a? tangpc cos^a? , # , jR 2 sin' but, tang^a? = — hence, J cot a? = — — 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 sec^ = , we have cos x 7 R 2 d cosa? R sinxdx a sec x = = . - cos^r cos w a? . R sin x , R 2 but, = tang x, and =sec^; cos x cos x , 7 sec a? tanga?c£r hence, a sec a? = =2 . R 2 R 2 69. Since cosec # = — : , we have sin a? * R 2 d sin 5? 7? cos 5? da? a cosec a? = — ■ — = — - — ■ — ; , , cosec a? cotxdx hence, d cosec x — R 2 70. If we make R-\, Arts. .63, 64, 65, 66, 67, will give, d sina?= cosxdx (1), dcosx = — sinxdx (2), d ver sina?= sinxdx (3), <* tanga;= i (4)> cZcotO? = r-r- (5). simr 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 7 Rdu Rdu ax cos x Vli 2 —u 2 If we make cosx = u, we have (Art. 64), _ Rdu Rdu sin a? \/W^u* If we make ver-sina? = u, we have (Art. 65), , Rdu ax = — : . But, s'mx—-\/R 2 —cos' i x, and cosx=R — u, therefore, cos 2 a? = R 2 — 2 Ru + u 2 , hence, sin a? == ^2Ru — u 2 , Rdu and consequently, dx = V2Ru — u 2 > If we make tang x — u, we have (Art. 66) 7 cos 2 x du ax 2 > R . cos x R , cos 2 a7 R 2 R but — =r— = , hence J? sec a? ' J? 2 sec 2 a? J^+tang 2 ^' # 2 dw hence, - da? R 2 + u 2 ' 7£ ELEMENTS OF THE Now, if we make R—l, the four last formulas become , du , du dx = — . dx = Vl-U 2 ' Vl-U 2 ' 7 du , du doc= =, dx — ; V211-11 2 l + u 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. s'm~ l u — the arc of which u is the sine, cos -1 it= the arc of which u is the cosine, tang -1 w — the arc of which u is the tangent, &c. &c. &c. If, for example, we have du x — sin l u, then, dx Vl-u 2 73. We shall now add a few examples. 1 , Let us take a function of the form Make cos x — z, and sin x = y ; then, u = z y , and (Art. 62); du = z v Izdy + yz y ~ x dz: DIFFBRENTIAL CALCULUS. 73 also, dz=—smxdx, and dy = cosxdx: hence, du = z v flz dy + -¥-dz\ = cosrf iax (lcosxcosx )dx, \ coso? / 2. Differentiate the function . _, 7 mdu x — sm mu, ax = Vl — m 2 u 2 3. Differentiate the function #? = cos l (uV 1 — u 2 ) d - (—1 + 2u 2 )du ' V(I37+m 4 )(1-w 2 )' 4. Differentiate the function _ x u j 2du x = tans — , 007 = - 5 2' 4 + w 2 5. Differentiate the function a? = sin 1 (2w Vl — u 2 \ dx = — \ / Vl-u 6. Differentiate the function , x * ydx — xdv u = tang" — , du = Z — —Z-. V y 2 + a? 74. We are enabled by means of Maclaurin's theorem and the differentials of the circular functions, to find the 7 74 ELEMENTS OF THE value of the principal functions of an arc in terms of the arc itself. Let u = F(x) = sin a?: then, du d 2 u . d 3 u — = cosa?, — — .— — smo?, _ .. = -— cosa?, dx dxr dx 6 d*u . d 5 u — — = sin x, ——■ — + cos x. dx* dx If we now render the differential coefficients independent of x, by making x — 0, we have (Art. 49), 17=0, £* = !, u>' = o, £/";=- i, £/"" =0, 17""= + 1: hence, sina? = * 3 | * &c. 1 1.2.3 1.2.3.4.5 75. To develop the cosine in terms of the arc, make u — F(x) = cos x ; then, du . cPu d 3 u — =— sma?, -— = — cos a?, -rr = sma?,- aa? aar oar — —- = COSa?, — — r=— SUIS?, d# 4 dx 5 and rendering the coefficients independent of a?, we have £7=1, 17=0, 17"= -1, £/"'=0, £/"""= 1, 17""'= 0: hence, cos* = 1 - ^ + j-^-^ - &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), du Vl-u 2 developing by the binomial theorem, we find dx i , 1 2 , i-3,,4 , 1.3.5 6 ; - — = H u H u H u 6 4- (fee. du 2 2.4 2.4.6 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 x = Au + Bu 3 + Cu 5 + Du 7 + &c; and the differential coefficient is ^ = A + 3Bu 2 + 5Cu* + 7Du*+6cc du 76 ELEMENTS OF THE But since the differential coefficients are equal to each other, we find, by comparing the series, A = l, **4 C = -i#-, D h35 2.3' ~~2.4.5 ~2.4.6.7' hence, u , 1 u 3 , 1.3u 5 , 1.3.5 7 , . 3C=sm~ 1 u= H — w+ &c. 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), doc 1 du 1 + w 2 which gives = {l + U>) , du hence the function x must be of the form x = Au + Bu z + Cu 5 + Du\ and consequently $• = A + ZBu 2 + 5 Cw 4 + 7Dw 6 ^ DIFFERENTIAL CALCULUS. 77 and by comparing the series, and substituting for A, B, C, &c. their values, we find _! u u 3 , u 5 u 1 ■ „ a? = tanff w = h &c. * 1 3 5 7 If we make x — 45°, u will be equal to 1 ; hence, arc45°=l- — + — - — + &c. 3 5 7 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 — ( — *. * Let a represent the arc whose tangent is — . Then (Trig. Art. XXVI), : 2 tans a 5 tang 2 a ■. tang 4a = 1 — tang 2 a 12 ■ 2 tang 2 a 120 1— tana 2 2a 119 The last number being greater than unity, shows that the arc 4 a ex- ceeds 45°. Making 4c=w?, 45 Q = B, the difference, 4 a — 45° = A — B = b, will have for its tangent , . _ _ % tang A — tang B 1 tang 6= tang (Jl—B) a ■■ . ° — ^—=r = 7^r ; s sv } 1 -f tang A tang B 239* hence, four times the arc whose tangent is -r-, exceeds the arc of 45° by an 1 ' ■ arc whose tangent is ■—- -. 78 But tang ELEMENTS OF THE ♦ -1 1 l tang 1 - r = 1 1 + .i 1 5 5 3.5 3 5.5 5 7.5 7 1 1.1 1 f &C.„ 239 239 3(239) 3 5(239) 5 7(239)' 7-f &c; hence, arc 45° = J A 5 3.£ 1 1 5 3.5 3 5.5 5 7.7 1 (± !_ +_J L_ + >> \239 3(239) 3 5(239) 5 7 (239V / (239) 3 ' 5(239) 5 7(239) 7 Multiplying by 4, we find the semi-circumferenee = 3.141592653. 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 tivo or more variables, and 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 = F(x,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 SO ELEMENTS OF THE and if we suppose x to remain constant and y to vary, the differential coefficient will be d ^ = F'{x,y). (2). 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 — - dx = F f (x, y) dx, and —^dy = F" (x, y) dy. The expressions, du 7 du 7 T* dx ' %**• 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 j(du\ _ d{du) \dx) dx which we designate by d*u t dx * hence, -^ — =- = F"' (x, y). dxdy 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 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 dx n dy m ' indicates that the function u has been differentiated n-\-m times, n times with respect to x, and m times with respect to y. 83. Resuming the function u = F(x,y), if we suppose y to remain constant, and give to x an arbi- trary increment h, we shall have from the theorem of Taylor, r,, , , N duh , d 2 u h 2 , d 3 u h 3 , e 82 ELEMENTS OF THE ■, . , du d 2 u d 3 u mwluch ' "■ s; as? rf?' are functions of a?- and ?/, and dependent on the constants which enter the F(x,y). If we now attribute to y an increment k, the function M, which depends on y, will become die and the function — will become aa? du , d?u k , d 3 u k 2 , d 4 u k 3 (fa? dxdy 1 7 dxdy 2 1.2^ dxdy 3 1.2.3 ' d?u and the function — r-^-, will become (far c^w , d 3 u k , d% k 2 , d 5 w /c 3 . p L Arc da* (far% 1 <fa*% 2 1>8"T cfa 2 ^ 3 1.2.3 * d 3 u and the function -j-j-, will become d 3 u d l u k_ d 5 u k 2 d 6 u k 3 „ ~dx J + dx T dyT + dx 3 dy 2 1.2 + <fa 3 <fy 3 1.2.3 / &c. &c. &c. &c. Substituting these values in the development of F(x + h, y), DIFFERENTIAL CALCULUS. 83 and arranging the terms, we have r/ . , . 7N du k . dht k 2 d 3 u k 3 s F(oc+h,y+k)=u+- — — h— \- &c, dull <Pu lik * d?u hk 2 o dx 1 dxdy l.l dxdy 2 1.2 da? 2 1.2 (fo^y 1.2 (ir 3 1.2.3 + -r^T-^r+&c.; 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 now transpose u = F(x, y) into the first member, and pass to the limit, Ave find d[F (x, y)] = du = — dx-{- j-dy. 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 F (x -f h, y -f 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 OP THE If then an increment I be attributed to z, there will be four terms of the development of the form du , du 7 du 7 u i T h -> ~T k * ~T L ax ay dz If u were a function of four variables, as u — F (x, y, z, s), there would be five terms of the form du 1 du 1 du 7 du u, — h, —k, —I, —on; ax ay dz as and a new variable introduced into the function, would introduce a term containing the first power of its increment into the development. If we transpose u into the first member, and pass to the limit, we shall have d[F{x >y' z)] =TJ x+ Zy d!/+ ^ d *' and , r x du 7 du 7 du , du , d[F{x,y,z,s)} = - [x dx + -dy + Tz az + -d S , from which we may conclude that, the total differential of a function of any number of variables is equal to the sum of the partial differentials. 86. The rule demonstrated in the last article is alone sufficient for the differentiation of every algebraic function. 1. Let u = x 2 -{- y 3 — z; then du —dx=2xdx. 1st partial differential ; ax DIFFERENTIAL CALCULUS. 85 du — dy — 3y 2 dy, 2d partial differential J dy ^dz=-dz, 3d dz hence, du — 2x dx + 3y 2 dy — dz. 2. Let u = xy ; then, du 7 7 —-ax — y ax 3 dx --du = xdy : dy J hence, du — ydx 4- xdij. 3. Let u = x m y n ; then, — dx = ?nx m ~ l y" dx, d> — dy == m/ n ~ l x m dy : hence, du = mx m ~ l y n dx + ny n ~ x x m dy — x m - x y n ~\mydx + nxdy\ x 4. Let u — — , then, y dx = —, ax y du , xdy — dy=: f- dy y 2 , 7 ydx — xdy hence, du = 2 £„ f s 86 ELEMENTS OF THE 5. Let u — - .. ■' = ay (x 2 -\-y 2 ) 2 ; then, Vx 2 + if du 1 ai/xdx -dx = 1 |f (x 2 +y 2 ) du_j_ ady arfdy y (x 2 + y 2 Y (x 2j ry 2 Y , 7 ayxdx — ax 2 dy hence, du = - — -. 6. Let u — xyzt ; then, du = yztdx + xztdy + xytdz -f xyzdt. 7. Let u — z y ; then, pdy = z y lzdy (Art. 55), pdx = yz y ~ l dz (Art. 32): hence, du = z y lzdy -\- y z v ~ x 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 a DIFFERENTIAL CALCTTI/l T S. 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 7 du , du , du — — ax -\- — dy. dx dy Since — and — are functions of x and y. the dx dy du du differentials ~rdx, ~r-dy, must each be differentiated dx dy J with respect to both of the variables; dx and dy being supposed constant: hence, d(-dx\-—dx 2 A- dhl dxdv C \dx dX )-dar dX +d^dy dXdy > j/du 7 \ dhi 7 9 , dhi , 7 and since the second differential of the function is but the differential of the first differential, we have d u d hi d "u dhi = —^ dx 2 + 2 -= — — dx d\j -f -— dy 2 . dxr dxdy dy 1 If we differentiate again, we have d 3 ii, 3 dhi dx 2 dy, d( ( ^Ldx 2 \- U dx* + — d \dx 2dX )~dx^ dX + dx* d{<lp-dxdy\ = 2-^-dxhly + 2-^- 2 dxdy\ v dxdy J dxrdy dxdy z 00 ELEMENTS OF THE and consequently, It is very easy to find the subsequent differentials, by observing the analogy between the partial differentials and the terms of the development of a binomial. We also see that, a function of two variables has tivo partial differential coefficients of the first order T three of the second, four of the third, &c. 88. There are several important results which may be deduced from the general development of the function of two variables (Art. 83). 1st. If we make x = 0, and y — 0, u and each of the differential coefficients will become constant, and we shall have t-i/7 7n I rdu 1 , du 1 \ +T^w + 2 ^V df k ) + &c, which is the development of any function of two variables in terms of their ascending powers, and coefficients which are dependent on the constants that enter the primitive function. 2d. If, in the general development, we make y = 0, and k = 0, we shall have DIFFERENTIAL CALCULUS. 89 _L. 7X , du h , d 2 u h 2 , d 3 u h? , s r(;r+ , )=M+ _ T+ __ + „___ +&c ., which is the theorem of Taylor. 3d. If we make y — 0, k = 0, and x — 0, we have „,-.. du h , d 2 u h 2 , d?u h 3 , s which is the theorem of Maclaurin. Lnplicit Functions. 89. When the relation between a function and its variable is expressed by an equation of the form y = F{x) in which y is entirely disengaged from x, y has been called an explicit, or expressed function of x (Art. 5). When y and x are connected together by an equation of the form F(x,y) = 0, y has been called an implicit, or implied function of x (Art. 5.) It is plain, that in every equation of the form F(x,y) = 0, y must be a function of x, and x of y. For, if the equation were resolved with respect to either of them, the value found would be expressed in terms of the other variable and constant quantities. 90 ELEMENTS OF THE 90. If in the equation u = F(x,y)-0 K 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, % is a function of x and y when they vary in succession. The value, however, which u assumes, when x or y varies, will reduce to when such a value is attributed, to the other variable as will: satisfy the equation F(x,y) = 0.: Now the partial differential du -. -Y-ax^ ax represents the limit of the change which takes place in the- function u under the supposition that x varies (Art.- 81); and the partial differential du , ts the limit of the change which takes place' in the function u under the supposition that y varies. But the change which takes place in u when x and y both vary is :. iience,, -^dx + —dy = &. dx ay 91. In discussing the equation F{x y y)^O h DIFFERENTIAL CALCULUS. 91 it is often necessary to find the differential coefficients of one of the variables regarded as a function of the other, and this may be done without resolving the equation. For, from the last article, du j , du , — dx-\- — dy = 0: dx dy or, hence, du du dy _ dx dy dx du dy _ dx dx du dy 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 of y, taken with a contrary sign. Let us take, as an example, the equation of the circle F(x,y) = a? + y 2 -R 2 = u = 0; a. du , du then, r— =. 2x 3 and — - = 2y : dx dy J hence, dy _ x dx y 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 circle, we have y= VlV-x 2 , 92 ELEMENTS OF THE dy _ X hence, dx Vj^ 2 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 q, the third by r, &c, we shall have dp dq . z. 93. To find the second differential coefficient in the circle, we have dy x dx y ■ j (dy\ — ydx -f- xdy t Kdx) ~ y 2 , dy d?y " V ^ X Tx i hence, -^ == - » dx 2 y 2 and by substituting for —■ its value - X , we have y d 2 y x 2 -f y 2 dx 2 ~ y 3 1. Find the first differential coefficient of y y in the equation y 2 — 2mxy -f- x 2 — a 2 — u = 0, du , rt du —- = — 2my + 2x< — — 2v — 2mx: dx y dy y DIFFERENTIAL CALCULUS. 93 1 fy _ r—2my-\-2x~\_ my — a? dx L 2 y — 2 ??za? J y — ma? 2. Find the first differential coefficient of y in the equation y 2 + 2077/ + a? - a 2 = 0. dx 3. Find the first and second differential coefficients of y> in the equation y 3 — 3 axy -f x 3 = 0, du dx = Sx 2 o du n - 3 ax, hence, <&2 dx 3 x 2 — 3ay ay — a? 3y 2 —3ax y 2 — ax For the second differential coefficient, we have , d v oJ\ f Mi ^f^Jy dhj (y 2 — ax) (a -^ — 2a? J — (ay — x 2 ) (%y C -f- — a\ or, by substituting for -f- its value, and reducing, dx 2 (y 2 — axf dy dx d?y 2 a?y 4 — 6 ax 2 y 2 + 2 yaf_ + 2 a 3 a?y dx 2 ~ (y 2 - ax) 3 ' 2xy ( y 3 — 3 tfajy + a? 3 ) + 2a 3 a?y _ (y 2 - ax} 3 but from the given equation y 3 — 3axy -f a? 3 = 0. , d 2 v 2a 3 xy hence, -±4 = - — — ?— aar (y — axy \i 94 ELEMENTS OE THE Differential Equations of Curves. 94. The Differential Calculus enables us to free an equation of its constants, and to find a new equation which shall only involve the variables and their differentials. If, for example, we take the equation of a straight line y = ax-\- b, and differentiate it, we find dy _ dx and by differentiating again, The last equation is entirely independent of the values of a and b, and hence, is equally applicable to every straight line which can be drawn in the plane of the co- ordinate axes. It is called, the differential equation of lines of the first order. 95. If we take the equation of the circle x 2 + y 2 = R 2 , and differentiate it, we find ocdx + ydy ~ 0. This equation is independent of the value of the radius R, and hence it belongs equally to every circle whose centre is at the origin of co-ordinates. f= 2Rx -x?; 2R _f + x\ J X DIFFERENTIAL CALCULUS. 05 96. If the origin of co-ordinates be taken in the circum- ference, the equation of the circle (An. Geom. Bk. Ill, Prop. I, Sch. 3) is from which we find and by differentiating, x ( 2 ydy -f- 2 xdx ) — ( y 2 + x 2 ) dx or by reducing (x 2 — y 2 ) dx-\-2xydy = 0, which is the differential equation of the circle when the origin of co-ordinates is in the circumference. The last equation may be found in another manner. If we differentiate the equation of the circle, y 2 = 2 Rx — x 2 , we have, after dividing by 2 ydy = Rdx — xdx ; . „ ydy + xdx hence, R — y J 7 . dx If this value of R be substituted in the equation of the circle, we have (x 2 — y 2 )dx + 2xydy == ; the same differential equation as found by the first method. 96 ELEMENTS OF THE 97. If we take the general equation of lines of the second order (A.n. Geom. Bk. VI. Prop. XII, Sch. 3), y 2 — mx + nx 2 , and differentiate it, we find 2ydy = mdx + 2nxdx ; differentiating again, regarding dx as constant, we have, after dividing by 2, dy 2 + yd 2 y = ndx 2 . Eliminating m and n from the three equations, we obtain y 2 dx 2 + x 2 dy 2 — 2xy dxdy + yx 2 d 2 y = 0, which is the general differential equation of lines of the second order. 98. In order to free an equation of its constants, it will be necessary to differentiate it as many times as there are constants to be eliminated. For, two equations are neces- sary to eliminate a single constant, three to eliminate two constants, four to eliminate three constants, &c. : hence, one constant may be eliminated from the given equation and the first differential equation ; two from the given equa- tion and the first and second differential equations, &c. 99. The differential equation which is obtained after the constants are eliminated, belongs to a sjiecies or order of lines, of which the given equation represents one of the species. Thus, the differential equation (Art. 94), DIFFERENTIAL CALCULUS. 97 belongs to an order or species of lines of which the equation y zzzo.x -r b, represents a single one, for given values of a and b. The equation of a parabola is if = 2px, and the differential equation of the species is _ 2 xdy — ydx — 0, or dy 2 -f yd-y = 0. 100. The differential equation of a species, expresses the law by which the variable co-ordinates change their values; and this equation ought, therefore, to be indepen- dent of the constants which determine the 77iag7iiiv.de, and not the nature of the curve. 