I 9;~~~~Z ~~~~U j~~~ I~~~~~ ELEMENTS OF THE CALCULUS. ELEMENTS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS. PREPARED BY ALBERT E. CHURCH, LL.D., PROFESSOR OF MATHEMATICS IN THE U. S. MILITARY ACADEMY. REVISED EDITION, CONTAINING THE ELEMENTS OF THE CALCULUS OF VARIATIONS. NEW YORK: PUBLISHED BY A. S. BARNES & BURR, 51 & 53 JOHN STREET. 1866. Entered according to Act of Congress, in the year 1861, BY ALBERT E. CHURCH, In the Clerk's Office of the District Court of the United States for the Southern District of New York. RENNIE, SHEA & LINDSAY, ByUauOTYPIRIs AND ELiCTNROTYPr f, GOX)RGE W. WOOD, PnumuB 81, 83 & 85 CENTRE-STREIT, No. 2 Dutch-st., N Y. NEW YORK PREFACE. AN experience of more than twenty-five years, in teaching large classes in the U. S. Military Academy, has afforded the Author of the following pages unusual opportunities to become familiar with the difficulties encountered by most pupils, in the study of the Differential and Integral Calculus. The results of previous endeavors to remove these difficulties, were given to the Public in former editions. Prepared, as these editions were, solely as aids to himself, in the instruction of his own pupils, he is gratified to know that they have proved acceptable to many other teachers, and that he has thus aided in extending a knowledge of this important branch of Mathematics, now absolutely necessary to thorough analytical research in the higher branches of Physical Science. These editions have been carefully revised, and such changes introduced in the arrangement of the matter, and in the modes of demonstration, as the Author's prolonged experience has shown to be improvements. Such new matter has also been added, as he deems necessary to the Vi PREFACE. perfection of the work as an elementary text-book, for those who will be satisfied with nothing short of a thorough knowledge of the subject. The pains which have been taken to secure accuracy in the algebraic work and in the language of the text, and a clear and neat typography, will, it is hoped, render' the present edition acceptable alike to teachers and pupils. U, S. MILITARY ACADEMY, West Point, N. Y., January 1, 1861. CONTENTS. PART I. DIFFERENTIAL CALCULUS. rAGE Definition and classification of functions............................ 1 " of the differential, and differential coefficient..6........... 6 Rules for obtaining them....................................... 8 Expression for the new or second state of a function............... 11 Manner of making first term of a series greater than the sum of all the others........................................................ 12 Differential coefficient of an increasing function is always positive, &c.. 12 Equal fundtions of the same variable have equal differentials......... 13 Differentiation of the product of a constant by a variable.......... 13 Differential coefficient of one variable the reciprocal of that of the other. 15 " " of an implicit function........................ 16 Differexitiation of the sum or difference of several functions....... 18 " of the product................................... 19 r" of the power of a function...................... 21 " of radicals......................................... 22 " of a fraction.................2............ 23 " of miscellaneous examples...................... 25 Successive differentiation........................................ 27 Maclaurin's Theorem..........................................831 Definition and property of functions of the sum of two variables...... 85 Taylor's Theorem..................................8............. 37 Failing case of Taylor's Theorem...................... 40 Development of the second state of a function of one variable.........41 Deduction of Maclaurin's from Taylor's formula.................. 42 tilll CONTENTS. FAGE Differentiation of logarithmic functions........................ 43 Deduction of the logarithmic series............................... 46 Differentiation of the transcendental function as................ 47 of complicated exponential functions................ 48 " of the circular functions......................... 50 " of the arc in terms of its sine, cosine, &c............. 53 Development of thesine and cosine in terms of the arc.............. 56 " of the arc in terms of its sine, and tangent............ 57 " of the second state of a function of two variables...... 59 " of the second state of a function of any number of variables 64 Differentiation of functions of two or more variables.............. 65 Development of any function of two or more variables.............. 68 Differential equations............................................. 69 Immediate differential equations;............................ 72 Elimination of constants, particular functions, &c................... 72 Differential equations of species of lines 3.................... 73 Partial differential equations.X....................... 75 Change of the independent variable............................... 6 Vanishing fractions.......................................... 80 General rule for determining their value..........................., 83 Practical rule for determining their value......................... 85 Maxima and minima of functions of a single variable............. 90 General rule for finding the maximum or minimum states of a function of one variable.......................................... 91 Practical rule.................................. 95 Solution of problems in maxima and minima............... 102 Maxima and minima of functions of two or more variables.......... 108 Geometrical signification of a function of a single variable, and of its differential coefficient................................... 114 General equation of the tangent line............................... 118 4 " of the normal................................ 119 Expressions for the subtangent, subnormal, tangent, &c......... 120 Mode of determining whether a curve is concave or convex.......... 121 Asymptotes................................... 123 Advantages of regarding the differential as infinitely small....... 127 Differentials of-an arc, plane area, surface and volume of revolution.. 129 Tendency of curves to coincide................................. 134 Manner of finding the order of contact of lines...................., 136 it the equations of osculatory curves............... 138 Equations of condition for the osculatory circle............. 142 General expression for radius of curvature.......,......... 146 CONTENTS. ix PAGE Definition of curvature..........................................'148 Value of radius of curvature of the conic sections................... 150 Evolutes....................................................... 152 Rule for finding the equation of the evolute....................... 155 Envelopes..................................................... 157 Application of the Differential Calculus to the construction and discussion of curves................................................ 163 Definition of singular points..................................... 163 Points of infiexion........................................... 165 Cusps.......................................................... 168 Isolated or conjugate points............................ 170 Multiple points............................................ 172 Logarithmic curve..................................... 174 Cycloid........................................................ 178 Polar curves................................................... 184 Spiral of Archimedes...................................... 188 Parabolic spiral............................................... 189 Hyperbolic spiral............................................... 189 Logarithmic spiral............................................... 191 Application of the Calculus to surfaces......................... 192 Maximum inclination or slope of surface................... 195 Equation of tangent plane to surfaces................... 196 " of normal line.............................. 198 Distance from any point of the normal to the point of contact....... 198 Projection of any plape area......................... 199 Partial differential of any surface or volume....................... 201 Osculatory surfaces............................................ 201 Circles of least and greatest curvature.............................. 204 PART II. INTEGRAL CALCULUS. Object and first principles..................................... 209 Integration of monomial differentials................... 211 " of particular binomial differentials..................... 213 " of fractions, in which the numerator is a constant into the differential of the denominator................................. 215 Discussion of the arbitrary constant and integration between limits... 217 X CONTENTS. Integration of differentials of circular functions and arcs......... 220 " of rational fractions............................... 223 " by parts.................................... 234 " of certain irrational differentials....................... 236 " of those containing -va + bx ~ cx2.................... 239 " of binomial differentials............................... 245 Formulas A, B, C, D, and E.................................... 249 Integration by series.......................................... 254 Series of Bernouilli............................................ 258 " for integrating between limits.............................. 259 Integration of transcendental differentials.......................... 260" of differentials of the higher orders................ 268 of partial differentials........................ 2 71 " of total differentials of the first order containing two variables.................................................... 273 Integration of total differentials of the first order containing three or more variables.................................................. 276 Integration of the same when homogeneous........................ 277 Mode of differentiating all indicated integral.......27...9..... 279 Separation of the variables in differential equations............ 280 Integration of linear equation dy + Pydx = Qdx................... 285 " of certain equations which may be made homogeneous... 287 Of the factors by which certain differential equations are rendered integrable..................... 288 Differential equations containing the higher powers of dy. 292 dx. Singular solutions............................................. 297 Integration of differential equations of the second order............. 299 of differential equations of the higher orders............ 305 " of linear equations.................................... 306 " of partial differential equations of the first order......... 311 Application of the Calculus to the determination of curves with particular properties............................................. 314 Rectification of curves........................................... 320 " of spirals.........3............. 325 Quadrature of curves......................................... 326 " of spirals................................ 333 Area of surfaces of revolution.................................. 337 " of curved surfaces generally................................. 340 Cubature, of volumes of revolution............................... 342 " of volumes bounded by any surface.,.................. 345 CONTENTS. xi PART III. CALCULUS OF VARIATIONS. PAGE First principles...................................... 349 Variation of the differential equal to the differential of the variation.. 352 " of the integral equal to the integral of the variation...... 353 General expression for the variation of a function............... 353 " for the integral of the variation of a function..... 355 " " for the variation of fvdx...................... 358 Maxima and minima of indeterminate'integrals................... 358 Conditions of maxima and minima................................ 359 Problems in maxima and minima. 362 Method of reducing the number of independent variations............ 367 PART I. DIFFERENTIAL CALCULUS. DEFINITION AND CLASSIFICATION OF FUNCTIONS. i. IN the branch of Mathematics here treated, as in Analytical Geometry, two kinds of quantities are considered, viz., variables and constants; the former admitting of an infinite number of values in the same algebraic expression, while the latter admit of but one. The variables are generally designated by the last, and the constants by the first letters of the alphabet. 2. One variable quantity is a function of another, when it is so connected with it, that any change of value in the latter necessarily produces a corresponding change in the former. Thus in the expressions u = bx, au.2 = cx5 u is a function of x, and x is also a function of u. Likewise, the expressions ay', (b - y)', are functions of y, and in each case y is a function of the expression. One of these variables is usually called the function, and 1 2 DDIFFERENTIAL CALCULUS. the other the independent variable, or simply the variable; since to one, any arbitrary values may be assigned, and from the connection between the two, the corresponding values of the other deduced. This relation is expressed-generally thus, -u = f(x), u = (x),, or f (u,) = 0, f and g being mere symbols, indicating that u is a function of x. The first two expressions are read, u a function of x, or u equal to a function of x; and the third, a function of u, and x equal to zero. The result, obtained, by assigning a particular value to the variable, is called a state or value of the function, and each function has an infinite number of such states. Thus, if we have the function (a - x)9 a, O, a, 4 a' &c., are states of the function corresponding to the particular values of x, O, a, 2 a, 3 a, &c. 3. Functions are Increasing and Decreasing: Increasing, when they are increased if the variable be increased, or decreased if the variable be decreased: Decreasing when they are decreased if the variable be increased, or increased if the variable be decreased. In the expressions u = aX3, u = (x + a)', u is an increasing function of x. In the expressions y = = (a - X)3, x y is a decreasing function of x. In the expression z - (a - y)2 DIFFERENTIAL CALCULUS. 3 z is a decreasing function for all values of y less than a, but increasing for all values greater than a. 4. Functions are also Explicit and Imzplicit: Explicit, when the value of the function is directly expressed in terms of the variable: Implicit, when this value is not directly expressed. In the examples U = (a - x)2 y = Va2 - u and y are explicit functions of x. In the examples au + bx = cx, y2 a2 _ x2 or au2 - bX - cx- =, y2 +- x - a2 = 0, they are implicit functions of x. The relation between an implicit function and its variable may be expressed, either by a single equation, as above, or by two or more equations, as u ay2, 2 = bx, in which u is an implicit function of x. The first relation is indicated generally by f(', x)= o, and the other thus. =f(y), y = p (x). 5. Functions are also Algebraic and Transcendental: Algebraic, when the relation between the function and variable can be expressed by the ordinary operations of Algebra, that is, by addition, subtraction, multiplication, division, the formation of powers denoted by constant exponents, and the extraction of roots 4 DIFKEiENTIAL CALCULUS. indicated by constant indices: Transcendental, when this relation cannot be so expressed. In the examples u = log x, u = sin (a - x), u = a', Tu is a transcendental function of x. If the variable enter the exponent, the function is called Exponential. The logarithm -of a variable expression is a Logarithmic function. In the expressions u -= sin x, u = cos x, u = tan - u is said to be a Circular function. 6. Functions are often mixed, being formed by the anion of different kinds of simple functions, as in the expressions log x + sin x, ax2 + am, 7. Functions are also Continuous or.Discontinuous: Continuous, when every state obtained by substituting values of the variable between the least and greatest which give real values of the function, is real: Discontinuous, when any of such states are imaginary. In the expressions b V at a y in the first is continuous, in the second discontinuous; as in the one all values of x between - a and + -a give real, while ian the other they give imaginary values for y. 8. A quantity is a function of two or more independent vari ables, when it is so connected with them that it will change it either variable be changed, as in the examples DIFFERENTIAL CALCULUS. 5 u = ax -- by, Z = axy2 - uVx denoted in general thus, = f (x, y) z = F (x, y, u). If in a function of a single variable, the latter be made equal to zero, the function reduces to a constant, as in the examplesu = ay', u = C + bx'; if y O, we have u = O; if z = O u = c., If in a function of two or more variables any one be made equal to zero, all the terms containing it will disappear, and the result will be entirely independent of this variable, as in the example u = az + by' + Cz3 + d, z = 0 gives =u = ax + by' + d = f (x, y); z = 0 and y - 0 give -u = ax,- d =f (x). If all the variables be made equal to zero, the result will be constant, as in the same example, z = O, y = 0, and x = O, give u = d = a constant. Likewise, when the variable which is made equial to zero is a factor of all the terms containing any of the others, as in the example u = c + ax'y + bzy' =f (x, y, z), y = 0 gives U -- C. 6 DIFFERENTIAL CALCULUS. DEFINITION AND PROPERTIES OF THE DIFFERENTIAL AND DIFFERENTIAL COEFFICIENT. 9. To explain what is meant by the dieffrential of a quantity or function, let us take the simple expression'u = ax2....................(1) in which u is a function of x. Suppose x to be increased by another variable h; the original function then becomes a (x + Al2; calling this new state of the function u', we have u'=- a (x + h)2 = ax2 + 2axh + ah2. From this, subtracting equation (1), member from member, we have u'- u = 2axh + ah............(2). The second member of this equation is the difference between the primitive and new state of the function ax', while h is the difference between the two corresponding states of the independent variable x. As h is entirely arbitrary, an infinite number of values may be assigned to it. Let one of these values, which is to remain the same, while x is independent, be denoted by dx, and called differential of x, to distinguish it from all other values of Ah. This particular value being substituted in equation (2), gives for the corresponding difference between the two states of u, or ax, U' — u; = 2ax.dx + a (dx)2. Now, the first term of this particular difference is called the differential of u, and is written du = 2ax. dx. The coeficient (2ax) of the diferential of x, in this expression, is called the differential coefficient of the function u, and is evidently DIFFERENTIAL CALCULUS. 7 obtained by dividing the differential of the function by the differential of the variable, and is in general written du -= 2,ax. d.r Pesuming the expression u! — u = 2axh + ahl, and dividing by h, we have - 2 2ax + ath. In the first member of this equation, the denominator is the variable increment of the variable x, and the numerator the corresponding increment of the function u; the second member is then the value of the ratio of these two increments. As h is diminished, this value diminishes and becomes nearer and nearer equal to 2 ax and finally when h - 0, it becomes equal to 2 ax. From this we see, that.as these increments decrease, their ratio approaches nearer and nearer to the expression 2ax, and that by giving to Ih very small values, this ratio may be made to differ from 2ax, by as small a quantity as we please. This expression is then properly, the limit of this ratio, and is at once obtained from the value of the ratio, by making the increment h = O. It will also be seen:that this limit is precisely the same expression as the one which we have called the differential coefficient of the function u. What appears in this particular example is general, for let -f (), u being any function, of x, and let x be increased t y h, then u' = f (x + h). Suppose f (x + h) to be developed, and arranged according to 8 DIFFERENTIAL CALCULUS. the ascending powers of h, and u to be subtracted from both members, we then have u' — u = Ph + OQh + Rh3 + &c........ (3), P, Q, R, &c., being functions of x, and every term of the second member containing h, because u' - u must reduce to 0 when h = 0. Substituting for h the particular value dx, and taking the first term for the differential of u, we have du = Pdx, and d = P. Dividing both members of equation (3) by h, we have ul- u - = P + Qh + Rh2 +, &c........(4). Obtaining the limit of this ratio by making h = 0, and denoting it by L, we have L = P du the same value found above for -; henceh, the diyerential coefficient of a function is always equal to the limit of the ratio of the increment of the variable to the corresponding increment of the function. 10. The differential of a function of a single variable may then be thus defined. If the variable be increaied by a particular value, called the differential of the variable, and the difference between the new and primitive states of the function be developed according to the ascending powers of the increment; that term oj this difference which contains the first power of the increment is the differential of the function. It will in general be found most convenient to obtain first the differential coefficient, for which we have the following rule: DIFFERE NTIAL CALCULUS. 9 Give to the variable a variable increment, find the corresponding state of the function, from which subtract the primitive state, divide the remainder by the. increment, obtain the limit of this ratio by,making the increment equal to zero, the result will be the differential coefficient: This, multiplied by the differential of the variable, will give the differential of the function. The object of the Differential Calculus is, to explain the mods-> of obtaining and applying the differentials of functions. 11. Let the preceding principles be illustrated by the following Examples. 1. Let i = bx3. For x substitute x + h, then, u'= b (x + h)3 - bx3 + 3bx'h + 3bxh2 + bh3, U' - u -= 3bx2h + 3bxh2 + bh3, - 3bx2 + 3bxh + bh2; passing to the limit, and denoting it by L, we have du L = 3bx - d; whence du 83 bxdx. 2. Let u = axn -- cx. Substituting x + h for x, and subtracting, we have 10 DIFFERENTIAL CALCULUS. u' - u = 2 axh + ah2 - ch, U - U'- = 2ax + ah - c; h Imaking h = 0, we have du L = 2ax - c = d whence du = 2 axdx - cdx. 3. Let a then = a a a - ah ux +h x +x 2 +4 xh' u- -u - a h x2 + xh' and a du L - = dx' whence adx du = - d 4. If u = 3axS - mx4, du = (9ax' - 4mxs) dx. DIFFERENTIAL CALCULUS. 11 12. Equation (4), article (9), may be put under the form h' = P + h (Q + Rh + &c.), and if the expression Q + Rh + &c. (which is a function of and h) be represented by P', this becomes P + Pf............ (1); whence U'-= u + Ph + P'h2; that is, the new state of the function is equal to its primitive state, plus the diff'erential coefficient (f the function into the first power of the increment of the variable, plus a function of the variable and its increment into the second potwer of the increment. This expression for the neew state of the function being an important one, should be carefully remembered. 13. If we resume equation (3), Art. (9), divide by h and transpose P, we have u - P =Qh + Rha 2 + &c. Since when h 0, the expression for the ratio / reduces to P, Art. (9), if h be infinitely small, we shall have < 2P, or P < P; h whence Qh + Rhi 2 + &c. < P, and multiplying by h, Ph > Qh' + Rh2 + &c. 12 DIFFERENTIAL CALCULUS. That is, in a series arranged according to the ascending powers of an infinitely small quantity, the first term is. numerically greater than the sum of all the others. 14. If u be an increasing function of x, its new state u' will be greater than u, and h P.- P'h....Art. (12), will be positive for all values of h. If u be a decreasing function, the reverse will be the case, and the ratio be negative for all values of h. But we see, by the preceding article, that when h is infinitely small, the sum of all the terms that follow P, in the above equation, will be less than P, and therefore the sign of P will be the same as that of the ratio; that is, positive when u is an increasing, and negative when u is a decreasing function. But P is the differential coefficient of u, Art. (9). Hence, the differential coefficient of an increasing function is always positive; and of a decreasing function, negative. It should be observed, that the signs of the differential and differential coefficient are always the same. 15. Let U = —, u and v being functions of the variable x, which are equal to each other for every value of x. If x be increased by h, and u' and v' be the new states of u and v, we have tUt = VPt, UPd - lb i -Uut- v' - v, uuvvh h DIFFERENTIAL CALCULUS. 13 Passing to the limit of these equal ratios by making h = 0, we have, Art. (10), du dv or du = dv; that is, if two functions of the same variable are equal, their differ. entials will also be equal. 16. But if u = v:1: C, u and v being functions of x, and C a constant, and x be increased by A, we have Uv u' V' - v.' = vI' it: C, U'- - UP =U v' -,= and passing to the limit dzu dv Td T d or du= d (v:t C) = dv; that is, if two dierentials are equal, it does not follow that the expressions from which they?tere derived are equal. We see also, that a constant connected by the sign i with a variable, disappears by differentiation. In fact, the differential of a constant is zero;'since, as it admits of no increase, there is no difference between two states, and of course no differential, Art. (10). 17. Let u - Av, then =o'- Av', UP - Av__- v 14 DIFFERENTIAL CALCULUS. and passing to the limit, du dv -d Ad - or du = d(Av) = Adv; dx dx' tlhat is, the ditferential of the poduct of a constant by a variable f;It(ction, is equal to the constant multiplied by the differential qf the fyuction. 18. When two variable quantities are so connected that one is a fhnction of the other, either may be regarded as the function, and the other as the independent variable. Thus, fiom the ex)prssion u -= ax2, we readily obtain x =; in which x a mnay be considered a function of the variable u. In general, let u =f(x)...........(1); then by deducing the value of x, = f' ()..........(2). In this last expression, let the variable u be increased by any variable increment u' — u- = k, x will receive the corresponding increment x' - x, and the ratio of these increments will be x' x k............(3) If the increment x' - x be denoted by h, and we substitute x + h for x, in equation (1), we shall obtain, Art. (12), u'- u = Ph + P'h = k, and substituting these values of x' —x and k in expression (3), we have x'- x h 1 k Ph/ + P'h2 P + P'h' DIFFERENTIAL CALCULUS. 1 Passing to the limit by making k, the increment of u, equal to 0, in which case h = 0, we have dx 1 1 du P du' dx du since P - d; that is, the differential coefficient of x regarded as a function of u, is the reciprocal of the differential coefficient of u reqarded as a function of x. It should be observed that du in the first member of the above equation is constant, u being the independent variauble, Art. (9), while dx is variable. In the second memrber, the reverse is the case, dx being constant, and du variable. To illustrate, take the example -u ax2; whence a du In article (9) we have found = 2 ax; then dx 1 1 1 1 dx_ 1 1 1 du du -.2 2aVx - 2-1/au 19. Let u be an implicit function of x of the second kind, Art. (4), as u f(y)........(1), y = (x)........(2). If x be increased by h, y will receive an increment y' -, which we denote by k; and these increased values of y and x in the second members of (1) and (2) will give, Art. (12), DIFFERENTIAL CALCULUS. u'= u + Qk + Q'k2, y'= y + Ph + P'h2; whence - = Q + Q'k, P + P'h, k= h tnd by multiplication, u'- u y'- y XY h A QP + Q'Pk + QP'h + &c.; or, since y'- y = k,: = QP + Q'Pk + QP'h + &c. h Passing to the limit by making h- O, which gives k- 0, we have du -- = QP. dx But Q d' and P d;' whence du du dy dx dy X dx' that is, the diJerential coefficient of u regarded as a function of x, is equal to the differential coefficient of u regarded as afunction of y, multiplied by the differential coefficient of y regarded as a function of x. If -f (x). (), a = (x)........ (.4), DIFFERENTIAL CALCULUS. 17 in which case u is evidently an implicit function of v, we find from equation (4) x -'(v)..........(5); and applying the preceding principles to equations (3) and (5), we have du du dx d- =x X....... (6). But dx 1 -dxz 1 -.....Art. (18), dv -d. dx which value in (6) gives du du dx cdv A dv dx that is, the differential coefficient of u regarded as a function of v, is equal to the differential coefficient of u regarded as a function of x, divided by the differential coefficient of v regarded as a function of x. PARTICULAR RULES FOR THE DIFFERENTIATION OF ALGEBRAIC FiJNCTIONS. 