101. The terms of an equation may be freed from their exponents, by differentiating the equation and then com- bining the differential and given equations. Suppose, for example, P and Q being any functions of x and y. By differentiating, we obtain nP n - l dP=dQ: by multiplying both members by P, we have nP n dP = PdQ, and b}' substituting for P" its value, 7iQdP = PdQ. 9 98 ELEMENTS OF THE The same result might also have been obtained by taking the logarithms of both members of the equation For, we have and (Art. 57). P n =Q. nlP = lQ, dP dQ hence, nQdP = PdQ. Of Vanishing Fractions, or those which take the form — . J 102. It has been shown in (Alg. Art. Ill), that — is sometimes an undetermined symbol, and that its value may be 0, a finite quantity, or infinite. This symbol arises from the presence of a common factor in the numerator and denominator, which, becoming for a particular value of the variable, reduces the fraction to the form — . If we have, for example, a fraction of the form P{x-a) m Q(x-a) n ' in which P and Q are finite quantities, and make x = Oy we shall have P{x-a) m _ Q Q{x-a) n ~ 0' DIFFERENTIAL CALCULUS. 99 The value of this fraction will, however, be 0, finite or infinite, according as m>n, m = n, ?n<n, for under these suppositions, respectively, it takes the form P(x-a) m - n P P ~Q J Q' Q{x-a)"- M ' Let the numerator of the proposed fraction be desig- nated by X, and the denominator by X', and let us sup- pose an arbitrary increment h to be given to x. The numerator and denominator will then become a function of a? -f h, and we shall have from the theorem of Taylor Y . dx h ; d 2 x h 2 , (Px K 3 ; . , dX h , d 2 X' h 2 d*X h 3 + dx 1 ' dx 2 l/2 + dx 3 1.2.3 +&C ' If the value of x = a, reduces to the differential coefficients in the numerator as far as the ?nth order, and those of the denominator as far as the nth. order, the value of the fraction will become, d m X h" dx m 1.2.3.4.. h &c, . .171 d n X If + &C. If we make h = 0, the value of the fraction will be- come 0, finite, or infinite according as m>n, ?n=zn, m < n, and hence, if the value x = a, reduces to the same number of differential coefficients in the numerator and 100 ELEMENTS OF THE denominator, the value of the fraction will be finite and equal to the ratio of the first differential coefficients which do not reduce to 0. 103. Let us now illustrate this theory by examples. 1. If in the fraction l-x n 1-x' we make x— 1, we have — . But dX n l dX = -nx n \ -— = - ax ax -i; in which, if we make x= 1, we have dX , dX — — = — n 9 and —r— = ax ax ~h , dX hence, — , — ■ dx — n. dX dx 7 therefore, the value of the fraction when x = 1, is +ft. 2. Find the value of the fraction ax 2 — 2 acx -f- ac 2 , — r-r-. when x — c, bx 2 — 2bcx-\-bc 2 4^ = 2ax-2ac, ^- = 2bx-2bc, dx dx both of which become 0, when x = c. Differentiating again, we have d?X <M' , -aW= 2a > -^r =2h; hence, the true value of the fraction when x = c is ~. DIFFERENTIAL CALCULUS. 101 3. Find the value of the fraction x 3 — ax 2 — a 2 x + « 3 i , when x — Ans. 0. -, when oo = a. or — a 2 4. Find the value of the fraction ax — x 2 , — 5 g -, when x=za. a i —2a 3 x + 2ax 3 — x* Ans. oo. 5. Find the value of , when x = 0. x Ans. la — lb. 6. What is the value of the fraction 1 — sin# + cos.a? , nnn - ., when 37 = 90°. sin^-h cos x — I Ans, 1. 7. What is the value of the fraction a — x — ala -f- alx , -j — , wnen x = a. a— V2ax — x 2 Ans. — 1. 8. What is the value of the fraction *-, when a? =1, \ — x + lx Ans. — 2. 9. What is the value of tke fraction Q n /£« ^ =-, when,a? = a. ia — Iz Ans. nc? e 9* 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 x renders every coefficient either or infinite. Such functions are of the form (a? -a 8 )- (x - a ) n ' in which m and n are fractional. In functions of this form we substitute for x, a-\-h 9 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 x, we have in arranging according to the ascending powers of h, F{a + h) _ Alt + Bh h + Ch* + &c, F\a + h)~ A'lt' + Bh b ' + Ch* + &c. Now there are three cases, viz. : when a > a\ a = a', a < a' . 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 x, in the fraction i. ' (x-ay DIFFERENTIAL CALCULUS. 103 , (2ah + h 2 y ,_ , ,J. we have \ — J — = (2a-\-h) 2 , and by making h = 0. which renders x = a, the value of the fraction becomes (2a)\ 2. Required the value of the fraction a (a?-3ax+2a 2 y . ; — - — when x = a. (x'-cfy Substituting a + h for x, we have .1 1 1 1 /** (3a 2 + Sah + S 2 ) 8 (3a 2 + 3a/* + h 2 )* which is equal to 0, when h—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 X T' to become infinite by the supposition of x = a. The fraction can be placed under the form 104 ELEMENTS OF THE X X 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 PQ=: which becomes — when x = a. 108. Let us take, as an example, the function (l-tf)tang~K-a?; in which t designates 180°. If we make x=l, the first factor becomes 0, and the second infinite. But 1 1 tang — 7tx = 2 ♦ 1 COt 7TX 2 hence, (1 — a?) tang — ttx = , cot — a-a? 2 2 the value of which is — when # = 1. it DIFFERENTIAL CALCULUS. 105 CHAPTER V Of the Maxima and Minima of a Function of a Single Variable. 109. If we have u = F(x), 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 f , u, u". Now, when u is greater than both u f and u f/ , u is said to be a maximum : and when u is less than both v! and u" y 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 follows. 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 u = F(x), the variable x be first increased by h, and then diminished by h, we shall have (Art. 44), r,r ''. x\ (lull d 2 U ll 2 , d 3 U h 3 , s v ; dx 1 dx 2 1.2 dx 3 1.2.3 ,, ^, 7X du h , c/ 2 ^ It" dhi li 3 , r, u"= Fix — li) — u — - h —7-7- -r-r — : h &c; v ; dx I dx 2 1.2 cte 3 1.2.3 and consequently, du h d?u It? j_ d 3 w A 3 - o du h d 2 u h 2 d 3 u h 3 o dxT d~aFT^2 ~~ ^ 3 1.2.3 Now, if w has a maximum value, the limits of v! '—u and u" — u, will both be negative ; and if u is a minimum, the limits of u'—u and u"—u will both be positive. Hence, in order that a may have a maximum or minimum value, the signs of the limits 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, have contrary signs, it follows that the limits of the developments will have contrary signs (Art. 44) ; hence, the function u can neither have a maxi- mum or a minimum unless ^ = 0; 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 limits of the developments will depend on the sign of tl.e second differential coefficient. But since the signs of these limits are both negative when u is a maximum, arid both positive when u is a minimum, it follows that the second differential coefficient will be negative when the function is a maximum, and positive when it is a minimum. Hence, the roots of the equation '- = o, 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 fol- lows that there may be more than one maximum or one minimum. But if the roota of the equation ax reduce the second differential coefficient to 0, the signs of the limits 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 maximum or 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 I OS ELEMENTS OF TIU 7 = o, ax 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 particular 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 render the product or quotient a maximum or a minimum, and hence the constant may be neglected. 2. Any value of the variable which will render the func- tion a maximum or a minimum, will also render any root or power a maximum or a minimum ; and hence, if a func- tion is under a radical, the radical may be omitted. DIFFERENTIAL CALCULUS. 109 EXAMPLES. I . To find the value of x which will render y a maxi- mum or a minimum in the equation of the circle f + x 2 = R\ dy __ x y dx x making — 0, gives x — 0. The second differential coefficient is d?y _ x 2 -;- y 2 dx 2 ~ f ' and since making x = 0, gives y = R, we have dx 2 ~ 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 -f x% differentiating, we find g=-6 + 2 ,, and 5 = 2, making, — b + 2 a? = 0, gives w = — ; A and since the second differential coefficient is positive, this value of x ™;il render y a mir> ; ^um. The minimum 10 110 ELEMENTS OF THE value of y is found by substituting the value of x t in the primitive equation. It is 3. Find the value of x which will render the function u — a* + b*x — c 2 x 2 , a maximum or a minimum, du ,. n ., , IP — = b J — 2c x. hence x = — g ; dx 2c 2 ' and, --— = — 2c 2 : ciatr hence, the function is a maximum, and the maximum value is 4. Let us take the function u = 3 a 2 x 3 — ¥x -\- c 5 , du b 2 we find -Y- — 9 a V — Z> 4 , and # = ± — . ax da The second differential coefficient is d 2 u 10 2 V-5 = 18 era? a a?* Substituting the plus root of a?, we have LTFFERENTIAL CALCULUS. Ill which gives a minimum, and substituting the negative root, we have which gives a maximum. The minimum value of the function is, 1 2b" and the maximum value 9a' 9a 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 x, 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 simpliiied. For, if — = A4 , dx we shall have dhi _ XdX XdX' dor dx dx But by hypothesis A" reduces to for that value of x which renders the function u a maximum or a minimum : d 2 u X'dX benC6 ' M = -dx~ ; 112 ELEMENTS OP 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 themth power of one of the parts multiplied by the nih power of the other shall be a maximum or a minimum. Designate the given quantity by a and one of the parts by x, 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 — x) n y whence, -^ = mx m ~ x (a - x) n - nx m (a - a?)- 1 , = (ma — mx — nx}x m ~ l (a — a?) B_ V and by placing each of the factors equal to 0, we have ma . x — , x — 0, x — a. m-\-n The second differential coefficient corresponding to the first of these values, found by the method just explained,, is °^-=-{m + n)x m ~ l {a - x) n ~ x ; ax and substituting for x its value, it becomes i+n-2 (m-\- n) 1 ifn- 3 hence, this value of x renders the product a maximums 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 yi _ 2vixy + x 2 — a 2 — 0. The first differential coefficient is dy _ my — x < dx y — mx ' x hence, my — x = 0, or y = — . m Substituting this value of y in the given equation, we find ma VT3 ,2 and the value of y corresponding to this value of x is a Vl — m l To determine whether y is a maximum or a minimum, let us pass to the second differential coefficient. We have -f- — {iny — x)(y — mx)~ x ; ax d?y (-2-0 hence, dx 2 y — mx 10 114 ELEMENTS OF THE and since -f- = y we have ax dx 2 y — mx and by substituting for y and x their values, we have *y = i . 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 = xy = the area of the rectangle. But b : h : : y : h — x: _, bh — bx lience, y — — T and consequently,. bhx — bx 2 b ,, ™ u — - = —(kx - Of) o /i a and omitting the constant factor, d u -l o ^ — = h — 2x. or #=: — : <fc 2 hence, the altitude of the rectangle is equal to half the altitude of the triangle : and since do? the area is a maximum. 2, DIFFERENTIAL CALCULUS. 115 8. What is the altitude of a cylinder inscribed in a given right cone, when the solidity of the cylinder is a maximum ? Ans. One third the altitude of the cone. 9. What are the sides of the maximum rectangle in- scribed in a given circle ? Ans. Each equal to R V2. 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. 3 12. To find the maximum right-angled triangle which can be described on a given "rrae. Ans. When the two sides 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. 116 ELEMENTS OP THE CHAPTER VI Application of the Differential Calculus to the Theory of Curves. 113. It has been shown in (Art. 13), that every relation between a function and a single variable on which it depends, may subsist between the ordinate and abscissa of a curve. Hence, if we represent the ordinate of a curve by a function ?/,• the abscissa may be represented by the variable x. Of Tangents and Normals. 114. We have seen (Art. 15), that if y represents the ordinate and x the ab- scissa of any curve as CP, the tangent of the angle PTA, which the tangent forms with the axis of ab- scissas will be represented R N by dy dx~' dy and dx being the differentials of the ordinate and ab- scissa of the point of contact P. DIFFERENTIAL CALCULUS. 117 But wc have (Trig. Th. II), 1 : TR :: tangT : RP; that is, 1 : TR :: &■ : y: hence. T.R — y—- — sub-tangent. 115. The tangent TP is equal to the square root of the sum of the squares of TR and RP ; hence, \/l TP = y\/ 1+^= tangent. 116. From the similar triangles TPR, RPN, we have TR : PR : : PR : RN, dx hence, ^ t— : y : : y : RN, dij consequently, RN = y~ = sub-normal. 117. The normal PN is equal to the square root of the sum of the squares of PR and RN ; hence, PN = y V 1 + -^1 = normal. 118. Let it be now required to apply these formulas to lines of the second order, of which the general equation (An. Geom. Bk. VI, Prop. XII, Sch. 3), is, y 2 — mx + nx 2 . Differentiating, we have dy _m-\-2 nx _ m -\- 2 nx dx 2y 2Vmx + nx 2 118 ELEMENTS OF THE substituting this value, we find , „, r> dx 2(mx + nx 2 ) sub-tangent TR = y— = -± - ' ° y dy m + 2nx | mx TP = y\/l + ~~=\/mx + nx 2 + v ay* y \_m + 2nx sub-normal RN = y-f = ■: J dx 2 PN = y\/l + ^=\/: mx + nx 2 + — (m + 2nx) 2 . 4 By attributing proper values to m and n, the above formulas will become applicable to each of the conic sections. In the case of the parabola, n = 0, and we have TR = 2x, TP=V mx -f- 4 x 2 , RN =f, PN w mx -\- 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 x' ', y' . Then, for the common point P, we shall have x = x f , y = y'. DIFFERENTIAL CALCULUS. 119 If we represent BG, the increment of the abscissa, by h, we shall have, from the theorem of Taylor (Art. 44), CG-PB=CF=^-^A + d -ft-^-+ &c, dx 1 dor 1.2 dx 3 1.2.3 CG-PB=CF=%± + %^%-^- + &c; do/ 1 da/ 2 1.2 3Pl>2.3 hence, by placing the two members equal to each othgr, and dividing by h, we have dy d 2 i/ Ji_ '• dy r d?y f h - <fc + ^1.2 ' '~"Z7' + "^" 1.2 + If we now pass to the limit, by making h = 0, we shall have cAr da/ 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 at a common point, when the co-ordinates and first differential coefficient of the one, are equal to the co-ordinates and, first differential coefficient of the other. 120. The equation of a straight line is of the form y — ax + b, hence, -f- — a. dx But the equation of a straight line passing through a given point, of which the co-ordinates are a/' \ y /f , is (An. Geom. Bk. II, Prop. IV), y-y"=a(x-a/'), 120 ELEMENTS OF THE or by substituting for a its value, we have, for the equation of a straight line passing through a given point, y-y"=||(*-*"). This line may be made tangent to a curve at any point of which the co-ordinates are x ;/ , y" , by substituting for ~% the first differential coefficient found from the equation ax of the curve, and making x'\ y l/ , equal to a/ ,: \ y" of the curve. 121. Let it be required, for example, to make the line tangent to a circle at a ooint of which the co-ordinates are a/', y" . Since the co-ordinates of this point will satisfy the equation of the curve, we have x" 2 + y" 2 = R 2 and by differentiating, dx" " x" and by substituting this value in the equation of the line, and recollecting that x' n -\-y ff2 ~ R 2 , we have yy"+xx"=R\ which is the equation of a tangent line to a circle. 122. A normal line is perpendicular to the tangent at the point of contact, and since the equation of the tangent is of the form DIFFERENTIAL CALCULLS. 121 the equation of the normal will be of the form (An. Geom. Bk. II, Prop. VII, Sch. 2), and this line will become normal to a curve at a point of dx" which the co-ordinates are x' f , y", if the value of — — be ay" found from the equation of the curve, and substituted for — , and the co-ordinates a/ f f y" of the straight line be ay made equal to x /f , y n of the curve. The equation of the normal in the circle will take the form, y» 123. To find the equation of a tangent line to an ellipse at a point of which the co-ordinates are a#, y ff , we have, Ay /2 + ^V /2 = A 2 5 2 . By differentiating* we have dy" _ B 2 x ,f m do/' ~~~AY~ ; hence, we have which becomes, after reducing, A 2 yy ff -\-B 2 xx"=A 2 B 2 . The equation of the normal is ii 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 y rr2 z=z mx" + nx" 2 . By differentiating, we have m + 2 nx" dx" 2y' hence, the equation of the tangent to a line of the second order is t// y-y = — ^77 — (*--*"), and the equation of the normal 2y" y-y"=- m + 2 nx 1 ' -,{x-x"\ 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 E the co-ordinate axes, and y-y'^%^-^' the equation of any tan- gent line, as TP. DIFFERENTIAL CALCULUS. 123 If in the equation of the tangent, we make in succes- sion y = 0, x=0, we shall find x-Al-x y dy//i y-AJJ-y -oC d ^. 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 it and Y, and will coincide with them when the co-ordinates of the point of tangency are infinite. In order, therefore, to determine if a curve have asymp- totes, we make, in succession, x = oo and y = oo in the values of A T, AD. If either of these 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- di/ mined by finding the value of -^-, under the supposition that the co-ordinates are infinite. 126. Let us now examine the equation y 2 = moc -\- nx 2 , 124 ELEMENTS OF THE of lines of the second order, and see if these lines have asymptotes. We find A rr 2if — mx AT—x - = , m-\-2nx m-\-2nx . r. mx 4- 2 nx 2 mx AD = y- 2y 2 Vmx + nx 2 which may be put under the forms AT = ~ m , Al)± + 2n 2\J- + n and making x = qo , we have AR=-^-, and i^=-^=. 