20. In order to deduce a particular rule for the differentiation of any species of expressions, we have simply to apply to the representative of the expression, the general rule for obtaining the differential coefficient, as given in article (10), multiply by the differential of the independent variable, and then translate the result. 2 18 DIFFERENTIAL CALCULUS. Let u v - W I Z......(1), in which v, w, and z are functions of x. Increase x by h, then U= u' v=' -' I — z; subtracting (1), member from member, and dividing by h, u- U V- v W' - W - z h h h hA Passing to the limit of these ratios, we have du dv dw dz dx dx d dx' and multiplying by dx, du = dv q- dw ~- dz; that is, the differential of the sum or difference of any number of functions of the same variable, is equal to the sum or dieffrence of their differentials. taken separately. Thus, if u - ax' - bx3, du = d (ax') - d (bx3) = 2axdx - 3bx'dx...Arts. (9 & 11). 21. Let r = uv be the product of any two functions of x. If x be increased by h, we have r'= u'v' = (u+ Ph + P'h') (v + Qkh + Q'h2)....Art. (12), or performing the multiplication, subtracting the primitive product, and dividing by h, DIFFERENTIAL CALCULUS. 19 - - vP + uQ + terms containing h. h Passing to the limit, dr d= vP + uQ; whence dr = d (uv) = vPdx + uQdx = vdu + udv, since Pdx = du, and Qdx = dv, Art. (10); that is, the differential of the product of two functions of the same variable, is equal to the sum of the products obtained by multiplying the diferential of eachfunction by the other. 22. Let uvs be the product of three functions. Place uv = r, then uvs rs, and d(uvs) = d(rs) = rds + sdr........(1). But since r = v, dr = udv + vdu; hence, by substitution in equation (1), d (uvs) = uvds + sudv + svdu. If we have the product of four functions uvsw, we may place sw = r, and, by a process precisely similar to the above, obtain d(uvswv) = uvsdw + uvwds + uwvsdv +- vwsdu........(2); ~20 D1N'E'~DIFFERENTIAL CALCULUS. and we readily see, that by increasing the number of functions, we may in the same way prove, that the diferential of the product of any number of functions of the same variable, is equal to the sum of the products obtained by multiplying the differential of each into all the others. Thus, if UV = ax2. bx, d (Uv) = axe.d (bx) + bx.d (ax) = ax'. bdx + bx.2axdx = 3abx Idx. 23. If we divide both members of equation (2) of the preceding article by uvsw, we have d(uvsiw) dw ds dv dC+ + + _ + -- + uvSW W s v u and we should have a similar result for any number of functions whence we may conclude in general, that the differential of the product of any number of functions divided by the nproduct, is equal to the sum of the quotients obtained by dividing the di'erential of each function by the function itself 24. Let U = vq, v being any function of x, and m any number, entire or fractional, positive or negative. Increase x by h, then = v'i = (v + Qh + Q'h)m........Art. (12), or placing in the binomial formula, (x + a)m = ox + maxm- + (m-) ax"-2 + &c., v for x, and (Qh + Q'hI) for a, DIFFERENTIAL CALCULUS. 21 we have a' [v + (Qh + Q'h2)]" = vt + m(Qh + Q'h2)v"' +- &., U - h = m(Q + Q'h)v"-' + &c., cach of the following terms containing h as a factor. Then du dx du dv- = mnv- Qd$ _ mvv"-dv........(1), since Qdx = dv, Art. (10). That is, to obtain the differential of any power of a function: Diminish the exponent of the power by unity, and then multiply by the, primitive exponent, and by the differential of the function. Example.s. 1. If u = ax4 then, Art. (17), du = a.dx' = a.4x'dx - 4ax3dx. 2. If Cu = bXb, 2, 2 2 bdx du = bx -1dx =- bx-k dx 2b 3 3 = 3. If a = CX-3 3 cdx du = — 3 cx4dx - - - 22 DIFFERENTIAL CALCULUS 4. If u = (aX - x2)5, du = 5(ax - x) d(az - ), blut d(ax - X) = adx - 2xdx.......Art. (20); hence du = 5(ax-x')4(a- 2x) dx. 25. If in equation (1) of the preceding article we make m -- - we have 11 -n dv dv -vn dv -- v " dv n n' n-i or dv d n/ v = If it = 2, we have dv 2VvJ that is, the differential of a radical of the second degree, is equal to the differential of the quantity under the radical sign divided by twice the radical. If n = 3, we have dv and in general, the differential of a radical of the nth degree, is equal to the differential of the quantity under the radical sign divided by n times the (n - 1)th aower of the radical. DIFFERENTIAL CALCULUS. 23 Exnamples. 1. If u = V, daxS 3 axdx 3 du - =-= - a..dx. 2 v/ax 2 V/ax3 2 2. If u = b x, du = dx 3aV(b x)2 3. Let u -- /bx. 4. Let u= /2ax- x2 26. Let u svs and v being functions of the same variable, then, Art. (21), du = v-'ds + sdv-' = v-'ds - sv-'dv, or ds sdv du = - -2 v V2 whence, by reducing to a common denominator, s vds - sdv v (1); that is, the differential of a fiaction is equal to the denominator into the di.ferential of the numerator, minus the numerator into thedifferential of the denominator, divided by the square of the denominator. If the denominator be constant, dv = 0, and equation (1) becomes vds ds du = -- —. v2 V 24 DIFFERENTIAL CALCULUS. If the numerator be constant, ds = 0, and equation (1) becomes sdv du = - v In this last case, it is evident that u. is a decreasing function of v, and that its differential, when expressed in terms of dv, should be negative, Art. (14). Examples. 1.If u = a - x (a - x) dx- xd (a-x) _ (a - x)dx +rxdx adx = (a- _) 2 -(a - x) - (a - x) ax4 2. If = dax4 4 ax' dx du b b 3. If C ax cdax 3 cdx = (axa)2 ax4 27. By a proper application of the preceding principles every algebraic function may be differentiated. Let them be applied to the following DIFFERENTIAL CALCULUS. 25 zitscellaneous Examples. 1. If u1 = (a + bxn)P, du = p(a + bx")P-'d(a + bx')........Art. (24); but d(a q+ bx") = nbx"-'dx; hence du = bnp(a + bx")P-'xn-'dx. The solution of this example and many others may be simplified by applying the rule of article (19) thus; make a + bx"- z, then U = zP, dz nx]du nbx"-', dIt =; dx Twhence du dz dz adx = — X Pbxn-I ='n2 (a + bxn)P-'xn-1and du = bnp (a + bxx")P-lx -' dx. 2. If u - (1 -x2)3, dut = 3(1 - x6)2d(1 - x2) = - (1 - x2)2xdx. 3. Let ax U = x x+ va + x 2 26 DIFFELRElNTIAL CALCULUS. Place y - + a2+ x/, then u = - y dy dx + xxdx du aydx - axdy -Va + x2 y2; nence a {(x + V/a + x2)dx - x (dx + /a Ix du _ (x Jr Va + )x2 or, after reduction, 7 a'dx.au_ __ _ (x + Va + x2)va + x 4. If u = (b + x) du - (x- -b')dx x Xs 5. u = /a - xm du - - n -(a -x M) dx. 6. $2 du - x(a-X2)-3dx. ya - -' 3 x d- dx 7. = -- du — U -x/1-x' /1 - - x(x - /1x2)2 8. Let u = (a- 2). 9. Let = ( + (1 +x).' x2 q- 1 — c 1'/1 -+ / — 12. u -- 13. u -- 1 Vcx'+ i + 1 +- Vq-X DIFFERENTIAL CALCULUS. 27 SUCCESSIVE DIFFERENTIATION. 28. It is readily seen from what precedes, that the differential coefficient of ae function of a single variable is, in general, a function of the same variable. It may then be differentiated, and its differential coefficient obtained. Thus in the example, du u =ax, d = 3ax2....(1), dx 3 axI is a function of x, different from the primitive function. If we differentiate both members of equation (1), we have d ( ) = 6axdx. But since dx is a constant, Art. (26), d (du d (.du) d u dx - dx -dx the symbol d'u (which is read second differential of u) being used to indicate that the function u has been differentiated twice, or that the differential of the differential of u has been taken. Hence d'u d2u d 6 axdx, or 6ax, dx dx2 in which dx' represents the square of dx, and is the same as if written (dx)'. The expression, 6ax, being the di'erential coefficient of the first ditferential coefficient, is called the second differential coefficient. To make the discussion general, let u = f(x) and p be its differential coefficient, then du d=............ (2). TX~~~~~) 28 DIFFERENTIAL'CALCULUS. Since p is usually a function of x, let it be differentiated and its differential coefficient be denoted by q, then dp dx = q ffi(3) In the same way let q be differentiated and its differential co efficient be r, then dq d = r....... By differentiating equation (2), we have du\ d2u d d - - dp, or d dp and by the substitution of this value of dp in (3), diu dx d2u dx q, or d q (5), which is the second differential coefficient of the function. By differentiating (5), we have d3u i= dq, and by the substitution of this value of dq in (4), d3au dx2 d3u = r, or dx = r; which is the differential coefficient of the second dilferential coeficient, and is called the third diferential coefficient. In the same way the fourth, fifth, &c., may be derived, each fioln the preceding, precisely as the first is obtained from the primitive function. DIFFERENTIAL CALCULUS. 29 For this reason, the successive- differential coefficients are often called, derived functions, and are designated thus, du d2u u = f(), - = f'(x), f "(), &c., f(x) being the primitive function, f'(x) its first derived function, f"(x) its second derivedfunction, &c. From the differential coefficients or derived functions, we may at once obtain the corresponding differentials, by multiplying by that power of the differential of the variable, which indicates the order of the required differential, thus, dau d'u = dx2 -/,(x)dx", d"udx dn dx" - f?(z)dx, &c. Exacmples. 1. Let u - ax', n being a positive whole number, then du n d~u d = nax'-' d,= n(n-I)axI-1 du_ n(-n 1)- ()ax-, dx3 = (n -.. (. - 2)a.. -.. d=u, — n(n -'1) (n - 2)........2.1. a. 30 DIFFERENTIAL CALCULUS. Since the last differential coefficient is constant, its differential will be 0, and we hlave 0. 2. Let u = (a -)then du d2u d= (a-x)-, d = 2(a-x)-3, d'u dz - 2.3......n(a - )-("+. 3. Let u = (a - x2)By examining the successive differentialscoefficients in the above examples, it will be seen that by each differentiation the exponent of the power is diminished by unity. When this exponent is entire and positive, it will finally be reduced to 0; and, if there be no negative or fiactional exponents in the expression, the corresponding differential coefficient will be constant. Thfe next in order, as well as all Which follow, will then be 0, and there will be a limited number. If the exponent be fractional, by the continued subtraction of unity the result can never be O0,but will finally, if the differentiation be continued, become negative; the successive differential coefficients will then always contain x, and there will be an infinite number. So also if the exponent be negative. And, in general, if all the exponents of an algebraic expression are entire and positive, there will be a limited number of differential coefficients. If any are negative or firactional, this num ber will be unlimited. DIFFERENTIAL CALCULUS. 31 MACLAURIN'S THEOREM. 29. The object of this theorem is, to explain the manner of developing a function qf a single variable, into a series arranged according to the ascending powers of the variable with constant coelcients. Let u = f(x), and let us assume a development of the proposed form, u = B + Cx + Dx2 + Ex3 + &c.........(1), in which B, C, D), &c., are entirely independent of x, and depend upon the constants which enter into the given function. It is now required to determine such values for the constants B, C, &c., as will cause the assumed development to be a true one, for all values of x. Since these constants are independent of x, they will not change when we make x = 0. If then in (1) we suppose x - 0, and denote by A what f(x) or u becomes under this supposition, we have A = B. Differentiating (1), and dividing by dx, we have du - = C + 2Dz +- 3Ex12 +- &c.........(2); dx du making x = 0, and denoting by A' what d reduces to il this -ase, we have A'- C. Differentiating (2), and dividing by dx, we have dT2? d — 2 D + 2.3EEx + &c.; dx 32 DIFFERENTIAL CALCULUS. making x - 0, and denoting by A" what d becomes, we have A"l A"- = 2 D; whence D = 1.2' d'u d4u In the same way, denoting by A"', A""', &c.,what d — &c., become when x = 0, we shall find A"' A"" E = 1.2.3 F - &c. -.2.3 = 1.2.3.4' Substituting these values in equation (1), we have u =f(x) = A + A'x + A" 1 23A.'- - + &c...(3), + 1.2.3..... in which the general term, or the one which has n terms before it, is what the nth differential coefficient of the function to be developed becomes when the variable is made equal to 0, multiplied by the nth power of the variable, and divided by the product of the consecutive numbers from I to n inclusive. This formula is often written thus, u = f (x) = u=o + +(d)o 1......; or u =f(x) =.f(o~)+f'(o) (o)....... ) + &c.; 1.2 in which the coefficients of the different powers of x are symbols denoting the same quantities as the letters A, A', A", &c., in formula (3). DIFFERENTIAL CALCULUS. 33 Exampl1e. 1. Let u -= (a + x)m. This, when-z = O, reduces to am; hence A = am. By differentiation, &c., we obtain = m(a + x)m-, d, = m(m - l)(a + )m-2)" d3u dz= m(m - 1) (m - 2) (a + x)m-3, &c. Making x = 0 in each of these differential coefficients, we have,'-= ma "', A"= — m(m —l)am-', A"'=.n(m - 1) (m —2)a- 8, &c. Substituting these values in the formula (3), we have m (m - ) a"-x2l (a + X)m a~ + mamlx + 1+ &c. 1.2 2. Let a u = - a(b — x)-' By differentiation, &c., we have du a d'u 2a du= = a(b- x)- x' 2a(b- x)- -(b-) d'u 2.3.a dx3 - - 2.3a(b - x)4 -(b- - )4 &c. Making x = O in the original function, and in each differential coefficient, we. have 3 34 DIFFERENTIAL CALCULUS. a a 2a A =, A' b A" These values in the formula (3) give a a a a a _a J a aIx2 +............ x"... b-x- = + v 2+ b"+ +. 3. Let u 4. Let u 1+x'. /l4xt' 1 + X 5. U 6. 6. u (1 +xI)3. Whenever the function to be developed contains the second or higher power of the variable, the work will be much abridged by substituting for this power a single variable, then making the development, and in the result resubstituting the power. Thus, in example 6, by putting z for x2, we have. = (1 + x2)3 - (1 + ~ )3 which is easily developed according to the ascending powers of z. 30. Functions which become infinite, when the variable on which they depend is made equal to O; or any of the differential coefficients of which become infinite, under the same supposition, cannot be developed by Maclaurin's formula, as in such cases, either the first or some succeeding term of the; series would be infinite, while the function itself would not be so. u = log., u = cotx, u = ax2, are examples of such functions. In the first two A, and In the third A', would be infinite. DIFFERENTIAL CALCULUS. 35 DEFINITION AND PROPERTY OF FUNCTIONS OF THE SUM OF TWO VARIABLES. 31. A quantity is a function of the sum of two variables, when in the algebraic expression for the function, a single variable may be substituted for the sum, and the original function thus reduced, without a change of form, to a function of the single variable. Thus u = a(x + y) is such a function, for if in the place of x + y we substitute z, the function becomes u' = az", a function of z of the same form as the primitive function. log (x- y), is also such a function of the two variables x and- y, which, when for x - y we put z, becomes log z. If in such a function either variable be made equal to 0, the result will be a function of the other variable, of the same form as the primitive function, since the effect of this is to substitute a single variable for the sum of the other two. Thus, in the first of the above examples, if x be 0, we have ay e; if y be 0, we have axn; two functions, one of y, the other of x, of the same form, which become identical if x be changed into y, or the reverse. 32. Let' = f(x + y). 36 DIFFERENTIAL CALCULUS. For x + y substitute z, then' =f(z). If we differentiate this, first as a function of x, y being regarded as a- constant; and then as a function of y, x being in turn regarded as constant, we shall have, Art. (19), du'- du' dz du' du' dz dx - dz dzx' dy dz dy But, since z = x + y, when y is regarded as constant, dz = dx; when x is constant, dz = dy, and the second factor in the second member of each of the above equations reduces to 1, and we have du' du' du' du' dx -dz' dy dz' hence dur du' That is, if a function of the sum of two variables be dif'erentiated as though one of the variables were constant, and then the samze function be differentiated as though the other variable were constant, and the diferential coefficients be taken, these two coefficients will be equal. To illustrate, let'= (X + y)'n, then du' = n(x + y)/-' d(x + y), which if y be regarded as constant becomes du' du' = n (x y) dx; whence d ( u y)XFdx And if x be regarded as constant, the same expression becomes du' = n(x +y)-,'dy; whence du'= n( +y)' = & x+y)' DIFFERENTIAL CALCULUS. 37 TAYLOR'S THEOREM. 33. The object of Taylor's Theorem is, to explain the manner of developing a function of the algebraic sum of two variables, into a series arranged according to the ascending powers of one of the variables, with coefficients which are functions of the other and dependent also upon the constants which enter the given function. Let us write a development of the proposed form,' =f(x + y) = P + Qy + Rya + +Sy + &C.....(1), in which P, Q, R, &c., independent of y, are functions of x. It is required to determine values for them, which substituted in equation (1) will make it true for all values of x and y. If we regard x as constant, differentiate both members of equation (1) with respect to y and divide by dy, we obtain du' = Q + 2Ry + 3 Sy + &c. If we regard y as a constant, differentiate equation (1) with respect to x and divide by dx, we obtain du' dP dQ dR =-~ Y+ 3 + ~ + &c. dT. d +.d dx du' du' But by the preceding article we have; therefore dP dQ dR Q - 2Ry + 3Sy +- &c. = - -y + y &c.; and since, by the principle of indeterminate coefficients, the coefficients of the like powers of y in the two members must be equal, dP dQ dR Q=....(2 2R.........3), 3S (4). dx ~ ~~~~~) 8s-,..., x.....() 38 DIFFERENTIAL CALCULUS. If in equation (1) we make y = 0; f(x + y) will reduce to a function of x, Art. (31), which we denote by u. Then U - P. Substituting this value of P in equation (2), we have du d.r This value of Q in equation (3), gives d dx du du 2R dx dxi whence.2.dx and this value of R in (4) gives ~.1. 2. dx d3u d'u 3S ) d; whence S - 1..3. _ dx I.2. dx' 1.2.3. dx3 By the substitution of these values of P, Q, R, &c., in equation (1), we have Taylor's formula; du y d2u y2 d"u y",' f(x + Y) = u + d- xY.2 +..... r I dx2 1.2. dx. 1.2. 3....n By an examination of the several terms of this formula, we see that the first (u) is what the function to be developed becomes, when the variable, according to the ascending powers of which the series is to be arranged, is made equal to 0. The second (d Y) is the first differential coefficient of the first term, multiplied by the first power of this variable; and the general term is the nth dizfferential coefficient of the first term, multiplied by the nth power of the variable, and divided by the product of the consecutive numbers from 1 to n inclusive. DIFFERENTIAL CALCULUS. 39 The development of f(x - y) is obtained from the formula by changing + y into - y; thus du d'u y I d3u Y3 -TX dx- 12 - 2 + 1. 2 E.xamples. 1. Let U = (x + y)m. Making y = 0, we obtain u = x", and thence by differentiation, du d'u -.d. = mx-', d2 - m(n - 1)x-2, d X' dxu d =m(m-1) (m- 2) x"-3, d-u-= m (m= l)...(m-n_+) x.These values being substituted in the formula, give M-I n (m - 1) x"-' y2 (x + y)'" = "- + mx"-'y +-.-.......... 1.2 m (m - 1).......... ( - n1 + 1 ) xm-n y 1.2...........n If it were required to develop the function in terms of the ascending powers of x, we should make x = 0, and obtain ym for the first term, from which the other terms are derived as before. a 2. Let u'- x =- Y a Making y = 0, we obtain u - - for the first term; thence'X 40 DIFFERENTIAL CALCULUS. du a d2u 2a d= ~ d'' cwl dSu 2.3.a d"u_ 2.3......na 3=x - — X4' ddx" x+ These values being substituted in the formula, give a a a a x +y x x-'~y. -k 3. Develop u' = according to the powers of - y. - X - Y)i a 4. Develop u' = - 2 according to the powers of x. 34. Since in the formula of Taylor, the coefficients of the different powers qf one variable are functions of the other, it is plain that if such a value be assigned to the other, as to reduce any of these coefficients to infinity, the second member will become infinite, and the formula fail to give a development for this particular value; as, in this case, the first member will become a function of the first variable, which function is not necessarily equal to infinity for a particular value of the second variable, on which it in no way depends. Thus, in the example u'- = a + x + y, which, when developed according to the ascending powers of y, gives =+, 1 1.............. =2Va 8-t% / (a + x) 3. the particular value x = - a reduces the coefficients of the DIFFERENTIAL CALCULUS. 41 powers of y to infinity, while the original function is reduced to ~. We should thus have /y = co, which cannot be. For every other value of x, however, these coefficients will be finite and the development true. The difference between this failing case and that of Maclaurin's formula is marked. In this, the failure is only for a particular value of that variable which enters the coefficients, all other values of both variables giving a true development; while in the former case, if the formula fails to develop a function for one value of the variable, it fails for every other value, 35. If u = f(), and x be increased by h, we have for the second state u'= f(x + h), and by changing y into h in Taylor's formula, we obtain du d'u lh' = f (+ r h) = U + W- h J+ dx 1 2+ &c.....(1), which is the development of the second state of a function. du d2u Otherwise, by substituting for u, d-, dTx'' &c., the expresi sions f(x), f'(x), f"(x), &c., as in Art. (28), we have,' = f(x + h) = f(x) -F f'(x)h + f,,"(x).2 + &c.; that is, the new state of the function is equal to its primitive state, plus its first derived function into the Jfirst power of the increment, plus its second derived function into the second power of the increment, divided by 1.2, plus, &c.; and this is but another form oi Taylor's formula. 42 DIFFERENTIAL CALCULUS. If the second state corresponding to a particular value of x, as x = a, be required, we have simply to substitute a for x, nd obtain hA f(a + h) = f(a) + f'(a)h + f"(a) -2 + &c.; in which f(a), f'(a), f"(a), &c., are symbols denoting what the primitive function and its successive differential coefficients, or derived functions, become when a is substituted for x. From (1) we have du d2u ha d4u h3 U - U h = + 1 + + &C. TX dX7 1.2 dx3 1.2. 3 If we now put for h the particular value dx, we have u'- U d + d1- _ 2 1.2 t- &c. 1.2 1.2.3 36. If in the development of f(x ~- y) by Taylor's formula, we du d2u suppose x - 0, and represent by A, A', A", &c., what u, dx' dx2' &c., become under this supposition, we have f(y) = A +- A'y +.AY2 +- 12 + &c. 1.2 1.2.3 A, A', A", &c., being constant, and since y is the only variable, we may write x for it, and thus have Al+xZ A"'xS /(x) = A + A'x + - 1.2. +- &c., which is identical with Maclaurin's1.2 1.2.3 which is identical with Maclaurin's formula. DIFFERENTIAL CALCULUS. 43 DIFFERENTIATION OF LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 37. Let u - log v, in which v is any function of x, and the logarithm is taken in any system. Increase x by h, then u' = logv', u' - U = log v'- log v log-. Substituting for v' its value, Art. (12), this becomes'-u = log+ Ph + P'h log(1+ Ph + P'h)....().' —- = log... P+ + P'hA By placing -+ for y in the formula (Davies' Bourdon, Art. 236), log (l + y) = M y- + - &c we obtain log + fPh + P'h = M Ph+P'h2 - h (P~P'h) log t. + &c. and this in equation (1), after dividing by h, gives Ur - U M (P ~ Plh h (P _ Ph)_ a' — M(P+___ h(P+'A)' &c) h b v2 whence, by passing to the limit, 44 DIFFERENTIAL CALCULUS. du P Pdx dv = M- du = M- -, do v v v since Pdx = dv. For the Naperian system, M = 1, and this expression becomes dlu = du The differential of the logarithm of a quantity is, then, equal to the modulus of the system into the differential of the quantity divided by the quantity; and this in the Naperian system, becomes the differential of the quantity divided by the quantity. Exam2ples. 1. If u = l(ax3), dax3 3 ax'dx dx -x ax = -. ax3 X3 2. If (a = x / a ( adx d____ (ax)' dx du= -- a a a - x a - X a - x The differentiation of logarithmic functions will be much simplified by the application of the principles for multiplication, division, &c., by means of logarithms. Thus, in the above example, * Throughout the book, the symbol 1 before a quantity will indicate the Naperian logarithm of that quantity. DIFFERENTIAL CALCULUS. 45 = I(a) = I - l(a-x), da du = dla - dl(a —x) _ a, -x B. Also, if u = z[(a + x)2/a x], = 21(a + x) + l1(a- x), 2dx 1 dx (5a - 7x)dx a + x 3a - x - (a —x') 4. Let u = I( + ). \2/1 ~+ x2 -' Multiply both terms of the fraction by the numerator; then u = l(I/1 + x2 + x)2 = 21(/ 1-+ x + x), du =2d (/l1 + x2 + x) _ 2dx (~+~ + X x+ $ /-q 5. If u - -x 2du - (et u I ) 7. Let 1(a ) () x) 8,,e,-I I~a - 29. 1 ( + XI. 46 DIFFERENTIAL CALCULUS. 10. Let u = (/x)"; then, Art. (24), n (Ix)"-' dx du _ (lX),_dix n -(x)-dx 11. Let u = l(lx); dlx dx then du - d lx - xlx 38. It has been seen, Art. (30), that log x cannot be developed according to the ascending powers of x. To obtain a logarithmic series, let us take u = log (a + x), and develop it by Maclaurin's formula. By differentiation, &c., Mdx du M du; - = M(a+ x)-'; a J- x dx a + x d2u - M d'zu 2 M =-a - M(a + x)-2 dx 3 - (d +xx)3 Making x = 0, we have for the values of A, A', A", &c., in the formula, M M 2M A = log a, A' A" -- A "' =- &c.; a aa Mwhence in which the logarithm of a quantity is expressed by a series, arranged according to the ascending powers of a quantity less by ca. If a =, since log a = log 1 = 0, the above series be comieS, DIFFERENTIAL CALCULUS. 47 X2 X3 X u = log(l +-x) = M(x — +.......... 2 3 n the ordinary logarithmic series. 39. Let us now take the exponential function, u = a', in which v is any function of x. Taking the Naperian logarithm and differentiating both members, we have du =u = v la, - = ladv, du - u la dv, or du = ca' = aladv; that is, the differential of a constant raised to a power denoted by a variable exponent, is equal to the power, multiplied by the Naperian loyarithm of the constant into' the differential of the elponent. Examples. 1. Let U = a b"7 du = abz la d.bx= - 2ba"laaxdx. 2. Let U a>. 40. If the successive differential coefficients of the function v, = a'c be taken, we have 48 DIFIERENTIAL CALCULUS. du jd2 u dlut d = a laa (la)d d a(la)3, &c. dx dx' dx' If in the primitive function and in each of these differential coefficients we make x = 0, we have for the values of A, A', &c., in Maclaurin's formula, A = a5 _ 1, A' = la, A" = (la)'.,,. An' = (la)' and these in the formula give = a0 = 1 + (a) +(la)x + (a) + (la)3 T.-.3 (la)'2 U.2 + 12..... 41. By the aid of logarithms we may simplify the differentiation of complicated exponential functions. For example: 1. Let U =Zs, z and y being any functions of the same variable. Take the Naperian logarithms of both members, then Iu = lZ = ylz; and by differentiation du dz - = dylz + Y-; whence du = u(dyz + y- zvlzdy + yzv'dz, which is evidently the sum of the differentials, taken by first.re gardirn y as the only variable, and then z. DIFFERENTIAL CALCULUS. 49 2. Let U = a. Taking the logarithms of both members, lu = bola, du la.db0= lab-lb dx, u du - a b'la lb dx. 3. Let then du.r dz u- = tlzX, - lz(tltds + st'-'dt), du. - z= Vt + - zltds + t lzdt). 4. Let u = xa. 5. u = (- -,X 6. u =4~X. 7. _ =DIFFERENTIATION OF THE CIRCULAPR FUNCTIONS. 42. Since any arc of a circle, when less than 90~,: is greater than its sine, and less than its tangent, we must have for all values of y less than 90~, sin Y < I and sin y sin y Y tan y y 4 50 DIFFERENTIAL CALCULUS. But ~ sin y sin y tan y sy whence ts os.... Making y = 0, cos y becomes 1, and we have foi the limit of the ratio (1), L = 1; sin Y sin y and since sin cannot exceed unity, nor be less than it Y tan y must, for all small values of y, be included between them; and sin y sin y. as - approaches the limit 1, must' approach the same tan y y limit; that is, the limit of the ratio of an arc to its sine is unity. 43. Let u = sin x. Increase x by h, then uw = sin(x +'h), a',- u - sin(x + h) - sn x, or by placing x + h for p and x for q in the formula, sin p - sin q = 2 [sin (p - q) cos-( + q) ], 61- u- = 2 sin.h cos (x + jh). Dividing both members by h, and then both terms of the fraction in the second member by 2,' - it sin Ih h-' -.= Si. h (cos(x + ih), and passing to the limit, since DIFFERENTIAL CALCULUS. 