2n ^ 2 Vn 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, AE, become finite. If we substitute for n its value — , we shall have A z AR=-A, and AE=±B. DIFFERENTIAL CALCULUS. 125 Differentials of the Arcs and Areas of Segments of Curves. 127. 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. 16), N and in which PQ = h, MQ = (P+P / h)h; dx' R Hence, PM= '\f^+{P-\-P'hfK 2 =h-y/l^{P-{-P , h) 2 . We also have NQ = Ph ; heiice, PN = Vh 2 + P 2 h 2 = h Vl + P 2 , NM= NQ -MQ=- P f fc hence, we have PN+MN hVl+PZ-P'h 2 Vl + P 2 -P'h PM Wl+(P + P'h) 2 Vl + (P -f P'hf ' 126 ELEMENTS OF THE of which the limit, by making h — 0, is VT+W = 1 VI -f P 2 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 PM ~ ' and consequently, the differential of the arc is equal to the differential of the chord. But when we pass to the limit of the arc and chord, PM becomes the differential of the chord, and PQ and QM, become the differentials of x and y ; hence, if we represent the arc by z, we shall have dz = -y/dx 2 + df~: 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. 129. To determine the differential of the arc of a circle of which the equation is x 2 + y 2 = R 2 , xdx we have xdx -f- ydy = 0, or dy — ; / x d Or dx ———^—— -whence, dz=y dx 2 + — y~ — — Vx 2 + y 2 , Rdx _ Rdx Vi^ 2 -ar ^, DIFFERENTIAL CALCULUS. 127 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). G M 130. Let B CD be any segment of a curve, and let it be required to find the differential of its area. The two rectangles DCFE, DGME, having the same base DE, are to each other as DC to EM ; and hence, the limit of their D E 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 DGME, and greater than the rectangle DCFE : hence, the limit of its ratio to either of them will be unity. But, DCME DCME DEFC_ DCME _j^n DE ~ DE X DEFC ~ DL * DEFC y ' X or by representing the area of the segment by s and the ordinate DC by y, and passing to the limit, we have ds dx or ds = ydx ; 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 o? + f = R\ and y = VR'-x 2 ; hence, ds — dx yR z — x 2 . The differential of the segment of an ellipse, is B ds = -—dx^/A 2 —x 2 , A. and of the segment of a parabola ds = dxV2px. 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. 15). We now propose to show the relation between a curve and the second differential coefficient, the ordinate being regarded 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 DIFFERENTIAL CALCULUS. 129 axis of abscissas. Then will M'Q = NQ r , and we shall have PM = y, dy h d 2 y h 2 P"M"=y-\ y +T x T dor 1.2 d»/2h ' d 2 y ilt dx\ dx 2 1.2 + &c, + &c, P'M- PM = M'Q =^A+i^JL+ &c ., dx 1 dx 2 1.2 ■p"M !, -P r M' = M"Q= ^h + 4^— + &c. ax dor 1 . 2 M 1 ' Q' - M> Q = + JIT^iV - -^r # + &c. ptar 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 if we pass to the limit, 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 ordinate s will be negative, and it is easily seen that Ave shall then fiave P P' I Ml A Q M r 5 **^^ M> Q' N M" \ M ,r Q f -MQ dx 2 130 ELEMENTS OF THE Now, since the first member 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. 133. Let us now con- sider the curve CMM'M", which is concave towards the axis of abscissas. We shall have, N / M" M y- ^ Q m/ / C Q A P 1 3/ J u runir? , dy h , d 2 y h 2 , s dx 1 dx 2 1.2 tv/ tit// , dy2h , d?y 4/* 2 , p * ■ dx I dx 2 1.2 P ff M ,f -P'M f = M fr Q f = C ^-h-{-~l-~ + &c, dx dx- 1 . 2 M'Q-MQ = - NM"=-^X-h 2 + &c. dor 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 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>Q + NM // =^K i -\- &c; a-ar 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 of a curve, as M. The first differential of y is the limit of the difference between the ordinates PM, P f M f , or the difference between two consecutive ordinates. The second differential of y is the limit of M~ f N, and is derived from M''Q or dy, in the same way that dy is derived from the primitive function. The abscissa x being supposed to increase uniformly, the difference, and conse- quently the limit of the difference between PP' and P'P !f is : therefore its second differential is 0. The co-ordi- nates x and y, and the first and second differentials deter- mine three points, M, M, M" , consecutive with each other, 135. Remark 2. When the curve is convex towards the axis of abscissa, the first differential coefficient, which 132 ELEMENTS OF THE 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 distin- guished by some particular property not enjoyed by the points of the curve in general : such as, the point at which the tangent is parallel, or perpendicular to, the axis of abscissas. 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 ax 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 _ doc _ dx dy DIFFERENTIAL CALCULUS. r33 will give lire abscissas of all the points at which the tan- gent is perpendicular to the axis of abscissas. 13S. 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 d 2 y rPi/ dx> = °> 0T -dJ = ™ will give the abscissas of the points of inflexion. 139. Let us now apply these principles in discussing the equation of the circle x 2 + y 2 = R 2 , vt e nave, u y u J S> and placing dy dx; X > y - X y =o, we \ haye x = 0. Substituting this value in the equation of the curve, we have y = ±R; 12 134 ELEMENTS OF THE hence, the tangent is parallel to the axis of abscissas at the two points where the axis of orclinates intersects the circumference. If we make dx X y ~ 00, or _.y,= X 0, we have y = : ; subs 3titutino[ 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 positive 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 = b±c(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 dx J- </__ dar ??w(x — a) T m(?n — l)c(x — a) m 2 . dy If we place the value — = 0, we find x = a, and sub- dx stituting this value in the equation of the curve, we find y = b : hence, x = a, y = b, 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 differential coefficient will be posi- tive for all values of cc. The curve will therefore be convex towards the axis of A", 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 A", but cuts the axis of Y at a distance from the origin expressed by y = b + ca m . 136 ELEMENTS OF THE 141. 2d. When c is negative, and m entire and even. We shall have, by differentiating, ( ^.= - 7 nc(x-a) m - 1 doc and ^ = _ W j( wl -l) c ( a? -fl)»- • dx 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, fenders 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-, ^-=±mc{x-a) m -\ dx <Py and — J = ±m{m — \) c(x — a) T ' 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 the exponent m — 2 is uneven, the factor (x— a) m ~' 2 will be negative when x < a, and positive when x > 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 x < a, and positive for x > 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 2 fraction having an even numerator, as m = — . By differentiating, and supposing c positive, we have 2c dy 2 y ~- = — c\x — a) dx 3 v ; 3(x — ay d?l_ dx 2 ~ 2c a » 9(x-ay If we make x = a, the first differential coefficient will become infinite ; and the tangent will be perpendicular to 12* 138 ELEMENTS OF THE trie 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 t 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=zb + c (x — a) 3 , and make x = a -\- h t and x = a — h, we shall have, in either case, y — b-\- ch 3 ; and hence, y will be less fov_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 y — b — c(x — a) 3 ; and we should have, by differentiating, dy 2c and 3{x — a) 3 &y__ 2c aar 9{x-ay 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) 3 , we make x — a -f li, and x — a — h, we shall have, in either case, y = b — ch 3 ; 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 — a, 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 characterized by g=°> the other by dy dx 00 . 140 ELEMENTS OF THE In the first, we determine whether the function is a maximum or a minimum by examining the subsequent differential coefficients ; 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 eolation y — b ± c{x — a) 3 , 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 , M, 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 vositive or negative and m a 3 fraction having an even denominator, as m = — . Under this supposition the equation of the curve will become 3 y = b± c(x — a) 4 , and by differentiating, we have dy _ 3 c dx 4(a?-a)V - &y 3c a - 4.40r-a)7 DIFFERENTIAL CALCULUS. 141 M The curve represented by this equation 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 l\~ convex. Every value of x less than a will render y imaginary. The co-ordinates of the point My 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- 3^ 5 Under this supposition the equation will become y-- — b±c(x -ay, and by differ entiating, we have dx 3 c 50- % 9 -a) 5 d?y dot?' 3 2c 7 5.5(,t — a)' 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 a? = a, and that the curve will be concave for x > a. Ef 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 (y — x 2 ) 2 = x 5 , - furnishes an example of this kind. By extracting the square root of both members and transposing, we have d 2 ij dx 2 2 2 and by differentiating -£ = 2x± — x 2 . 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 „<!!*-, and concave for x>-^- and ,<l. At the last point the curve passes below the axis of abscissas and becomes convex towards it. If Ave 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. At this point the differential coefficient of the second order is positive for both branches of the curve, hence they 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 z=b±(x — a) Vx — c. By differentiating, we obtain d?/ , x — a dx 2^x-c We see, from the equa- tion of the curve, that y will be imaginary for all values of x less than c. For x — c, we have y~h ; 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 4- Va- 144 ELEMENTS OF THE the other, an angle whose tangent is — yfl — c ' 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 Gif — x 3 -\- bx 2 = 0, will be satisfied for the values oc= ± 0, y = ±0 ; 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 and hence, y will be imaginary for all negative values of x, and for all positive values between the limits x = and oc = b. For all positive values of x greater than b } the values of y will be real. The first differential coefficient is dy _ x(Sx—2b) <te~2Vax 2 (x-b) ' DIFFERENTIAL CALCULUS. 145 or by dividing by the common factor x, dy _ Sx-2b df ~ 2 Va{z — b) and making x = 0, there results dy_ 26 dx 2 V—ab ' which is imaginary, as it should be, since there is no point 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 conclusions from the preceding discussion. 1st. The equation ^ = 0, determines the points at ax which the tangents are parallel to the axis of abscissas. 2d. The equation -~ == qd , determines the points of 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 -5-^ = 0, or -^-=00 determines n dx 2 dx 2 the points of inflexion. 4th. The equation -f- = an imaginary constant, in- dicates a conjugate point. 13 146 ELEMENTS OF THE CHAPTER VII. Of Osculatory Curves — Of Evolutes. 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 Oy 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 general 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' , and the co-ordinates of the third EPF by x' f , y" . If we designate the common ordinate PR by y, if, y" , we shall then have >/ '__ , dy h d 2 y If 9 R' = y + + dhj do: 1 da?1.2 dx 3 1.2.3 4- &c., sR* dy' h &y' h 2 <Pi/ h 3 da/T da? 2 1.2 dx r3 1.2.3 &c; nR! y V dx" 1 * d?y" h 2 _^ d 3 y" h? dx' 2 1.2 da?" 3 1.2.3 &c. But since the curves are tangent to each other at the point P, we have (Art. 119), 148 ELEMENTS OF THE y=y'=y"> and %=%=%■■ hence > A-. a -^( $"y <Py'\ h 2 ( d 3 y dy \ h 3 „ q \dx 2 da/ 2 ) 1.2^ \dx 3 (fa/Vl.2.3 + ' * U 2 ^Vli 1 ^^ dx"V\.2.2>^ Now, in order that the first curve APB shall approach more nearly to the second CPD than to the third EPF, we must have d<d', and consequently, h 2 h 3 l? h 3 A— + B-^— + &c, < A'— + B'-^— + &c, 1.2 1.2.3^ '-> i:sT 1.2.3 '. in which we have represented the coefficients in the first seiies by A, B, C, &c.-, and the coefiicients in the second by A', B' 9 'o\ &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 h 3 h 2 h 3 6—^— + &c, < A' — + B r -±— n + &c, 1.2.3 1.2 1.2.3 and by dividing by h\ B_^_-j-&c.,<A' — + B f -^—- +&c, 1.2.3 ' 1.2 1.2.3 and by passing to the limit DIFFERENTIAL CALCULUS. 149 But when A = Q, we have dy _ jy da? ~ do/ 2 ' 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 13* 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 + 1 constants, a contact of the mth order. It is plain, that in each of the foregoing cases the highest order of contact is determined. The line which 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 osculatory to a curve, at a given point of which the co-ordinates are a/ r , 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 a and b as to cause the line to fulfil the conditions, x = a/>, y = y", and % = %■ DIFFERENTIAL CALCULUS. 151 By differentiating the equation of the line, we have ax and since the line passes through the point of osculation y — y" — -j-(oc — xf f ). Substituting for ~ its value -tit, we have for the equation of the osculatrix. In the equation of the circle a/ f2 + y" 2 = R\ dy" cc" we find ±T=-yr hence, the equation of the osculatrix of the first order, to the circle, is x" y-y ! '=--rM--x"\ or by reducing yy" -f xx" = R 2 . 154. If et, and £ represent the co-ordinates of the centre of a circle, its equation will be of the form ( X -ccf-\-(y-^f = R 2 . If this equation be twice differentiated, we shall have, (x — «)dx + (y — /3) dy = 0, dx 2 -i-dy 2 + (y-^(Ty = 0; 152 ELEMENTS OF THE and by combining the three equations we obtair dx 2 -f dy 2 y-P = d 2 y ' dy /dx 2 4 dy c< dx \ dhj R=± {dx 2 4- dy 2 ) ] dxd 2 y 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" f we have only to substitute in the three last equations, the values of dy dx dy^ dx" d 2 y _ ffy" "dx 2 ~dx 772 deduced from the equation of the curve, and to suppose, at the same time, the co-ordinates x and y in the curve to become equal to those of x and y in the circle. If we suppose x ;/ , y", to be general co-ordinates of the curve, the circle will move around the curve and become osculatory to it, at each of its points in succession. 155. If the circle CD be osculatory to the curve EF, at the point P, we shall have qsz=Cx + W 1.2.3 for h positive ; and h 3 q's' Cx - 1.2,3 &c. &c, DIFFERENTIAL CALCULUS. 153 for h negative: hence, the two lines qs, c/s', have contrary signs. The curve, therefore, lies above the oscillatory cir- cle on one side of the point P, and below it on the other, and consequently, divides the oscillatory circle at the point of osculation. Hence, also, the oscillatory circle separates the tangent circles which lie without the curve from those which lie within it (Art. 151). In every osculatrix of an even order the first term in the values of qs, qW, will, in general, contain an uneven power of h ; and hence the signs of the limits of their values will depend on that of It. The curve will therefore lie above the osculatrix on one side of the point P and below it on the other ; and hence, evert/ osculatrix of an even order will, in general, he divided by the curve at the point of osculation. 156. The first differential equation of Article 154, (x — a)dx + (y — f)dy — may be placed under the form dx If we make the circle oscillatory to the curve we have a? — x f/ , y = ij" , and dx dx" . -r — -r^j\ hence, dy dy n dx" which is the equation of a normal at the point whose co- ordinates are x" y" (Art. 122). But this normal passes through the point whose co-ordinates are * and /3. Hence, the normal drawn through the point of osculation, ivill contain the centre of the osculatory circle. 157. It was shown in (Art. 155) that the osculatory cir- cle is, in general, divided by the curve at the point of oscu- 154 ELEMENTS OF THE lation.. The position of the curves with respect to each other indicates this result. For, the osculatory circle is always symmetrical with respect to the normal, while the curve is, in general, not symmetrical with respect to this line. If, however, the curve is symmetrical with respect to the normal, as is the case in lines of the second order when the normal coincides with an axis, the curve will not divide the osculatory circle at the point of osculation ; and the condition which renders the second differential coefficients in the curve and circle equal to each other, will also render the third differential coefficients equal, and the contact will then be of the third order. 158. The radius of the osculatory circle dxd?y is affected with the sign plus or minus, and it may be well to determine the circumstances under which each sign is to be used. If we suppose the ordinate to be positive, we shall have (Art. 133) — ~, and consequently d 2 y negative when the curve is concave towards the axis of abscissas, and positive when it is convex. If then, we wish the radius of the osculatory circle to be positive for curves which are concave towards the axis of abscissas, we must employ the minus sign, in which case the radius will be negative for curves which are convex. DIFFERENTIAL CALCULUS. 155 159. If the circumferences of two circles be described with different radii, and a tangent line be drawn to each, it is plain that the circumference which has the less radius will depart more rapidly from its tangent than the circum- ference which is described with the greater radius ; and hence we say, that its curvature is greater. And gener- ally, the curvature of any curve is said to be greater or less than that of another curve, according as its tendency to depart from its tangent is greater or less than that of the curve with which it is compared. 