5i1 sin-h 1* du dx = cos x; whence du - d sin x = cos x dx. If u = cos x, du - d cos = d sin (900 - x) = cos (90~ - x) d (90~ - x); whence d cos x - sin x dx. If zu = ver-sin x, du - d ver-sin x d(l - cos x) = - d cos x; whence d ver-sin x = sin x dx. If fu = tan x, sin x du - d tan x = d Cos X (cos xd sinx - sinxind cos x) _ dx(cos2x ~ sin2x) COS 2 X COS2 whence dx d tan x = COS X * This notation indicates that the expression for the quantity within the parenthesis becomes unity when h = 0. 52 DIFFERENTIAL CALCULUS. If tu = cot x, dU = d cot x = d tan (90 - ) (90 ~- x) Cos2 (900- ) dx whence d cot x =- If u - secx, 1 sin x dx du - d secx = d - COs x Cos2x whence tan x dx d sec x - = tan x se x dx. I~fu u- osec x, du = d cosecx z d see (900 — x) = cot x.cosee x d(90~ - x) whence d cosee x =- cot x.cosee x dx. If any other radius than 1 be used, it must be introduced into these formulas, by rendering them homogeneous, as in Trigonometry. Thus the formulas for the differential of the sine and cosine become cos x dx sin x dx d sinx= R - d cosx - R E'xaples. 1. if u- = sin — a 6x bx b bx du = cos - - - cos - dx a a a DIFFERENTIAL CALCULUS. 53 1 2. If u = cos, dI. 11 1 1 du --- sinld - - I1 sxn-d- =.~-sin-dx. 3. If u = tan(a - x), du d(a-x)' _ 2 (a - x) dx Cos (a- x) Cos2(a - X)' 4. If u = cot'x,'2 cot x dx: du -2cotxdcotx 2 Otcot x dx 5. If U = (COS X)u " = make cos x = z, sin x = y; then u = zO, and Art. (41), du = z9lzdy - yz-'dz = dx (cos x) C(cos X i cos s x os - / Cos x sin (1 - x) 6. Let u — 7. Let u-tan(-nvx). 44. In the preceding article we have found the differentials of the sine, cosine, &c., in terms of the arc as an independent variable; let it now be required to find the differential of the arc, in terms of its sine, cosine, &c. If u = sin x, then x = sin-lu,* dut du = cosx dx, and CosX. The notation sin -l u, tan-l u, &c., is used to designate the are whose sine is u; whose tangent is u, &c. 54 D;FFERENTIAL CALCULUS. If now x be regarded as the fuinction, and u as the independent variable, we have, Art. (18), dx 1 1 du - du cos x' Clx and since cosx = v/1 - sin2x = -V/1 -- u' whence dx - If du. Cos = C- lsin x; du d1 1 d - sin x, and = si w; sin x = (2-ver-sin - ) ver-sin x (2- u), whence (u)u dx — u - ver-sin x, X -- ver-sin-l t, du dx 1 sin x, and dx d-u - sin x; or since sin x -V (2 ver-sin x) ver-sin x - V(2 -- ) u, dx 1 du whence dx - DIFFERENTIAL CALCULUS. 55 If = tan x, x tan-u, d dx - Cos2X' dx 1 1 du cos x = sec2x 1 1+ tana2x whence du dx- d 1 + u~ When required, the radius may be introduced into these formulas, as in the preceding article. Thus, the last formula will become R2du dx = R -- u2 Examples. 1. If x = sin-'2uV 1/-, d(2u'V - u') 2du dz /i - (2v,' 1 - u2a)2'1L 2. If x = tan'(- C), d(_ cdy dx -- 1a+-(~)2 c + y2 3. If U =_ Cos Y, du ady a - Y (a- y)va2 - 2ay 56 DIFFERENTIAL CALCULUS. 4. If u = ver-sin-, du = xV2x -1 45. To develop the sine and cosine of x, in terms of the ascending powers of x, we use Maclaurin's formula. Thus: du 1. u = sin, cos, d2, d3u d/ --- sin x, do = - cCOS x &c. dx-1" dx3 Making x =0, we obtain for the values of A, A', &c., in the formula, A = 0, A' = 1, = 0, A"' = - 1, &c.; thence x X3 x5 X + U sin 1 1.2.3 1.2.3.4.5 &C. 2. U = cos XI du d2u d3u' d - -sin = - cos sin x, &c.; in which, making x = 0, we obtain A 1, A' 1 Al, A",' = O &c.; and thence X2, 4 X x2 u =- cos X 1 — + &C. 1.2 1.2.3.4 DIFFERENTIAL CALCULUS. 57 These series, for small values of x, are very converging, and will give with great accuracy the values of sin x and cos x for small arcs, and may therefore be used in the calculation of a table of natural sines, &c. Thus, R being unity, we have for the semicircumference or 7r, the number 3,14159.....; this divided by 18, and the quotient by 60, will give the length of the arc 1', which value, substituted for x in the series, will give the sine and cosine of one minute. 46. We can also develop the are in terms of its sine, tangent, &c. If dx 1 =- sin-'u, d....Art. (44), du /1 - u2 d'x d.x d2= - (1 t)- 3) / dU_ = (1-u) — 23+ 3u (1- - )-u, &e. Making u = 0, we obtain A = 0, A'= 1, A"= 0, A"'= 1,&c.; and by substitution in Maclaurin's formula, us 3U5 x = sin-'u -- u + + d &c. 1.2.3 1.2.4.5 If u -- = sin 300, this series becomes x = sin-' 1 300 1+ 1 _ _ 1.I 2 2 1.2.3.23 + 1.2.4.5.25 + by the summation of which, we find 30~ = 0,52359......... and multiplying by 6, 180~ -= = 3,14159.......... 58 DIFFERENT1IAL CALCULUS. 47. If x tan-u d - 1 u = (1 + u)-"....Art. (44), aid the development may be made as in the preceding article; or otherwise thus: Developing (1 + ut)-1 by the binomial 1ormula, we have dx du= 1 - u + u4 - u6 + &c.........1 and since, by differentiation, the exponent of u in each term is diminished by unity, we must have, before the differentiation, an expression of the form, x = Au + Bus + Cu5 + &c.; whence dx = A + 3Bu + CU4 + &c......... (2) Comparing the coefficients of the like powers of iu in (1) and (2), 1 A=1, 3B= -1, and B= 5C= 1, and C — &c.; whence Us US U x = tan-lu - u - - 7 + &c..(3). 3 5'17 If u = 1 = tan 45~, this series becomes 1 1 1 x = 45' = 1 - - + &c., 3 7 which is not sufficiently converging to enable us to determine the length of the are with accuracy. To obviate this difficulty, we DIFFERENTIAL CALCULUS. 59 will make use of the principle that the arc 45~ is equal to the arc whose tangent is 1, plus the arc whose tangent is. * From equation (3), by the substitution of ~ and I for u, we have,1 1 1 1 1 tan- - + &c., 2 2 3.2 5.25 7.2 &c., 1 I 1 1 tan - 3 + -'. &+. hence 1 1 450 = tan- - + tan-1' - 2 3 1 1 1 i 1 1 ~2 - - - &c. +- a - - &c. — 0,78539....., 2 3.23 5 25 3 333+ 531 6-&c=O78539 which, being multiplied by 4, gives 7r = 3,14159.......... DEVELOPMENT OF THE SECOND STATE OF A FUNCTION OF ANY NUMBER OF VARIABLES. 48. Heretofore our rules for differentiation have been limited to functions of a single variable; it is now proposed to extend them to functions of any number of independent variables. * To prove this principle, take the formula tan a + tan b tan (a q- b) 1 - tan a tan b Make tan a -, and a +- b = 45~; then, tan 45~ = 1 = + tanb whence tan b =; 1 -- = tan b; hence 450 = a +- b - tan-'- +- tan-'. 60 DIFFERENTIAL CALCULUS. Let u =/f(, Y); x and y being entirely independent of each other. The second state of the function will evidently be obtained by giving to both x and y variable increments. First let x receive the increment h; f (x, y) then becomes f(x + h, y), which (if y for a moment be regarded as constant) may be developed according to the ascending powers of h, by Taylor's formula; whence du du hA' d3u, h3 f(x+h,y) = U + ph +d l~ 2+ dx - 3 3+&c.....(1), in which du dx &c., are the differential coefficients of u = f(x, y), taken under the supposition that x alone is variable; and are evidently all functions of x and y. If in this development we now put y + k for y, we shall obtain in the first member f(x + h, y + k), which is the second state of the function u. Th% ~rst term of the second member (u), being a function of x and y, will, when for y we put y + k, become du dzu k.d3u k8 f(x, y + k)= + TYk + dy.2 + y 1.2.3 ~ &C. du In the same manner, Td' when for y we put y + k, may be developed, and will give, Art. (33), _ d() k d2( a) k (dudu du dx + &d k -- dy dy 1.2 or reducing, fdu\ J du d +u du + &c. dx dx + k + dx- 2 t &c.._,.~-, dx dxdy d+dy'1 2 DIFFERENTIAL CALCULUS. 61 Also, /d2UX d-2 d3l d4lu k dx~ dx l dk ddy + d 1. 2 &c., (d3u\ d3u d4u \ d3j/yy+-k EdX3 + dxady k + &c. These values being substituted in the second member of (1), give for the development of the second state of a function of two variables, du d'u kP d3u ks f(x +h, y+k) = u + k + d - + &C., dy dy 1.2 dy 1.2.3 du d'u d3u hk+ + -h + d + hk Jrx + &c., dx dxdy dxdy' 1.2.......... (2). d'u V d3u h'k dx.2 12 dy 1.2 & +d3 31 h 3 l: + &c. dx~ 1.2.3 in this development u is the original function; du is its differential coefficient taken under the supposition that y alone varies, and is called the parlial dijereltial coejicient of the first d'u d'u order, taken with respect to y; dy d / &c., are successive differential coefficients taken under the same supposition, and are called partial differential coeficients of the second, third, &c., older, du d'u d'u taken with respect to y. dx' dx'' dx' are obtained from the original function under the supposition that x alone varies, and .62 DIFFERENTIAL CALCULUS. are called partial differential coefficients of the first, second, &c., d2u order, taken with respect to x; dd- is obtained by differendu tiating d with respect to y and dividing the result by dy, and is called a partial differential coefficient of the second order, taken by differentiating first with respect to x and then with respect to y; and, in general, dx' n is a partial differential coefficient of dx dy' the m + nth order, and is obtained by differentiating first n times wvith respect to, and then nz times with respect to y. By an examination of these results, we see that from a function of two variables, there are derived two partial differential coefficients of the first order, viz.: du du -- and; dz dy three of the second order, viz.: dlu d2u d2u dx' dxdy' dy 2 four of the third order, &c. The expressions, du du d2u d2u dx d yu dy, d2 d, dxdy dxdy, &c., dz y Iz' dxdobtained by multiplying the several partial differential coefficients respectively by dx, dy, dx2, dxdy, &c., are called partial di'erentials, and are the results obtained by differentiating a function of two or more variables, as though, at each dif lerentiation, all the variables but one were constant. 49. If, instead of first increasing x by h, we increase y by k, we shall obtain DIFFERENTIAL CALCULUS. 63 du_ d2u k2 d3u k3 f (xy k) + + dy2 1.2 dy3l.2.3 &c.; and if in this we put x + h for x, we shall evidently deduce du d2u h2 f(x + h, y + k) = u + J h + d + &c., du d2U + -k + x kh + &c., dy dydx dlu k' d2 1 + &c., which development must be identical with the one in the preceding article; hence the terms containing the like powers of Ah and k must be equal to each other, and we must have d2u d2u d3u du u d +mu dn+mu dxdy dydx' dxdy2- y d' ""..........dx dy - dy"'dx' which shows that we shall obtain the same result, whether we differentiate first with reference to x and then with reference to y, or the reverse. 50. Let it now be required to develop the second state of the expression U = XMy....................(1). Differentiating with reference to x and y respectively, we obtain du du d - t- = x-. nxmyn...... (2)(3) dx dy Now differentiating (2), first with reference to x, and afterwards with reference to y, we obtain 64 DIFFERtNTIAL CALCULUS. dlu d'u dX2=m(m-1)x2"- y....(4) dxdY =mnxM'-Y..(5). In the same manner, by differentiating (3), first with reference to x, and then with reference to y, we obtain d'u dlu dlu dydX - mnxm"'ly- -- d1 xdy' 2y= n(n- 1)xmy"-....(6); and by continuing the differentiation of (4), (5), and (6), d'u d3u d. =.m(m-l)(m-2)xm-3y, dd", =m(m-l)nx,'-,y-,&c. Substituting these values in the formula of article (48), we have (x+h)"(y+k)" = x"y" + nxzy"-'k + n(n-1)xmyn-2 +&c. +- mx,"-y"h+ mnx"m-'yn-lAk + &c. + m(m- 1)xm-2yf - +~&c. 51. Let us now take the general case in which v, is a function of any number of independent variables; that is, let u = f(x, y, z, &c.). It is plain that we may deduce the development of the second state of this function in precisely the same way as in article (48), by first increasing x and y; then in the result thus obtained increasing z, and in the new result increasing one of the other variables, and so on until each shall have received an increment~: We shall thus find du du d &c. f(x+k, y+k, z+z, &c.) f(x,y, z, &c.) + Wli+ k+dz1+ &c. DIFFERENTIAL CALCULUS. 65 DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE VARIABLES. 52. If from both members of the last equation of the preceding article we subtract f(x, y, z, &c.), we have (x+sh, y+k, z+l, &c.) -f(x, y, z, &c.) = d d+uGk +d l &c., plus other terms which will be of the second degree at least, with reference to the increments h, k, I, &c.; and this is the development of the difference between the new and primitive states of a function of any number of variables. If in this development we substitute for h, k, 1, &c., the constants dx, dy, dz, &c., and take the sum of all the termis of the first degree with reference to these constants for the differential of the function, thus extending the definition in Art. (10), to functions of any number of variables, we have du d du du = df (x, y, z, &c.) = d + dy + -dz + &c. dx dy dz The first member, which is the symbol for the differential of the function, is often called the total difjerential of the function, to distinguish it from the terms in the second member, each of which is a symbol for a partial differential. From this we see that the differential of a function of any number of variables is equal to the suImn of the partial diTferentials of the function. It is important, in all operations, to preserve the notation as given for the partial differentials, as we thus not only distinguish them from the total differential du, but know in each case with reference to which variable the partial differential is taken. 5 66 DIFFERENTIAL CALCULUS. Examples. 1. If u = ax'y3, du du d'x = 2axy'dx, d-dy =- 3ax2ygdy; hence du = 2axy3dx + 3ax'y2dy. 2. If _ (a - ) du= 2s (a - x 2) [2xydx J+ (a x) dy]. 3. If u- = axyz3, du- = ay'z3dx + 2axyz'dy + 3axylz'dz. 4. If u = tan- d yd - dy y' y' +- x2 5. Let u 6. Let u = x'x2 + y' 53. Having obtained the first differential of a function of two variables, we may from this at once derive the successive differentials. Since du du d, = y dx + d dy, d du ( du dx /y DIFFERENTIAL CALCULUS. 67 du Differentiating dx, first with reference to x, and then with reference to y, we have d dx) = dx + dxdy; d-yzydxdy and in the same way, d dy) = d' dy + d2 whence, since d2u d 2u dxdy dArt. (49), d'u = x+ 2j dxdy + dy2. Differentiating this result, since - d( dx2) = j a-d.Jc3 + daxdl dxady, (d2u d\ d3UdXd d d3u dXd d \ddy dd) x2dy x dY + dy, TXT2 dx' dX2dy d (d dy2) = d2dX dy dx d dy we derive d3U ~ 3d3u 3d3u d3u dlu Pdu3 dx 3+ d-' dx2dy + - dxdy2 + d dy3. dx3 dX2dy xdyxdy d3y In the same way the differentials of a higher order may be derived; and in like manner we may deduce the successive differentials of a function of any number of variables. 6 8 DPEDIFFERENTIAL CALCULUS. DEVELOPMENT OF ANY FUNC~rION OF TWO OR [MORE VARIABLES. 54. If in the development (2), article (48), wemake both x and y equal to 0, the first member will become a function of h and k; the first term of the second member, and the different coefficients of h and k will, under the same supposition, become constants. Denoting by A what u or f(x, y) becomes when x and y are made 0; by B and B' what the partial differential coefficients of the first order; by C, C', and C" what those of the second order; and by D, ID', I)", and D"' what those of the third order become under the same. supposition, we obtain' f(h, k) =A + (BK + B'k) + 1 (CAh2 + 2C'hk + C"n") ~1~ ~1 + 1.2- (Dh + 3D'A'k + &C.) or since we may change h and k into x and y, we have for tbe general development of any function of two variables, 1 Cx~ f(x, Y) = A + (Bx + B'y) + 12 (O' + 2C'xy + C('y 2) + 123 (Dx -+ 3D'x'y + &c.). If in development (2), above referred to, we make y and k each equal to O, u becomes a function of x alone, and we have du d'u h' d3u P f(x + a) -' dx + dX2 1.2 dx 1.2.3 which is Taylor's formula. DIFFERENTIAL CALCULUS. 69 In the same development, making x, y, and k, each equal to 0, du d2u and denoting by A, A', A", &c., what u, d, d 2 &c., reduce to under this supposition, we obtain f(h) = A + A'h + A" - A"' + &c.; 1.2 1.2.3 or changing h into x, x x f() = A + A'x + A"' 1 2 3 + &c.9 = i2 +.2., which is Maclaurin's formula. 55. By making x, y, z, &c., each equal to 0, in the development of Art. (51), and then changing h, k, 1, &c., into x, y, z, &c., wo may deduce the development of a function of any number of variables. DIFFERENTIAL EQUATIONS. 56. The most general form of an equation containing the two variables x and y, is f(x, y) = f'(x, y)........(1). Since y, in this case, is an implicit function of x, Art. (4), we may suppose its value in terms of x to be substituted in equation (1). Each member will then be an explicit function of x.; and since these functions are equal, their differentials will be equal, Art. (15). Hence, to obtain the differential equation of a given equation containing two variables, or the equation expressing the relation between the variables and their diferentials: Differentiate each member as a function of a single variable, and place the two results equal. ,70 DIFFERENTIAL CALCULUS. Should either member be 0 or a constant, the differential of the other will be equal to 0. Since every term of the differential equation thus derived will contain dx or dy, we may, by transposition, place it under the form, Pdx + Qdy = 0........ (2); from which, after dividing by dx, we may at once obtain an cxdy pression for the differential coefficient, Yx' It is also manifest that the first member of the above equation (2), may be obtained by transposing all the terms of the given equation into the first member, and taking the sum of the partial differentials, as though x and y were independent variables. It should be observed, however, that owing to the relation between x and y, dy is not constant, but will in general be a function of x. 57. If an equation contain three variables, one will necessarily be a function of the other two; and each member may be regarded as a function of two independent variables, and may be differentiated as ill Art. (52), and the two results placed equal to each other. In accordance with the same principles, and in precisely the same manner, the differential equation of one containing any nuinber of variables may be derived. If the differential equation derived by one differentiation be again differentiated, the new differential equation will be of the second order; and if this be differentiated, we shall have one of the third order, and so on. Examples. 1. If we take the equation of the circle y' = 2 - X2.........(1) DIFFERENTIAL CALCULUS. 71 differentiate each member, and equate the results, we have 2ydy = - 2xdx........(2); from which, after dividing by dx and 2y, we obtain dy - X. (3). dx y Dividing equation (2) by 2, and then differentiating, x, y, and dy, being variable, we have dy2 + ydgy - - dx2; whence dy - - 1+ + 2 2 + X2. dgy xI Y_ dx' - y Y Y3 since dY2 equation (3). Equivalent results may be obtained by differentiating the ex pression y = /R1 - x', deduced from equation (1). 2. If yg - 2mxy + x2 - a2 =........(1), 2 ydy - 2mxdy - 2 mydx + 2xdx = 0........(2); -whence dy my - x dx y - mx Differentiating (2), and dividing by 2dx2, we obtain (y _- m dx) d y dy - 2m - + 1 = 0; dX 2 dX 2 jdz' d 72 DIFFERENTIAL CALCULUS. from which, after the substitution of the expression for dW, we may obtain the expression for the second differential coefficient. 3. Let y3 - axy + x3 = 0. Equations derived as above, immediately from the primitive equation by differentiation, are named immediate dilfferential equations. 58. Differential equations arise, not only from simple differentiation, as in the preceding article, but from the combination of the successive immediate differential equations with each other and the primitive equation, in such a way as to eliminate certain constants, or particular functions, which enter the primitive equation. Thus, if we take the equation of the right line, y = ax + b......... (1); differentiate, and divide by dx, we have dy dx a.............. (2), a result which is the same for all values of b. By the substitution of this value of a in equation (1), we have ydx = xdy + bdx, which is the same for all values of a. Differentiating (2) and dividing by dx, we obtain d'y hich is entirely independent of both a and b. which is entirely independent of both a and b. DIFFERENTIAL CALCULUS. 73 2. Take also the equation of the conic sections y2 = 2px + r'x'.......(3). By two differentiations, we get 2ydy = 2pdx + 2r'xdx, dy2 + yd2y = r2dx2. By combining the three equations, 2p and r' may readily be eliminated, and an equation obtained which will be entirely independent of them. The result of this elimination is y2dx +r + x dy' yx'd 2y - 2yxdydx- 0. 3. By differentiating the equation y3 - 2ax' + ag = O, and eliminating a, we obtain 16yx2dx' - 24x3dydx. + 9y2dy2 = 0. And, in general, all the constants of any equation may be eliminated by differentiating it as many times as there are constants. The differential equations thus obtained, with the given equation, make one more than the number of constants to be eliminated; an equation may therefore be derived which will be freed fiom these constants. Equations thus obtained are properly the differential equations of the species qf lines, one of which is represented by the given equation, since, being independent of the constants, they are evidently the same for all lines of the same kind referred to the same co-ordinate axes. 4. Let y = (a2 + _x)n, t7 DIFFERENTIAL CALCULUS. dy = (a: + x2)"- 2xd = m( + )d nra= n (an + ( x); r substituting y for (a' + x2)", and clearing of fractions, n(a' + x')dy = 2mxydx; a differential equation free from the irrational function. 5. Let y = a sinx - b cosx, dy = a cosxdx + b sin xdx, day = - a sinxdx' + b cosxdx', d2y = - ydx-; which is free from the circular functions. 6. y = e 0cosx. 7. y = I(sinx). 59. The Differential Calculus enables us also to eliminate, from an equation containing three variables, an arbitrary function of either two, the form of which may be entirely unknown. Thus, it u = F[f(x, y)], the form of the function designated by the symbol F being arbitrary, we can find a differential equation expressing a relation du du between x, y, and the partial differential coefficients d dWx' dy' which will be the same, no matter what the form of the function F may be. DIFFERENTIAL CALCULUS. 75 Make f(x, y) z..........(1), then u = F(z). Differentiating this, first with reference to x, as the independent variable, and then with reference to y, we obtain, Art. (19), du du dz du du dz dX dz dx' dy - dz'dy' dividing these equations member by member, and then clearing of fractions, we have, du dz du dz._.-.(2). dz dy - dz d z By substituting in this the values of d and taken by differentiating equation (1), we shall have the required differential equation. Such equations are called partial diferential equations. To illustrate, suppose 1. f (x, y) = ax + by, and u = F(ax + by). Place ax + by = z, then dz dz - a, and = b. dxr dy These values in equation (2), give du du b - - a o0. dx dy 2. Let f(x,y) = x2 + y = z, and u = F(x.+ y'). Differentiating z, we find 76 DIFFERENTIAL CALCULUS. dz dz 2x, and 2y. dx dy These values in equation (2), give du du - = 0. 3. Let f(x,y) -, and u = F-. 4. Let u = F(exsiny). CHANGE OF THE INDEPENDENT VARIABLE. 60. In the discussion of expressions containing the successive differentials or differential coefficients of a function, it is often desirable to change the independent variable, and to regard the primitive function, or s-o e.other variable quantity, as the independent one. This has been done in Art. (44), and is simple in cases like this, when the first differential coefficient alone is considered. Should d2y the second differential coefficient gd- enter the expression, we must remember that it was obtained, as in Art. (28), by differentiating Y as a fraction with a constant denominator, thus dJ obtaining d2y d.r If we now consider both dy and dx as variable, and differentiate d as in Art (26), e have dx DIFFERENTIAL CALCULTS. 717 d (dy\ dxd 2? - y dZx dx - (dx and this should replace d whenever it enters an expression, if we desire a result in which neither dx nor dy is regarded as the independent variable; or for d we should substitute the above expression divided by dx, that is dxd2y - dyd'x d3........(. dx( d3y If we recollect, also, that d Y is obtained by differentiating dPy and dividing by dx, we shall obtain an expression to replace dx2 it by differentiating expression (1), without regarding any of the differentials as constant, and dividing by dx. Thus, differentiating. and reducing, we have (dxd3y - dyd3x) dx. + 3 (dyd2x - daxdy) d'x dx....(2) and in a similar way, by differentiating this expression and dividy4 ding by dx, we shall obtain n expression to replace dy4. d2y d3y Whenever we have any expression containing 3 & C in which x has been regarded as the independent variable, and it is desirable to change to a more general one, in which neither x d21 d3y nor y is independent, we have simply to substitute for dxyl dxy&c., the expressions (1) and (2). If in the result we desire to make y the independent variable, we must place day = O, d3y = O, &c.; 78 DIFFERENTIAL CALCULUS. in which case the particular expressions (1) and (2) reduce to dyd'x 3dy(d2x)2 - dydxdx - dX3 (1)' dx5M..(2), which may be used directly when we wish to change from x to y. Example. If we take the equation dry dy2 Y - +d.a + 1 - 0, and sbstitute expressiond (1), we have, after reduction, c-/xf expression (1), we have, after reduction, y(dxd2y - dyd2x) + dy2dx + dx3: 0, in which neither x nor y is regarded as the independent variable. If y be regarded as the independent variable, d2y = O, and we have, after dividing by dy3 ahd reducing, d2x dx3 dx Ydy' - dy - dy 61. If we have a differential equation containing x, y, and the successive differential coefficients of y; and we also have x given as a function of another variable, or x and y as functions of two other variables, and desire to deduce an expression, in the first case, independent of x and its differentials, and in the second, independent of both x and y and their differentials, we must first transform the given equation into its most general form, as indicated in the preceding article, and then deduce the values of dy, dx, dcy, and d2x, from the equations expressing the relation between x and y and the new variables, and substitute them in the general form. DIFFERENTIAL CALCULUS. 79 1 Let us have d2y x dy Y_- o_ dx2 1 - dx + 1....(1) and x = cos 0................. (2). Substitute in (1), for ~d2Y expression (1) of the preceding article, and we obtain the general form, d2ydx - d'xdy x dy Y dx 1 - x2 dx 1 - Differentiating equation (2), regarding 0 as the independent variable, we have dx = - sin OdO, d2x = - cos Od2. Substituting these in (3), and recollecting that 1 -- 2 = 1 - COS20 = Sinl 0, we have, after reduction, d2y do 2 + Y = 0, independent of x and its differentials. 2. Let xdy - y dx x r cos v(2) dz =.Y.x. (1);.. rc(2). ydy + xdx' y = r sin v | Differentiating equations (2), we have dx = cosvdr - r sinv dv, dy = sinvdr + r cosvdv. 80 DIFFERENTIAL CALCULUS. Substituting these in (1), and reducing, we obtain rdv dr' in which either -r or v may be regarded as the independent variable. VANISHING FRACTIONS. 62. In the discussion of the results obtained by the application of the Calculus, we often meet with expressions which, for a particular value of the variable, become o. This, although in general the algebraic symbol of an indeterminate quantity, does not indicate such a quantity in the particular cases referred to. As in the example a 2 - w2 which becomes ~ when x = a; if we divide both numerator and denominator by the common factor a - x, we obtain x a + x' and this, when x = a, reduces to i, which is the true value of the fraction in the particular case. Expressions of this kind are called vanishing fractions, and reduce to o in consequence of the existence of a factor common to both terms; which factor becomes 0 under the particular supposition. All such fractions may be represented generally by the expression P(x- a)C)fx - a)" DIFFERENTIAL CALCULUS. 81 in which P and Q are functions of x, not containing the factor (x - a). There are three cases: 1. When mn = n, the fraction becomes P (x - a)m P (x - a)"' - which, when x = a, becomes a finite quantity, 2. When m > n, it may be put under the form P(x - a)- -" Q m - n being positive; and this, when x = a, becomes 0 3. When m < n, the fraction may be put under the form 8. When m < n, the fraction may be put under the form Q ( - a)n- M' n - m being positive; and this, when x = a, becomes Puma = 00 We see from the above, that if we can, by any process, ascertain the relative values of m and n, we shall know the true value of the fraction when x = a. 6 82 DIFFERENTIAL CALCULUJS. 63. Whenever the common factor can be readily discovered, the simplest method of obtaining the true value of the fraction is to strike it out, and then put for the variable its particular value. But as in most cases it is not easy to detect this factor, other nethods becoiie necessary. r Let - be a vanishing fraction, r and s being functions of x, and let a be the particular value which, substituted for x, reduces the fraction to 0. It is plain that, if we substitute a + h for x, and, after reduction, make h = 0, it will amount only to the substitution of a for x. Suppose this substitution made, and that in the result both numerator and denominator are arranged so that the exponents of h shall increase fiom left to right; we then have (r\ AA" +- Bhm +- &c. kS L-a+A =A'h'n + B'h' +* &c.' in which A, A', B, B', m, n, &c., are constants. After reducing this fraction to its lowest terms, by dividing both numerator and denominator by that power of h which is indicated by the smallest exponent, we shall have one of three cases. 1. If m = n, A + Bhm'-"' + &c. 1 x-a+A A' + B'h"'- + &c. 2. If m > n, (r\ Ahm'- + &c. \S -a+A A' + &c. 3. If m < n, Ar\ A + &c. xS -Pa+ A'hn-m + &c. DIFFERENTIAL CALCULUS. 83 Now making h - O, we have for the true value in the three cases, 1. () A 2. (s= A' - o. tr\ A 3, - 00. a Whence we derive the general rule: For the variable, substitute that value which causes the fraction to reduce to -~, plus an increment; reduce the result to its simplest form, and then make the increment equal to O. The final result will be the true value of the fraction for the particular value of the variable, and may be finite, zero, or infinite. The effect of the application of this rule is evidently, by the reduction of the fraction to its lowest terms, to cause the common factor to disappear. To illustrate, take the fraction 3 (x - a) which becomes ~ when x C a. For x, put a + h; the primitive fraction then becomes (2oah + h2)2 3 Dividing both terms by ha, we obtain (2a + h)2; which, when h = 0, becomes (2a), the true value. 8.4 DIFFERENTIAL CALCULUS. In this case the common factor (x - a)Q is evident; strikin~ it out, we have (x + a)2, which becomes (2a), when x = a. 64. The application of the general rule of the preceding article which is strictly algebraic, will, in most cases, give rise to compbl cated-algebraic work. We may, however, by the aid of the Dii forential Calculus, deduce a practical rule of much more eas. application. Thus, if the vanishing fraction, as in the preceding article, be u -- -, then r = us, dr - uds + sdu; in which, if we make x = a, we shall have (since s,= - 0), (drXa - whence = r (dr ). (..'.ee (1), for the true value of the fraction in the particular case. If (dr). a -= O, this value is 0. If (ds).-.a = 0, it is o. If both are 0 at the same time, the second member of (1) dr becomes 0, and is a new vanishing fraction; then, as above, we take the differentials of both its terms, put a for x, and thus obtain DIFFERENTIAL CALCULUS. 85 (ds 2)r If this again becomes o, we continue the same process, and have and so on. The rule may then be thus(d) Take the and so on. The rule may then be thus enunciated: Take the differentials of the numerator and denominator; in, each, substitute that value of the variable which reduces the original fraction to O; i'f both do not reduce to 0 or ifinity, what the forminer becomes divided by what the latter becomes, will be the true value of the fraction. If both reduce to 0, take the second diferentials, and make the same substitution; or continue the differentiation, dc., until two differentials of the same order are obtained, both of which do not become 0 or infinity; what one becomes divided by what the other becomes, will be the true value of the fraction. it should be observed, that the effect of the application of this rule is, at each differentiation, to diminish by unity the exponent of the factor which causes the fraction to reduce to -, Art. (28). If the exponents of this factor in the numerator and denominator are fractional, and not contained between the same two consecutive whole numbers, it is plain that the least one will be reduced to a negative number, by a less number of differentiations than will be required by the other. The differential of that term of the fraction which contains it, will then, by the substitution of the particular value of the variable, reduce to infinity, while that of the other reduces to 0, and the true value of the fraction will be either oo 0 -_= co, or - 0. If, however, these exponents are contained between the sanme two consecutive whole numbers, they will become negative by the same number of differentiations, and the differentials of both terms 8(6 DIFFERENTIAL CALCULUS. of the fraction reduce to infinity at the same time, as will the successive differentials. In this, the only failing case of the rule, we shall not be able, by its application, to obtain the true value of the fraction, but must fall back upon the general rule, Art. (63). As an illustration of this, we may refer to the example in article (63), in which the second differentials, and all -which follow, become infinite when x = a. Examples. 1.If r' - 1 which becomes ~ when -- 1, drl = nx-I dx, ds - dx; making x = 1, in each of these, we have (dr),_l = ndx, (ds),1 = dx, and )' = n. 2. If r sin x' = COS X which becomes -o when x dr - - cosxdx, ds = - sin adx; DIFFEREN'TIAL CALCULUS. 87 making x = - in each, we have 2 (dr) = 0 (ds) r - x, the quotient of which is 0, the true value of the fraction. 3. If' ax2 - 2 acx + ac2 s bx - 2bcx + bc2' dr = (2ax - 2ac)dx, ds (2 bx - 2bc) dx, both of which reduce to 0, when x = c. Differentiating again, d2r = 2adx2, ds = 2bdx2, and (r\ a 4. Take a - hen x - O. Ans. la-lb. ml(l+ ) x 0. Ans. -. x a 1 - sinx + cosx 7r 6. x sinx + cosx -2 1. a - x — ala + alx a - V2ax _ x2 X - X 8. X+x = 1. 1 - x -t 1$ 88 DIFFERENTIAL CALCULUS. x 2- sinx 9. x -Z 0-. x sin x x - sin x x 11. x 0. 1 - cos mx 65. We sometimes meet with the product of two factors, one of which becomes 0, and the other oo, for a particular value of the variable. Let rt be such a product, in which r becomes 0, and t infinite. It may be written rt' t which, for the particular value, becomes 0. Its value may then be determined as in the preceding articles. Examrple. Let rt - (1 —) tan 2 when x- 1. Writing it under the proposed form, we have I — x 1 - x 1 rTX - - cot tan - 2. the true value of which, when x - 1, is 2 AT DIFFERENTIAL. CALCULUS. 89 66. The fraction s may become W, in which case it may be written r 1 8 _r X -a 8 which becomes oo x - oX 0 and may then be treated as in the preceding article. 67. Sometimes, also, we find expressions which become co -o. 1 1 Let r s be such an expression, r and s becoming 0. It may be written 1 1 s - r r S rs which will reduce to A.'For an example, take. 77 cot x 2 cos x' which becomes oo - oo, when x =- 2. By reduction we obtain ir xsinx - - the true value of which is - 1, when x = -. 90 DIFFERENTIAL CALCULUS. MAXIMA AND [MINIMA OF FUNCTIONS OF A SINGLE VARIABLE. 68. Let u = f(x), and suppose x to be increased by insensible degrees from its least value, until we obtain a corresponding state of the function which is greater than the state which immediately precedes it, and greater also than that which immediately follows it; this state of the function is called a maximum. If we obtain a state which is less than both of these consecutive states, it is a minimum. We say, then, that a function of a single variable is at a maximum state, or a maximum, when it is greater than the state which immediately precedes, and greater also than the state which immediately follows it; and a minimum, awhen it is less than both of these states. 69. If u is a function of x, and x supposed to be increasing, it is evident that when passing from the preceding states to its maximum, u must increase as x increases, that is, be an increasing function of x; and when passing from its maximum to the succeeding states, it must decrease as x increases, that is, be a decreasing function of x. In the first case, Art. (14), the sign of its first differential coefficient must be positive, and in the second, negative; therefore at the maximum state the first differential coefficient must change its sign from plus to minus, as the variable increases. For a similar reason at a minimum state, the first differential coefficient must change its sign from minus to plus; and these changes of sign, in the first differential coefficient, are respectively the a,.alytical characteristics of the maximum and minimum states of a function. But, as a function which is continuous can change its sign only by becoming zero or infinity, it follows that no value of the variable will give a maximum or minimum value to the function, unless the same value reduces the first differential coefficient to zero or infinity. DIFFERENTIAL CALCULUS. 91 The roots of the two equations, dau du dx dx-~.... (1), and d -- 0 or d = 0....(2), dx — x'a will then give all the values of x which can possibly make ut a maximum or a minimum. After having obtained these roots, let each, first with an infinitely small decrement and then with an infinitely small increment, be substituted in the given function; the results will be the states which immediately precede and follow the one obtained by substituting the root itself; if both are less than this, the latter will be a maximum; if both are greater, a minimum. Or, as will in general be more convenient, let each of these roots, with an infinitely small decrement and increment, be successively substituted in the first differential coefficient; if the first result be positive, and the second negative, the root will make the function a maximum; if the reverse, a minimum. If the two results have the same sign, the root under consideration will give neither a maximum nor a minimum. Since equations (1) and (2) may give several roots which will fulfil the required conditions, there may be more than one maximum or minimum state of the same function; and, therefore, the maximum state is not necessarily the greatest state, nor the minimum the least. Examples. i. If u = a + (x- b)............... (3), da dx 1 2 (x - b), and d-, x 2(-),du -2 (x -b)' du Placing 0 = 0, we have 2 (x - b) =-; whence x = b. 92 DIFFERENTIAL CALCULUS. If in equation (3) we substitute first b - h for x, and'then b + h, denoting the corresponding states of the function by u" and u', we have u"-= a + h2, and u' = a + h", both of which are greater than u = a, the result obtained by substituting b for x; hence u = a is a minimum. dx The only value of x which will reduce lu to 0, is x =; there is then no finite value of x which will satisfy this condition, hence x = b gives the only minimum state, and there is no maximum. 2. If u = a - (x —b))...... (4), du - 2 dx - 3(x —3)3 (x )' and - b)3 du Placing d = 0, we obtain x -= o, which gives no finite solution. dx Placing 0, we have 3 (x- b) - 0; whence x = b. If, then, in (4), we substitute first b - 1, and then b + h, for x, we have 2 2 =Ul a - h3, and U' = a - ]3 both of which are less than u = a, the result of the substitution of b for x; u = a is then a maximum and the only one, aind there is no minimum. DIFFERENTIAL CALCULUS. 93 If, in the first differential coefficients in the above examples, we substitute b - h and b + h for x, we obtain in the first, for b - h a negative, and for b + h a positive result; and in the second the reverse, as it should be. 70. When the states which immediately precede and follow the maximum or minimum state of u, can be deduced from Taylor's formula, a more convenient practical rule may be applied. To demonstrate it, let = f(x); then let x + h be substituted for x, and the difference between the two states be developed, as in Art. (35), and we shall have du dzlu he d3u h3 dA dx2 1.2 +dX 1.2-.().3. If, in this, h be infinitely small and negative, u' will be the state immediately preceding u; and if h be positive, u' will be the state immediately following u; and in both cases, the first term of the second member will be greater numerically than the sum of all the others, Art. (13), and the sign of the second member will be the same as that of its first term. Now, if u be a maximum, it must be greater than u', whether h be positive or negative; that is, u' - u must be negative in both cases; and if u be a minidu mum, u' - u must be positive. But - h evidently changes its sign as h changes from negative to positive; u cannot, theredu fore, be either a maximum or a minimum, unless the term -d h disappears, which, sinqce h is not zero, requires that d-= 0o...... *....,..(2). dx 94 DIFFERENTIAL CALCULUS. The roots of this equation will then, in the case under consideration, give all the values of x which can possibly make u either a maximum or minimum. Let a be one of these roots; to ascertain whether it will make v a maximum or minimum, substitute it in equation (1), and we have, since (d x - dxcl - -a ld U\ 2 c2 + I3 \ hd\ ('- )0 = (d2d) 1 2 + dI ( 2 3+&c 3 (3). The sign of the second member will now be the same as - d2), since h/2 is positive. If /d ___ is negative, then u will be greater than u', whether h be positive or negative, and ux=a will be a maximum. If ( be positive, u= will be a minimum. If Pdit= = O, then the sign dX 2 xa h' wh ich of (3) will depend upon the sign of (i 1 which \dX IX-a 1.2.3' evidently changes its sign as h changes; and there can be neither a maximum nor mininmum for x = a, unless (d).0. In this case the sign will depend upon that of ( d 4 ) = and there will be a maximum when this is negative, and a minimuln when it is positive, and so on; if the first differential coefficient which does not rednte to 0, is of an odd order, there will be no maximurn nor minimum for x- r; if of an even order, there will be one or the other, according' as its sign is negative or positive. If the first differential coefficient which does not reduce DIFFERENTIALT CALCULUS. 95 to 0, becomes infinity, this is a failing case of Taylor's formula, Art. (34), and the rule tius demonstrated fails with it. Whence, to determine the maximum or minimum states of a given function: Find its first di Jerential coetlcient and place it equal to 0; substitute each of the real roots of the equation thus formed, in the second diferential coelcient. Each one which gives a necgative result will, when substituted in the function, make it a maximum; and each which gives a positive result, will make it a m2nninizunm. If either reduce the second differential coefficient to 0, substitute in the third, fourth, &c., until one be obtained which does not reduce to 0. If this be of an odd order, the root will correspond to neither a maxinum nor mnivmumn; if of an even order and neyative, there will be a corresponding maximzum; if positive, a miin)imu. Substitute the root in the function; the result will be the corresponding:maximum o0r minimum. To illustrate, take the example 3 U- = -p ar2,. 3 —a2 du d2u x- + 2ax - 3a, 2x - 2.... (1). a~x dx. du Placing the expression for - 0, we have X2 + 2ax - 3a -= 0O the roots of which are x- = a, and x -- 3 a. The first substituted in (1) gives 4a, which being positive, indicates a minlimum. The second substituted ini (1) gives - 4a, which indicates a maximum. Substituting the rootg in the given futnction, 5a3 we have for the minimum u - and for the naxilnum u = 9a3. 96 DIFFERENTIAL CALCULUS. ABBREVIATIONS IN THE APPLICATION OF THE RULES FOR MIAXIMA AND MINIMA. 71. Let v = Au, u being any function of x, and A a positive constant. By differentiation, &c., we have dv du d v d2U A Adx dx' dx' dx2 from which it appears that'those values of x, which make du dv du = 0, will also make d- 0, and the reverse. Also, that = dx any of these values which will make d- - negative, will make d 2v d2u d'd d a negative; and any which will make d-2 positive, will make positive. Ience every value of x which will make dx u a maximum or minimum, will make v or Au a maximum or minimum. Therefore a constant positive factor may be omitted during the search for those values of the variable corresponding to a maximum or minimum. To illustrate, take the example 2bxc4 + a3bx b V - - a2 - 2(2x4 + a3x). Omitting the constant factor, we may write u = 2x4 + a3x, du d2U - 8x1 + a' dx 24........ W dxe dxl DIFFERENTIAL CALCULUS. 97 du Placing the expression for d = 0, we have 8x3 + a3 = 0; whence x 2 This value in (1) gives 6a', and indicates a minimum, which is 3a4 3alb u -; whence v - - 72. Let v = u, u and v being functions of x, and n entire. Then dv du - fnu TX = nul"-1dy2 + n(n -1)u" 2dx du Now every value of x which will make _=, will also dv make = 0; and if the same value makes nun-' positive, d:v d'u du2 it will give to d the same sign as d-~ (since d- 0); that is, if it makes u a maximum or minimum, it will make v a maximum or minimum. If it makes nu-' negative, it will d~v d2 It give to dx2- a sign contrary to that of d; that is, if it makes uf a maximum, it will make v a minimum, and the reverse. All values of x, however, which will make v-d- a maximum or minimum, will not necessarily make u a maximum or minimum, for the equation 7 98 DIFFERENTIAL CALCULUS. dv du - = nun =- O dx d O may be satisfied by making either du nu —1 =, Or or Those values of x which satisfy the first, and not the second of these equations, will make u neither a maximum nor'minimum, but may make v = u" a maximum or minimum. As in the example v = (a3 - x3)2 = U dv du dv = 2udu, 2 2 dv WVe may make d= 0, by placing either 2u - 2(a - xS3) = 0, whence x = a; du or = 3x2 = 0 whence x = 0. dx The value x = a evidently makes v a minimum, but as it du does not reduce -=- 3x2 to 0, it will make u neither a. maximum nor minimum. The value x = 0 answers to neither a maximum nor a minimum. As the corresponding power of a radical expression is formed by omitting the radical sign, we may, in accordance with the above principles, omit it, and seek those values of the variable which will make the power a maximum or m7irmum. We are sure thus to get all the values which will make the root a maximum or minimum. Care should be taken, however, not to use any of those which belong only to the power. DIFFERENTIAL CALCULUS. 99 To illustrate, take the example = -,/ax3 - 3X2 Omitting the radical sign, we have v - ax3 - 3x2 Taking the first differential coefficient of v and placing it equal to 2 0, we find the two roots, x = 0 and —. The first gives a maximum value, 0, for both v and u. The second gives a minimum value for both v and ze, viz.: 4 3 4 V = - -tU = - -a':* a a It would not be proper to extract the root of a function before applying the rule, as those values which make the power, and not the root, a maximum or minimum, would thus be excluded. 73. In a manner similar to the above, it may be shown that any value of the variable which will render u a maximum or minimum, will also render log u and au a maximum or minimum; and also any value which will make u a maximum, will make - a minimum, and the converse. 74. It often happens that the first differential coefficient is composed of two or more variable factors, each of which, when du placed equal to 0, will give real roots of the equation d = 0. In this case we may easily ascertain what the second differential coefficient reduces to, by the substitution of any one of these roots, 100 DIFFERENTIAL CALCULUS. without deducing the expression for the second differential coefficient itself. Thus, let du d XX' dx be such a coefficient, X being 0 when x = a. Then d2u dX' dX;T _ -d + X' dx2' dx d+ x or, since X = 0 when x = a, d druX dX That is, to obtain the corresponding value of the second differential coefficient: Multiply the differential coefficient of that factor which is 0, by the other factors, and then substitute the particular value of the variable. To illustrate, let u = X2($ -a) d- 2$x(x-a)(2 -a), which is equal to 0, when 2x 0; whence x -- (1). (x — a) = 0; x a....(2). (2x -a) = 0;,x ~ =...(3) Taking the first factor, 2x, and multiplying its differential coefficient by the other factors, we obtain the expression 2 (x - a) (2x - a); DIFFERENTIAL CALCULUS. 101 from which, by making x = 0, we obtain dx' J -o which indicates a minimum. Multiplying the differential coefficient of the second factor by the other factors, and making. = a, we obtain 2a2, which indicates a minimum. Proceeding in the same way with the third factor, we obtain - a, which indicates a maximum. 75. As the principles and rules for maxima and minima, in the preceding articles, are demonstrated independently of the nature or kind of the function, they are equally applicable to all kinds of functions of a single variable. To apply them to anll implicit function, we have then only to find its first and successive differential coefficients, by the rules previously given, Arts. (19) and (56), and then proceed precisely as in the foregoing examples. To illustrate, take the example y2 _ nzmxy + x - a' = 0....... (1), and let it be required to find the value of x which will make y a maximum or minimum. By differentiating as in article (56), we obtain 2ydy - 2 mxdy - 2mydx + 2xdx = 0; whence dy my - x dx$ y- antmx.~~~ ~(2). Placing this equal to 0, we have my - x = 0; whence x.= my, which, in equation (1), gives 102 DIFFERENTIAL CALCULUS. a ma y =.; whence x - Differentiating the factor my - x, equation (2), dividing by dx, and multiplying by - Art. (74), we obtain the y -x expression Y__ (m —m- 1), y - m T d which, by the substitution of the values of y and x (since then dy dy 0), becomes a /1 - m2 and indicates a maximum. SOLUTION OF PRACTICAL PROBLEMS IN M[AXIMA AND 3MINIMA OF FUNCTIONS OF ONE VARIABLE. 76. The only difficulty in the application of the preceding principles to the solution of problems, consists in obtaining a convenient algebraic expression for the function whose maximum -or minimum state is required. No general rule can well be given by which this expression can be found. In order to indicate as clearly as possible the methods to be pursued, we will give, in detail, the solution of several cases differing from each other. 1. Required the dimensions of the maximum cylinder which can be inscribed in a given right cone. Suppose a cylinder inscribed, as represented in the figure. Let DIFFERENTIAL CALCULUS. 103 VA = a, BA = b, VC = x, CO = y; then AC -a- x, and the volume of the v cylinder, which we denote by v, is equal to iry'(a - x).......(1). From the similar triangles VCO and VAB, we have the proportion E bx 5)x.. ".., x: y:: a: b; whence y -. Substituting this value in (1), we have e = a x(a - x.3......(2). Omitting the constant factor, Art. (71), we may write u = ax2 - X3whence du d'u 2ax - 3x, d 2 2a 6x....(3). du Placing = 0, we find the roots, x = 0 and x = 2a. The second value of x in (3) gives - 2a, and therefore will make 4r7rab' v a maximum, which is 27 For the altitude of the maximum cylinder, we have a - x = a, and for the radius of the base, y = 3b. The first value of x in (3) gives 2 a, which indicates a minimum, which is evidently v = 0. 2. Required to draw a tangent to the given quadrant ABD, so that the triangle CFG shall be a minimum. 104 DIFFERENTIAL CALCULUS. Let CB R FB = x, BG y; then FG = x + y. The area of the triangle is equal to ICB x FG, which, since ~CB is constant, will be a minimum when FG is a minimum, Art. (71). In the a A F right-angled triangle CFG, since CB is perpendicular to FG, we have R' = xy; whence y = -, and FG = u = x —, x du R2 xs — R' dx- X- - 2. which, being placed equal to 0, gives x = R, and y = R. Hence the angle BCF = 450. Obtaining the corresponding d2 u 2 value of d-W, as in Art. (74), we find for a result 3. The whole surface of a right cylinder being given, it is required to find the radius of the base and the altitude, when the volume is a maximum. Let m2 = the surface, x = the radius of the base, and z = the altitude; then v = Irx z. But rn = 2rrxz + 27rx2; whence z - 2 ~'x therefore Knx V = - -?TX 2 DIFFERENTIAL CALCULUS. 105 and x - 1, and z 2 when v is a maximum. 4. Required the minimum distance from a point without a given circle to the circumference. Let 0 be the centre of the given A circle, OM its radius - R, C the given point, CO = a. Join C and 0, and take OC as the axis of X,. P /C the origin being at O. Denote the co-ordinates of M by x and y, and the distance CM by u; then u =VR2 + a2 - 2ax. Omitting the radical sign, we have u2 = v = R + a2 _ 2ax....... (1). dv d - - 2a, dx which cannot be 0 or oo. We should not, therefore, conclude that CM admits of no minimum value; for it is evident, from the inspection of the figure, that CA, corresponding to the value x = R, is a minimum. Cases of this kind are remarkable, but readily explained by a reference to either demonstration of the rule; for it depends entirely upon the principle, that the function is expressed in terms of a variable which admits of a value both less and greater than the one which corresponds to a maximum or minimum. Now, in the case under consideration, there is no value of x greater than R, which corresponds to a real state of the function. Both rules must therefore fail in cases of this kind. The remedy is, to deduce an expression for the function, in terms 106 DIFFERENTIAL CALCULUS. of some other variable which will admit of proper values. Thus, if the value of x = %/R2 - y2 be substituted in equation (1) we have v = R2 + a-2 2a -/R' -- y3, dv = 2ay, 2 ay -, y 0; doy /RZ _ y2 which gives x = R, and x — R. The first value corresponds to a minimum, and the second to a maximum value of CM. The same result might have been obtained directly, by taking any other line than CO as the axis of X, in which case x would have admitted of proper values. 5. Required the minimum isosceles triangle circumscribing a given circle. Let 0 be the centre of the circle, ABC the circumscribing triangle, AC and BC being the equal sides, OM = R, PC = x, A13 = 2y, and denote the area by u; then u = xy.......( 1). From the similar triangles COM and CPB, At~F-a.l-"" X~-' we have PB: OM:: CB: CO, or y R" vx + y2 x -R; whence (x-R)y = R/Vx2 + yi, and y = R/ x x- 21' DIFFERENTIAL CALCULUS. 107 Substituting this in (1), we have u - Rx R. x- 2R x- 2R Omitting the constant factor R, and the radical sign, Arts. (71) nd (72), we have X3 dv (x - 2R)3x2 - x3 =x - 2R' d (x - 2R)2 dv Placing dv = 0, (x - 2R)3x2 - = 0, 2x3 - 6Rx2 = 0; whence x = 0, x = 3R. The value x = 3R gives a minimum, which is the circumscribed equilateral triangle. The value x = 0 corresponds to no maximum nor minimum, as there is no real state of the function immediately following the state corresponding to x _ 0; all states, from x = 0 to x = 2R, being imaginary. 6. Required to divide a given quantity, a, into two parts, such that the mth power of one, multiplied by the nth power of the other, shall be a maximum. If x = one of the parts, then x - + 7. In a given triangle, it is required to inscribe a maximum rectangle. The altitude of the rectangle'= - altitude of triangle. 2 108 DIFFERENTIAL CALCULUS. 8. A certain quantity of water being given, it is required to find the relation between the radius of the base and the altitude of a cylindrical vessel, open at the top, which shall just hold the water and have its interior surface a minimum. The radius -= the altitude. 9. Required the maximum rectangle which can be inscribed in a circle. Each side =R /2. 10. Required the maximum cone which can be inscribed in a given sphere. 11. Required the minimum cone circumscribing a given sphere. 12. Required the minimum triangle formed by the axis, the produced ordinate of the extreme point, and a tangent to the are of a parabola. 13. Required the maximum cylinder that can be inscribed in a given ellipsoid of revolution. 14. Required the axis of the maximum parabola that can be cut from a given right cone. 15. Required the minimum value of y in the equation y = x'.'MAXIMrA AND MINIMA OF FUNCTIONS OF TWO OR MORE VARIABLES. 77. A function of two or more variables is a maximum when it is greater, and a minimum when it is less, than all of its consecutive states. Let - f(, y), then u' = f( + h, y + k), DIFFERENTIAL CALCULUS. 109'- u = (p +p't) + -r (q + 2q't + q"t') + &c....(1); 1. 2 after placing in the development of article (48), k ht du du P d2u d'u du dx2 - q' dxdy dy2 The sign of this series, when h is infinitely small, will depend upon the sign of its first tenn. We shall obtain all of the consecutive states of u by giving to h and k proper infinitely small values, both positive and negative; and therefore, when u is either a maximum or a minimum, the sign of u' - u for all these values of h and k must be the same. But the first term of the series (1) evidently changes its sign when the sign of h changes; there can, then, be neither a maximum nor a minimum, unless A(p r pit) -_ O, or p + P't - O; and since this must be 0 for all values of t = -, we must have separately p = O, and p' = 0, dit du or - = 0....(2), = (3). The values of x and y, deduced from these equations and substituted in the second term of series (1), (h. and k being infinitely small), should make it negative for a maximum, End positive for a minimum. This term may be put under the forml ( r f+ - ~t + t', 11.0 DIFFERENTIAL CALCULUS. which, if there be a maximum or minimum, must not change its sign for any value of t; that is, the quantity within the parenthesis must be positive for all values of t. By adding and subtracting -,, we may write - i r 2q' t + -' (t q,,- q,,,...