160. The curvature is the same at all the points of the same circumference, and also in all circumferences described with equal radii, since the tendency to depart from the tan- gent is the same. In different circumferences, the curva- ture is measured by the angle formed by two radii drawn through the extremities of an arc of a given length. Let r and r' designate the radii of two circles, a the length of a given arc measured on the circumference of each ; c the angle formed by the two radii drawn through the extremities of the arc in the first circle, and d the angle formed by the corresponding radii of the second. We shall then have 2*r : a . : : 360° : c, hence, c also, 2*r 7 2*r / : a :: 360° : d, hence, c' = ?^: 2*-/ ' and consequently 156 ELEMENTS OF THE that is, the curvature in different circumferences varies inversely as the radii. 161. The curvature of plane curves is meas- ured by means of the osculatory circle. If we assume two points P and P 7 , either on the same or on dif- ferent curves, and find the radii r and r' of the circles which are osculatory at these points, then curvature at P : curvature at F : : — : —7; that is, the curvature at different points varies inversely as the radius of the osculatory circle. The radius of the osculatory circle is called the radius of curvature. 162. Let us now determine the value of the radius of curvature for lines of the second order. The general equation of these lines (An. Geom. Bk. VI, Prop. XII, Sch. 3), is y 2 = moo + noo 2 , which gives, , |_ (m + 2nx)dx ^ , 2 = [4y 2 +(m + 2^) 2 ]^ 2y ' J 4z/ 2 J2 _ 2n y d% 2 —(m J \-2noo)dxdy_[Ani/ 2 —(m-\-2 nxf] doo 2 ay ~ Ty 2 ~— ^3 • DIFFERENTIAL CALCULUS. 157 Substituting these values in the equation . R= {dj+df) dxd l y .we obtain 3 21 „ _ [4(?nx 4- nor) -+- (hi + 2«a;) 2 ] a which is the general value of the radius of curvature in lines of the second order, for any abscissa x. i 163. If we make x = 0, we have K = — m = — - ; 2 A that is, in lines of the second order, ^c radius of curva- ture at the vertex of the transverse axis is equal to half the parameter of ilia t axis. If be required to find the value of the radius of curva- ture at the extremity of the conjugate axis of an ellipse, we make (An. Geom. Bk. VIII, Prop. XXI, Sch. 3), 2J5 2 B 2 m == — — , n— -5 , and x = A, A A A which gives, after reducing, hence, the radius of curvature at the vertex of the conju- gate axis of an ellipse is equal to half the parameter of that axis. In the case of the parabola, in which n = 0, the general value of the radius of curvature becomes 14 158 ELEMENTS OF THE p __(m 2 4- 4 ma?) 2 ~ 2 m 2 164. If we compare the value of the radius of curvature with that of the normal line found in (Art. 118), we shall have „ (normal) 3 that is, the radius of curvature at any point is equal to the cube of the normal divided by half the 'parameter squared : and hence, the radii of curvature at different points of the same curve are to each other as the cubes oj the corresponding normals. Of the Evolutes of Curves. 165. If we suppose an os- culatory circle to be drawn at each of the points of the curve APP'B, and then a curve ACQ ' C !l to be drawn through the centres of these circles, this latter curve is called the evolute curve, and the curve APP f B the invo- lute. 166. The co-ordinates of the centre of the osculatory circle, which have been represented by «> and /?, are con stant for given values of the co-ordinates x and y of the DIFFERENTIAL CALCULUS. 159 involute curve, but they become variable when we pass from one point of the involute curve to another. 167. We have already seen that the osculatory circle is characterized by the equations (Art. 154) (x-«Y + (y-Pf = R 2 , (1) (x-*)dx + (y-P)dy = 0, (2) da?+dif-\-(ij-Q)d 2 i/ = 0. (3) If it be required to find the relations between the co- ordinates of the involute and the co-ordinates of the evolute curves, we must differentiate equations (1) and (2) under the supposition that * and /3, as well as x and y, are variables. We shall then have (x - *)dx + (y-p)dy-(x- a)d* - (y - p)dp = RdR, dx 2 -j- dy 2 -f (y — /3) (Py — dadx — dp dy = 0. Combining these with equations (2) and (3), we obtain - {y - p)dp - {x - «)dx = RdR } (4) — dccdx — dp dy = 0. The last equation gives dp _ dx f s T,-~~dy [D) But equation (2) may be placed under the form which represents a normal to the involute (Art. 1 22), and which becomes, by substituting for — -^ its value -7-, dy da 160 ELEMENTS OF THE y-*=s;(*-«)> ( 6 ) or ,8 - y = -£(« - x) (Art. 120). dec This last is the equation of a straight line passing through a point whose co-ordinates are x and y, and tan- gent to the curve whose general co-ordinates are # 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 ( a; -«) = (y-/ 3 )J ) (7) or by squaring both members, combining this last with equation (1) we have mM^t?-^ (8 ) Combining equations (4) and (7), we have -(y-fi)dfi-(y-fi)^- = RdR, or -^ — — >-(y-P) = RdR; DIFFERENTIAL CALCULUS. 161 or by squaring both members Dividing this last by equation (8), member by member, we have (dR) 2 = d« 2 + dfi 2 or dR = VdJ + dp. But if s represents the arc of the evolute curve, of which the co-ordinates are a and /3, we shall have (Art. 128), ds= Vd«? + d!i 2 ; 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, R = s -\- a is the form of the equation which expresses the relation between them. 14' 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, R = s + a, and for the second B! = s f + a ; hence, R'-R = s'- s =C / C // ; and hence, the difference betiveen the radii of curvature at any two points of the involute is equal to the part of the evolute curve intercepted betiveen 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 — m = 0-\-a or m and for any other point of the curve R = s + —m. DIFFERENTIAL CALCULUS. 163 Either of the evolutes, FE, FE', F'E', or F'E, corres- ponding to one quarter of the ellipse, is equal to (Art. 169) B A 171 . The e volute curve takes its name from the connexion which it has with the corres- ponding involute. Let CC'C" be an evolute curve. At C draw a tan- gent AC, and make it equal to the constant a in the equa- tion R = s + a. Wrap a thread ACQ ' C" around the curve, and fasten it at any point, as C". Then, if we begin at A, and unwrap or evolve the thread, it will take the positions PC f , P'C" , &c, and the point A will describe the involute APP f : for PC'-AC = CO and P'C" -AC= CC'C", &c 172. The equation of the evolute may be readily found by combining the equations do? + dy 2 x with the equation of the involute curve _ dy{dx?+dy 2 ) dxd?y 164 ELEMENTS OF THE 1st. Find, from the equation of the involute, the values of — and d?y, dx and substitute them in the two last equations, and there will be obtained two new equations involving «s, /3, 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 evolute curve. 173. Let us take, as an example, the common parabola of which the equation is y 2 — mx. We shall then have dx 2? <?,,= and hence m 2 dx 2 „ _ a - 4 yy4y 2 + ?n 2 x _ 4y 3 + m 2 y _ 4£ y P ~ m 2 \ 4z/ 2 )~~ w? ~ w?^ y and by observing that the value of x — * is equal to that of y — £ multiplied by — j--> we nave 4z/ 2 -f- m 2 m u> — «* -— hence we have, 2m m 4y 3 , 2iP m — j8 = -i- and a: — <* = -— m 2 m 2 DIFFERENTIAL CALCULUS 165 substituting for y its value in the equation of the involute y = m' 2 x~, we obtain 3. If! m 2 = — 2x m and by eliminating x, we have 27 ??i\ 2 J' which is the equation of the evolute. If we make ^ = 0, we have 1 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 1 and consequently 2 ' 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 corresponds to the part AP of the involute, and CO' 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 — ljc. DIFFERENTIAL CALCULUS. 167 176. If we designate the base of a system of loga- rithms by a, we shall have, (Alg. Art. 241) and if we change the value of the base a to a r , we shall have It is plain, that the same value of x, 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 — x, will enable us to describe the curve by points, even with- out the aid of a table of logarithms. For, if we make y ; &C, we shall find, for the corresponding values of x, x = 1, x = -\fa, x — a -y/a, x = tfa &c. 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 x~0 and oc = AE = 1, 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. If we represent the modulus of the system of logarithms by A, and differentiate, we obtain (Art. 56), dy = . dx X dy dx _A X or But r-«£ represents the tangent of the angle which the ax 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 1o the curve. The tangent which is parallel to the axis of X is not an asymptote: for when x— co , we also have y = oo . 179. The most remarkable property of this curve be- longs to its sub-tangent 7 Y i? / , estimated on the axis of logarithms. We have found, for the sub-tangent, on .he axis of X (Art. 114), DIFFERENTIAL CALCULUS. 1C9 ™=g* stnd by simply changing the axes, we have T'R'^cc = A: dec hence, the sub-tangent is equal to the modulus of the system of logarithms from ichich the curve is constructed. In the Naperian system AT = 1, and hence the sub-tangent will be equal to 1 = AE. Of tlie Cycloid. G b \ A N L ISO. 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 enly to this part when speaking of the cycloid. 181. If we suppose the point P to be on the line AL at A, it will be found at some point, as L, after all the 15 170 ELEMENTS OF THE 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 BM, 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, AR = x, PR = NQ=y, we shall then have PQ = 4%ry^y\ x = AN-RN= arc NP - PQ ■ hence, the transcendental equation is x = ver- sin~ l y — y2ry — y 2 . DIFFERENTIAL CALCULUS. 171 1 63. The properties of the cycloid are, however, most ^^asily deduced from its differential equation, which is readily found by differentiating both members of the trans- scendental equation. We have (Art. 71), rdy J(ver-sin \y) — d( — V% ry — y 2 ) hence, V ' 2ry — y rdy-ydy . V2ry—y 2 dx ~dy rdy — ydy or dx V2 ry — y 2 V y d y 2ry- y 2 V2ry — if 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 = v2ry, sub-normal RN = \2ry — y 2 , tangent P T = l ^DL— ^ sub-tangent TR - V2ry — y 2 \2ry — y 2 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. 1 72 ELEMENTS OF 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 y where the parallel cuts the semi-circumference, draw the supplementary chords pn, pg, and then draw through P the parallels PN, PG r 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 a y* ... V%ry — y 2 which may 1 be put under the form dy _ dx ■y/2ry — y 2 a /2r y v y -i. If we make y, — 0, we shall have dx and if we make y = 2r, we shall have ax 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 dx _ y d y -y/2 ry — y 2 regarding dx as constant, we obtain o=^ + dy)vw^-- ydy{ ;' dv - yd / ) -, V2ry — y 2 or by reducing and dividing by y, = {2ry-y 2 )d 2 y + rdy\ whence we obtain J 2ry — if 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 R ^ (da? + dy 2 f dx d 2 y the value of cPy found in the last article : we shall then have R = 2 2 (ry) 2 = 2i/2ry: hence, the radius of curvature corresponding to the ex- tremity of any ordinate y, is equal to double the normal. 15* 174 ELEMENTS OF THE The radius of curvature is when y — 0, and equal to twice the diameter of the generating circle for y = 2r: hence, the length of the evolute curve from A to A' is- equal to twice the diameter of the generating circle. Substituting the value of d 2 y in the values of y— p f as — *■ (Art. 172), we obtain y — p=:2y, x — «>——2^2ry — y 2 ; hence we have y=— j3, x — *— 2V—2rp-~p 2 . Substituting these values of y and x in the transcen- dental equation of the cycloid, we have ver-sm /*+ V-2r/3- 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 f , and change at the same time the_2y direction of the posi- tive abscissas : that is, instead of estima- ting them from the X' left to the right, we will estimate them from the right to the left. Let us designate the co-ordinates of the evolute, referred to the new axes A! M,. A f X f , by *' and fif+ DIFFERENTIAL CALCULUS. 175 Since A'X' — AM — the semi-circumference of the gene- rating circle, which is equal to rr, we shall have, for the abscissa A r R r of any point P', A!K! —^ — rTT — x, hence, <*. ■=. m — *! \ and for the ordinate, we shall have R'P'= #~ RE - P'E = 2r- (- /3) = 2r + 0, hence, P = — 2r + p f , or — fl = 2r — p'. Substituting these values of * and £ in the transcen~ dental equation of the evolute, we obtain rw — *'= ver-sin- 1 (2r — p') -f- V2r/3' — /3 /2 , or (/== rv — ver-sin" 1 (2 r - £') - V2rp'—& 2 . But the arc whose versed-sine is 2r — ,$', is the supple- ment of the arc whose versed-sine is jS', hence «' = ver-sin l £' — V 2 r& — /3 /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. 176 ELEMENTS OF THE Let ABC 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 f the point will have described the curve AaDD\ 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 f , 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 of 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 DIFFERENTIAL CALCUXUS. 177 and t, may in general be expressed by the equation u = at n , in which n depends on the laiv 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 u — at'", or u ~r> in which we shall have U — 00 for t = o, 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. x 191. If we make n = 1, the equation of the spiral be- comes u — at. If we designate two different radius-vectors by v! and u", and the corresponding arcs by t' and t' f , we shall have v! — at' and u" = at'' 178 ELEMENTS OF THE and consequently if to the measur- This spiral is that is, the radius-vectors are proportioned ing arcs, estimated from the point B. 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 lx± a t. 2t, and consequently a = But since t is the circumference of a circle whose radius is unity, we shall have 1 a 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 BV , and any ordinate as ha, will take the direction of the normal Ab'a' . \.. _..-«-'' The curve Bale' , 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 f , c c' , &c., are propor- tional to the abscissas or arcs BU, Be . DIFFERENTIAL CALCULUS. 179 If we designate the distances, estimated from the pole A, by u, we shall have Va! — u — r: hence, (u — r) 2 = 2pt, 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 u = at?, we may, if we please, make t = loga. 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, 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, 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 Pilf = smJ? we shall have 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 = oc ; 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 revolve 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 DC 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 subtangent of the spirals. The subtangent is the projection of the tangent on a line drawn through the pole and perpendicular to the radius-vector passing through the point of contact. The equation of the spirals may be written under the form u = F{t\ in which we may suppose t the independent variable, and its first differential constant. Let AO =1 be the radius of the measuring circle, PTa tan- gent to the spiral at the point P, and A T drawn perpendicular to the radius- vector AP, the sub- / tangent. Take any other point of the spiral as P', and draw AP' . About the centre A describe the arc PQ, and draw the chord PQ. Draw also the secant PP' and prolong it until it meets AT, drawn parallel to QP, at T. T T From the similar triangles QPP' t A T'P\ we have PQ : QP' :: AT' : AP' ; hence, QP' PQ AP f AT' But when we pass to the limit, by supposing the point P' to coincide with P, the secant VPP' will become the tangent PT, and AT will become the subtangent AT. 16 182 ELEMENTS OF THE > <£ But under this supposition the arc NN f 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 PQ> we have 1 : NN' : : AP . : arc PQ; hence, 1 : c?i : : u : arc PQ, and PQ = udt. Substituting these values, and passing to the limit, we have du ■. u ~u~di~~lT : , hence, we have the subtangent u 2 dt AT du 198. If we find the value of u 2 and du from the gen- eral equation of the spirals u = af", we shall have AT=—r+ l , n DIFFERENTIAL CALCULUS. 1S3 In the spiral of Archimedes, we have n = 1, and a= — ; 2*" i 2 hence, AT= — . 2*- If now we make t = 2x-= circumference of the mea- suring circle, we shall have AT — 2*- = circumference of measuring circle. After m revolutions, we shall have t~ 2«wr, and consequently, A T = 2 ?n 2 7r = 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~~dhl 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 t=]ogu, we have dt — A — ; u . A T udt . hence, ——=——— A; AP du 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 -f is the Naperian logarithm of u, the angle will be equal to 45°. 201. The value of the tangent in the spirals is 202. To find the differential of the arc, which we will represent by z, we have PP^^qF /2 +QP 2 ; or, by substituting for QP f and PQ their values, and passing to the limit, we have <k = Vdu 2 + ic 2 dt 2 . DIFFERENTIAL CALCULUS. 185 203. The differential of the area ADP when referred to the polar co-ordinates, is not an ele- mentary rectangle as when re- ferred to rectangular axes, but / is the elementary sector APP 1 . \ The limit of the ratio of the sector APP' with the arc NN' t will be the same as that of either of the sectors APQ, AP"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 u 2 dt ds di APxPQ _u 2 VNN' ~ 2 ° r ds which is the differential of the area of any segment of a spiral. Of Tangent Planes and Normal Lines to Surfaces. 