(4) q __ If q/ -,,, which is entirely independent of t, be negative, such values may be assigned to t as to make expression (4) either positive or negative. To render the entire expression positive, q and q" must then have the same sign, and q q'2 A, q -,,2 -> 0, or = 0, that is, we must have ~qq I_- q 2 > 0, or qq", -q2 = 0. The conditions then are yd22l \ d2u d2u d2u d2u dxdy < X or = - X d2u d2u and also that d-2 and have the same sign, after the du values of x and y deduced from the equations d - 0 and du 0 — 0 have been substituted. And since the sign of the dc~~~~~~~~~y~~2 second term will then. depend upon q", the sign of d must be negative for a maximum, and positive for a minimum. DIFFERENTIAL CALCULUS. 11 If the second term becomes 0, we must substitute the values of x and y in the third, which must also be 0, and the sign of the fourth negative for a maximum, and positive for a minimum; the discussion of the several conditions of which, although complicated, may be made in a manner similar to the above. Exalples. 1. Required to divide a number a into three parts, such that the cube of the first, into the square of the second, into the first power of the third, shall be a maximum.'Let x = the first part, and y = the second; then a - x - y = the third, and u xy'(a - x — y), du du dx -'y2(3a - - 4x), dy x y(2a - 3y- 2x). Placing these equal to 0, we have 3c - 3y - 4x = 0, 2a - 3y - 2x = O a a whence x -=, y -= 3 We have also d'u q- dX2 = 2xy2(3a - 3y - 6x), s2' q'. = - y(6a - 9y - 8x), dxdy d2u dy'Y" 112 DIFFERENTIAL CALCULUS. which, for the particular values of x and y, become a4 a4 a4 9' 12' 8" Hence -a< a and d2u - q'2 ~ -- i 44 < qq" = -- and 144 72' dy2 8 u is therefore a maximum when its value is 432' 2. Make the preceding proposition general, by putting for the cube, square, and first power, the mth, nth, and rth powers. Then u = Sxmy (a - x - y)' ma na m +- n + r' Y + n + r 3. Required the shortest distance from a given point to a given plane. Let the equation of the plane be placed under the form z cx + dy +- g, and the co-ordinates of the given point be x', y', and z'; then u= (x - X't)2 + (Y - YT) + (Z -z)9 or putting for z its value, = V( - x')2 + (y - y')2 + (cx + dy + g _- z)2. Calling the radical, R, we shall have DIFFERENTIAL CALCULUS. 113 du y - y' + (cx + dy + g - z')d dy R du. _ x - x' + (cz + dy + g - z') dx R Placing these equal to 0, and solving the resulting equations, we may obtain the values of x and y, and thence of z. Or other wise, putting for cx + dy + g its value z, we have y — y' + d(z - z') = O, and x - x' +- c(z - z') - 0, which are evidently the equations of a perpendicular to the plane, and if combined with the equation of the plane will give the values of x, y, and z. 4. The volume of a rectangular parallelopipedon being given, required its three edges when its surface is a minimum. 5. Required the maximum rectangular parallelopipedon which can be inscribed in a sphere. 78. In order that a function of three or more variables be a maximum or a minimum, we must have du du du d = o, d o &c.;= and the relation between the partial differential coefficients of the second order must be such, that the second term, in the development of the difference u' - u shall remain of the same sign, for all the consecutive values of the function. 8 114 DIFFERENTIAL CALCULUS. GEOMETRICAL SIGNIFICATION OF A FUNCTION OF A SINGLE VARIABLE, AND OF ITS DIFFERENTIAL COEFFICIENT. 79. Let y be any function of x expressed by the symbol y = f(x), and let any value be assigned to x, and the corresponding value of y be deduced; these two values may be regarded, the first as the abscissa and the second as the ordinate of a point which nmay be constructed. Any number of values may thus be assigned to x, the corresponding values of y deduced, and a series of points thus constructed, through which, if a line be traced, y will be its variable ordinate and x the abscissa. Hence, we conclude that every ftnction of a single variable may be regarded as the ordinate of a line, of which the variable is the abscissa. 80. Let BMM' be a curve, the equation of which is y = f(x), and M any point of this curve, the co-ordinates being x and y. Increase the abscissa AP or x, by the variable increment PP' = h; M denote the corresponding ordinate P'M' by y'; draw the secant M'MT', the tangent MT, and MQ parallel -r T A- P PI to AX. Then M'Q = P'M' - PM = y'- y = Ph + P'h2....Art. (12) From the triangle M'MQ, we have tan M'MQ M = tan IT'X, DIFFERENTIAL CALCULUS. 115 and placing for M'Q and MQ = PP', their values, this becomes tin MT'X = + Ph' P + P'h.......(1). Now, if h be diminished, the point M' approaches M, and the secant M'T' approaches the tangent MT, and finally, when h = 0, the point M' coincides with M, and the secant with the tangent. If then, in (1), we make h = 0, we have tan MTX =P = d that is, the tangent of the angle which a tangent at any point of a line makes with. the axis of X, is equal to the first dsiferential coefficient of the ordinate of the line. To show the application of this principle, let us take the equation of a circle, x2 + y2 = R=; whence dy x = - -..................(2), dx y(2) for the general value of the tangent of the angle, made by a tangent at any point of the circumference, with the axis of X. If the particular value at a point whose co-ordinates are x" and y" be required, for x and y let x" and y" be substituted; then dy"_ d". * The symbols dY"X d-y," &c., are used to indicate what the first, second, &c., differential coefficients become, when for the general variables x and y the particular values x" and y" are substituted; and are called the first, second, &c., differential coefficienits of the ordinate of the curve taken at the point x", y". 116 DIFFERENTIAL CALCULUS. Take also the equation of the conic' sections, y- = 2px +- 2 2X; whence dy p + r'x p + 1r2x dx y V2px + rqx2 For the particular point y" and a", this expression becomes dy" p + rx" dx- - = 2px" + r2x"z2 81. If it be required to find the point of a given curve, at which the tangent line miakes a given angle with the axis of X, we know that at this point the first differential coefficient must be equal to the tangent of the given angle. Calling this tangent a, we must then have dy dx and this, combined with the equation of the curve, wiI1 give the particular values of x and y, for the required point. If the tangent line is to be parallel to the axis of X, then for dy the point of tangency, 0; and if perpendicular, Co. We will illustrate each of these cases by an example. 1. Let it be required to find the, point on a given parabola, at which the tangent line makes an angle of 450 with the axis. The equation of the parabola is y2 = 2px, by the differentiation of which, &c., we have dy _ p dx y But as tan 45~ = 1, we have, for the required point, DIFFERENTIAL CALCULUS. 117 dy p dx y and, combining this with the equation y2 = 2px, we fin, - Y = p. The tangent at the extremity of the ordinate passing through the focus, will then fulfil the required condition. 2. Let y = a + (- x),.......() represent a curve; then dy dx= - 2 (c - x), dx which is equal to 0, when x = c; and this value of x in (1) gives y = a. These are then the co-ordinates of the point at which the tangent is parallel to the axis of X. 3. Let y = a + (c -x) represent a curve; then dy 1 d 2(c-x) 2 which is equal to infinity, when x = c. x = c and y- a are then the co-ordinates of the point at which the tangent is perpendicular to the axis of X. 118 DIFFERENTIAL CALCULUS. EQUATIONS OF TANGENT AND NORMAL. EXPRESSIONS FOR SUB-TANGENT, SUB-NORMAL, &C. 82. If x" and y" represent the co-ordinates of a given point on a given curve, whose equation is y = -f(x), the equation of a straight line passing through this point will be y - = a(x - x")......(1), a being indeterminate. In order that this line be a tangent at the given point, a must be the first differential coefficient of the ordinate of the curve taken at this point, Note, Art. (80); that is, for a we must substitute We thus obtain dx"' dy" Y" = ~,' (x - x")..... By differentiating the equation of an ellipse, a2.y2 + blx' = a2b., we deduce dy blx dy" b2x" dy = - b~x; whence d" dx a y dx"'- a y" and this value in (2) gives, for the equation of a tangent to an ellipse at the point y", x", b2x" Y - - a2,(x - x")t which, by reduction, becomes a2yy" + bxx" = a2b-2. DIFFERENTIAL CALCULUS. 119 83. If the equation of a tangent be required, which shall be parallel to a given line, or make a given angle with the axis of'X, we may determine the co-ordinates of the point of contact as in article (81); and knowing these, the equation may be deduced as above. Thus, if a tangent to a circle be required to make with the axis of X an angle whose tangent is 2, we must have for the required point, equation (2), Art. (80), x y From this, we find y =-', which, combined with the equation of the circle, gives 2R R x = -._ " y - -y"; V/5 /5 and equation (2), Art. (82), becomes, when we use the upper signs, y + 2- 2(x- or y = 2x - R V. 84. Equation (1), Art. (82), will become the equation of a normal at the point x", y", if for a, we substitute 1 dxi' - = dy (Analyt. Geo.); do" and we thus obtain for the general equation of a normal, dx" -y" - dy,,= 120 DIFFERENTIAL CALCULUS. 85. The right-angled triangle MTP (Figure of Art. 80) gives PM PM = PT tan MTP; hence PT = tan MTP' or Subtangent = Y dx dx Also, MT - MP - PT or Tangent = + dx y1 / + d.2 dy2 = ~ i The right-angled triangle PMR gives PR = MP tan PMR; but PMR = MTP; hence, dy PR = MP tan MTP, or Subnormal = y Also, MR = /MP + PR; hence, /dy2 + dy2 Normal = /y + Y - d/ 2' To apply these formulas to a particular curve, it is only necesdx dy sary to substitute in each the expression for d yor - deduoed from the differential equation of the curve. The results DIFFERENTIAL CALCULUS. 121 will be general for all points of the curve. If the values for a given point be' required, in these results let the co-ordinates of the point be substituted for x and y. For example, take the general equation of Conic Sections, y 2 = 2px + r x2; whence dy _ p + r'x dx /2px +- r 2x dx V/2pxZ r'x2' dy p + r2x These expressions substituted in the formulas, give PT r PR -p +- rpx, v + r2x MT = /2px + r2x2 +( 2 r + r'x)2 MR = V2px + r2x2 + (p + r2x). For the parabola r2 = 0, and these expressions become, PT = 2x, PR p, MT = -/2px + 4x2, MR = 1/2px + p2. CONVEXITY AND CONCAVITY OF CURVES. 86. A curve, at a point, is convex towards another line, when, in the immediate vicinity of the point, its tangent lies between it and the line. It is concave when it lies between its tangent and the line. 122 DIFFERENTIAL CALCULUS. If a curve be convex towards the axis of X, and the ordinate jositive, as in the annexed figure, it is plain, that as the abscissas AP, AP', &c., increase, the angles MTX, M'T'X, &c., will increase, and the reverse; consequently their tangents will also increase as x increases, or decrease as x decreases. Since these tangents are represented by the corresponding values of the first differential coefficient of the ordinate (dY),' it mzust be an increasing function of x, and its d2y differential coefficient, i. e., d must be positive, Art. (14). differential coefficient, i. d If the curve be still convex, and the ordinate negative, the angles A STX, S'T'X, &c., and their tangents dy plainly decrease as x increases; d is a decreasing function of x, and d'y must be negative. dxq If, then, a curve be convex towards the axis of abscissas, the ordinate and its second differential coeficient, taken at the different points, will have the samre sign. If the curve be concave, and the ordinate positive, as in the figure, the angles MTX, M'T'X, po, then pn' > pc';' - ~ ~ or if p< p)o, then pn' < po'; hence, in this ease, in the vicinity of the point M the circle lies entirely within or entirely without the curve. In these cases it will be found that the order of contact of the circle is odd, and higher than the second; for, unless A"'-= 0 the circle must intersect, as shown by the preceding demonstration. Since the osculatory circle has a more intimate contact with a curve at a given point than any other circle, it will necessarily separate those circles which are tangent without the curve from those which are tangent within. 105. The curvature of a curve at a given point is its tendency to depart frogm its tangent at that point; or is the angular space included between the curve and its tangent. Thus, of the two curves DIFFERENTIAL CALCULUS. 149 AC and AB, having the common tangent AD, b. the former has a greater tendency to depart firom the tangent and has the greatest curvature, since the angular space DAC > DAB. The curvature of the circumference of a circle is evidently the same at all of its points; but of two different circumferences, that one curves the most which has the least radius; as in the figure, the tendency of abd to depart from the a tangent is greater than that of ab'd', and this tendency plainly increases as the radius decreases, and the reverse; that is, the curvature in two different circles varies inversely as their radii. This being the case, the expression m lay be taken as the measure of the curvature of a circle whose radius is R. Since the contact of the osculatory circle with a curve is so intimate, its curvature may be taken for the curvature of the curve at the point of osculation; and the two in the immediate vicinity of this point may be- regarded as one and the same curve; hence, to compare the curvatures at different points of a curve, we have - only to compare the curvatures of / V the osculatory circles drawn at these points. Thus, in the curve MM', / 1 1 curvature at M: curvature at M': - r rl 106. The radius of the osculatory circle at a given point of a curve is called the radius of curvature, at that point; and the centre of the circle is the centre of curvature. A general expression for this radius is given in article (99), and it may be found for any particular curve by differentiating the equation of the curve, and 150 DIFFERENTIAL CALCULUS. substituting the derived expressions for dy and d'y in the formula, - - _ (dx2 + dy 2) ~R _ dxd2y If the value at any particular point of the curve be required, for x and y in the expression just deduced, substitute the coordinates of the particular point. As only the absolute length of the radius of curvature is required in determining the curvature of curves, we may use eitlier the plus or minus sign of the formula. It is best, in general, to use that which, taken with the sign resulting from the expression, will make R essentially positive. Let it now be required to find the general expression for the radius of curvature of Conic Sections. Their equation is y x= 2 + r2x; whence dy (p + rx)d Y dx~2 + dy. - [y + (P'+' x2.)2]dX2 2 r ydx2 - (p + r2x)dxdy [r2y2 - (p + r2x)']dx2= Tyh e Y~ These expressions substituted in the formula give, after reduction, [2px + r'x2 + (p + r2x)]2] 1 R= - [p ~ ( (1); the numerator of which is the cube of the normal, Art. (85). Hence, the radius of curvature at any point of a conic section, is equal to the cube qf the normal divided by the square of half the DIFTFERENTIAL CALCULUS. 151 parameter, and the radii at different points are to each other as the cubes of the corresponding normals. If in (1) we make x = 0, we have, at the principal vertex, R _ p = one-half the parameter, which for the ellipse and hyperbola is The radius of curvature at the vertex of the conjugate axis of the ellipse, is obtained by substituting in (1), p- = r -- 6 and x - a. The result is R = - = one-half the parameter of the conjugate axis. It may be readily shown that p is the least value of R; therefore the curvature at the principal vertex of a conic section, is greater than at ally other point. Likewise, - is the greatest value of R in the ellipse; hence, the curvature of the ellipse is least at the vertex of the conjugate axis. The curvature of the other two curves diminishes as we recede from the vertex. For the parabola r' = 0, we then have - (2px + pl) p2 152 DIFFERENTIAL CALCULUS. EVOLUTES. 107. If, at the different points of a given curve, osculatory circles be drawn, and a second curve traced through their centres, the latter is called the evolute of the:- " " former, which is the involute. Thus, /z7 \CC" is the evolute of the involute MM". Points of the evolute may always be constructed by drawing normals at the:~\,!, —-— ~-..~Nk different points of the involute, and on each of these normals laying off the /it X corresponding value of R, deduced as in article (106). 108. If a and 3, the co-ordinates of the centre of the osculatory circle, be regarded as variables, they will determine all the points of the evolute; but a, 3, and R, are functions of x and y, the co-ordinates of the points of osculation; and the relation between these five variables is expressed by the three equations of Art. (99), which may be written thus, (x — a)" + (y-) R....(1) (x- a)d + (y- /) dy = 0..... (2), (y - )d'y + dy' + dx2 = o.....(3). If we differentiate (1) and (2), regarding all the quantities, except dx, as variables, we obtain (x - a) dx + (y - P)idy - (x - a) da - (y -,)do = RdR, dx2 +* dy2 r (y- P3)d2y - dxda - dyde = 0, DIFFERENTIAL. CALCULUS. 153 and these, by means of equations (2) and (3), are reduced to - (x - a)da - (y - P)d = RdR.......(4), - dxda - dyd3 = 0......... (). Equation (5) gives dx do dy.de..MMMM(6). dx - is the tangent of the angle which a normal to the involute at the point (x, y) makes with the axis of X, Art. (84), and dfi da is the tangent of the angle which a tangent to the evolute at the point (a, p) makes with the same axis; hence, these angles are equal. But the normal at the point (x, y) passes through the point (a, /), Art. (103); therefore the normal and tangent form one and the same line; that is, the radius of curvature is normal to the involute, and tangent to the evolute. The evolute may therefore be constructed by drawing a curve tangent to the normals at the different points of the involute. From what precedes,. it is plain that the evolute may be regarded as formed by the intersections of the consecutive normals to the involute, and that the point of intersection of any two consecutive normals may be taken as the centre of the osculatory Ocircle, which passes through the two consecutive points of the involute at which the normals are drawn. 109. Equation (6) of the preceding article, combined with (2), gives da - a - d (Y f). 154 DIFFERENTIAL CALCULUS. Substituting this value in (1), we have, after reduction, (y_ -f3)2(da, + dEl/).=..(v) (Y + P)2.... dfp Substituting the same value in (4), reducing and squaring both members, we obtain (Y- ) ( d 2 R dR2. d/T3 Dividing this by (7), member by member, and taking the root, A/da I + df3 = dR. But if z represent the arc of the evolute, we have dz = V/dact + dfa........ Art. (90); hence dR = dz, and R = z + c......Art.(16). 110. If any two radii of curvature be drawn, as one at M' and the other at M"; the first being denoted —... by R, the second by R', and the corre-'MCX ~ 1 sponding arcs CC' and CC" by z and z',.//. we have ij R - z + c,' -= z' + c; whence Clt )/ \R' - R = z' - z; that is, the dci.Ference between any two radii of curvature is equal to the arc of the evolute intercepted between them. DIFFERENTIAL CALCULUS. 155 If in the equation R = z +,c, we make z = 0, and denote by r, the corresponding value of R, we shall have = O + c = C; that is, the constant c is always equal to the radius of curvature which passes through the point of the evolute, from which its are is estimated. If we estimate the evolute of the ellipse M from the point C, we have c -- MC = -. Art.(106); hence b2 R= - z + -. a Also, since M'C' =, a bM'C' - MC - =CC'. b a If the evolute and one point of the involute be given, and a thread be wound upon the evolute and drawn tight, passing through the given point MI, when the thread is unwound or evoleed, the point of a pencil first placed at M will describe the involute; for by the nature of the operation, CC' is always equal to M'IC' - MC. 111. The equation of the evolute of any curve may be found thus: Differentiate the equation of the involute twice; deducet the expressions for dy and d2y, and substitute in the equations (2) and (3), Art. (99), 156 DIFFERENTIAL CALCULUS. dy a- - dx (y - P).........Mf dX + dy(2 d2y combine the results, which will contain the four variables, a, / x, and y, with the equation of the involute, and eliminate x and y; the final equation will contain only a, 13, and constants, and will therefore be the required equation. As an example, let it be required to find the equation of the evolute of the common parabola. The equation of the involute is 2 = 2px, whence d _ dx - y' pd____ pdx 2 dy2-' y2, d~y Substituting these values in (1) and (2), and reducing, we have y2 x - - a _ P........(3), y3 _ y~ y- = + y, whence -= p...(4); p p and putting for y, in (3) and (4), its value V'2px = (2p)-; &x we have p2 DIFFERENTIAL CALCULUS. 1a7 Tlee value of x = - ( - p) taken from the first equation, and substituted in the last, gives - = 7: (a -p)3 27p which is the required equation. If we make / = 0, we have a == p, and laying off AC = p, C will be the point at which the evolute meets the axis of X. If we transfer the origin of co-ordinates to this point, M we have cc=p + t, o e i~ 3- a'f P a='; hence / 2 = 8 3 27p Since every value of a' gives two values of /3', equal with contrary signs, the curve is symmetrical with the axis of X. If a' be negative, A3' is imaginary, and the curve does not extend to the left of C. The branch CC' belongs to AM, and CC" to AM'. ENVELOPES. 112. Let It = f/(, y, a) = O........(1), represent a curve given in kind only, a being the only arbitrary constant in the equation. If a be regarded as variable, and be changed so as not to change the form of the equation, we may 158 DIFFERENTIAL CALCULUS. obtain an infinite number of curves of the same species as that represented by (1). If a be increased by da, we shall evidently obtain the curve of the species which is consecutive with the first. fBy increasing a again by da, we shall obtain the next consecutive curve, and so on. In general, these consecutive curves will intersect each other two and two, and by their intersections form a niew curve, which is called the envelope of the species represented by equation (1). 113. To explain the mode of obtaining the equation of this envelope, we substitute, in equation (!) of the preceding article, a. + -da for a, and obtain' =f(x, y,, + dX) = 0; or by Taylor's theorem, Art. (.35), du d''t cda2' u + - + t- +&c. O........ (]1). da da 1.2 Since u = f(., y, a) _ 0, and since, da being infinitely small, all terms after da may da be rejected, equation (1) becomes du du dau d OX or d - 0........(2), da da which is the equation of the first consecutive curve. If this be combined with equation (1) of the preceding article, the values of x and y, in the result, will be the co-ordinates of the points of intersection of these two curves; and if they be comnbined in such a way as to eliminate a; x and y will be the coordinates of the points of intersection of any two consecutive curves of the same species, or the general co-ordinates of the DIFFEREINTIAL CALCULUS. 159 curve formed by these intersections. To obtain, then, the equation of the envelope of a curve given in kind, we comrbine its equation with its differential equation, taken with respect to the arbitrary constant, and eliminate the constant; the result will be the required equation. 114. This elimination may be effected by deducing the expression for a, in terms of x and y, from the equation:du d= o..........(1), and substituting it in equation (1) of article (112). This expression may be represented by ca - p (xr, Y). If, then, in equation (1) of Art. (112),' a be regarded as equal to g (x, y), that equation will represent the envelope. If, under this supposition, the "equation be differentiated, we have, for its differential equation, du du, du O dx+ - dy + d 0, which, since - 0 reduces to. da -dx+ -dy 0; dx dy an equation identical with that obtained b y differentiatinig equady tion (1), Art. (112), when a is constant. The expressions for dy dx' deduced from these two equations, will then be the same; hence, at the point of intersection of two consecutive curves, -the tangent to the envelope will be the same as the tangent to the first curve; or, the envelope is tangent to, the different curves of the species, hence its name. 160 DIFFERENTIAL CALCULUS. 115. We may illustrate by the following examples: 1. Deduce the equation of the envelope formed by the intersections of the consecutive right lines given by the equation ut = y - ax - = 0.,.......(1), a when a varies. Differentiating with respect to a, we have du b a =-x + a 2-= 0, whence a _:t1 Substituting this value in (1), reducing, and squaring both members, we have Y?'_ 4bx, the equation of a parabola. 2. Deduce the equation of the envelope of the parabolas, given by the equation (1 + a')x2 - 2apx + 2py = 0, when a varies.. 116. If the equation of the curve have two constants, we may limit the species of curves represented by it, by requiring an equation of condition to exist between these constants, such as to make one dependent upon the other. In this case, the expression for one, in terms of the other, may be obtained from the equation of condition and be substituted in the equation of' the curve, and then the equation of the envelope of the species be deduced as in the preceding article. Or otherwise, the given equation may be dif DIFFERENTIAL CALCULUS. 161 ferentiated, regarding one of the constants as a function of the other; the equation of condition may also be differentiated under the same supposition, and then, by the combination of the four equations, the'differential coefficient and'onstants be eliminated, thus giving the equation of the envelope. In the same wa7y the equation of the envelope, when there is any number of constants with a proper number of equations of condition, may be deternmined. ~'xamp7es. 1. Find the equation of the envelope of the different positions of a right line of given length, whichW is moved with its extremities in two rectangular axes. Let I be the length of the lipe, a and b the distances cut off from tile axes of X and Y respectively.! The equation of the line may be put under the form -i x y - + 1........ x a 6 From the condition of the problem, we also have a2 + b2 = 1..........(2). Differentiating, regarding a as a function of b, we-have da + ydb a-da u b —- O, ada -- bdb = 0. da Deducing from these the expressions for y-, and equating, 162 DIFFERENTIAL CALCULUS. a2y b 1 a2y -Yx -' whence - = bx - a' a b-ac Substituting. this in (1), we have ay Y= or J- 1, or (a2 + 2)y b3y and since a2 + b2 = 1, b3 =y12, b =/yl2. In the same way, we find Substituting these expressions in (2), and reducing, we have, for the equation of the envelope, 2 2 2 2. Find the equation -of the envelope of a series of ellipses having the same centre, same co-ordinate axes, and same area. Let the equation of the ellipse be put under the form, Y2 X.I. 1(1). Since the areas are the same, we must have ab = c'............ (2), c2 being constant. By differentiating, &c., as in the preceding article, we find for the envelope, Xy = _the equation of an equilateral hyperbola, referred to its asymptotes. DIFFERENTIAL CALCULUS. 163 APPLICATION OF THE DIFFERENTIAL CALCULUS TO TIlE CONSTRUCTIPN AND DISCUSSION OF CURVES. SINGULAR POINTS. 117. The most general division of curves is into the classes, Alqebraic and Transcendental. When the relation between the ordinate and abscissa of a curve can be expressed entirely in algebraic terms, Art. (5), it belongs to the first class; and when such relation cannot be expressed without the aid of transcendental quantities, it belongs to the second class. We have seen, in Analytical Geometry, the mode of constructing and discussing curves when their equations are given. By the aid of the Differential Calculus, this discussion may not only be simplified but much extended, and the nature, form, and properties of the curve be thus more fully ascertained. On many curves points are found, at which there exists some remarkable property not enjoyed by the other points of the curve. These are called singular points. They are entirely independent of the system or position of the co-ordinate axes, and are easily discovered and characterized by: the Calculus. A detailed discussion of the general equation y b + c (x a)............ (1) in which m is any positive number, will illustrate these principles. First: Let m be entire and even. Since every value of x, positive or negative, gives one real value for y, the curve is continuous, and extends indefinitely in the direction of the axis of X. By the differentiation, &c., of (1), we have 1].64 DIFFEMIENTIAL CALCULUS. dydy dx- = (m 1)..........e c Placing d= 0, we obtain X = a. This value of x, when substituted in (I), (2), (3), &, c gives y = b, and reduces the successive differential coefficients to 0, as far as the rnth, which, if c be positive, beeomes a positive constant, and is of an even order; hence, y = b is a minimum ordinate, Art. (7I0). Since for z=a, we have -Z 0, the tangent lime at the extremity of this minimum ordinate is parallel to the axis of X; and since (m and i - 2 being even) for all values of x except x = a, d2y y an dX2' are positive, the curve at all of A,, its points is convex towards the axis of X, Art. (8G). If c be negative; the mth differential coefficient will be negative; and x - a and y = b will be the co-ordinates of a point at which the ordinate is a maximum. In this case, the second differential coefficient for all values of x, except x = -a, is negative, and the curve, for all positive values of y, concave, and for all negative ~ — values of y, convex, towards the axis of X. l!'k~~~~~~~Y DIFFERENTIAL CALCULUS. 165 118. Second: Let im be entire and odd. In this case, each value of x gives one real value for y; and each value of y, a real value for x; hence, the curve is unlimited in either direction. When x - a, the first differential coefficient, as before, is equal to 0; as also the second, third, &c. The mtth, however, if c be positive, is a positive constant, and of an odd order; there is then in this case, neither a maximum nor a minimum, Art. (70). By examining the second differential coefficient, we see (since m- 2 is odd), that for every value of x < a, it is negative; that for x -a, it is 0; and when x > a, it is positive: hence, for all values of x < a, which give y positive, the curve is concave towards the axis of X; and for all values of x > a, it is convex, as in the figure. A P Therefore, at the point whose co-ordinates are x = a and y = b, as x increases, the curve changes from being concave, and becomes convex, towards the axis of X. If c be negative; the reverse will be the case, and as in the second figure, at the point M, whose co-ordinates are z = a and y = b, there is a change from convexity to concavity towards the axis of X. Such points are singular, and are called points of inflexio,. In both cases the tangent line at the point of M inflexion is parallel to the axis of X, and also cuts the curve., 119. Third: Let m be a fraction, the numerator and denominator of which are odd, as s. Then y - b +- c(X- a)] dy 3c d2y 2 3c dx (-)5 (x - ) d2 5( - a),' 166 DIFFERENTIAL CALCULUS. X = a gives dy d'y y = b, d' d~Y= A, &c. dx dx& If c be positive; dxY' for all values of x < a, will be positive, and for all values of x > a, negative; hence, for all values of x less than a which give y positive, the curve will be convex, and for all values / M of x greater than a it will be concave towards the axis of X, as in the figure. If c be negative; the reverse is the case, as in the secohid figure. The point M, whose co-ordinates are x = a and y = b, is in both cases a point of inflexion at which the tangent line is perpentdicular to the axis of X Whence we may say: A point of inflexion is one at which, as the abscissa increases, a cturve changes fron being concave towards any right line, not A passing through the point, and becomes convex, or the reverse. If the right line be taken as the axis of abscissas, this point will always be characterized by a change of sign in the second differential coefficient of the ordinate. For, since the curve on one side of the point is concave, and on the other convex, the second differential coefficient in one case has a different sign from that of the ordinate, and in the other the same; hence, at the point the sign must have changed. In order that this may be the case, the second differential coefficient must be equal to zero, or infinity. The roots of the two equations, d2y d2y dx.0 - O, and d2 will then give all the values of the variable which can belong to points of infiexion. DIFFERENTIAL CALCULUS. 167 It sometimes happens that a point of inflexion lies on the axis of X, as in the second case above discussed when b = 0. In this case x - a gives dyn y = 0, and d - and the corresponding point M is a point of inflexion, at which both the second differential coefficient and ordinate change their signs. It is evident, firom the preceding discussion,'A/ that if any right line be drawn through a point of inflexion, the curve on both sides of the point will either be convex towards the line, or concave. 120. Fourth: Let ret be a fraction with an even numerator, as a. Then y - b + c(x- a)3, dy 2c d'y 1 2c d 3(2X-a)3 3 3(x - a)3 x = a gives dy d'y y = b, dx =0 dx = dy If c be positive; for x < a, d- will be negative, and for dx b x > a, it will be positive; hence at the point whose co-ordinates dy are x = a and y = b, dy must change its sign from minus to plus, which change indicates a minimum ordinate, Art. (69). If c be negative; the reverse will be the case, there will be a change of sign from plus to minus, and the ordinate will be a maximum. 168 DIFFERENTIAL CALCULUS. In the first case, the second differential coefficient for all values of x is negative, and the ordinate positive; the curve is therefore concave towards the axis of X, as represented in A fig. (a). In the second case, d-Y is always positive. For all positive values of y the curve will then be convex, and for all negative values of y concave, as in fig. (b). The tangent iM at the point M is in both cases perpendicular to' 9(z) the axis of X., < - The point M is singular, and is called a cusp. It is a point at which the curve, uwhen interrupted in its course in one direction, turns immediately into a contrary one. 121. Fifth: Let m be afraction with an even denominator, as 3. Since the denominator of the fraction indicates that the square root is to be taken, the double sign i — must be placed before (x - a), and we then have y = b -E c(x-a),dy 3 4 d2y 3e dY= i 2 ( x - a.) 2, i 4Every value of < a gives y imaginary; - a) gives Every value of x < a gives y imaginary; x - a gives y = b, and x > a gives two values, one greater and the other less than b. There is then no point on the left of that one whose co-ordinates are x = a and y - b; but on the right of this point the curve must extend indefinitely, and consist of two branches. DIFFERENTIAL CALCULUS. 169 dy x = a gives 0; the tangent at M is then parallel to the axis of X. Each value of x > a gives two values for M d y, the one positive corresponding to the.A greater value of y, and the other negative; hence, the upper branch is convex, and the lower, until it crosses the axis of X; concave, as in the figure, and the point AI is a cusp. 122. Let us now take the equation (y X2) = from which we deduce y = x2:~i X, dy 5 d2y 5 3' =2x i _X = 2 i= X dx 2' dx2 2 When x - 0, we have y = 0. If x be negative, y is imaginary. For every positive value of x, there are two real values of y, both offwhich are positive as long as x2 > X 2 or x < 1; after which, one is positive and the other negative. dy When x = 0, = 0; also when 5 1 16 2 i: -x2 O, or x = 2 25 170 DIFFERENTIAL CALCULUS. hence the axis of X is tangent to the curve at the origin; and the tangent to the lower branch, at the point whose abscissa is 16 16, is parallel to the axis of X. d2y The first value of d belongs to the upper branch, and is always positive. The second value is also positive as long as 2 > 5232-, or < -6; after which it is negative. The origin is then a cusp, at which both branches lie on the same side of the common tangent, and is of the [A. ~ >- second species, those before discussed being of the first species. The point of the lower branch whose abscissa is 64 is a point of inflexion. 123. From the equation, ay - x23 + bx' = 0, we derive X x2(x-b) dy 3x- 2b o = 1 = = a' d 2 /a (x — b) Since x = 0 gives y = 0, the origin A is a point of the curve. All negative values of x make y imaginary, as also all positive values less than b; hence, A has no consecutive point. Such points, given by the equation of a curve, but having no consecutive points on either side, are singular, and are called isolated or co7jugate points. dy Substituting 0 for x in the expression dI, \ it reduces to =F b a/_ ah DIFFERENTIAL CALCULUS. 171 an imaginary expression; and, in general, at a conjugate point, one or more of the dijferential coefficients; of the ordinate must be imaginary, since y', the consecutive ordinate, when developed as in Art. (35), can only be imaginary under this supposition. If we take the equation, a4y = X6 _ a2x44 whence X2 ely- i 3x3 - 2a'x y-~V' -i - a', a d ix a2 V/Xa _ a,' x O0 and y = 0 will satisfy the equation, while no other value of x, numerically less than a, will give real values for y. The dy origin is then a conjugate point. In this case, for x = 0, reduces to O. If the second differential coefficient be taken, it will, for x = 0, reduce to an imaginary expression. 124. Take the equation, y = b i (x - a) V/ - c, atnd suppose a > c. By differentiating, we derive dy =i /x - c: x- a dx 2 /' - G For every value of x < c, except x = a, y is imaginary. d.y For x = c, y -= b, and d-X For every value of x > c, there are two real values of y. dy a - c, For x -- a, y = b, and dVac 172 DIFFERENTIAL CALCULUS. and at the corresponding point M there are two tangents, one making an angle, the tangent of which is + V'a - c,' and the other - Va - c. The point M is singtldar, and belongs to "RB < a class called multiple points, or points at which two or more branches A of a curve intersect. If but two intersect, the point is a double multiple point; if three, a triple; and so on. Since there will be a separate tangent to each branch, at one of these points, it will be characterized by two or more values of the first differential coefficient, for the same values of the variables. If a < c, x = a, and y = b give a conjugate point. 120. We will close the discussion of algebraic curves by constructing the curve given by the equation ay -- x3 + (b - c)x + bcxr = 0; whence v —4 - b -)(x +c) dy = 3 x 2 -(b - c)- b' d 2 Vax (x - b) (x + c) Each of the values, = O, x = b, x = - c, gives y-= 0. Every negative value of x, numerically greater than c, gives y imaginary; while every such value less than c gives two equal values of y with contrary signs. Every positive value of x < b gives y imaginary; and every such value greater than b, gives two equal values of y with contrary signs. The curve is then symnetrical with reference to the axis of X. dy Each of the values, x = 0, x b,.=- c, reduces: to w; hence, at the three corresponding points the tangent is perpendicular to the axis of X. DIFFERENTrAL CALCULUS. 173 By solving the equation, 3x2 - 2x(b- c)- be 0, we shall find two real values for x, one positive and the other negative, and thus determine the points at which the tangent is parallel to the axis of X. The positive value will be found to be less than b, and hence will give no point of the curve. The negative value is numerically less than c, and gives two points, one above and the other below the axis of X. The C curve may then be drawn as in the figure, in which AC = - c, and AB - b. If c =- 0, the equation becomes ay2 - X3 + b2 0, and the oval AC reduces to the conjugate point A, as in article (123). If b = 0, the equation becomes ay2 -- 3 C = 0, and the curve takes the form indicated in figure (b), the origin being a double multiple point, since dy becomes equal to. ~ dx a If b and c are both equal to 0, the equation becomes ay -; whence y =:j/, and the curve will be as in figure (c), the point A being a cusp of the first species. 174 DIFFERENTIAL CALCULUS. 126. One of the most important of the transcendental curves is THE LOGARITHMIC CURVE, so named because it may always be referred to a set of co-ordinate "axes, such that one co-ordinate will be the logarithm of the other. Its equation is usually written y = log x, or, if a be the base of the system of logarithms, x - aL. vY 71C The curve is given when a is known, and \ ~ may be constructed by laying off on the axis M,, of X the different numbers, and on the coriresponding perpendiculars the logarithlls of A IA/X)? these numbers. Or it may be constructed s/ "'.., from the equation x = a', by making y = -, 3, X, &c.; whence, the corresponding values of x are 4 -- X= Va, x=a ='a, - = A-c/ &c. When y = 0, x = 1. This being the case for all systems of lograritlims, shows that all logarithmic curves, when referred to the same axes, cut the axis of X, or axis of numbers, at a distance from. the origin equal to unity. If a > 1, and x > 1, y is positive, and increases as x increases; if x < 1, y is negative, and. increases numerically as x decreases, until x = 0, when y - o. If x be negative, there will be no corresponding value of y. The curve will then be of the form indicated by the full line in the figure. If a < 1, then reverse will be the case, and the curve will be represented by the dotted line. DIFFEREENTIAL CALCULUS. 175 127. If now we differentiate the equation y = log x, M being the modulus, we deduce dy M d2y M dx x' dx2 - 2 d M; When x = dy 0 dx 0 hence, the tangent at the corresponding point is the axis of Y; and since for x = 0, y = - oc, this tangent is an asymptote. dy l When x = do O. But x = oo gives y = oo; hence, there is no tangent parallel to the axis of X, at a finite distance from it. The value for the subtangent on the axis of X is dx x PT _ y logl x. If the subtangent be taken on the axis of Y, we have dy SS' = x; that is, the subtangent on the axis of logarithms is constant, and equal to the modulus of the system in which the logarithms are taken. If M =1, SS' = 1 = AB. d2y Since, when a > 1, d2x is negative for all values of x, the part BM is concave towards the axis of X, and BM3I' convex. d'2y When a < l, Mi is ne~gative, -Y will be positive, the part Bnt' convex, and Bmn concave. 176 DIFFERENTIAL CALCULUS. 128. The curve given by the equation = xlx, is remarkable. Each value of x gives but a single value of y. For values of x > 1, y is positive; for values of x < 1, y is negative; and for small values of x, decreases numerically and approaches the limit 0, which it reaches when x = 0. Negative values of x give no values for y. The origin is then a point of the curve at which it is interrupted in its course, but does not turn into a contrary one as at a cusp. Such points are called terminating points., Y The value x = 1 gives y =0; hence, the curve cuts the axis of X at a distance from the origin equal to 1. Differentiating the equation, we have'-~ - <- = lx +, = - adr dx dx2 x Placing lx + 1 = O, we have Ix = - 1, 1' 1 or, e- 2,71 e 2,71...... which corresponds to a minimum value of y, Art. (70). At the point B, xdy becomes equal to 1, and the tangent makes an angle of 45~ with the axis of X. Between the points A and B the curve is concave, and from the point B it is convex towards the axis of X, Art. (86). 129. Let 1 y -e X. Each value of x gives a single positive value of y. x = gives y = 1. As x decreases, y decreases, until x = 0 gives DIFFERENTIAL CALCULUS. 177 y = 0. If x be negative and infinitely small, y is infinitely great, and as x increases numerically, y decreases, until Y x = - o gives y- l. At the origin the curve is interrupted, as in the pre- o ceding article. Differentiating, we have 1 1 dy _ e- dy e-(l - 2x) d -- 2' — x4 4 dx X dx X For x = 0, becomes oo, and the axis of Y is an asympdx tote of the left-hand branch of the curve. For x = oo or - a, dy becomes 0, and the line OS at a distance from the axis of X dx equal to 1, is an asymptote to both branches. For all negative d y values of x, d is positive, and the curve convex towards the axis of X. Also, for all positive values of x < 1. For x =, d becomes 0, and is negative for all values of x >' The dx' point M, whose abscissa is 1 is a point of inflexion, Art. (119), and beyond this point the curve is concave towards the axis of X. 130. Another singular point is given by the equation, y = b + x tan-l X' whence dy 1 z d = -tan - X' dx X X2 + 178 DI'E RENTIAL CALCULUS. Each value of x gives but a single value for y. For values of x which are numerically equal, one positive and the other negative, the corresponding values of y are equal; hence, the curve is symmetrical with respect to the. axis of Y, and x- 0 gives y= b. For all positive values of x, L- is positive, and as x is diminished to O, x dy d increases to dx dy\ 1 (dJ) = tan - = tan'l o -. dx/. 0 2 dy For negative values of x, dy is negative, and increases numerically as x is thus decreased to 0, when we have ~d-)- tan(- tan ) dX 0 0 2 We thus have two branches terminating at the point MI, not tangent to each other as at a cusp. This, which is but a particular case of a multiple point, is called a salient point. 131. The most remarkable transcendental curve is THE CYCLOID, which is generated by a point in the circumference of a circle, when the circle is rolled in the same plane, along a given straight line. DIFFERENTIAT. CALCULUS. 1'79 Let AB be the given line, and suppose the circle to have been placed upon it, so that the generating point was at A, and then to have been rolled to the position RME. The generating point now at M, has generated the are AM. / E/. D BlI AP C. E Take the origin of co-ordinates at A, and let AP = x, PM = y and RE, the diameter of the generating circle = 2r; then AP = AR - PR........ (1). But since every point of the circumference from M to R, as the circle was rolled, came in contact with AR, we have AR - are MR = ver-sin-'RN - ver-sin-'y. Also, PR = MN = ~-RN x NE = V'y(2r -y) = a/2ry- y. Substituting the values of AP, AR, and PR in (1), we have x = ver-sin- /2ry - - y -........ (2), which is the equation of the Cycloid. After the circle has been rolled over once, every point of the circumference will have been in contact with AB, and the generating point will have arrived at B;: we have then AB = circumference of generating circle = 2 -rr. 180 DIFFERENTIAL CALCULUS. The given line is called the base of the Cycloid, and the line CD -- 2r perpendicular to AB at its middle point, is the axis. If the rolling of the circle be continued beyond the point B, an infinite number of arcs, each equal to ADB, will be generated. Every negative value of y in equation (2) makes x imaginary; heuce there is no point of the curve below the axis of X. y = 2r, gives x = ver-sin-12r ='r = AC. Every value of y > 2r makes x imaginary; hence the greatest ordinate of the curve is equal to the diameter of,the generating circle. For the points of each branch between D and B, the essential sign of the radical must evidently be plus. By differentiating (2) we have, Art. (44), after introducing the radius r, ddx ry rdy - ydy V/2ry - y V 2ry - yi or reducing dx, - -.....,(3), V/2ry- Ywhich is the differential equation of the Cycloid. 132. Substituting the preceding value of dx in the formulas of article (85), and reducing, we have Subtangent, PT = Y V2ry 2- y Tangent, MT _ - 2 — /2ry - y" Subnormal, PR = /2ry_ y-2, DIFFERENTIAL CALCULUS. 181 Normal, MR = =V2/uy. Since the subnormal PR = V/2ry - y = MN, the diameter ER and normal MR intersect the base at the same point. Hence, to construct the normal at a given point, join it with the point at which the corresponding position of the generating circle is tangent to the base. Or, upon the greatest ordinate CD as a diameter, describe a circle, and through the given point M draw a line parallel to the base; from the point F in which it cuts the circle, draw the two chords CF and DF to the extremities of the diameter; a line through the given point parallel to CF will be the normal, and one parallel to DF the tangent. If it be required to draw a tangent parallel to a given line, as T'T", draw the chord DF parallel to the given line, from F draw FM parallel to the base; the point M is the point of contact, through which draw a line parallel to T'T". 133. From equation (3), article (131), we have dy _ V'/2 ry - - 1. which becomes 0 when y =2r, and when y 0; hence, at the extremity of the greatest ordinate, the tangent is parallel to the base; and at the points A, B, &c., where the curve meets the base, it is perpendicular. If we square both members of equation (1), we have dy2 2r dx~ y Differentiating both members of this, we have 2dydy 2 rdy d2y r dZ 2 y 2- or = dx2 y2 dxy y, 182 DIFFERENTIAL CALCULUS. This second differential coefficient being negative for all values of y, the curve is concave towards the axis of X, Art. (86). 134. Substituting the values of dy and dly in the expression (dx2 + dy2)2 dxd'y we obtain 2 rydx2 = 2 r R _ 2 2 y 2 /21 y Y 2 or since V/2ry is the expression for the normal, Art. (132), tahe radius of' curvature is equal to twice the normal at the point of osculation. If' y - O, R = 0; and if y = 2r, R - 4r; hence, the radius of curvature at A (see figure in next article) is equal to 0; and at D is 4r; therefore, Art. (110), the arc AA' = 4r. 135. To obtain the equation of the evolute, let us substitute the values of dy and d'y in equations (1) and (2) of article (111). After reduction, we find y - = 2y, x -- a =a - 2 /2ry - y2; whence y = - 3, x = a - 2 V- 2rf3 - (3. DIFFERENTIAL CALCULUS. 183 These values, in the equation of the involute, Art. (131), give a = ver-sin-'( — ) + /- 2t3 - p/......(1), far the required equation. If we produce DC to A', making CA' = DC, and then transfer the origin to A', the new axes being A'X' and A'D, and the new co- A. ordinates a' and O', we shall have for any point, as M',.f AG - a, GM' =- B, A'P' a', P'M'= /3'. Since AC = rrr, and CG = A'P', a T rr - a'; and since GP' = 2 r, GM' = 2r - /', or - P = 2r - A'. Substituting these values in (1), we have wr - a' = ver-sin-'(2r — /')+ ~ /2rp' -- /32, whence a' — = rr - ver-sin-'(2r- /') - / 2r/3'-'32; But rr - ver-sin-'(2r - /') = ver-sin-l''; hence, the last equation becomes a' = ver-sin-l' - /2or3' - /3V', 184 DIFFERENTIAL CALCULUS. which is the equation of the evolute referred to the new axes, and is of the same form and contains the same constants as the equation of the involute; therefore the two curves are equal. Since are AA' = 4r, its equal AD = 4r, and ADB - 4.2r; that is, equal to four times the diameter of the generating circle. POLAR CURVES. SPIRALS. 136. In Analytical Geometry we have seen, that we may obtain the polar equation of any curve, given in terms of rectangular co-ordinates, by substituting for these co-ordinates their values, in terms of the polar co-ordinates, taken from the formulas x - a 4+ r cosv, y = b + r sinv....(l). Also, if we have the differential equation of the curve, or any expression containing the differentials of the variables, we at once pass to the corresponding equation or expression in terms of polar co-ordinates and their differentials, by substituting for x and y the above expressions, and for dx and dy the expressions below, obtained by differentiating equations (1). dx = cosvdr - r sin v dv, dy = sinvdr + r cosvdv. 137. If a right line be revolved uniformly, in the same plane, about one of its points, a second point of the line continually approaching, or receding from the fixed point, in accordance with some prescribed-law, will generate a curve called a spiral. DIFFERENTIAL CALCULUS. 185 The fixed point is called the pole or eye of the spiral. The portion of the spiral generated while the line makes one revolution, is called a spire; and since there is no limit to the number of revolutions, the number of spires is infinite, and any straight line drawn through the pole of the spiral will intersect it in an infinite number of points. For this lreason, the relation between the ordinate and abscissa of a spiral cannot be expressed algebraically, Art. (117). r' The system of polar co-ordinates is generally used to determine the different points of a spiral, and its equation may be represented by r = f(v), in which r denotes the radius vector, and v the variable angle. 138. Before discussing the particular spirals, it will be necessary to determine general expressions for the subtangent, &c., and the differentials of the are and area, in terms of polar co-ordinates. The subtangent, in such case, is the part of the perpendicular to the radius vector of the point of contact, intercepted between thle pole and the point where the tangent meets this perpendicular. Thus, if A be the pole, and MT the tangent, AT perpendicular to AM is the subtangent. To find the expression for it, let the are c receive the increment PP' (AP being = 1); describe MC with the radius AM - r; draw the chords MC and c MM', and the line AT' parallel to MC,; and produce MM' to T'. From the similar triangles MM'C and M'AT', we have'MC x AM': M'C: MC::AM': AT'; AT' =MC....(I) M'C 186 DIFFERENTIAL CALCULUS. Also, from the similar sectors APP' and AMC, 1 PP':: AM: arc MC; arc MC - AM x PP'. Now, suppose the increment PP' = dv, then M'C = dr, \rt. (88), M' becomes consecutive with M, the secant M'T' coincides with the tangent MIT, AT' - AT AT, AM A r, and chord MC = arc MC - rdv. Making these substitutions in (1), we have r~ dv AT = subtangent = dr.... From this we deduce AT AT r dv. —- -A --- tanAMT. r AMi dr The tangent MT = /AM2 Jr AAT2 = I r 1 L + d The similar triangles AMT and AMR, give r2 dr AT::: r: AR; AR -- - d = subnormal. When M' is consecutive with M, MM'C may.be regarded as a triangle, right-angled at C; hence, MM',- + MC-'. But MM' is the differential of the are; therefore dz = v/dra + r'd2'. If ADM be any segment; AMM' will be its increment when v is increased by dv. Calling the segment s, AMM' will then be ds, DIFFERENTIAL CALCULUS. 187 and may be measured by the sector AMC. But the alea of -the sector, I1~~~ r'dv AMC -= AM X are MC = 2 2 2 hence, r'adv ds -- 139. An equation from which the particular equations of most of the spirals may be deduced, by assigning particular values to' a and n, is r = av". Ifn be positive, v = 0 will give r = 0, and the spirals represented by the equation have their origin at the pole. If n be negative, v = 0 will give r = co, and the spirals have their origin at an infinite distance, continually approach the pole, and r becomes equal to 0 only when v = o. 140. Let n = 1, then r = av, and if r' and v', r" and v", represent the co-ordinates of any two points of the spiral, we shall have rf = av', r- = a-v; whence rf: rr"!v: V", or the law in accordance with which the generating point must move is, that the radius vectors shall be proportional to the correspondizg angles. 188 DIFFERENTIAL CALCULUS. The curve thus generated is the Spiral of Archimedes. If we take for the unit of distance, the length of the radius vector after one revolution; then r = 1, v = 27r and the equation gives 1 = a.27r, a - and the primitive equation becomes V _ ldv r - - _~-~; whence dr - 27r 27r This spiral may be constructed by dividing a circumference into any number of equal parts, as 8, and the radius AB into the same number of equal parts. On the radius AC lay off one of these parts; on AD two, AE three, &c.; on -.f'-I-..T... AB eight, then again on AC nine, &c. da /i The distances thus laid off will be proportional to the angles BAC, BAD, &c., and: B the cur've through their extremities the'if i,\< |required spiral.: Substituting the values of r and dr in equation (2), Art. (138), we have AT = subtangent = 2-. If v = 27r, that is, if the tangent be drawn at the extremity of the arc generated in one revolution, we have AT = 2 r = circumference of measuring circle. If v = m.27r, or the tangent be drawn at the extremity of the arc generated in m revolutions, AT = m'.27r = m.2mr; DIFFERENTIAL CALCUTLUS. 189 that is, equal to m times the circumference described with the radius vector of the point of contact. For the subnormal we find dr 1 AR== dv 2r' 141. If n = 2, the general equation becomes 2 r = av2 or r = av. This equation being of the same form as that of the parabola, the curve given by it is called the Parabolic Spiral. It may be constructed by first constructing the parabola whose equation is y2 - ax, and then laying off from P to B, C, D, &c.,, 7 along the circumference, any assumed ab-'"'| scissas, and from A to M, M', &c., the corre- i sponding ordinates; the points M, M', &c., \ will be points of the spiral, since for each o we have y2 = ax, or r = a2v. 2r3 The subtangent at any point is AT a 142. If n = - 1, r = avw becomes a r =a1 = a or rv = a, and the spiral thus given is called the ZHyperbolic Spiral. 190 DIFFERENTIAL CALCULUS. If r' and v', r' and v", be the co-ordinates of any two points a a of the spiral, we have r'-, and r" = whence 1 1 *r" *: or the radius vectors are inversely proportional to the ang~les. c __ If M be any point of the spiral, AM- r, MAP -v..... ~ The right-angled triangle MAP, MP gives r = sin v Substituting this value of r in the equation rv = a, we find sin v MlP = a v As v is diminished, this value approaches nearer to a, and since n v ) i, when v = O, we have MP = a. v /ro If then, at a distance, AC - a, a line be drawn parallel to AP, it will continually approach the curve, and touch it at an infinite distance. r2 dv The subtangent AT = d =- a dr a It is then constant, and equal to AC. Also, rdv d- _ tan AMT = - v; do' DIFFERENTIAL CALCULUS. 