204. Let u = F(x,y,z) = Q t 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/ r , y", z" . 16* 186 ELEMENTS OF THE Through this point let a plane be passed parallel to the co-ordinate 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" , z ff , are * = *"=«", y-y" = ^(*-*"); dz 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 y = y" = h", cc - a/' '== ^(z - z") 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 doc — represents the tangent of the angle which the projection CbZ 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 — -r- dy + — dz = ; dy J dz DIFFERENTIAL CALCULUS. 187 du . dy dz hence, -~- = — — : dz du dy 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 du dz y-?/' = ■ g* *">; dy and the equation of the projection of the second tangent on the plane of ZX is du x-x" = • Tz (z z") ■ du {Z Z h dx The equation of a plane passing through the point whose co-ordinates are a/' ', y", z u is of the form A(x -x") + B(y - y") + C(z - *") = 0, C in which— —will represent the tangent of the angle which the trace on the co-ordinate plane YZ makes with the Q axis of Z, and — T" tne tangent of the angle which the A. trace on the plane of ZX makes with the axis of Z. 1S8 ELEMENTS OF 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 € dz B~ du' dy du C dz A~ ~~du~' dx hence, — B = % n. ~~du~ ' dz du hence, — A = doc c du dz Substituting these values of B and A in the equation of the plane, and reducing, we find which is the equation of a tangent plane to a surface at a point of which the co-ordinates are a/', y !l , 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 x -x" = ~{z - z"\ y-y = -^L(z- z"\ du du dz dz ELEMENTS 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 the expression. x m dx. We have found (Art. 32), that d{x m + l )=(m + l)x m dx, dx m+l ,/tf OT+1 \ whence, xdx — — - = a ( — — - ), m + \ \wi + l/ 190 ELEMENTS OF THE x m+l and consequently — , 1 J m + l' is the function of which the differential is x n dx. The integration is indicated by placing the character / before the differential which is to be integrated. Thus, we write fx m dx m + 1 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 vaiiable. 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 m dx as the sum of all the products which arise by multiplying the mth power of x, for all values of x, by the con- stant dx. dx 210. Let it be required to integrate the expression — . x We have, from the last rule, f—'^fdxx- 3 J x 3 J x~ 3+l x~ 2 _ i -3 + 1 -2~ 2^* a similar manner, we find 2 fdx\/oc 2 = fx~* dx : 2 x1 + l 5 x's 6 3x3 f« ~5~* 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, x m + x fax m dx — afx m dx — a Wi + 1 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), that every con- stant quantity connected with the variable by the sign plus or minus, will disappear in the differentiation ; and hence, the differential of a -j- x m , is the same as that of x m ; viz. mx m ~ x 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), -f. = a, or dy = adx. ax 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 192 ELEMENTS OF THE fdy-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 + b, we shall have, for x = 0, y = -f- b, and consequently, C = -f b ; and the true integral will be y = ax -f 6. 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 — —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, / xdx-\- fydy = 0, or x 2 -\-y 2 = 0, or finally, x 2 -f y 2 + 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 x = 0, y 2 =-C = R\ and hence, C= — R 2 ; and consequently the true integral is ar> 4. f -R 2 = 0, or Z* + f = R\ 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 tJic variables. 214. There is one case to which the formula of Art. 208 does not apply. It is that in which in = — 1 . Under this supposition, r m1 x m +' x~ l + l x° 1 / x ax = = = — a= — = oo. J m-fl -1 + 1 But when ?n = — 1, .dx fx m dx = fx~ l dx = f x and f— = logx+C. (Art. 57). ./ x 215. Since the differential of a function composed oi several terms, is equal to the sum or difference of the diffe- rentials (Art. 27), it follows that the integral of a differen- 17 194 ELEMENTS OF THE tial expression, composed of several terms, is equal to the sum or difference of the integrals taken separately. For example, if bdx du-~ - adx X — -\-x-\/x dx, we have fdU: —J (adx — bdx -f- x <\/x~dx), ani U — ■x*+C. 216. Every polynomial of the form {a -f- bx -}- ex 2 4- &cc.) n dx, in which n is a positive and whole number, may be inte- grated by the rule for monomials, by first raising the poly- nomial to the power indicated by the exponent, and then multiplying each term by dx. If, for example, we make n — 2, and employ but two terms, we have f(a + bxfdx —f(a 2 dx + 2abxdx + b 2 x 2 dx), 7,2 3 = a 2 x + abx 2 + ?^-+C. Integration of Particular Binomials. 217. If we have a binomial of the form du — {a-\- bx n ) m x n ~ l dx ; that is, in which the exponent of the variable without the parenthesis is less by unity than the exponent of the vari- able within, we may make INTEGRAL CALCULUS. 195 a + bx n = 2, which gives dz nbx n l dx = cfc, or a?" ' c&r = — T ; no dz z m + l whence du — z m ^-, or u = — -r- ; no {m-\-\)no and consequently u . Ja + borr^ c (m-j- l)nb Hence, the integral of the above form, is equal to the bino- mial factor with its exponent augmented by unity, divided by the exponent so increased, into the exponent of the vari- able within the parenthesis into the coefficient of the variable. For example, f(a + 3a: 2 ) 3 xdx = <1±1^1 4. Q; and - m 1 f(a + bx 2 ) 2 mxdx =z—j-(a-{- bx 2 ) 2 + C. 218. A transformation similar to that of the last article will enable us to integrate certain differentials correspond- ing to logarithmic functions. If we have an expression of the form adx du bx dz make c-\-bx = z, which gives dx = —, and by sub- stituting, we have /adx Cadz a C dz a , ~ 196 ELEMENTS OP THE and by substituting for z its value In a similar manner, we should find C at J c— 4.iog(fr-foo+c, bx b in which the integral is negative, since d( — x) = — dx. We can find, in a similar manner, the integral of e very- fraction of which the numerator is equal to the differential of the denominator, or equal to that differential multiplied by a constant. If, for example, we have , _ {b + %cx) mdx # a-\-bx -\- ex 2 make a-\-bx-\- ex 2 = z, which gives, bdx + 2cxdx = dz, and hence, mdz du = , or u — mlogz, z ' ° ' and by substituting for z its value u == mlog(<2 -h ia? -f ca, 12 ). Of Differentials whose Integrals are expressed by the Circular Functions. 219. We have seen, Art. 71, that if x designates an arc and u the sine, to the radius unity, we shall have du INTEGRAL CALCULUS. 197 du hence, / , ~x-\- C ; J Vl-U 2 or adopting the notation of Art. 72, du / Vl-u 2 = sin u-\- C If the arc expressed in the second member of the equa- tion be estimated from the beginning of the first quadrant, the sine will be 0, when the arc is ' 0, and we shall have, for u = Q du /■ ■0, and consequently C = 0, and under this supposition, the entire integral is / du Vl-u 2 To give an example, showing the use of the arbitrary constant, let us suppose thatgthe arc which is to be ex- pressed by the second member of the equation, is to be estimated from the beginning of the second quadrant. This supposition will render du / VT^u~ 2 = for u = 1 . But when u=l, sin 1 w = — ?r ; hence, -L*+C=0, or C=——ir: 2 ' 2 and we have, for the entire integral, under this supposition, / du . , 1 = SWT l U -*\ Vl-u 2 2 17* 198 ELEMENTS OF THE 220. It frequently happens that we have expressions to integrate of the form dz Let us suppose, for a moment, that a is the radius of a circle, and z the sine of any arc of the circle ; and that u is the sine of an arc containing an equal number of degrees in a circle whose radius is unity : we shall then have, u : a : z; hence, u = and consequently, du /du /• and r dz a , dz du — — ; a v J Va dz hence. /du f ^ dz . _j z the arc being still taken in a circle whose radius is unity. 221. We have seen (Art. 71), that if x designates an arc, and u the cosine, to the radius unity, we shall have du dx = Vi hence, or adopting the notation of Art. 72, t du -l , ^ / , = cos l u+C. INTEGRAL CALCULUS. 199 If the arc be estimated from the beginning of the first quadrant, it will be equal to — tt for u — 0; hence, the 2 1 first member of the equation becomes equal to — n when I 2 u = 0. But under this supposition, cos -1 u— — *■ : hence, C = 0, and the entire integral is du f Vl-u 2 222. By a method analogous to that of Art. 220, we should find r dz _, z J 7 — COS , J Va 2 -z 2 o the arc being estimated to the radius unity. 223. We have seen (Art. 71), that if x represents an arc, and u its tangent, to the radius unity, we have 7 du ax = ; 1 + u 2 ' hence, / r = x + C : J l + u 2 or, adopting the notation of Art. 72, du / \ + u 2 tang 1 u-\-C. If the arc is estimated from the beginning of the first quadrant, we shall have /* du tang _1 = 0, when /- 5 = 0; hence, C = 0, J 1 + w 200 ELEMENTS OF THE and the entire integral is / du tang" u. l + u 2 224. To integrate expressions of the form dz let us suppose for a moment that a is the radius of a circle, and z . the tangent of any arc, and that u is the tangent of an arc containing an equal number of degrees in a circle whose radius is unity : we shall then have 1 : u : : a : z ; z z dz hence, u = — , u 2 = — -, and du = « — ., a a 1 a and consequently, /du r dz U z hence, by dividing by a, /, dz 1 , z = — tang — . d 2 + z 2 a "a the arc being estimated to the radius unity- 225. We have seen (Art. 71), that if x represents an arc, and u the versed-sine, to the radius of unity, we have 7 du ax = . — ; V2u — u 2 hence, / . = x = ver-sin" 1 u + C : J V2u — u 2 INTEGRAL CALCULUS. 201 and if the arc is estimated from the beginning of the first quadrant, C = 0, and we shall have /du V2u — u 226. To integrate an expression of the form dz V~2 az — z Suppose, as before, a to be the radius of a circle, and we shall have (Art. 224), dz a a and consequently, du r dz /du r V2u-u* ~ J V2* \az to the radius unity. Integration by Series. 227. Every expression of the form Xdx, in which X is such a function of x, that it can be developed in the powers of x, may be integrated by series. For, let us suppose X == Ax a + Bx h + Cx e + Dx d -f &c, then, Xdx = Ax a dx + Bx h dx -f Cx c dx + Dx d dx + &c., fXdx=^—x a + l -\--^-x b + l + — ^ + 1 -f-5-^+ 1 + &c. J fl+1 b + l c+l ^d+i ^ 202 ELEMENTS OF THE Hence, the integration by scries is effected by develop- ing the function X in the powers of x, multiplying the series by dx, and then integrating the terms separately. (J /Y> Let us take, as a first example, — , a-{- x — = dx x == dx(a + a?) -1 , a -\- x a + x a a 4 a 6 ar and consequently, ydx /V 1 7 xdx x 2 dx x 3 dx , D \ — — =± I (—dx 2-H 3 — T -4-&c.): a 4~ x J \ a a a ar J and integrating each term separately, we obtain /aX X XX X , p i /^r /dx — — log (a + x) (Art. 218), a ~r~ X we have X u X X X 2 log (a + x) = — - " + -^o - f4 + &c + C. sv ; a 2a 2 3a 3 4a 4 To determine the value of the constant, make x= 0, which gives log a = 4- C, or C = log a ; hence, log( a + .) = loga + |-£- 2 + g-£ ; +&c. ) log(«+«) - log«=log (l + £) = | - ^ + ^- &c, a result which agrees with the development in Art. 58. INTEGRAL CALCULUS. 203 dx l + x 2 228. Let us take, for a second example We have, dx _ =dx{l + x 2 )- 1 ; l^j-ir and by developing and integrating, /* dx x 3 , x 5 x 1 , - , ^ / — x h &c + C. J 1 + x 2 3 5 7 When we make a? = 0, the arc is ; hence, tang #=:a? -j h&c.: 6 3 5 7 a result which corresponds with that of Art. 78. 229. If, in the expression — | — - , we place x 2 in the 1 — p" x first term of the binomial, and then develop the binomial x 2 + 1 , we obtain rj^ = /74-^ + i-i.+&c.)<fo; J .'<r + 1 J \xr or x b x b J and by integrating, we have tang l x— -f —3 — — ^ + &c. + C. x 3x° 5x 5 To find the value of the constant C, let us make the arc = 90° = -«■. This supposition will render the tan- gent x infinite, and consequently every term of the series will become 0, and the equation will give — *• = <)+ C, or C = — sr. 2 ' 2 204 ELEMENTS OF THE t Making this substitution, we have, for the true integral, /-. dx . _! 1 1.1 1 tang X x = —-7T f- _ _j- & c . and 2 i i '> are > as tne y sh 011 ^ be, essentially the same. a? 2 +l to 2 a* 3cc 3 5x 5 230. The two series, found from the expressions — ■ — - dx x 2 -\-l For, the tangent of an arc multiplied by its cotangent, is equal to radius square or unity (Trig. Art. XVIII). Hence, if we substitute for x, in the first series, -, we x shall have, for the complemental arc, tana; 1 -— = ^4- x x 6 x Q..3 5a fi > and subtracting both members from -v, 1 ,1,1 11 1 — 7F — tang — = tang x~--7r .4- ___ -__j_ & c . 2 b x ° 2 a? 3^ 3 5a? 5 231. We have found (Art. 71), dx sin '^/vCT^ 1 -^'- and by developing, we find (l-a*)~* = 14-— ** + — .— x* 4-i-.JL.JL ajp+dcc; v ' 224 246 multiplying by c&r, and integrating, we obtain, . . , 1 x 3 1 3 x 5 , 1 3 5 a? 7 , & sm a? = a? 4 • • — • h &c, 232 452 4 6 7 INTEGRAL CALCULUS. 205 the constant being when the arc is estimated from the beginning of the first quadrant. If we take the arc of 30°, the sine of which is equal to half the radius (Trig. Art. XIV), we shall have • -lonn 1,111,13 1113 5 11. sin * 30° =- + -.-,-5^ — .-.-. — + -.-.-. -.-r-4- &c.; 223 2 3 2 45 2 5 2467 2 7 hence, n . _, onn n( \ 1.1.1 1.3.1.1 1.3.5.1.1 , - \ w = 6sin *30° = 6(-H ,-f -A =+&c.J, V2^ 2.3.2 3 ^2.4.5.2 5 ' 2.4.6.7.2' ) and by taking the first ten terms of the series, we find *■ = 3. 1415962, which is true to the last decimal figure, which should be 5. 232. We will add a few more examples. 1. To integrate the expression - — ^ Vx-x 2 By making s/~x~ = u, we have dx dx 2 du V x — x 2 VxVl—x V 1 — u 2 But from the last series r 2du / lv? , 1 3m 5 , 1 3 5u 7 \ J Vl-u 2 \ 2 3 245 2467 / T ' hence JV^? V 23 24 5 T 246,7 ;vt 2. da?-*/ 2a# - a? 2 = {2a)*x*dx(l - — ) ¥ . 18 206 ELEMENTS OF THE But V 2a) } n'1 1 x 1 1 x 1.1.3 a! 3 . 2 — 1 ^-g < 2aJ 2 2a 2*4 4a 2 2.4.6 8a 3 £ 7 |__1_ j?_^___J_ i_ 2 a? 8 2 ' 5 2a 2 '4 ' 7 4a 2 hence Jdxy 2ax — x 2 — ( — x 1 1 3 2 tf 2 \ . — . — . r-— &c. ) *J2a-\- C: 2 4 6 9 8a 3 ) y J and consequently J_ _1_ 3_ J_jr 3 2 ' 4"6"9 8( g ^- &c.)2a?</2aff-h C, If the radius of a circle be represented by a, and the origin of co-ordinates be placed in the circumference, the equation will be (An. Geom. Bk. Ill, Prop. I, Sch. 3), y 2 — 2ax — x 2 \ hence y—yllax — x 2 , and consequently (Art. 130) dx V 2 ax — x 2 = ydx is the differential of a circular segment. If we estimate the area from the origin, where x = we shall have C — 0. If then we make x = a, the series will give the area of one quarter of the circle, if we make x = 2a, of the semicircle. J V1 + lx 3 1.3a? 5 1.3.5a? 7 , s ^^-23- + 2^y-2^6Y +&C - + a INTEGRAL CALCULUS. 207 f d * - !x l 13 L3 - 5 fee I C J V^^lT 2.2^ 2.4.4a; 4 2.4.6.6* 6 ° 6C - + °' Integration of Differential Binomials. 234. Differential binomials may be represented under the general form v_ x m ~ x dx(a + bx n )% in which, without affecting the generality of the expres- sion, m and n may be regarded as entire numbers, and n as positive. For, if m and n were fractional, and the binomial of the form JL 1 L x 3 dx{a + bx*y make x = z 6 , that is, make the exponent of z the least common multiple of the denominators of the exponents of x, and we shall then have x' 6 dx(a + bx' 2 ) q — 6z 7 dz(a + bz 3 )* t in which the exponents of the variable are entire. If n were negative, we should have, x m ~ l dx(a 4- bx~ n y, and by making x — — f we should obtain p_ z- m - l dz(a + bz n y, the same form as before. 208 ELEMENTS OF THE Furthermore, the binomial p x m ~ l dx(ax r + bx n y may be reduced to the form x m * dx(a + bx n ~ r y, by dividing the binomial within the parenthesis by x r , ant ; pr multiplying the factor without by x q . 235. Let us now determine the cases in which the binomial x ni ~ l dx{a -f hx n y has aai exact integral. Make a -f bx n ~ z q ; we shall then have* m n z q — a . ; , £ „ m (z*—a\ n x n = —j-— f (a + baT)* =:z p , x m = (——) t and by differentiating, no lience x m ~ l dx = rZ-^lZ-Z) dz ; nb \ b J l dx(a -f bx n y = ±s?+^dz^-j^j , which will have- an exact integral in algebraic terms when — is a whole number and positive (Art. 216). If — is n n negative see Art. 260. Hence, every differential binomial has an exact inte- gral, when the exponent of the variable without the paren- thesis augmented by unity, is exactly divisible by the exponent of the variable within. Thus, for example, the expression p_ x 5 dx{a+bx 2 y INTEGRAL CALCULUS. 209 has an exact integral. For, by comparing it with the general binomial, we find 171 m~Q, ?i — 2, and consequently, — = 3, and the transformed binomial becomes 26 \ b J 236. There is yet another case in which the binomial zf n - l doc(a-\-bx n )i has an exact integral. If we multiply and divide the quantity within the paren- thesis by x n , we have 2. L ac m ~ l dx(a + bx n Y = x m - l dx[(ax- n -f &)#"]« p_ np =zx m ~ l dx{ax~ n -[-bYx^ = a? « dx{ax~ n -\- b)" y Now, if we add unity to the exponent of x without the parenthesis, and divide by — n, the quotient will be — ( (- — )> 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 + vdu, 2J0 ELEMENTS OF THE whence, by integrating, uv =fudv -\-fvdu, and, consequently,. fudv — uv —fvdu. Hence, if we have a differential of 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 y JPQdx = 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 dx(a+bx n ) If now, we multiply by the two factors x n and. x \ the value will not be affected, and Ave obtain x m - n x n - l dx(a + bx n y. Now, the factor x n ~ 1 dx(a-\- bx n ) p is integrable, whatever be the value of p (Art. 217) ;, and representing this factor by dv, we have V -~T , i \ i> > and u=zcc > (p+l)nb and, consequently, oc m - n (a-\-bx n Y* 1 m-n {p+l)nb (p+l)nb fx m - l dx{a + bx n ) p = ■fx m - n - l dx{a + bx n y+ l . INTEGRAL CALCULUS. 