191 that is, the tangent of the angle made by the tangent and radius vector, is equal to the arc which measures the' angle made by the radius vector and fixed line. We may apply these properties to the construction of the curve by points, thus: With A as a centre and radius = a,,, describe a circle; join any, f point T with A, draw the! \ J indefinite radius vector AM perpendicular to AT. Make x, / AD - arc PN; join D and N, and draw TMI parallel to DN, M will be a point of the curve; for by the construction AD = tan AND = tan AIT = arc NP. 143. The spiral represented by the equation v = log r, is called the Logarithmic Spiral. Differentiating, we find Mdr dv = Md whence tan AMT rdv that is, the angle formed by the radius vector and tangent is con stant, and the tangent of this angle is equal to the modulus of thl system of logarithms used. If the Naperian system be chosen, M -= 1, and AMT = 45~ 192 DIFFERENTIAL CALCULUS. Since v is the logarithm of r, if it be increased uniformly, so that the different arcs v, v', v", &c., shall be in arithmetical progression, then r, r', r", &c., must be in geometrical progression, and the curve may be constructed thus: /-'..-......... With AO = 1 describe a circle, and divide,'f~ 1)s the circumference into any number of equal i ~ _~, ~parts, and draw the lines AO, Ap, Ap',./ &c. The distances laid off on these lines.\. / are to be in geometrical progression, since the arcs Op, Op', Op", &c., increase by the constant difference Op. To find the ratio of this progression, let v = 0, then r = AO = 1. Now make v = the arc Op, and find the corresponding value of r in the system of logarithms used, which lay off to m, then Am the ratio. AO On Ap', Ap", &c., lay off Am', Am", so that AO: Am: Am': Am": Am"': &c., m, nM', mn", &c., will be points of the curve. APPLICATION OF THE CALCULUS TO SURFACES. 144. Since the equation of every surface expresses therelation between the co-ordinates of its points, it must contain three variables, and may be generally written = F(x, y, z) = 0........(1); or since either two of these variables may be assumed at pleasure, and the remaining one determined from the equation, the latter may be regarded as a function of the other two, they being en DIFFERENTIAL CALCULUS. 193 tirely independent of each other, and the equation of the surface be thus otherwise expressed, =f/(x, y)......... (2). By the same course of reasoning as that in Art. (79), it may be proved that every function of two variables may be regarded as the ordinate of a surface of which the variables are abscissas. In the equation of every surface considered, z will be regarded as a function of x and y; and the co-ordinate planes will be taken at right angles to each other. The differential equation of a surface may then be obtained, either by differentiating equation (1), as in article (57), or by differentiating equation (2), as in article (52). By the latter method, we obtain dz dz dz dx + dy......(3). 145. Let M be any point of a surface, a portion of which is d represented in the annexed fig- N ure. The co-ordinates of this /, point are z-Ab, y=Ac, z = M. M. I Let a plane be passed through i / MI, parallel to YZ. For every' point of this plane,.. X = Ab = x". If, then, in the equation of the'l surface, we make x = x", and suppose z and y to vary, they can only belong to points in the curve dMd', the intersection of the plane and surface. 13 194 DIFFERENTIAL CALCULUS. In the same way, if y - y", in the equation of the surface, and z and x vary, we shall have the curve eMN. If x and y vary at the same time, and receive the increments bb' = h and cc'- k, we have M'P' = (x' + 71, y + k), which may be developed as in Art. (48). When x = x", equation (3), Art. (144), gives dz do Z dz day -p'dy or d l -P'dy (4); equations which evidently belong only to the section tdMd' parallel to YZ. If y = y", the corresponding equations fbr the section parallel to XZ are dz dz dz = dx = pdx, or -...... dx d. The value of d-X equation (4), is the tangent of the angle which a tangent to the section dMd', at any point, makes with dz the axis of Y, or with the plane XY; and -Tx, equation (5), the corresponding expression for the section eMN; and since these angles are the same as those made by the curves, at the point of contact, with XY, they give the inclination or slope of the surface in the direction of these curves. 146. If it be required to find the slope of the surface at any point, as M, along the section MM' made by the plane MM'PP', we take the equation of this plane, y = ax +.... (1), z indeterminate; DIFFERENTIAL CALCULUS. 195 a being the tangent of the angle made with the axis of X by the trace PP', and equal to dy - Now, in order that z shall represent only the ordinates of points in the section MM', the relation expressed in equation (1) must exist between the variables x and y, and we must have dy _ adz, which, in equation (3) of article (144), gives dz - (p + ap') dx. Mi'P' - MP The limit of the ratio pp, is evidently the tangent of the angle (S) which the tangent, and consequently the curve at the point M, makes with PP', or with the plane XY. But since PP-= V/pQ2 + pQ2 = h ~1 + a, we have M'P'- MP z'- z PP' - h +- 2 the limit of which is 1 dz p + p xp' X -= - - tan S. 1/1 -Jr- a dx /1 +- a2 To find the direction in which the section MM' must be made, in order that the slope at a given point M, along the curve cut out, be greater than along any other, it is only necessary to obtain that value of a which will render the expression p + ap' /1' +.' 196 DIFFERENTIAL CALCULUES. a maximum, the values of p and p' being taken at the given point AI. Differentiating the expression with reference to a, and placing the result equal to 0, we have' —P~ = h0; (1 -- a+)whence 2 — Pa = O a0 This value of a substituted in equation (1), (/ being first determined by the condition that the line PP' shall pass through P), will give an equation which, combined with that of the surface, will determine the line of greatest slope. EQUATIONS OF TANGENT PLANE AND NORMAL LINE. 147. The co-ordinates of a given point M, being x", y", and z", the equations of a tangent to the section parallel to XZ at this point, will be -z - X"), y.( and to the section parallel to YZ, - z dz" = dP...2) The equations of a plane passing through the same point, Analyt. Geom., Art. (64), will be - " = c ( - x") + d(y y )..... (3). DIFFERENTIAL CALCULUS. 197 The intersection of this plane by the plane through M parallel to XZ, will be represented by z - "= c( - x"), y = y"; and the intersection by the plane parallel to YZ, by Z - z' = d (y- y"), x = x". If the plane (3) is tangent to the surface at M, these lines should be tangent to the sections of the surface, and therefore identical with those represented by equations (1) and (2), and we must have dz" dz'" c - d dy"; and equation (3) becomes the equation of the tangent plane, z - Z d (+,, X - ) d+ "d ( - y 4) To illustrate, take the equation of the ellipsoid x yy z2 a + b2 + P 1, from which, by differentiating, first with reference to x, and then with reference to y, and substituting z", x", and y", we obtain dz" C2;x t dz" C2ytdx" - a2/z" dy" - 2z These expressions in equation (4) give, after reduction, XX"t yY zz" a + —z + --- = 1. 198 DIFFERENTIAL CALCULUS. 148. The equations of a straight line passing through the point M, are - = a(z - "), y - y" = b(z - "). This will become perpendicular to the tangent plane, if we have the conditions, Analyt. Geom., Art. (59), dz" dz" a = - c, b - d, or a - b and we thus deduce the equations of a normal line, x x- ( -,- (Z- "), Y Y dy" (- )' By substituting these expressions for x -'" and y - y" in the general expression D = V/(X — x") + (y- y")2 + (-z Z-")', we obtain for the distance from any point of the normal to the point of contact, D = (z-z")/ + 1 + ((d:)2 If z = 0, ( dz" (dz"' for the distance from the point where the normal pierces the plane XY to the point of contact, the minus sign being omitted, as the numerical value only is required. z" divided by this distance, gives the sine of the angle which the normal makes with the plane DIFFERENTIAL CALCULUS. 199 XY; and this angle is the complement of the angle made by the tangent plane with the plane XY; hence, we have, denoting this angle by 3, cos - 1 d+ + (da)2 PARTIAL DIFFERENTIALS OF A SURFACE AND VOLUME. 149. Let BMM' be any curve in space, and B'PP' its projection on the co-ordinate plane XY. Let the plane of the curve MM' make an angle j3 with the plane XY, and let its intersection with that plane be taken for the axis of X. Then, if the ordinate of /y - the curve be denoted by y, the / / ordinate of its projection by y', A..' the area of the curve by s, and that of its projection by s', we have, Art. (92), ds = ydx; ds' = y'dx. The right-angled triangle MPQ gives y' = y cos /; hence, ds' = cos3ydx = cos/3ds, and the sum of all the values of ds' is equal- to the sum of all the values of ds multiplied by cos /3. But the sum of all the values of ds' is the area s', Art. (88), and the sum of all the values of ds is the area s; hence, s' = cos/s; 200 DIFFEiRENTIAL CALCULUS. that is, the projection of any plane area is equal to the area multi. plied by the cosine of the angle included between its plane and the plane of projection. 150. Now, let u denote the area of any surface, as ZeMd, and M any point of the surface, whose co-ordinates are x, y, and z. Since the equation of the surface gives z in terms of x and y, the area u is man-'Z ifestly a function of x and y. Let x be increased by bb'- dx, y remaining the same, the increment of the surface will be -/ [i/If! t'~ k \iMdfN, which will be the partial differential of u taken with respect to x; that is d;i~* du;'j' ~MdfN - dx........ If now in this, y be increased by PQ' = dy, and x remain the same, the increment MNM'N' will be the partial differential of (1) taken with respect to y; that is, d U MNM'N' -ddy. dx dy The same result may be obtained by first increasing y and then x. The infinitely small area MNM'N' may be regarded as a plane area in the tangent plane at M, and will, by the preceding article, be equal to the area of its projection PQP'Q' = dxdy, divided by cos fl. f3 being the angle made by the tangent plane with XY; hence dy C. d 1 - -....A A, ( DIFFERENTIAL CALCULUS..201 d p -~~/, ldz\2 /dz\2 or d2u = dxdy y1 + ( d\ ) + \d/)......(2), in which it should be remembered that d'u is the partial differential of the second order, obtained by differentiating first with respect to one variable, and then with respect to the other. 151. Let v represent any volume limited by a surface, and the co-ordinate planes as AbPc-MZ. It will be a function of x and y. If x and y be increased in succession by dx and dy, as in the preceding article, we obtain first the increment volume dv bb'QP-Nd = - dx, and for the increment of this, the volume d2v PQP'Q'-M'M = d dx dy. dx dy But this infinitely small volume does not differ from the parallelopipedon whose base is PQP'Q' = dxdy, and altitude MP='z; hence d'v dx dy = z dxdy, or d2v = zdx d... (1); dx dy in which d2v is a partial differential. 152. One surface is osculatory to another, when it has with it a more intimate contact than any other surface of the same kind; and the conditions which must exist in order that a surface, given in kind only, shall be osculatory to a given surface at a given point, can be determined by a method similar to that pursued in 202 DIFFERENTIAL CALCULUS. article (95). But from the nature of the case, these conditions are more numerous and complicated, and their determination more difficult; so much so as to render osculatory surfaces of little use in the measure of curvature; hence another method has been devised which will now be explained. Let M be any point of a surface, at which it is proposed to examine the curvature. Let this point be taken as the origin of co-ordinates, and let the normal at this point be the axis of Z, the axes o of X and Y having ally position in the tangent plane XMY. The equation of the surface, Art. (144), will be tv~~ - = f(x,y).....(1). /y~' —""""""... ^Through the normal let any plane ZMIX', making an angle (p with the plane ZX, be passed; it will cut from the surface a curve MRO. For any point of this curve, as O, denoting the abscissa MX' by x', we shall have X ='x cos, y = x' sin g.......(2), and these values, substituted in equation (1), will evidently give the equation of the curve referred to the two axes MZ and MX'. Now, by varying the angle A, all the normal sections at the point.M may be obtained, and by examining the curvatures of these different sections at the given point, an accurate idea of the curvature of the surface may be formed. Differentiating equations (2), we have dx = dx' cos q, dy = dx' sin qp.....(3). -The general expression for the radius of curvature of one of the normal sections, Art. (106), is DIFFERENTIAL CALCULUS. 203 R -(dx'2 + dz2)) IR = (dx d2z (4). Differentiating equation (3), Art. (144), we have, Art. (53), d~z 2d2z d2z dxz d + 2d dydxdy + - dy2...... (5). Substituting the above values of dx and dy in equation (3), Art. (144), and in (5), we have dz = (p coso p + p' sin q ) dx'............ (6), d2z = (q cos29p + 2 q' cos p sin p + q" singp) dx'2..(6'); p and p' representing the partial differential coefficients of the first, and q, q', and q" those of the second order of the function z. If these values of dz and d2z be substituted in expression (4), we shall have the expression for the radius of curvature of any one of the normal sections. But as we only desire this for the point M, we may first substitute the co-ordinates of this point, which are x = O, y" = 0, z"- 0; and since the normal at this point coincides with the axis of Z, we must also have, Art. (148), dz" dz"I - - 0, -, or p = O, p'= O. Substituting these values in equations (6) and (6'), and the results in equation (4), we obtain 1 qcos2p + 2 q' cos qp sinQ + q" sin'2p in which q, q', and q" are what the partial differential coefficients 204 DIFFERENTIAL CALCULUS. of the second order of the function z become, when 0 is substituted for x, y, and z. 1 2 Dividing by cos'ps, and recollecting that I- = 1 + tan this value may be put under the form R ~ ~1 + tan2q) q + 2q'tang + q"tan'q. () We have taken the positive value of R, Art. (106), since, as the surface is represented in the figure, the sections are above the axis of X', and convex towards it; d, — must therefore be positive, Art. (86), and the value of R positive, as it should be when laid off from M above the plane XY. If the section at the point M lies below the plane XY, it must still be convex towards this tand2Z gent plane; daz will be negative, and R negative, and must therefore be laid off from M below XY. By assigning all values to qp from 0 to 3600 in equation (8), we shall obtain a value of R for each normal section. Among these values there must be one which is greater, and another which is less than all the others. The values of qp which will give these principal values of R, will be obtained as in Art. (69). Differentiating equation (8), we have dR 2(q'tan2q - + (q - q")tanp - q') d tan p (q + 2 q' tang + q- " tanc q ) If the denominator be placed equal to 0, we shall obtain values of the tan p which, when real, will reduce the value of R to infinity. The curvature of the corresponding section will then be zero, and the section itself a right line, or the point M a singular point, Art. (118), cases which do not occur in all surfaces. Let us then place the numerator equal to 0; we thus have DIFFERENTIAL CALCULUS. 205 tan qp q q tanq - 1 = 0.......(9). This being either of the first or second form of equations of the second degree, the roots will always be real, and their product'equal to - 1, that is, denoting them by tan q' and tan p", tanpg'tanq" + 1 = 0; hence, the normal planes in which the greatest and least radii of curvature are found, must be perpendicular to each other. The sections by these planes are called principal sections, and their exact position will be determined by solving equation (9). The values of tan q' and tan q" being determined, and the traces of the normal planes conZ structed as in the figure; let us take MX" as a new axis of X, and MY" as a new axis of Y, and suppose the surface to be M referred to them, with MZ as an axis of Z. Then we must have for these new axes,/ x" tan g' = 01 tan qp" = oo, tan g' + tan q" = o, which requires in equation (9), that q' = O. Substituting this value of q' in equation (7), we have cR 1 os......... q.(10). Substituting in this the values of p, corresponding to the maximum and minimum radii as above determined, viz., q = 0 and q = 900, and denoting the values of the principal radii thus determined by R' and R", we have '206 DIFFERENTIAL CALCULUS. 1 1 R' = - 1,,, and finally, fiom equation (10), q cos2p + q" sin' q= - cos29 +4' sin qll, -whlich expresses the reciprocal of-the radius of curvature of any normtal section, in terms of the principal radii and the angle p. If R' and R" are both positive, all values of R will be positive, and the greatest of the two will be a maximum, and the least a in nimrn m; and all the normal sections at the point M will lie above the plane XY. If R' and R" are both negative, the sections will lie below XY. I' one is positive and the other negative, a part of the values of R will be positive and a part negative, and a part of the sections will be above and a part below the plane XY; and this plane will cut the surface at the point M, giving a poinlt analogous to the point of inflexion, Art. (92). If R' R- R", all the values of R become equal to R' or R", and the curvature of all the sections will be the same; as at any point of a sphere, or at the vertex of a surface of revolution. 153. If R"' be a value of R, in any section perpendicular to the one which makes the angle p, we may obtain its reciprocal by 1 substituting in the last expression for 9t, 90~ + q for q; and since sin (90~ - q+ ) = cos, cos(90~ + 0) = - sin, this expression will become 1 1 1 " R7in 1' Cos q DIFFERENTIAL CALCULUS. 207 Adding this to the expression for R, member to member, we have 1 1 1 1 that is, Art. (105), the sum of the curvatures of any two normal sections at the same point, which are perpendicular to each other, is constant, and equal to the sumnz of the curvatures of the piincipal sections. PART II. INTEGRAL CALCULUS. FIRST PRINCIPLES. 154. WE have seen that the sum of all the values of a differential is the function from which it is derived, Art. (88). By whatever process this function may be obtained from its differential, it amounts to the summation of the infinite number of its elements, or infinitely small values of the differential. This process is called integration, and its symbol is f, which always indicates an operation the reverse of differentiation; thus fdu = u. The object of the Integral Calculus is to explain how to pass from differentials to the functions from which they may be derived; or in any particular case, to find an expression which, if it be differentiated, will produce the given differential. This expression is called the integral of the differential. 155. We have found, article (17), dAu = Adu; therefore fAdu = fdAu = Au = AJfdu. 210 INTEGRAL CALCULUS. From which we see that a constant factor may be placed without the sign of integration, without affecting the value of the integral; thus, b(a - x2)dZ = b(a - x2)dx, 1x dx4. Also, in article (20), we have d(u + v + &c.) - du + d dv: &c.; hence f(du + dv: &c.) =d(u v: &c.) =& + v &., - fdu + fdv ~ &c.; that is, the integral of the sum or defference of any number of differentials, is equal to the sum or difference of their respective integrals. Also, in article (16), we have d(u +C) = du, no matter what the value of the constant C may be; hence an infinite number of expressions differing from each other in a constant term, when differentiated will produce the same differential. For this reason, to complete the integral immediately found, we add a constant; thus, fdu - u + C. INTEGRAL CALCULUS. 211 INTEGRATION OF MONOIAL DIFFERENTIALS. 156. By article (24), we have cdxzm+ = c(m + 1)xmdx; and from this, cx, dx. cd" d" —- - - -m+l 1mq 1' hence m+E -1 CXm+I fcxMdx Jf cd 1 - + Therefore, to obtain the integral of a monomial differential: Multiply the variable, with its primitive exponent increased by unity, by the constant factor, if there is one, and divide the result by the new exponent. Excamles. 1. If du = xdx, fdu — fxdx= -+- C. x3dx X 4 2. If du fdu - fxddx +0C. ~C c 4c 5 5 2 tbx 3 bxW 3. If du bx 3dx, u = 5 =, +C. _ _ t-m 4. If du u =- -. Ce'e(n-nm) 212 INTEGRAL CALCULUS. adx 3x- dx c2x4cdx adx 3./B Ydx_ 2X4CJ...XArt.(155). u. If +% J-..A() The application of the above rule does not give the proper integral when m = - 1, as in this case we have - 1+1 1 Jx-ldx — = i +1 = 0 o -; whereas fx'dx- = d. = x + C..Art. (3').'dx This result was to be expected, since J or lx cannot be expressed in algebraic terms, Art. (5). a dx du= =o x C or -= log x + C, the logarithm being taken in the system whose modulus is 157. If we have an expression of the form du = (a + bx + ex2 + &c.)mxndx, in which m is a positive whole number, the integral may be found INTEGRAL CALCULUS. 213 by raising the quantity within the parenthesis to the mth power, multiplying each term by x"dx, and then integrating it as in the preceding article. Examples. 1. Let du = (a + X2)2x3dz, or du = (a= + 2ax" + x4)X3dx; then a2 x4 2axX6 s u = f(a'x'dx Jr 2ax5dx + x'dx) = a + 2a - C. 4 6 8 2. Let du = (b -2 x2) 2xdx. 3. Let du = (b - cxP) x- dx. INTEGRATION: OF BINOMIAL DIFFERENTIALS OF PARTICULAR FORMS. 158. Many expressions, by the introduction of an auxiliary variable, may be transformed into monomials, and then integrated as in the preceding article. I. Let du = (a + bx")"c'x"-Idx. Place a + bx" = z, then nbx"- dx = dz, xn-' dx - 6n 214 INT:EGRAL CALCULUS. Substituting in the given expression, and integrating, we have fdu =P dz z' - d 1fb = -fzmdz = bnm +; and replacing the value of z, we have, finally, c'(a + bx)m+ + U (m'+t 1)nb that is, to integrate a binomial differential when the exponent of the variable without the parenthesis is one less than that within: Multiply the binomial, with its primitive exponent increased: by unity, by the constant factor, if there is one; then divide this result by the product of the new'exponent, the coefficient, and the exponent of the variable uivthin the paxenthesis. Examptes. 1. If dr = (a + bx') excx, u, b. 2 + + C. 2. + 2. If du = (2 — 3x5)- 3x4dx, 2 - W) )2 +C. 3. If du = (a - bx)-2x-dx, u = b + C. m p m 4. Let du = a(b - cz-f)-z —'dz. Iax"- ld II. Let du =- -x" b: x~ INTEGRAL CALCULUS. 215 Place b 4. x = z; then dz nx-"' dx = dz, x"-'dx -: and. j;adz = al a+ C. nz n In the same way, wermasy find the integrals of the following expressions: I. II = + L2)dx a + bx +- cx' Place a + bx + cx2 = z, then (b + 2cx)dx = dz, dz u =.mf mi = n mlz l(a + bx + C2) + C. 2.If dM _ 2 dy 21(a + C a y 3. If du (2 + 2 x-)- dx U 1(2x + 2) + C. 2x+ x2' u=(2x+x)+C. 4. Let du- = 2 z. 1 - z Since, in general, radu al we see that in all cases wlieie th-e numerator of an expression is the product of a constant and the differential of the denominator, its integral will be the product of the constant and the Naperian logarithm of t.e denominator. 216 INTEGRAL CALCULUS. 159. Every expression of the form du = Axz(a + bx)"dx, can be integrated, when either m or n is a positive whole number. If n be positive and entire, we may integrate as in article (157). If m be positive and entire, n being either fractional or negative, place Z - -a a +- b = z, then x = - b' dz z__-_a__ d dx d du = A = Z z) n b' b Tb A -a b Jstb )dz, which may be integrated as in article (157). The value of z being then replaced, the integral will be expressed in terms of x. Ezaamples. 1. Let du = bx2(a - x)2dx. Place a- x = z, then x = a - z, dx =- dz, 1 2 3 4 2 7 u = f- b(a- z)lz2dz = ba- 2t- + baz - bz and finally, by replacing the value of 4, 2 4 4 2 U 6 bal (a -x)2 ca(a - x) 2 - b(a - x)2 + C 3 7 INTEGRAL CALCULUS. 217 2. If du = 2xd (1 - 3x) it may be placed under the form' 2 r du = 2x(1 — 3x)-adx; whence u - (1 - z) z- dz and finally, 4, 4 u = - 9(1 -3x) + 2j(1-3x) + C. x'dx ydy 3. Let du - 4. Let du = c;1 -' (3-2y)' If du - (Axm + BxP + Cxq + &+.)d (ax + b)' we may place it under the form Axmdx BxPdx (ax + b) + (ax +- b)n + and may then integrate each fraction as above, if m, p, q, &c., are entire and positive. USE OF THE ARBITRARY CONSTANT. INTEGRATION:BETWEEN LIMITS. 160. To complete each integral as determined by the preceding rules, we have added a constant quantity C. If, in the particular case under consideration, we happen to know what the integral must be for a particular value of the variable, this constant can be determined. Thus, if 218 INTEGRAL CALCULUS. fXdx = X' + C........(1), X' representing the function of x, obtained at once by the application of the rules for integration, and we know the integral must reduce to N when x - a, we have N - X'.=. + C, C = N -Xt, In general, however, this constant is entirely arbitrary,, since; whatever value be assigned to it, it will disappear by differentiation, Art. (16). This arbitrary nature of the constant enables us to cause the integral to fulfil any reasonable condition. Thus if, in equation (1), it be required that the integral reduce to the particular expression M, when x = a; we may determine the value which must be assigned to C, by writing M for fXdx, and substituting a for x in the function X'. Calling the result of this substitution A, the equation reduces to M = A + C; whence C = M - A, and fXd = X'+ M- A...... (2), which will fulfil the Required condition. If M = O, C =- A and f Xdx =X'- A. The integral.fXdx = X' + C, before any particular value has been assigned to C, is called a complete, or indefinite integral,. and expresses the indefinite sum of all the values of the differential. After a particular value has been assigned to C, as in equation (2), it is called a particular integral, and expresses the sum of all values of the differential, commencing at the or igin of the integral, that is, at that particular value of the integral which is 0. It, in this particular integral, a particular value be given to x, the result is called a Ldefinite inteyral. We should thus have, when x = b, INTEGRAL CALCULUS. 219 fXdx = B + M - A......(3), B representing X',,; and this expresses the definite sum of all values of the differential, from the value at the origin to that which corresponds to x = b. That value of the variable which causes the integral to reduce to 0, and belongs to'the origin, is always found by placing the particular integral equal to 0, and solving the resulting equation. If in(1) wemake xz a, and then x =b, we have f(Xdx).- = A + C, f(Xdx)_a = B + C, whence, by subtraction,.(Xdx)g_. - f(Xdx)._. = B - A. This is the integral taken between the limits a and b, and is usually written.b Xdx - B -A, the limit corresponding to the subtractive integral being placed below. This expresses the definite sum of all values of the differential, between those which correspond to x = a and x = b. This integral between limits may always be obtained from either the complete or particular integral, by substituting, in succession, those values of the variable which indicate the limits, and subtracting the results. If a, b, c...... k, 1, be several increasing values of x, and we have Xdx- A', Xdx B',... Xdx'; a b 1h then evidently JXdx = A' + B' + C'.......... +'. =A 220 INTEGRAL CALCULUS. Example. f6x'dx = 2x'3 + C, is a complete or indefinite integral. If it be required that this reduce to 4, when x = 1, we have 4 = 2 + C, C 2, and f6xdx = 2x3 -- 2, the particular integral. For the integral between the limits x = -0 and x = 3, f(6xB2dx),,_ = 2, f(6x'dx)_,3 = 56; hence, J6xdx = 54. The value of x corresponding to the origin of the particular integral, is obtained by placing 2x3 + 2 = O; whence X3 -- 1, - 1. INTEGRATION OF THE'DIFFERENTIALS OF THE SIMPLE CIRCULAR FUNCTIONS, AND OF CIRCULA AR RCS. 161. By a reference to article (43), we see that 1. fcosxdx = sin x. 2. f — sinxdx = cosx. 3. fsinxdx = ver-sinx. 4. Jd = tan X. dx d =cot._ 6. ftanx.secxdx = secx, &c. 5.J-sin ~ INTEGRAL CALCULUS. 221 and from these we readily derive the integrals of the expressions, 7. Jf2x cos x2dx = fcos x%. 2xdx = sin x2. /, S- 1 I 1 dr 1 8. -- -sin -dx - sin-. - -- cos-. tX X1 X' X /x d ( a -- xa )._tan(a — ( x). / I cos 2(a- ) IcoS2(a-x2)= tan(a _2). These integrals will be completed by adding to each a constant. Hereafter, as in these examples, this constant will be understood, and written only when it is necessary for the discussion of the integral. 