211 But, fcc m - n - l dx(a + bx n ) p+l = Jx m - n - l dx{a + bx n ) p (a + bx n } = afx m ~ n - l dx(a + bx n ) p + bfx m ~ l dx(a + for")* ; substituting this last value in the preceding equation, and collecting the terms containing the integral Jx m ~ x dx(a + bx n y, we have x m - n (a + bx n ) p+l - a(m - n) f x m ~ n ~ x dx(a + bx) p m {p+l)nb ' whence, formula (A.) fx m ~ 1 dx(a + bx n ) p =z J\. x m - n (a + bx n ) p+l -a{m- n) J x m ~ n ~ x dx(a -f bx n y 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 + ta n )* ; and by a similar process we should find fx m ~ n - 1 dx(a-\-bx n y to depend on fx m -* n ~ x 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 n, the factor m — ?i, m — 2n, m — Sn, &c, will finally become equal to 0, and then the differential into which it is multiplied will disap- 21 2 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 - x dx(a + bx n ) p = foc m - l dx{a + bx n ) p ~\a + bx n ) = afx m - l dx(a + bx n ) p ~ l -f bfx m+n ~ l dx{a + bx n ) p ~\ Applying formula (A) to the second term, by placing m + n for m, and p — 1 for p, we have fx^ n - l dx{a + bx n y- l =: x m (a + bx n Y - amfx m - l dx(a + bx n y~ l b(pn-\-m) Substituting this value in the last equation, we hare H formula (B) , . .fx m ~ l dx(a + bx n ) p = x m (a + bx n y+pnafx^- x dx{a + bx n y~ 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 fx m - l dx(a + bx n ) p to fx m ~ rn - l dx(a-{-bx n y- 8 ; 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 -f bx' 3 y is reduced, by formula (A), to fx*dx(a-{-bx 3 y, and then to / xdx (a -f bx 3 ) 2 : INTEGRAL CALCULUS. 213 and by formula (B) fxdx(a-\- bx 3 ) 2 , reduces to fxdx(a -f- bx 3 ) 2 , and finally to fxdx(a + bx 3 ) 241. It is evident that formulas (A) and (B) will only diminish the exponents m — 1 and p, when m and p arc positive. We will now determine two formulas for dimin- ishing these exponents when they are negative. We find from formula (A) fcc m - r - l dx(a + bx n ) p = x m ~"{a + bx n ) p+l — b{m -f np)fx m - x dx(a + b af 1 ) 7 * t a(m — n) and placing for m, — m-\-n, we have formula (C) fx- m ~ l dx(a-\-bx n ) p = O x~ m {a + bx n ) p+1 + b(m — n — np)fx- m + n - l d^(a + fo n ) p — awi 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), fx m - l dx(a + bxy- l = x m (a + bx n ) p — (m + np)Jx n ~ x dx{a + bx n ) p pna ' writing for p, — p + 1, we have formula (D) fx m - l dx(a -f &c*)- p = '1) x m (a + bx n )- p + l - (m + n - np)fx m ~ 1 dx(a + bx n )~ p+l (p— \)na 214 ELEMENTS OF THE This formula does not apply to the case in which p = 1. 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 both terms of the binomial. We shall therefore determine a particular formula for integrating the binomial x q dx x q {2ax — x 2 ) ^dx~ V2i „2 i CIX — X This binomial may be placed under the form fx 1 *dx(2a-x)~ 2 , and if we apply formula (A), after making ™ = q+— , ri-=h JP=-— , a = 2a, Z>=-.1, we shall have fx ~dx(2a—x) 2 = q q J y > ' and if we observe that i .i 3 i q — — q-l -i- ? — — 7 — 1 — — x 2 =x X 2 X * —X X 2 , and pass the fractional powers of x within the parentheses, we shall have E formula (E) f— r * — — x q ~W2ax-x 2 (2q-l )a C x q ~ l dx q q J V2UX-X 2 ' 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 /dx x /n 2 =s ver-sin -1 — . (Art. 226). V2ax — x 2 a Integration of Rational Fractions. 244. Every rational fraction may be written under the form Px n ~ l + Qx n ~ 2 + Rx + S , Fx 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 cue 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 Fx n +Q'x n - 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, a? — &,#—- 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. adx 245. Let us take, as a first example, Hi — CO By decomposing the denominator into its factors, we have adx adx x 2 — a 2 (x — a)(x -\- a)' and we may make adx (x — a) (x + < ») = ( A + B ■ \x — a x -\- a dx, in which A and B are 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 + Bx — Ba) dx (x — a){x J r 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 x, (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 A ~2' ■- 2' Substituting these values for A and B, we obtain adx -odx ~_dx x 2 — a 2 x — a x-\-a* and integrating, we find (Art. 218) I A = 1 los(x - a) - ¥°° {x + a) + c ' and, consequently, T ft3 H - bX 2 246. Let us take, as a second example, — ; ~dx. arx — or The factors of the denominator are x and a 2 — x 2 ; but a 2 — x 2 = (a + x) (a — x) : hence, the given fraction becomes a?(a — a?) (a -{-a?) Let us now make a 3 4-ba? A . B C x(a — x)(a + x) x a — x a-\-x 19 218 ELEMENTS OF THE reducing the terms of the second member to a common denominator, we have a 3 + bx 2 Aa 2 - Aa?+ Bax + Bx 2 + Cax -Cx 2 x(a — x) (a + x) x(a — x){a J r-x) and, comparing the like powers of x (Alg. Art. 208), B-A-C^b, Ba+Ca = 0, Aa 2 = a 3 . From these equations, we find A = a, B=«±±, C=- a -±*, 2 ' 2 ' and substituting these values, we obtain a 3 + bx 2 7 d x , a + b 7 a -f b 7 — ~^rdx~a 1 - -ax - -ax ; a z x — x 6 x 2{a — x) 2(a-\-x) and integrating (Art. 218), Ca 3 + bx 2 , , a + b, ( -^log(a + X )+C = a log a; — ^— — [log(a - x) + log(a + x)] + C = alogx -log(a — x) {a + x) + C = alogx log(a 2 — x 2 ) + C — alogx — (a + 2>)log <\/a 2 — x 2 +C. 247. Let us take, for a third example, -5 — ~- -dx. INTEGRAL CALCULUS, 2K* Resolving the denominator into the two binomial factors (Alg. Art. 142), (a? — 2), (a?— 4), we have 3^-5 A , B . H -, nence x 2 — 6x + 8 x — 2 ^-.4' 3x-5 _ Ax-4A + Bx-2B x 2 - 6a7+8 ~~ # 2 -6^ + 8 ' and by comparing the coefficients of x, we have -5= -4 A- 2B, 3 = A + B, which gives B = — , A = — -, 2' 2' and substituting these values, we have dx J x 2 — 6x4-8 ' 2 J x — -2^~ 2 J x — 4 = ^log(x-±)-±-log(x-2)+C. "248. Let us take, as a last example, xdx aP + ^ax — b*' Resolving the equation a? + 4«a? — #=«, ■we find x=—2a4- V4a 2 + £ 2 , #= — 2a- V4a 2 + 6 2 , and consequently, for the product of the factors, (x+2a+ V4a 2 +b 2 )(x4-2a- ^±a 2 4-b 2 )=x 2 4-4:ax-fi l . 220 ELEMENTS OF THE To simplify the work, represent the roots by — K and — L, and the factors will then be cc + K, x + L, and we shall have 1 — : hence a? + 4 ax — b 2 x + K x + L x Ax + AL + Bx + BK 3? + Aax — b 2 ~~ x 2 + 4:ax—l? ~~ whence, AL + BK = 0, A + B=l, and, consequently, K B= L K-L 7 K-L hence, / K log(a.+/0--A-log(a ? +L)+C. x 2 +±ax-b 2 K-L &v ' ' K-L 249. In general, to integrate a rational fraction of the form Par-'+Qx™-* +Rx+S x m +QV- 1 .. .. -fi^+S' °°' 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 CALCULUS. 221 3d. Integrate each partial fraction separately, and the sum of the integrals thus found will be 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 Px' + Qx 3 -\- Rx 2 -{- Sx -]- T (x — a) (x — b) (x — c) (x — d) (x — e) A + * +- c -+^-, + E x — a x — b x — c x — d x — e ' if several of these roots are equal, as for example, a = b — c, the last equation will become Px*+Qx 3 +$ce. A + 5+C D E (x — af (x — d) (x — e) x—a x — d x — e* in which A + B + C may be represented by a single con- stant A! . 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 x, 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 x — a x — b' x — c which have the same denominator, we go through the 19* 222 ELEMENTS OF THE process of reducing them to one, their sum may be placed under the form A! + B'x+Ox 2 (x-af ' or, by omitting the accents, A + Bx+Cx 2 (x-af ' 251. Let us now make x — a = z, and consequently, x = z -+- a ; we shall then have A + Bx + Cx 2 A + Ba+ Ca 2 + Bz+2 Caz + Cz 2 (x-a] A + Ba+Ca 2 , B + 2Ca , C substituting for z its value, and representing the numera- tors by single constants, we have A + Bx+Cx 2 A' \ B' C + 7- IV2 + (x — af (x — af (x — af x — a" 1 the form under which the fraction may be written. Since the same reasoning will apply to the case in which there are m equal factors, we conclude that Px m - l +Qx m -* +Rx+S _ (x-af A A! A! 1 . A"../ (x-a) m ^(x-a)^- l ^(x-a) m ~ 2 ' x-a 252. In order, therefore, to integrate the fraction Px*+Q f a? + &c. (x — af (x — d)(x — e) dx, INTEGRAL CALCULUS. 223 place it equal to A . A' A!' D . E (a? — af (x — df 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 a A f A" D E ■.dx -f —dx + dx -\ dx -\ dx. (x—a) 3 (x—a) 2 (x—a) x—d x — e The first two fractions may be integrated by the method of Art. 217, and the three last by logarithms. Hence, finally, / P x*+Qx 3 + Rx 2 +Sx+T dx _ A A! (x — a) 3 (x — d)(x — e) 2(x — a) 2 x — < -f A"\og(x —a) + Dlog(x - d) + Slog (a? - e) + C. 253. Let it be required to integrate the fraction 2 ax , 7 ^dx. {oc + a) 2 We have 2 ax A A! (x + a) 2 (x + a) 2 x + a 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! and A + A'a — 0: hence, A = — 2a 2 , and A! = 2a ; 224 ELEMENTS OP THE and, consequently, 2axdx 2a 2 dx 2adx (g-hd)> (* + «) 2 0* + a) ' hence, (Arts. 217 & 218), /2axdx 2 a 2 , ~ , , N (?T^ = ^+7 +2alog(iBH - a) - 254. Let us find the integral of aPdx x s — ax 2 — a 2 x -+- <r 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—an factor. Dividing by x — a, the quotient is x 2 — a 2 : hence, the fraction may be placed under the form a?dx a?dx (a? — a 2 ) (x - -a) (x + a) (x — a)(x - a?dx -a) Let us now make x 2 ~ (x — a) 2 (x + a)' B (x — a) 2 (x-\-a) (x — a) 2 (x — a) x-\-a Reducing the terms of the second member to a common denominator, we have x 2 _ A(x + a) + A'(x 2 - a 2 ) + B(x- a) (x — a) 2 (x-\-a)~~ (x — a) 2 (x-\-a) and developing, and comparing the coefficients of the like INTEGRAL CALCULUS. 225 powers of x, we obtain the equations A' + B=l, A-2Ba = 0, Aa- A! a 2 -f Ba 2 = 0. Multiplying the first equation by a 2 , and adding it to the third, we have Aa + 2Ba 2 = a 2 ; then multiplying the second by a, and adding it to the last, we have a 2 = 2Aa, and consequently, A = — a ; substituting this value of A, we find B = — and A' = — . 4 4 Substituting these values of A, A f , and B, we have aPdx adx Sdx dx (x — a) 2 (x-\-a) 2(x — af 4(x — a) 4(a? + a)' and consequently, /a?dx _ a 3 , . . a?-ax 2 -a 2 x + a*-~~2(x-a) + T i0g[X - a) + -^hg(x + a)+C. 255. We can integrate, in a similar manner, when the denominator contains sets of equal roots. Let us take, as an example, adx adx ( a? 2_l)2-( a7 _ 1 )2 (a7+l) 2- 226 ELEMENTS OF THE Make , ^ , B , 5' (^-1) 2 (^+1) 2 (a?— I) 2 ' oc-1 ' (^+1) 2 J?+l' reducing the second member to a common denominator, we find the numerator equal to A(^+l) 2 +A / (a7-l)(a?+l) 2 +5(^-l) 2 +5 / ( J r+l)(a?-l) 2 ; and comparing the coefficients with those of the numera- tor of the first member, we have the following equations : A' + B' = 0, A +A f + B-B f = 0, 2A -A'-2B-B' = 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 + 2B == a: hence, we have a — R — ^L A'——— B'— — ' consequently, the given differential becomes 1 r dx dx dx dx ~) T a L(a?-l) a + (^+l) 2 ~^=T + ^+TJ , and by integrating, Jx^rf^i-^i- ^-iog(.-i)+iog(, +1 )] + a 256. If an equation of the second degree has imaginary roots, the quantity under the radical sign will be essential!}/ INTEGRAL CALCULUS. 227 negative (Alg. Art. 144), and the roots will be of the form x = =f <2 + h -y/— 1, a?==Fa — h->J — 1, and the two binomial factors corresponding to the roots will be (a? ± a -6 y/^X) (# ± a + h V — 1) = «? ± 2a# + a 2 -f- 6 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, a = 0, x= -\-b-\/ — 1 , x= — b V — 1, and the factor will become x 2 + 6 2 . In the equation, a?-6cx+\Qc 2 = y the roots are, x = 3c -f c y r —T # = 3c — c V— 1 ; comparing these values of a? with the general form, we have a = — 3c & == c, and the given equation takes the form x 2 — 6c;r-f 9c 2 +c 2 = 0. Comparing the roots of the equation, a^-H4a7+12 = 0, with the values of x in the general form, we have a = 2,. b=</8, 228 ELEMENTS OF THE and the equation may be written under the form a? + 4a? + 4 + 8 = 0. 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?+Sx 3 + &c. (a? — a) (a? — b) .... (a? — h){x 2 -\- 2ax + a 2 -f- b 2 ) which may be placed under the form dx y Adx Bdx Hdx Mx + N , +" Z h ' ' ' ' + Z I + i'L^i „2 i S^' x — a x — b x — h x 2 -f 2aa? + ft2 + W 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 ., ,dx. {x + a) 2 + b z If we make x -f a — z, the expression becomes Mz + N—Mci j ¥+b 2 dZ ' and making N — Ma = P, it reduces to Mz+P, z l + b l which may be divided into the parts, Mzdz . Pdz z * + b 2 ' z 2 + b 2 ' which may be integrated separately. INTEGRAL CALCULUS. 229 To integrate the first term, we have zdz M f 2zdz_ b 2 fMzdz _ f zdz _M f 2zd J z 2 + b 2 ~ J z 2 +b 2 ~ 2 J z 2 + in which the numerator, 2zdz, is equal to the differential of the denominator: hence (Art. 218), fMzdz M, , ., , , 2 . z 2 + b 2 or by substituting for z its value, x + cl, fMzdz M, r/ . x2 , 7 21 M = ~log(x 2 + 2ax-\-a 2 + b 2 ) = M log i/a* + 2ax + a 2 + & 2 . Integrating the second term by Art 224, gives f Pdz P ,/z\ J^Th 2 = T l ^'(-b} or by substituting for z its value, x -f a, and for P, iV" — Ma, we have /' Pcfe N—Ma _,/a? + a\ ft^— — tan « (-r) ; and finally, */* 2 + 2aaH-a 2 + & 2 Mlog V^+2^ + « 2 + o 2 -f- ^— tang- 1 (^p ). 259. Let us take, as an example, the fraction 20 230 ELEMENTS OP THE in which, if + 1 be substituted for x, 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 c+fx (x-l){x 2 + x+l) dx, m which x 2 + x + 1 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 a?+2ax + a 2 + b 2 =0, we find a 2 ~V 4* We may place the given fraction under the form c+fx _ A Mx + N m (x— 1) (x 2 + x + 1)~~ x— 1 x 2 + x + 1 ' reducing the second member to a common denominator, and comparing the coefficients of x in the numerator with those of x in the numerator of the first member, we obtain Substituting these values of M and N, as also those of a and 6, in the general formula of Art. 258, and recollecting that /' Adx c+f r dx c+f, . . INTEGRAL CALCULUS. 231 we find j '^^d X = ^±l\og{x-l)-^±^h g V^+7+Y + /^ ta „ g -r^+r|+c. VI UVI 260. The equation which arises from placing the de- nominator of the fraction equal to 0. may give several pairs of imaginary roots respectively equal to each other In this case, the factor x 2 zt 2ax + a 2 -\- b 2 will enter several times into the denominator, or will take the form 2 +2^ + a 2 -f6 2 ) p ; and hence, that part of the fraction which contains the pairs of equal and imaginary roots, must be placed under the form (Art. 251) H+ Kx H' + K!x + (x 2 + 2ax + a 2 -{-b 2 y {x 2 + 2ax + d 2 + b 2 ) p - 1 II" + K"x H n + K n x (x 2 +2ax + d 2 + b 2 ) p - i x 2 + 2 ax + a 2 + b 2 Now, reducing to a common denominator, and comparing the coefficients, we find the values of the constants H, K, H\ K', iF, K" H\ K n . . . after which, multiply each term by dx, and then integrate the terms separately. Since all the terms are of the same general form, it will only be necessary to integrate the first term, which may be written under the form H+Kx [{x + af + b 2 y dx ; 232 ELEMENTS OF THE which, if we make x + a = z, will reduce to H-Ka + Kz A and making M—H— Ka> it will become M+Kz , Kzdz v , Mdz ■ dz = -— — dz ' (b 2 + z 2 ) p (b 2 + z 2 Y ' {b 2 + z 2 f The first term of the second member may be placed under the form Kf(b 2 + z 2 )-»zdz, and integrating by the formula of Art. 217, we have Kzdz IK 1 I; 5=i+C- (b 2 +z 2 y 2 (i- P )(b 2 + z 2 y It then only remains to integrate the second term Mdz By comparing the second member of this equation with formula (D), Art. 242, we see that it will become identical with the first member of that formula, by supposing m=l, a = b 2 , b = l, and n~2\ and hence, by means of that formula, the exponent — p may be successively diminished by unity until it becomes — 1, when the integration of the term will depend on that of dz W+z 2 ' But we have already found (Art. 224), f dz 2 1 - x fz\ INTEGRAL CALCULUS. 233 and hence the fraction may be considered as entirely in- tegrated. 261. It follows, from the preceding discussion, that the integration of all rational fractions depends on the follow- ing forms : x m+l 1st. fx m dx — . • m+1 == ± loff (a ± x). a±x 5V ; /dx 1 , / x \ 3d. Integration of Irrational Fractions. 262. The method of integrating rational fractions having been explained, we may consider an irrational fraction as integrated when it is reduced to a rational form. 263. Every irrational fraction in which the radical quantities are monomials, may be reduced to a rational form. Let us take, as an example, i_ ^ x — \a x % — \a ax, or — [ r . %f~x— Voc Having found the least common multiple of the indices of the roots, (whicii indices are the denominators of the fractional exponents,) substitute for x a new variable, z, with this common multiple for an exponent, and the frac- tion will then become rational in terms of z. 234 ELEMENTS OF THE In the example given, the least common multiple is 6; hence we have x = z 6 and tJx = z' 3 , ^/x — z 2 , dx = 6z 5 dz ; and substituting these values, we obtain an expression which may be integrated by rational frac- tions ; after which we may substitute for z its value, -\/x. 264. If the quantity under the radical sign is a polyno- mial, the fraction cannot, in general, be reduced to a rational form. We can, however, reduce to a rational form every expression of the form X( VA + Bx± Cx 2 ) dx, in which X is supposed to be a rational function of x. If we write a denominator 1, and then multiply the numerator and denominator by VA -f Bx =b Cx 2 , the expression will take the form X!dx VA + Bx± Cx 2 ' in which X r is a rational function of x : hence the two forms are essentially the same. If now, we can find rational values- for V A+ BxzLCx* and for dx, in terms of a new variable, the expression will take a rational form. There are two cases to be considered: 1st., when the coefficient of x 2 is positive ; and, 2d, when it is negative.. INTEGRAL CALCULUS. 235 Let us consider them separately. First, make VA + Bx+Gx*= Vcy/A + ^v + a* = V~C Va + bx -f x 2 , in which a = —-, b = —. G C In order to find rational values for dx and Va + bx+x 2 , place Va -f 6a; + x 2 = oc + 2:, (1) from which, by squaring both members, we find a + fo?=:2^ + ;2: 2 , (2) and hence, and substituting this value in equation (1), V a + bx + a? 2 = T — — - -f z ; b — 2z and by reducing to the same denominator, V a + b X + 3 ? =- Z °- bz + a . (4) — fiZ Let us now find the value of dx in terms of z. For this purpose we will differentiate equation (2), we then find bdx = 2xdz + 2zdx + 2zdz ; whence we have {b — 2z)dx = 2(x + z)dz ; 236 ELEMENTS OF THE and by subtracting equations (1) and (4), and substituting for x + z the value thus found, we have f-L o \j 2{z 2 — bz -\- a) , (b — 2z)dx = i— - —^dz 9 b — 2z , , 2(z 2 -bz + a) , ,_. 8nd **=- \b-2z? ^ (5) 265. Let us take, as an example, dx xVa + Bx+Cx 2 which may be written under the form dx V~C x x Va + bx + x 2 ' and substituting the values of Va -f bx + x 2 and dx } from equations (4) and (5), we have dx ■■' 2dz Va + bx + x 2 ~b-2z' and multiplying the denominator by the value of x, in equation (3), dx __ 2dz xVa + bx + a?~ * 2 -a' and then by Vc, we have dx dx 2dz or VC X x Va + to -f a? a? V-A+BaH- C* 2 (z 2 - a) VC which is a rational form, and may be integrated by the methods already explained. INTEGRAL CALCULUS. 237 266. Let us take, as a second example, dx Vh + c 2 x 2 which may be placed under the form dx v&* c c and comparing this with the form of Art. 264, gives c = -yJC, b = 0, — = a. & Hence, dx 1 r dx /dx 1 r d V7H- c 2 x 2 ~~~ J V^ Having placed i/iT+x 2 = z-{-x, we found, Art. 264, equations (5) and (4), hence dx r dz /ax C — 1 Va~+~x^~ J ~T~~ ° g z. Substituting for z its value, and multiplying by — , we c have and substituting for a its value, ~, we have 238 ELEMENTS OF THE / dx Vh + c 2 a? ~ log — ( Vh -r c 2 a? — ex) l_ c + c — log log( Vh + cV — ex) + C. But since the difference of the squares of the two terms within the parenthesis is equal to h, it follows that if h be divided by the difference of the terms, the quotient will be their sum (Alg. Art. 59). But the division may be effected by subtracting their logarithms. Let us, then, add to, and subtract from, the second member of the equa- tion, — \ogh. We shall then have, / 1, 1 1, -log loffA-| — log h loe( VA+cV— cx)4- C: Vh+<?a* c c c c c or by representing the three constants — log log Zf, c c c and C, by a single letter C, we have -log( VT+cV + ex) + C. / dx Vh+c^x*' 267. Let us take, as a third example, dxVm 2 + x 2 . Comparing this with the general form, we find a = m 2 and 6 = 0; hence (Art. 264), 2% I 772 VnF+x 2 = and dx 6Z 2z 2 INTEGRAL CALCULUS. 239 and consequently, 7 r^TT-^i (z 2 + m 2 ) 2 7 dx V m z + x z — — - ,. ' dz, which is rational in z ; and, having found the integral in z, substitute the value of z in terms of x. 268. Let us now consider the case in which the coeffi- cient of x 2 is negative. We have <y/A + Bx- Cx 2 = V^\/^- + ~--< = Vc" -yja -\-bx-x 2 . If now, we make as before, Va + hoc — x 2 = x -+■ z t and square both members, the second powers of x in each member will not cancel, as before ; and therefore, x can- not be expressed rationally in terms of z. We must, therefore, place the value of the radical under another form. We will remark, in the first place, that the bino- mial a + bx — x 2 , may be decomposed into two rational factors of the first degree. For, if we make x 2 — bx — a = 0, and designate the roots of the equation by a and <*', we have (Alg. Art. 142) (x 2 — bx — a) = (x — et) (x — *'), and consequently, by changing the signs, (a + bx — x 2 ) = — (x — *) (x — <x!) = (x — *) {«! — x) ; 240 ELEMENTS OF THE and placing the second member under the radical, we may make V{x — *) (*' — x) = (x — cc)z\ (1) squaring both members {x _ u ) [a! — x) = (x— ec) 2 Z 2 j and by suppressing the common factor x — #, «! — x = {x — *)z 2 , (2) whence, X- ct' + ctZ 2 ~ 1+Z 2 ' and X- — C6 r ct! + az 2 l+z 2 or by reducing, ,r — - C6 — a! — a, fh l+z"' which, being substituted in the second member of equa- tion (1), gives V(» -.)(*_ *) = ■£=£*; (4) and by differentiating equation (3), we obtain 2 («' — «) , ... dx =-{T+W () 269. To apply this method to a particular example of the form dx Va + bx — oe* INTEGRAL CALCULUS. 241 substitute the values of V a -f- bx — x 2 and dx, found m equations (4} and (5) : we find dx 2(x! — u)z , _ 2dz Va + bx-x 2 (l | z 2 ) 2 z ^'~ u) l + z2) \-\-z 2 hence dx f—= =-2t<mg- l z+C; J yci + bx — x 2 or, by substituting for z its value from equation (1), /' dx =r,-^-i( V ^-«^«[-^ ) J Va-i-bx-x 2 ^ («-«) J = C-2tang- 1 A / /?^. V x — cc 270. If, in the last formula, we make a = 1 and b = 0, the trinomial under the radical will become 1 — x 2 , and the roots of the equation x 2 — 1 = are « = — 1 and ce! = 1 . Substituting these values, and the general formula becomes and if we suppose the integral to be when x = 0, we enan have 0=C-2tang- x (l) = C - 2(450) (Trig. Art. VIII) = C-90°: hence C = — . 2 21 242 ELEMENTS OF THE Substituting this value, and we have /• dx * _ x /l—x I — — 2tang l \/ , 271. We have already seen (Art. 219) that /dx and hence, — 2tan£ -1 \/- — ° V 1 + should also represent the arc of which x is the sine To prove this, we have (Trig. Art. XXV) „ , 2tangA tang2A = - . 8 . . 6 1 — tang 2 A /l — x Substituting for tang A, \J , and reducing, we have Ung2A = y^ER; /l — cc that is, twice the arc whose tangent is i/ — — is equal » 1 ~~J~ x /— — rj to the arc whose tangent is Z x I- g But the arc whose tangent is 1 — # ? j s t jj e com . a? plement of the arc whose tangent is . , (Trig. Vl — x 2 Art. XVIII) ; and this arc has x for its sin*. Hence, either member of the equation INTEGRAL CALCULUS. 243 r dx a- _ /i_ x 1—7== = 2 tang \/ , J Vl-x 2 2 ° V l + ar' represents the arc whose sign is a?. 272. Let us take, as a last example, the differential dxy 2 ax — x 2 . In comparing this with the general form, we find (Art 268) cc — and ec ; = 2a; and Art. 268, equations (4) and (5), give 2az j _ 4<2.2r i + ?' (T+?) : ^/a:(2a — x) = t "~ 2 , dx = — ^ri^z. Substituting these values, we have 8a 2 z 2 dz dxy2ax — x 2 (i+* 2 ) : which may be integrated by the method of rational fractions. Rectification of Plane Curves. 273. The rectification of a curve is the expression of its length. When this expression can be found in a finite number of algebraic terms, the curve is said to be rectijiable, and its length may be represented by a straight line. 274. The differential of the arc of a curve, referred to rectangular co-ordinates, is (Art. 128) dz= V d ^ + dy 2 - 244 ELEMENTS OF THE Hence, if it be required to rectify a curve, given by its equation, 1st. Differentiate the equation of the curve. 2d. Combine the differential equation thus found with the given equation, and find the value of dx 2 or dy 2 in terms of the other variable. 3d. Substitute the value thus found in the differential of the arc, which will then involve but one variable and its differential. Then, by integrating, we shall find the length of the arc, estimated from a given point, in term; of one of its co-ordinates. 21 h. Let us take, as a first example, the common para bola, of which the equation is y 2 = 2px. Differentiating, and dividing by 2, we have ydy = pdx, and consequently, dx 2 = — „ dy 2 ; p 4 substituting this value in the differential of the arc. we have = —dyVp 2 + y 2 ; which, being integrated by formula (B) Art. 239, gives, by supposing m = 1, a =p 2 , b=l, n = 2, p = -r- INTEGRAL CALCULUS. 245 f , / • = iog( Vp 2 + y 2 + y) ; and integrating the second term by the formula of Art. 266, we have, after maKing h =p 2 , c 2 = 1, dy p 2 -\-y and consequently, If we estimate the arc from the vertex of the parabola, we shall have y — for s = : hence = -^\ogp+C or C=--|-logp; and consequently, z _ yVp 2 + y 2 + p lo r Vp 2 + y 2 + y \ . and hence, the value of the arc, for a given ordinate y, can only be found approximatively. 276. The curves represented by the equation y n =px m , are called parabolas. This equation may be placed under the form 1 m y^p n x T ' f or by placing p n ==p', and -~ = n f , we have y=p'x n '' y 21* 246 ELEMENTS OF THE or finally, by omitting the accents, the form becomes y=px\ By differentiating, we have dy = ripx n ~ l dx, and by substituting this value of dy in the differential of the arc, we have i z=f(\+nyx 2n - 2 fdx. The integral of this expression will be expressed in a finite number of algebraic terms when is a whole to 2n — 2 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+I — i, or n= 2n-2 7 2i ' and substituting for n, we have 2H-1. y—px 2i > or y 2i —p^x %i+l y 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 rinite and algebraic terms, viz. when o +1T is a positive whole number (Art. 236 and 235.) 3 277. If we make i = 1, we have n = — , and At y 2 =p 2 a?, which is the equation of the cubic parabola. INTEGRAL CALCULUS. 247 Under this supposition, the arc becomes (Art. 217) z=f(l +nYa>-^ = JL.(i + lf x f 4- 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 and consequently, 278. If the origin of co-ordinates is at the centre of *he circle, the equation of the circumference is R 2 = x 2 + y 2 , and the value of the arc, r dx If the origin be placed on the curve y 2 = 2Rx - x 2 , dx and z = R i V2Rx-a?' both of which expressions may be integiated by holies, and the length of the arc found approximative^. 279. It remains to rectify the transcendental cur"- s. The differential equation of the cycloid is (Art. 182) V 2ry — y 2 248 which gives ELEMENTS OF THE dx 2 _ y 2 dy' 2ry-y>* Substituting this value of dx 2 in the differential of the arc, we obtain & =v^S= rf V 2ry 2ry — y* 2ry-y l = d y V^7~ = ( 2r f( 2r - y)~^ d y> But (Art, 217) f(2r-y)-hy=-2(2r-yy + C; and hence, r z= -(2r) 2 2-v/2r-y+ C= — 2^/ r 2r(2r — y) + C. If now, we estimate the arc z from B, the point at which y = 2 r, we shall have, for z — 0, y = 2r; hence ^ = + C, or C = 0, and consequently, the true integral will be z= _^2V2r(2r — y) ; 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 jB£ = 2r — y. But JBG =BMxBE: hence .££ = V2r(2r — y), JNTEGRAL CALCULUS. 249 and consequently, BD = 2BG; 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) dz = Vdu 2 + ii z dt 2 . Taking the general equation of the spirals u = at", we have du 2 = n 2 aH Zn ~*dl? ; and substituting for du 2 and u 2 their values, we obtain dz = at n - l dtVn 2 + f. If we make n = 1, we have the spiral of Archimedes, (Art. 191), and the equation becomes dz = adtVl + ^; 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 t — hgu, 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 quadrdble, and may be represented by an equivalent square. 283. If s 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, ivhick values will he 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 = F(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 y n =px m : we shall then have and i. J? y = p n x n , — — 7VO n m + n 72 fF(x)dx=fp n x n dx=—±- — x n = ■ — xy+C\ J v ' J 1 m + n m + n J 1 m 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 y n =px m > dx = j — dy, we have and consequently, •J ^ " ~ m J ~ " ~~ m + n - m + n * ' p m p m n by substituting x for its value, — , which is the same re- p m 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 ihere- ^ m + n r fore quadrable. In the common parabola, n = 2, m=l, and we have fF(x)dx = — xy y 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 y n =px m , 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 function, 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 y 2 = 2px, and substituting in the equation ydx the value of dx -= ^-^ we have P d P INTEGRAL CALCULUS. 253 or, if the area be estimated from the vertex A, we have C = 0, and fydx — — — . Jy Sp P F f If now, we wish the area to terminate " at any ordinate PM = b, we shall then take the integral between the limits of y = and y = b ; and, to express that in the differential equation, we write 1 rb o 7 b 3 which is read, integral of y 2 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. : AMP \i\ yHy 3p' AM we then have PMM y P' = —f C y 2 dy = — : pj y y Sp 'P === AMM'P r - AMP = — fl tfdy __J_ b 3 _ 1 287. Let us now determine the area of any portion of the space included between the asymptotes and curve of an hyperbola. 22 254 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 PMM'P', of which PM=y and PP f = dx. If we designate the angle YAX=MPX by |8, we shall have area PMM'P = ydx sin i and substituting for y its value — , x the area BCMP by s, we have 7 nr • d°c as = MsmQ — , and representing C&x and 5 — M sin/3 / — = M sin/3 logx + C. J x If AC is the semi-transverse axis of the hyperbola, and we make AB=1, and estimate the area s from BC, we shall have, for x = 1 , 5 = 0, and consequently C = ; and the true integral will be s = Msinp\ogx. 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= 1, and consequently, s = sin /3 logo?. Now, since s, which represents the space BCMP for any abscissa x, 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 svsiem of which sin (3 is the modu- lus (Alg. Art. 268). Therefore, any hyperbolic space BCjIP is the logarithm of the corresponding abscissa AP, taken in the system ichose ?nodulus 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 siiiS =: 0.43429945, (Alg. Art. 272). 2S3. The equation of the circle, when the origin of co- ordinates is placed on the circumference, is y 2 — 2rx — x 2 , or y — V 2rx — x 2 , and hence, the differential of the area is dx y2rx — x 2 ; mid this will become, by making x — r — u, \_ -fdu(i 2 -u 2 y. If we integrate this expression by formula (B) Art. 239, 256 ELEMENTS OF THE we have - fdu{r> - u 2 f =-\u{r z - u 2 f - ±-r 2 fdu(?*-u 2 f* du — — u-y/ r 2 — u 2 -\ f 2 i 2 2 J ■ But we have (Art. 253) /—du _ _ i /u\ and placing for u its value fdxV 2rx — x 2 = (r — x) V 2rx — x 2 -\ r- 2 cos _1 ( ) 4- C ; 2 v ; 2 \ r J ? and taking this integral between the limits x — and x = 2r, we shall have the area of a semicircle. For x — 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 — x* reduces to 0, and the second term to -l^cos-X- l) = —r 2 *r (Trig. Art. XIV), z z and consequently, the entire area is equal to r 2 ^, which INTEGRAL CALCULUS. 257 corresponds with a known result (Geom. Bk. V, Prop. XII, Coi. 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 r> y = —V^Ax—x 2 , A. and consequently, the area of the semi-ellipse will be equal to fydx = — I dxyj 2Ax — x 2 . A J Integrating, as in the last example, between the limits * = 0, and x = 2A, and multiplying by 2, we find ABtt 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 dx = ydy V2ry - y 2 ' whence y 2 dy fydx= j- ■y/2ry — y 2 and applying formula E, (Art. 243) twice, it will reduce to f . dy - ; and (Art. 226) «/ V2ry-y 2 /V^== = ver-sin-<m. J V 2ry -y 2 v r / But we may determine the area, of the cycloid in a more simple manner by introducing the exterior segment AFKH, 22* 258 ELEMENTS OF TUB Regarding FB as a line of abscissas, and de- signating any ordinate as KH, by z — 2r — y, we shall have F K B \ B / f &"J >>. A M L But whence zdx d(AFKH) = (2r-y)ydy V27 -y-y ZC&T. dy^2ry~-y\ AFKH=fdyV2ry—y 2 -f a But this integral expresses the area of the segment of a circle, of which the abscissa is y and radius r (x4rt. 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 t and consequently, area AHBM = ,4 \F£M - i¥IGJ5. 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, axQa.AHBM=3MXGB r 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) 7 u 2 dt as == : 2 and placing for u its value at (Art. 189) 7 a 2 t 2 "dt , a 2 f n+1 ; n as — and s = h 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, f = 2?r, and we have _ a 2 (2*Y n+l S ~ 4:71 + 2 ' which is the area included within the first spire. 291. In the spiral of Archimedes (Art. 192) a — — and n = 1 ; 2* hence, for this spiral we have f s 24^ 2? which becomes — , after one revolution of the radius- 3 vector ; the unit of the number — being a square whose o 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 in any revolution, it will pass over, or redescribe, all the area before generated. Hence, to find the area at the end of the with revolution, we must integrate between the limits t = (m— 1)2*- and t = m.2?r, which gives m 3 — (?n — 1 ) 3 1 —7T. 3 If it be required to find the area between any two spires ? as between the with and the (m + 1 )th, we have for the whole area to the (m -f l)th spire equal to (m + 1 ) 3 — mf 3 " ; and subtracting the area to the mth spire, gives 0+l) 3 -2m 3 -{-0- l) 3 —7F — 2mir, 3 for the area between the with and (wi + l)th spires. If we make m =± 1, we shall' have the area between the first and second spires equal to 2t: hence, the area he- tiveen the mth and (m + 1 )th spires, is equal to m time's the area between the first and second. 292. In the hyperbolic spiral n = — 1, and we have ds =c 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 a 2 / 1 1 \ S =-2KT-V)' INTEGRAL CALCULUS. 293. In the logarithmic spiral t — \ogu : hence, dt 261 du l dt udu hence, 2 2 udu u 2 /udu ir n and by considering the area s — when u = 0, we have C = and Determination of the Area of Surfaces of Revolution. M M B 294. If any curve BMM f , 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- ridian curve. It is required to find the differential of this surface. For this A P P' X purpose, make AP == x, PM == y, and PP' — h : we shall then have PM = F(x) = i/, P'M' *(.+»)=, + £* + $£+ to. 262 ELEMENTS OF THE In the revolution of the curve BMM , the extremities M and M! of the ordi- nates MP, M'P f , 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 -J- circ.MP') 2 (2 7 rMP+2vM / P>) (■ x chord MM P P' X that is, to X chord MM' =v (MP +MP') X chord MM' '; and by substituting for MP, MP' their values, the expres- sion for the area becomes dy d 2 u h* If now we pass to the limit, by making h — 0, the chord MM' will become equal to the arc MM (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 — 27rydz, and by substituting for dz its value (Art. 128), we have ds — 27ry ydx 2 + dy 2 : 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 x. 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 b. 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 y — -—x and dy = —-dx. h h Substituting these values of y and dy, in the general for- mula, we have bx , /tt, — nr bar f27ry^dx 2 +dij 2 =f27r — dxVJ^+b 2 = 7 r--V}i 2 +b 2 -i-C, and integrating between the limits x = and x — h, we obtain surface of the cone = ?r b -\Jli- -j- li z = 2nb x 2 • AJ, CB = 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 line DC will be y = b : hence, dy = 0. Substituting these values in the general expression of the differential of the surface, we have f27ryVdx z +dy 2 = f27rbdx = 27rbx+C; 264 ELEMENTS OF THE and taking the integral between the limits x = x = h, we have surface = 2vbh = 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 , x 2 + y 2 = R 2 , and by differentiating, we have xdx + ydy = ; hence , XO/X -, 79 X OjX and ay' y y 2 Substituting for dy its value, in the differential of the surface ds — 27ry Vdx 2 -j~ dy 2 we obtain s = f2*y y dx 2 -+- ~dx 2 =/2irRdx = 2rrR X + C. If we estimate the surface from the plane passing through the centre, and perpendicular to the axis of X, we shall have 5 = for x — 0, and consequently C = 0. Now, to find the entire surface of the sphere, we must integrate between the limits x = + R 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 s = 2*R 2 ; and integrating between the limits x = and x = — R, there results s=-2*-# 2 ; hence, surface = 4*-Pc 2 = 2*Rx2R; that is, equal to four great circles, or equal to the curved surface of the circumscribing cylinder. 298. The two equal integrals s = 2~R~ and s=-2*R 2 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 which being differentiated, gives dx= y^y and ^m. p p Substituting this value of dx in the differential of the sur- face, it reduces to **&/(£££)*? = jydy vV+TT 23 266 ELEMENTS OF THE But we have found (Art. 217) fjydyV¥+f = ^(y 2 +P 2 f + C: hence, and if we estimate the surface from the vertex at which point y — 0, we shall have, ~ 2*P 2 „ %*P 2 . = -£-+C, whence, C= - -f- f and integrating between the limits y=0, y = b, we have s = f p [(V+P % T-y*\ 300. To find the surface of an ellipsoid described by revolving an ellipse about the transverse axis. The equation of the meridian curve is A 2 y 2 + B 2 x 2 = A 2 B\ whence T B 2 xdx B ocdx A " y A Va 2 -* 2 substituting the square of this value in the differential of the surface and for y its value ^VA 2 -x? we have : 2 7T- dx v/A 4 - ( A 2 - B V* INTEGRAL CALCULUS. 267 and «'="*« || -fA^&fdx \J-^p - °? ; and if we represent the part without the sign of the inte- gral by D, and make A 4 A 2 -B 2 we shall have R\ s = DJdxjW-o?. But the integral of das yR 2 — x 2 is a circular segment of which the abscissa is x, the radius of the circle being R. 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 s =fdx -y/R 2 — x 2 between the limits x = and at = A, and designate by D' the corresponding portion of the circle whose radius is R, we shall have — surface ellipsoid = D x D f ; hence, surface ellipsoid = 2D x D f . 301. To find the surface described by the revolution of the cycloid about its base. The differential equation of the cycloid is dx. ydy V2ry — ,2 268 ELEMENTS OF THE Substituting this value of dx in the differential equation of the surface, it becomes , _ 2^^2ry * dy -\/2ry — y 2 Applying formula (E), Ajt. 243, we have But, /v©=/^ = ^ (2! - yr ^- 2(2) - y)l; hence, l~ 2 — 8 - -1 s = 2^/2r —~y\/2ry — y 2 r{2r — y) 2 \+ C. _ 3 3 _J If we estimate the surface from the plane passing through the centre, we have C — 0, since at this point 5 = and y = 2r. If we then integrate between the limits y == 2r and y = 0, we have s = — surface = ^r 2 ; hence, 2 3 s— surface = Trr 2 , 3 that is, the surface described by the cycloid, when it is revolved around the base, is equal to 64 thirds of the generating circle. The minus sign should appear before the integral, since the surface is a decreasing function of the variable y (Art. 31). INTEGRAL CALCULUS. 269 Of the Cubature of Solids of Revolution. 302. The cubature of a solid is the expression of its volume or content. 303. Let u represent the volume or solidity generated by the area ABMP, when revolved around the axis AX. If we make AP = a?, PP' = h, we have MP'— F(x-Yh). Now, the solid gene- rated by the area ABMMP', will ex- ceed the solid described by ABMP, by the solid described by the area PMM'P' . The solid described by the area ABMP is a function of co, and the solid described by the area ABMM'P' is a simi- lar function of (x-\-h). If we designate this last by v!, we have dhi h 3 du 1 , d?u lr u-{-—-h + -— dx dx i 1.2 + dx* 1.2.3 hence, the solid described by PMM , P / is h 2 . d?u h 3 , du , <Pu U r —U = — h + t-5- dx dx 2 1.2 dx 3 1.2.3 + &c; + &c. Let us now compare the cylinder described by the rectan- gle P'M with that described by the rectangle P'C. The equation of the curve gives MP = y = F(x) M'P'=F(x + h); hence, since PP' = h, cylinder described by P'M— v [F(x)fh f cylinder described by P'C = * [F(x + k)fh; 270 ELEMENTS OF THE and the ratio of the cylinders is [F(x + h)] 2 [F{x)] 2 ■ the limit of which, when h — 0, is unity . But the solid described by the area PMM / P / is less than one of the cylinders and greater than the other ; hence, the limit of the ratio, when compared with either of them, is unity. Hence, du , , d 2 u h 2 , o du , d 2 u h c - T - h + -— - h&c. —r--\--rr + &c. dx dx 2 1.2 (fo? ^ 2 1 .2 ^[FO)] 2 /* "■[J , («)] ! the limit of which, when h — 0, is du dx I, whence and finally <2w = Try 2 dx ; the differential of the solidity iry 2 dx being a cylinder whose base is ^y 2 and altitude dx. 304. Remark. The differential of a solid, generated by revolving a curve around the axis of Y, is nx?dy. 305. Let it be required to find the solidity of a right cylinder with a circular base, the radius of the base being INTEGRAL CALCULUS. 271 r and the altitude h. We have for the differential of the solidity ny 2 dx, and since y — r, it becomes nr^dx ; and taking the integral between the limits x=0 and x = h, we have which expresses the solidity. 306. To iind the solidity of a right cone with a circular base, let us represent the altitude by h and the radius of the base by r, and let us also suppose the origin of co-or- dinates at the vertex. We shall then have r y = —x ana y J h y h XI and substituting, the differential of the solidity becomes .2 and by taking the integral between the limits x = and x = h, we obtain — r 2 -h = ^r 2 x — ; 3 3 that is, the area of the base into one third of the altitude. 307. Let it be required to find the solidity of a prolate spheroid, (An: Geom : Bk. IX, Art. 33). The equation of a meridian section is 272 ELEMENTS OF THE which gives hence the differential of the solidity is B 2 du — n-— 2 (A 2 — x 2 )dx, and by integrating u = ^( A 2 x--) + C If we estimate the solidity from the plane passing through the centre, we have for'a? = 0, u = 0, and consequently C = 0; and taking the integral between the limits x — Q and x = Aj we have 1 2 — solidity = —"B 2 x A ; and consequently 2 solidity ±= -— - nB 2 x 2 A. o But a-jB 2 expresses the area of a circle described on the conjugate axis, and 2A is the transverse axis : hence, the solidity is equal to two-thirds of the circumscribing cylinder. 308. If an ellipse be revolved around the conjugate axis, it will describe an oblate spheroid, and the differential of the solidity would be du = TcxPdy : INTEGRAL CALCULUS. 273 and substituting for x 2 , and integrating, we should find 2 solidity = —n A 2 x 2 B : o that is, two-thirds of the circumscribing cylinder. 309. If we compare the two solids together, we find oblate spheroid : prolate spheroid : : A : B. 310. If we make A = B, we obtain the solidity of the sphere, which is equal to two-thirds of the circumscribing cylinder, or equal to 4 1 R 3 = ^ 3 6 311. Let it be required to find the solidity of a para- boloid. The equation of a meridian section is if = 2px, and hence the differential of the solidity is du = 2 TTjyxdx ; hence u = TV par + C ; and estimating the solidity from the vertex, and taking the integral between the limits x = and x == h, and designa- ting by b the ordinate corresponding to the abscissa x = h, we have u = nph 2 z=z nb 2 x — ; r 2 that is, equal to half the cylinder having an equal base and altitude. 312. Let it be required, as a last example, to determine 274 ELEMENTS OF THE the solidity of the solid generated by the revolution of the cycloid about its base. The differential equation of the cycloid is y d y hence we have dx~ du V^ry — y 2 Kifdy V2nj-if which, being integrated by formula (E) Art. 243, and then by Art. 226, we find the solidity equal to five-eighths of the circumscribing cylinder. Of Double Integrals. 313. Let us, in the first place, consider a solid limited by the three co-ordinate planes, and by a curved surface which is intersected by the co-ordinate planes in the curves CB, BD, DC. Through any point of the surface, as M, pass two planes HQF and EPG respectively paral- lel to the co-ordinate planes ZX, YZ, and intersect- ing the surface in the curves IIMF and EMG. The co-ordinates of the point M are AP=x, PM=y, MM'=z INTEGRAL CALCULUS. 275 It is now evident that the solid whose base on the co-ordi- nate plane YX is the rectangle A QM'P, may be extended indefinitely in the direction of the axis of X without chang- ing the value of y, or indefinitely in the direction of Y without changing x. Hence, a? and y maybe regarded as independent; variables. If, for example, we suppose y to remain constant, and x to receive an increment Pp = h, the solid whose base is the rectangle AQM'P, will be increased by the solid whose base is the rectangle M'm'pP ; and if we suppose x to remain constant, and y to receive an increment Qq — /c, the first solid will be increased by the solid whose base is the rectangle Qqn f M f . But if we suppose x and y to receive their increments at the same time, the new solid will still be bounded by the parallel planes epg, hqf, and w T ill differ from the prim- itive solid not only by the two solids before named, but also by the solid whose base is the rectangle n'M'mJW . This last solid is the increment of .the solid whose base is the rectangle M'Ppm!, when we suppose y to vary; or the increment of the solid whose base is the rectangle Qqn'M', when we suppose x to vary. Let us represent by u the solid whose base is the rect- angle A QM'P ; u will then be a function of x and y, and the difference between the values of the increments of u, under the supposition that x and y vary separately ; and under the supposition that they vary together, will be equal to the solid whose base is the rectangle n'M'm'N'. By taking this difference (Art. S3) we have 276 ELEMENTS OF THE hence, solid n'N'm'M...M d?u 1 d 3 u , 1 d?u hk dxdy ' 2 cfo 2 dz/ 2 cfody 2 and passing to the limit, by making h = and h=0, the second member becomes -j — 7-. dxdy As regards the first member, the rectangle n'N'm'M 7 = hxh and the altitude of the solid becomes equal to M!M—z when we pass to the limit : hence d 2 u dxdy ~ 314. Although the differential coefficient &u dxdy ~ has been determined by regarding u as a function of two variables, we can nevertheless return to the function u by the methods which have been explained for integrating a function of a single variable. For we have d?u ~\dxJ «© dxdy dy hence and integrating under the supposition that x remains con- INTEGRAL CALCULUS. 277 slant, and y varies, we have du whence **/*+*• — dx = dx fzdy + JT c?07 ; aa? and if we integrate this last expression under the supposi- tion of x being the variable, and make j X! dx = X\ u = fdxfzdy + X + Y. It is plain that the constant, which is added to complete the first integral, may contain x in any manner whatever; and that which is added in the second integral, may contain y : the first luill disappear ivhen we differentiate with respect to y, and the second when ive differentiate with respect to x. The order of integration is not material. If we first integrate with respect to x, we can write d?u _ \dy) . d,xdy dx ( and by integrating, we find -——fzdxy u=fdyfzdx: hence we may write u — J fzdy dx, or u = ffzdxdy, which indicates that there are two integrations to be per- formed, one with respect to x, and the other with respect to y. 24 278 ELEMENTS OF THE 315. If we consider the differentials as the indefinitely small increments of the variables on which they depend, we may regard the prism whose base is the rectangle n'N'm'M', as composed of an indefinite number of small prisms, having equal bases, and a common altitude dz. Each one of these prisms will be expressed by dxdydz, and we shall obtain their sum by integrating with respect to z between the limits z — and z = MM', which will give / dx dy dz — zdx dy. 316. It is plain that zdx is the differential of the area of the section made by the plane HQF parallel to the co-ordinate plane ZX ; and consequently f z dx = area of the section HQF. Hence, (Jzdx)dy is equal to the elementary solid in- cluded between the parallel planes HQF, hqf t or f(fzdx)dy=ffzdxdy is equal to the solid which is limited by the surface and the three co-ordinate planes. If we consider a section of the solid parallel to the co-ordinate plane YZ, we have fzdy = area of the section EPG, and ffzdxdy '■==. solidity of the solid. 317. Let us suppose, as a first example, that _ 1 z ~ tf + y 2 ' we shall then have JJaZ + y 2, J J a? + y 2 J J J ap + y 1 INTEGRAL CALCULUS. 279 Let us now integrate under the supposition that x is con- stant ; we then have /*_^_ = .i tang -^ + jr, J x l -\- y x x in which X! represents an arbitrary function of x. If we now make fX / dx = X, and integrate again under the supposition that x is a variable, we have fdx P\ 9 dy "l =Jdx[— tang- 1 -^- + X'l J J x z + y z J Lx ° x J = fe ang -y +x J X X dx u The integral of — -tang -1 — is obtained in a series by xx substituting the value of (Art. 228), x x 3x 6 5x 5 7x 7 and since, in integrating with respect to x, we must add an arbitrary function of y, which we will represent by Y, we shall obtain r rdxdij _ x , Y _y_ + _£ l/Lljl. &c We shall obtain the same result by integrating in the in- verse order, viz., by first supposing y to be constant. Under this supposition r^^ = -tang- , ^+F ) J a? + y 2 y ° y 280 ELEMENTS OF THE then integrating with respect to x, '*/^=^[7 , '"< r 7 +I ''] -/ y y y y But by observing that (Trig. Art. XVIII), tang 1 _ = __tang l -±-, we shall have, after the second integration, and the addi- tion of an arbitrary function of x, and as we can include the term — logy in the arbitrary function Y, this result may be placed under the form ffj^y_ X+Y _fdy tJLt J J xr -\-y z J y x which is the same as the result before obtained, as may be shown by placing for tang -1 — its value, multiplying each term by — , and integrating. 318. When we consider ffzdxdy as expressing the solidity of a solid, it is necessary to con- sider the limits between which each integral is taken, and these limits will depend on the nature of the solid whose cubature is to be determined. Let it be required, for ex- INTEGRAL CALCULUS. 281 ample, to find the solidity of a sphere, of which the centre is at the origin of co-ordinates. Designating the radius by R, we have a? + y* + z 2 =zR 2 , and consequently, ffzdxdy =ffdxdy VR 2 -x*-y 2 . If now, wc suppose y constant, and make R 2 — y 2 = R /2 , and then integrate with respect to x, we have / dx VR" - x' z -if=fdx VR /2 - x 2 , and integrating this last expression, first by formula (B) Art. 239, and then by Art. 220, we have fdxVR /2 -cc 2 = — ^n^^+ — R f2 sm- l ~ + Y; and substituting for R' 2 its value, we obtain It should be remarked, that fzdx expresses the area of a section of the sphere parallel to the co-ordinate plane ZX t for an v ordinate y — AQ, and to obtain this area we must integrate between the limits x — and x — QF. But since the point F is in the co-ordinate plane YX t we have for this point z — 0, and the equation of the sur- face gives QF=:x=VrT^ 2 ; therefore, for every value of y the integral fzdx must be taken between the limits x = and x — VR 2 — y 2 . Inte- 23* 282 ELEMENTS OF THE grating between these limits we have fdxVR 2 -x 2 -f = —{R 2 -y 2 )sm" l {l) since, sin -1 (l) = — : hence, fdyfzdx = ^-fdy(R> - f) = ^y-t> ) + X, and taking this last integral between the limits y = and y = AC = R, we obtain 6 ' which represents that part of the 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?^~ — D 3 n. We might at once find the solidity of the hemisphere which is above the horizontal plane YX, by integrating between the limits x = — VR 2 — y 2 and x = + V R 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 = -f VR 2 — y 2 > INTEGRAL CALCULUS. 283 we have fzdx =— (R 2 — y 2 ) ; hence, between the extreme limits, we have fzdx = ?L(R*-tf). Then taking the integral Sdyfzdz = ^fdy(R*-f) p between the limits y=—R and y = 4- R, we find the solidity to be I** or the solidity of the entire sphere is, 3 THE END • ++ y \°^> \l' .\ vOq. ; ** v '^- •— ' : / V ^ ^ ^ %%*■ ^ V* 4* X° °xn - ^ ^ s