162. By a reference to article (44), we see that 1. d _ = sin -u. 2. = cos-'u V'1-u2 dV^/1 -u U V I -'- U" " d, a du 3. -_ = ver-sin-'u. 4. / s = tan-'u. And from these we derive the integrals of the similar expressions, du u Zadu - a Za - sin-U a a f 8L 3f / - - - 1 sin-'-. V2 -x V22 - x V2 :222 INTEGRAL CALCULUS. 2dx _ 2 dx 22 x 7. = in - 1/9 - 3x' 3 1 /3 -/3 x /3 du - du a da J. /~l - _ = f _ —-. cos-I-. /a - U2 uo 2 a Ct u 2du du 9. Co- - COS-. du 11. /.f.-ver-sin- - 2acu _ ~Ua a 1a 3 dx 3 dx _ 3 Ver-sin x3 -v/4 x -2x2 V/2S v/2x x x 2 du ddu 12 -t2 + 2 J - - W2 = - tan- I 1 2 d a quu a U a a' a a2 2dx 2 dx 2 z 13. 12 =- x 3 tan- J 2+ v6 3 3 INTEGRAL CALCULUS. 223 INTEGRATION OF RATIONAL FRACTIONS. 163. Every rational fraction which is the differential of a -function of x, will appear as a particular case of the general form (Axm + Bxm-1 + Cxm"-2 + &c.) dx A'x"n + B'x"-1 + C'x"-2 + &c. in which m and n are whole numbers, and positive. If,m be greater than n, the numerator may be divided by the denominator, and the division continued until the greatest exponent of x in the remainder is, at least, one less than in the denominator; the quotient will then consist of an entire and rational part, plus the remainder divided by the denominator, and may bb written (Axm + Bx'l + &c.) dZx (A"x"-' + B"x'-2 + &c.) dx -Xdx A'x" t+ Bx'I — + &c. A'x" + B'x"' + &c.' and the integral of the primitive fraction will be the sum of the integrals of the two parts. It will be necessary, then, t0 explain only the manner of integrating' the second part, or those rational fractions in which the greatest exponent of the variable in the numerator is at least one less than in the denominator. First, suppose the denominator to be divided into its simple factors of the first degree, and let them be represented by x - a, x - b, x - c, &c. 224 INTEGRAL CALCULUS. There will be four different cases, each of which will require a separate discussion. 1. When the factors are real and unequal; II. When they are real and equal; III. When they are imaginary, and no two alike; IV. When they are imaginary and alike, two and two. 164. I. As an example of the first case, let us take the fraction (ax + c)dx x _ b2 The two factors of the denominator are- x + b and x - b; then (ax + c)dx (ax + c) d X b2 - (x + b)(x -b) Place ax + c A A' x _ b2 = xI b -+ - x - b....... x2 b62x+ b x-b. (1'''''(1 ), A and A' being constants to be determined. For the purpose of determining them, clear the equation of its denominators; then ax +- c = Ax - Ab + A'x + A'b. Since this is true for all values of x, by the principle of indeterminate coefficients, we may place the coefficients of the like powers of x, in the two members, equal to each other, and have a = A + A', c = A'b - Ab, ab - c ab + c A = 2 A = 2b 2b INTEGRAL CALCULUS. 225 Substituting these values in (1), multiplying by dx, and prefixing the sign f, we have (ax r c) dx ab - c dx ab + c dx xJ -2 - b~ - 2b x b 2b x -- b ab - 2c xb 26 x- ab - Cl( + b) + b +- (- b). 26 2 b If there be n factors in the denominator of the given expression, there should be n corresponding fractions of the above form; and these, when reduced to a common denominator with the first member, will give an identical equation, containing n different powers of the variable (including the zero power), from which, as above, n equations may be formed and the values of the n numerators be determined. The method pursued above indicates the following rule for all similar expressions: Place the primitive fraction (omitting the differential of the variable) equal to the sum of as many partial fractions as there are factors of the first degree in its denominator; the numerators of these fractions being constants to be determined, and the denominators the several factors of the original denominator; clear the resulting equation of denominators, equate the coefficients of the like powers of the variable in the two members, and thence determine the constants; then multiply each partial fraction by the differential qf the variable, and take the sum of their integrals as in case II., article (158). 2. Integrate the expression (32 _ 1)d. X3 -- X The-factors of the denominator are, x + 1, x 1, and x; hen 15 226 INTEGRAL CALCULUS. 3xa _1.A A! A" $a3- - x x + 1 x - 1 x Clearing of denominators, 3X2'- 1 = Ax2 -_ x + A'x2 + A'x + A"x' - A"; whence 3 = A + A' + A", O =- A + A', 1 = A"' A = 1 -A' A", and j(3x- _) dx =j d 1 X- +dx ff - S X + [ $ - 1 x - I(x + 1) + l(x - 1) + x = I(X3 -), as may be seen at once, since the numerator of the given differential is the exact differential of the denominator. 3. Integrate the expression (1 - y) dy y2-_ 2y - 2' Placing the denominator equal to 0, we have y - -2y- 2 = 0; whence y = 1 V3,. and the corresponding factors are y- (1 + V-), y- (1-V/8), or y —m and y n. Finally, INTEGRAL CALCULUS. 227 2(y -m) - - (y-n). y2 _P- 2Y -- 2 ~ n - n n - 4. Integrate (2x + 3)dx X -. -- 2x Integrate (x3 — 1)dx X2 4 165. II. In the second case it may be remarked, that if all the factors of the denominator are equal, the fraction will take the form Ax"-' + Bxn-2 + &c. (x -a) dx, which may be integrated as in article (159). We need, then, only consider the case where a portion of the factors are equal. The rule of the preceding article is not appli-.cable here, as will be seen by taking the expression adx (x -b) (- c)' in which two of the factors are equal to x- b. By an application of the rule referred to, we should have a A A A' A" (x-b)'(x-c) x - b x b x - c A + A' A" B A" - + - - + x - b x - c x s- b x c. since A + A' must be regarded as a single constant. 228 INTEGRAL CALCULUS. If this equation be cleared of denominators, and the coefficients of the like powers of x in the two members placed equal to each other, we shall evidently form three independent equations, with only two unknown quantities, B and A". We obviate this difficulty by writing, for the equal factors, the B B' two fractions and thus have ( - 6)2( - ) - ( - b)2 - b - c which, being cleared of denominators, gives a = B(x- c) + B'(- b) (x - c) + A(x - b); whence B' + A - O, B -B'c -B'b - 2Ab - 07 B'b - Bc + Ab- = a, three equations with three unknown quantities, which can then be determined. And in general, if there be n equal factors, we should write n partial-fractions in the form B B' l('- )' 4 + 1* -+ (a-b)+ (a-b)-...... - b' the numerators of which are constants, and the denominators the different powers of the equal factor, from the nth down to the first power. After B, B', &c, are determined, each partial fraction, being first multiplied by the differential of the variable, will be integrated as in article (158). INTEGRAL CALCULUS. 229 Examprnles. (2 + x) dx 1. Integrate (2 - )2(x - Place 2 + x B B' A (x- 1)2(- 2) (x -1)2 x - 1 x - 2 Clearing of denominators, and equating the coefficients of the like powers of x, we have o = B'+ A, 1 =B —3B'- 2A, 2=-2B 2B'+ A, B =- 3, B'=- 4, A = 4; and finally: (2+x)d _ 3 (x- )(x-2) - - - 41(x -1) + 41(x -2). (x - 1),(x - 2) x - 1 ~,xdx 2. Integrate - dx xs -- x2 -- - 1 If there are different sets of equal factors, partial fractions must be written for each set; thus,' 2 A A' B B' (x- 1)'(x X- 1)' - (x- 1) - - - -) q -' 166. III. We know, from the general theory of equations, that imaginary roots are found only in pairs, and that for each pair we must have a factor of the second degree, which, placed equal to 0, will give the imaginary roots. Each pair of roots will always appear as a particular case of the general form 230 INTEGRAL CALCULI S. x.= a V 2 b2........(l), and the corresponding factor of the second degree will be X2 _ 2ax + a' + ab = [x —(a J+ / — bl)] [x — (a — ~/By a comparison of the imaginary factors, in any given with these general expressions, we determine the correspond' values of a and b. Thus, if the factor of the second degree be XI - 2x + 5, we place it equal to 0, and find the two roots x = 1 ~-i- - 4; whence, by comparison, a = 1, b2 - 4, b - 2. Now, in the third case, for each pair of imaginary factors, i partial fraction be written, of the form Mx + N Mx + N xz - 2.ax + a' + 2 q - (x - a)2 + bi' By clearing of denominators, &c., as in the preceding articles,.M and N may be determined. We shall have then to integrate the expression (Mx + N) dx (x - a)' + b' For this purpose, make x - a - z, then x _ z + a. dx -= dz. Substituting these, the original expression becomes (Mz + Ma + N)d za + b' INTEGRAL CALCULUS. 231 or, by making Ma + N = P, and dividing the expression into two parts, Mzdz Pdz z+2 b + z2 + b 2 The first part may be integrated as in case II., Art. (158). Thus, f Mzdz M / + =~z(~ +t bl) =I,z + +b = MI ~(x - a)' +.b'. The integral of the second part is dz P'Z PJ + b= btan-'..Art.(162); or, by substituting the values of P and z, j Pdz N + Matan (x a); z 2 +Pb b; and finally, f (Mx + N)dx M ( a) + N + Ma (x - a) +- -tan b.....(2). Take the particular example (x - l)dx x3 + x2 + 2x The factors of the denominator are x and xt + x + 2, the latter being the product of the two factors corresponding to the imaginary roots 232 INTEGRAL CALCULUS. 1'7 2 4 which, compared with (1), give a = —' - b, b = / Place x- 1 A Mx + N s3 -- xa - 2x x X - x -+ 2 Clearing of denominators, &c., we find 1 1 3 A M N 2' 2 Substituting these values of M, N, a, and b, in formula (2), fAdx I 1 dx 1 observing that J- = - - and rex 2 x 2 ducing, we have (x - )dx _ = - - 1 lx.+ /x' + x + 2 + — tan-' 167. IV. In the fourth case, where there are several imaginary factors, alike two and two, those of each pair multiplied together will give the same factor of the second degree; and if there be p such pairs, the denominator will contain a factor of the form (xi - 2ax + a' + b')). For this, we write p partial fractions; thus, Mx - +N M'x + N' M(P-')'x + N(P-I) [(x- a)2 b2]P [(x-a)2+ b2]P- (x —a)2 + b INTEGRAL CALCULUS. 233 Clearing of denominators, &c., the values of M, N, M', N', &c., may be determined as before; and since the several partial fractions, after multiplying by dx, are all of the same form, we have only to explain the mode of integrating any one of them, except the last, which is to be integrated as in the preceding article. Take the first, (Mx + N).d [(x - a)2 + b']p' and make x - a = z; the fraction then becomes (Mz + Ma + N)dz (z2 + b2)P or, placing Ma + N = P. Mzdz Pdz (z2 + b2) + (ZY + b2) The first part is integrated as in case I., Art. (158). Thus, Mzdz _ M(z2 + b2)-P+' M (z2 + b2)? — (-p + 1)2 2(1 —.p)(z2 +b2)pBy means of a formula hereafter to be determined [Formula D, Art. (181)], we shall find Pdz p(z) + C' d C' z ( b2 - + () ) + C' + b. 2 =?(z) +-b tan b; then J(Mz + P)dz M C' z ('2 +i b'2) 2(1 -p) (z- b2)p-+ q(z) +- tan after which, substituting the value of z, we shall obtain the complete integral of the primitive expression. 234 INTEGRAL CALCULUS. 168. By a review of the preceding discussion, it will be seen that all differentials which are rational fractions can be integrated, provided the factors of the denominator can be discovered; and that the integrals will depend upon one or more of the four forms, Ix + a' f d x2 + aINTEGRATION BY PARTS. 169. In article (21), we have found duv = udv + vdu; whence uv = Judv +.fvdu, and fudv = uv - fvdu......(1); from which we see that the integral of udv can be obtained, whenever we are able to integrate vdu. This method of integrating vidv is called integration by parts. To apply it to a particular example, divide the given differential into two factors, one of which shall contain the differential of the variable, and be capable of immediate integration; substitute this factor for dv, its integral for v, and the other factor for u, in formula (1). Examples. 1. Integrate the expression x3dx t/a - xi. This may be divided into the two factors, x2 and xdx Va - x2. INTEGRAL CALCULUS. 235 Place x = u and xdx a -x = dv; then d =2zdx, 2 v = fr.cx =-xa (- -( ) Substituting these in formula (1), we have 3 3 udv -- 3- 2 xdx; and finally, f dX Va - _ 2 (a - x2) 3 15 ( I X' )X 2dx 2. Integrate ( Place (1 ) = u, and = dv; then __ X'1 i J (1 ~~-Ld -- s in'x, xadx 3. Integrate 3' (a2 - -%) xa' dx 4. Integrate Va2 2/. __X. 236 INTEGRAL CALCULUS. INTEGRATION OF CERTAIN IRRATIONAL DIFFERENTIALS. 170. In the preceding articles, rules have been given by which every rational differential may be integrated, except the case referred to in article (168). It may then be taken for granted, that, in general, every irrational differential which can be made rational in terms of a new variable, can also be integrated. Let axk dx rn P bxn + cxq be a differential, all the irrational parts of which are monomials. Make x - kZn; then xk z= Zng in_ P x" = z"', x _ = zk'p dx = knqzkn"- dz. These values substituted in the given expression, evidently make it rational in terms of z and dz. It may then be integrated, after which the value of z in terms of x must be substituted. We may then enunciate the following rule for the integration of expressions of this kind: For the variable substitute another, with an exponent equal to the least common multiple of the indices of the radicals; then integrate by the known rules, and substitute in the result the value of the new variable in terms of the primitive. Examples. i 2 2x2 -3 xs 1. Let du = dx......(1). 5x~ INTEGRAL CALCULUTS. 237 The least common multiple of the denominators or indices being, 6, we place = z6, then dx 6 z5d, z = xA. Substituting in (1), we have 2z'3 - 3z4 12 18 du - - 6zdz = --- zdz 1 — z8dz 5z 5 5 and integrating, 12 18 3 4 2 3 - z z9 -- X -x2. 40 45i 10 5 3x2dx axdx 2. Let du 1 3. Let d = - 2x2 - x3 b -c V/X P 171. If the irrational parts are all of the form (a + bx)q, the expression may be made rational in terms of z, by placing a +- bx = zx r being the least common multiple of the indices of the radicals. WVe shall thus have = - a dx rzr-ldz b' b which, substituted in the primitive expression, with the value of a + bx, will evidently give a rational result. Take the examples: dx 1. du - (lx) + + (1 + X) Place 1 + x = Z2; then'dx 2 zdz, z = (1 +x). 238 INTEGRAL CALCULUS. These values substituted in (1), give d 2 zdz 2 dz du Z3 +- Z 1 + Z'2 whence u = 2fj daz 2 tan-'z = 2 tan-l (1 +-x) 2. Let du d x /i + X 3. Let du = (1 - x)W + (1 - x)4 172. Differentials of the form Xdx a Xda' + b' X being a rational function of x, may be made rational by placing a + bx +- = z', deducing the values of x and dx, and substia' + b'xz' tuting them. For example, lbt du - xdx()....... 1 q- a z83- 1 6zldz Place 1, then x — 3 dx - - z3t + 11 (zS + ) These values in (1), give =(d - 1)6z4dz which is rational. INTEGRAL CALCULUS. 239 173. Every radical of the form V'a + bx ~i cxz. can be written thus V/= + b_ i s2_ a+ +x +, C C a b after making - - a, and - 3. c c To render rational a differential, the only irrational part of which is a radical of the above form, it will then only be necessary to find rational values for x, dx, and V'a -+ OPx:i x2, in terms-of a new variable and its differential. I. Take the case in which the sign of xa is +, and place a + x+ = z - x......(1). Squaring both members, we have a + x- =z - 2zx; whence --. (2). By differentiating this value of x, we obtain dx _ 2 (z' + fz + a)dz dx -~~~~(3), ( +: 2z)'.... and by substituting the value of x in the second member of (1), =2 + Z + O4 /a d- ~ —+ x ~...(4o. + + 2z 240 INTEGRAL CALCULUS. These values of x, dx, and V/c + 3- +3 x2,. substituted in the primitive differential, will evidently give a rational expression in terms of z and dz. After integrating this, the value of z, taken from (1), must be substituted. Examples. 1. Let du dx d _ v/a + bx c+x -/c V-a + fx + x" Substituting for dx and V/a + O3x + x2 their values as found above, and reducing, we have dx 2 dz v';Va + px + x2 -V/ + 2z)' whence U-= ~ I(O + 20) = 1[-[+ 2(oa+pO+ x2+ x)]..(5). 2 Let du = /h + c'x2 c j/ h 2 By comparison with the similar expression in the preceding example, we see that h c =5, o= B. 2=. Substituting these values in (5), we deduce INTEGRAL CALCULUS. 241 U = / t zdx = 12( + XI +) 1 (V/h J+ C2X2 + Cx) 112 +(h 2 and, finally, after uniting the constant -I with the arbitrary C C constant, /. dD = f & _ (C+;c, + c,) d d/12x + x3. Let du = - d V 2. Comparing this with formulas (2), (3), and (4), we see that 52 0 = a, 2 X Al= p, x O =a, 2 -2 + 2z' 2dz 2(z9 + 2z) dz / +2za (2 + 2z)' 2 + 2z whence du = (z + 2)'dz z_(Z + ) ) 4. Let du = dX V'- dx 5. Let du d x v/x2 -- ax 16 242 INTEGRAL CALCULUS. 174. II. If the sign of x' be minus, it will be necessary to pursue a different method, and deduce other formulas; for if we write Va +t- x - Z the second powers of x in the squares of the two members will have contrary signs, and not cancel each other, as in the first case; and therefore the deduced value of x in terms of z will not be rational. Denoting the roots of the equation x2 - 3x - a = 0, by.6 and 6', we have x- _i - a = (X - 6)(x - 6'); or, changing the signs, a + 3x - x = ( - 6) (6-x), Vat + 3x - x = /(x - ) (i'- x). Now, if we make A/($ - 6)(6'- x) = ( - 6)z......(l), square both members, and strike out the common factor x - 6, we have -3' + 6z2' - = (x- 6)z', x = x- - z 6 +....(2). Substituting this value in equation (1), we obtain ~Va + Ofx - x2. =- V2'x-6-)(6'-x) - -'By quo (2), find By differentiating equation (2), we find INTEGRAL CALCULUS. 243 2(6' - )zdz (1 + z2)2. These values of x, /Va + OtX- x', and dx, substituted in the primitive expression, will make it rational. ExanvplIes. 1. Let du -- dx /a -I+ /3 - Ox By substituting the values of dx and 1/a -+ x - x, we obtain 2 dz dz +1 + z2, u --'2 dz _ 2 tan'-z; + z2 - and since, from equation (1), /a- x we have, finally, ~ =u'<~ + d = - 2 tan7' —46 + C. -1/a O3x - 2 - If in this we make f = O, c = 1, the expression reduces to Ud - /= - 2 tan-'I/ -; c by pcin 1+ X' since, by placing x2 - 1 - O, we find 24' INTEGRAL CALCULUS. z = I 1, or = - 1,'- 1. If we now introduce the condition that the integral shall be 0, when x = 0, we have O = C - 2 tan-1 1 C C -, 2' 2 and I x- i2 tan-l. o/J,1_ xa 2 1-f. x' The direct integral of the first member is sin-' x, Art. (162); hence, in - 7r 2tan- - x 2 -1 - x 2. Let du= d 2ax-' a:2 Placing 2ax$- x_ = O, we deduce x 0, and x = 2a; hence, 6 = 0, and 6' = 2a. Substituting these in the formulas, &c., we have dX ~2ax - x' 2z'dZ 2 - a 2 1 + z"' a simple rational fraction.'x& d&v'2 - x, 3. du = d 4. du = 7V4x - xa X INTEGRAL CALCULUS. 245 INTEGRATION OF BINOMIAL DIFFERENTIALS. 175. 1. If we have a differential of the form P x"-'dx (ax' + bxn), xt (r being supposed less than n) may be taken out of the parenthesis, and for the primitive expression we may write rp P p P xi-ldxx(a + bxn-) = m+ —'dx(a +- bx-f)v in which but one of the terms, in the parenthesis, contains the variable x, and the exponent of this variable will be positive. 2. If, after this, the exponent of x should be fractional, either within or without the parenthesis, or both, we can substitute for x another, variable, with an exponent equal to the least common multiple of the denominators of the given exponents, and thus get rid of the fractions; as in the example xTdx(a + bx-),, by making x z6, we obtain I I P P x3dx(a + bxo) = 6zldz(a + bz8)q, in which the exponents of z are whole numbers. Hence, every binomial differential can be placed under the form xm-ldx (a + b), in which m and n are whole numbers, and n positive. 246 INTEGRAL CALCULUS. 176. 1. The binomial differential being placed under the proposed form, if P is entire and positive, it may be integrated as in article (157); if P is entire and negative, we have P- z"-I'dx xm-'ldx(a + bx")' - (a + bx")Q which is a rational fraction. 2. If P is a fraction, either positive or negative, place a +- bx" - zq; then -_by?,() n= Id - a (a + bx") z....(1), b,x m a(mdx -I ( ydz..(2). The values (1) and (2), substituted in the primitive expression, give m-lxd(a + bx*)7 = zi-d (Z which is rational in terms of z and dz, when - is a whole n number. INTEGRAL CALCULUS. 247 Example. Let du x= dx(a - bx) 2, in which m rn- 1 = 3, n = 2, -_ 2, p =3, q = 2, b =-b, These values in equation (3), give x.dx(a - bx2) = z4dz(Z_ - a) in which 2 = a - bxe 3. If - is not a whole number, we may write n. _ a a x-ldx(a + bxn)9 - _"-dxZ[xf(, + b)] P = xm+.-'Idx(ax-n + b)f, and, in accordance with the preceding principle, this may be made rational if m + P _ - (m + s a whole number. -n n q To obtain the proper rational expression in terms of z, we need only make in equation (3), n+-., n- n, a b, b = a. q Thus, x'+'-'dx (ax-" 4b)- - q zP+- dz... (4) 248 INTE-GRkAL OALOULTUS. ESample. Let du = xdx (a + bx3), in which m p - 1 - 1, n = 3, = 1, q = 3, -+- = 1 These values in equation (4) give )3 ~ dz- 3 -_ -9 xd (a + bx') = in which Z3 - ax-3 + b. From what precedes, we see that every binomial differential of the proposed form can be integrated, if the exponent of the parenthesis is a whole number; if the exponent of the varliable without the parenthesis, plus unity, divided by the exponent of the variable within, is awhole number; or if this quotient, plus the exponent of the parenthesis, is a whole number. 177. Let us now write p for and then divide the expression x"-I dx(a + bx")P = "-"x"-'dx(a + bx")p, into the two factors Xm-n = u, and x"-'dx(a + bx")P = dv; whence (a + bx")p+' du = (m n) xi dx, v (p + 1) nb''Art.(158). INTEGRAL CALOULUS. 249 Substituting these values in the formula fudt = uv - Svdu.....Art. (169), and making (a + bx"l) - X, e have x'm-nXP+1 (m-+ )fn (p + ix)nbX (p + 1) nb. But since XP+l - XPX = XP(a + bx") = aXP + bxnXP, fSx"-"-Id.XP+l - a f xm — IdxXP + bfxm-ldxXP. Substituting this value in (1), and clearing of denominators, (p + 1) nbfSx"-dxXP = x"-XP+l - (m- n)[afxm-"-ldxXP +- bfxz-dxXL]; transposing, &c., we obtain Xd-nXp+l - a (m - n)fxm-"-IdxXP b (-pn + m) By a single application of this formula, we cause xam-i dxXP to depend upon xM"-n- dxXP, in which the exponent of the variable without the parenthesis is diminished by the exponent of the variable within. By an application of the same formula to fxm — ldxXP, it may be made to depend upon fx"'-'2-IdxXP; and finally, by repeated applications, fxm-'dxXP will depend upon the expression a (m - rn)fxm-"-'dxXP, 250 INTEGRAL CALCULUS. in which r represents the number of times mn will contain n. If mil is positive and an exact multiple of. n, then n - rn = O, the term containing the expression to be integrated disappears, and the integration is complete. If pn + n -= 0, the second member of the formula becomes infinite, and it fails to answer the purpose; but in this case p + _ - 0, which, substituted in equation (4) of article (176), gives an expression which may at once be integrated. 178. We may also write fx'dxXP =xm-dxzXP-'X = afx'-dxXP-' +. bfx,,-'dxXP-'. If nbw in formula A we change m into -n + n, and p into p- i, we have " = XP - amf x"- i dxXP,b (pn + mr) Substituting this value in the preceding equation, and reducing, we obtain Lxm-ldxXP = XmXP + p nafx"m-ldcXPlf x"- l dxXP' ---..~.B; by which the primitive expression is made to depend upon another, in which the exponent of the parenthesis is one less than before. By repeated applications, this exponent, if positive, may be reduced to a fraction less than unity, either positive or negative. 179. The use of the preceding formulas may be illustrated by the example f.r'dx(a + bx2). INTEGRAL CALCULUS. 251 Place a + bx' = X, m = 3, n = 2, p = 2; then, from formula A, 53 ~3 f xadxX XX - afdxX2 6b Applying formula 3B to the expression fdxX-, making - = 1 n -= 2, p =, we have 3 fJdxX = xX2 + 3 afdxX2 and by another application xX2 + a/ ff dxX2 Substituting these values, we have, finally, 5 3 1 i XX Y axX a" XX 2 a fx'dxX3 a d 6b 24b 16b 16b -' dx dx The expression - -- may be integrated X~ b'/a +bx as in Art. (173). 180. If, in the primitive expressions, m and p are negative, theeffect of the application of formulas A and B, would evidently be to increase them numerically. Other formulas are then required. From A, by transposition and reduction, we find xm —-ldXXP - xm_-"XP+l' - b(m + np)fx"m-ldxXP a(-m - n) 252 INTEGRAL CALCULUS. If in this we change m into - + - n, we have x- xXP+' - b (n -m + np) fJx —'-cldxXP C ~ x-1dxXP= C; -am by the application of which, - m will be numerically diminished by the number of units in n. m If n - m + np = O, or n + the part to be integrated will disappear, and the integration will be complete. If m = 0, the formula fails. 181. From B, by transposition and reduction, we find XXM-dxX-X + (m + np)/x^-'dxXP.pna If in this we change p into - p + 1, we obtain'X-p+' - (m n + n - np) fJ-'dxX-+l /xf'd-x-...... D;.a, -.(P-1) in which the exponent of X is numerically one less than in the primitive expression. If p - 1 = 0, the second member becomes infinite; but in this case p = 1, and the primitive expression reduces to a rational fraction. If m + n - np = O, or n = ~ the simple application of the formula gives the integral at once. INTEGRAL CALCULUS. 253 182. Let us illustrate the use of these formulas by the example fa-r'dx (2 - X2) - Making in, m = 1, a = 2, b — 1, n = 2, p =-, we have - X-I 2 fS-'dx(2 - X')-2 =- - + fdxX-...(1). By formula D, after making m = 1, n = 2, a = 2, b -=- 1, P- = -, we have 3 xX-2 fdxX —= 2 Making the proper substitutions in (1), we obtain, finally, x'-X TM xX fx-ldx(2 - XI) - + 2 in which X = 2 - X2. 183. By the aid of formula D, we are now able to integrate the expression d(-z'+ b = dz (z2 + Pb)-P'....Art.(167). (2z + b2), Bymaking m = 1, x = z, a = b2, b = 1, n - = 2, we cause 1f2 + 6 to depend upon the integration of another expression, in which the exponent is one less; and by repeated applications, we shall find that the integral will depend upon the expression dz 1 z.2 _-_b = tanl-'. 254 INTEGRAL CALCULUS. 184. For the expression x dx //2 cx - X we may write fxqdx (2cx - -2),2,fx'-dX (2c - X)-2 to which applying formula A, making t = q + 1, a 2c, b = - 1, p =- ~, n =1, and recollecting that xq = xq-x'x, and xq- = X —2, we obtain 2' V2cx _ x~ - - 2 (2q - 1 )cj x-dx q v cf —_ x —-/.2x... By repeated applications of this formula, when q is a whole number, we make the primitive expression depend upon - - =- ver-sin-'....Art. (161). d -2cx - x2 c INTEGRATION BY SERIES. 185. If it be required to integrate the expression Xdx, X being any function of x, it is often convenient and useful to develop X into a series by any of the known methods, generally by the binomial formula; and then, after multiplying by dx, to integrate each term separately. This is called ijezteir tizg by series; INTEAGIRAL CALCULUS. 205 since we thus obtain a series' equal to the integral of the given expression, from which, when the series is converging, we can, for particufar values of the variable, deduce the approximate value of the integral. 1. Let us take the example dx du - = dx(l + x)-' By the binomial formula, wve have ( l+x)- - = 1 x- + x2- X3 + &c. Multiplying by dx, and prefixing the sign f, I +l ~ = /(dx - xdx + x.2dx - xdx- + &c.); whence dx x2 x3 X4 I- - 2+ q - + +&c.; 1 X- 2 3 or since = 1(1 + X), 2 3 4' 1(1 + x) - + - + &c... Art.(38). 2. Let du -- x ( -'2)dclx. By the binomial formula, we have a'2 a4 a6 ( —x') -= 1 - - - &C. 2 8 1 6 &. 256 INTEGRAL CALCULUS. Multiplying each term by x dx, &c., 5 9,,. I~ X~dx x 2dx fx~ (1 -x) dx= -(X2 - &.d); x —)dx _ f(x dx 2 8 whence II fx2(1 - 2)4dx - 23I 11 x2 x (1 -2-') 2dx=- - X2 - &c. 3 7 44 3. Let du - adx. In article (40), we have found X Xr Xs =1- + + a + (la)2 X + (la) + &c.; 1.2 1.2.3 hence, faldx = + - + ( a)x' &c. If a — = e, then le = 1, and X X= x4 /e'dx = x + + -+ + c. dx dx 4. Let du —- -. Make x- = iu; then dx = 21/xdu, and dx 2du VxV I - X V1 - U which may be readily integrated, and we shall obtain INTEGRAL CALCULUS. 257 dx r Izm3x2 ___ —----- =2 sin-4 -a 2 f &c.. =x(l + 23 + 2.4.5 5. du = e- dx. 6. du = dx 2ax - x2. dx v/ - e'2x2 7. Let du - */1 --' Developing /1 — P ex - (1-e'gx2), we have./1 - e'2x 1 e - e - - -e'4x4 - &c.; hence a dx 1V'- eq'x 1-1 e,2 11 dx ~I 1 - = f(1 - 2x - -e"x - &C.) l - After the multiplication, each term of the second member will be of the form AJ/"', dx which, by formula A, may be made to depend upon = sin-I x. dx dx 8. Let du = -x(- = 2cb f 2cx-'2)(b-x2) V/2 ccx _2 -Vlb- b If we develop - (b - x)-2, and multiply by -/b - x dx _tLAx d.Z: each term will be of the form < V2cx- 2cx - xX x which may be reduced and integrated as in the preceding article. 17 258 INTEGRAL CALCULUS. 186. By the application of the formula for integration by parts, Art. (169), to the expression Xdx, we obtain fXdx = Xx - fxdX.............(1), and then to xdX, &c., dX x= dX _ X2d2X fdx 2 d3 2 d jx9d2X rdj Xx2dx xs d2X f X3 d3X J 2 d - J dx2 2 2.3, dx- 2.3 dx'.&c. Substituting in succession the values above deduced, equation (1) will become dX x2 d2X x'.fXdz = Xx - - &c., dx 1.2 dx 1.2.3 a series, expressing the integral of Xdx in terms of X, and its rdifferential coefficients; which has received the name of its distinguished discoverer, John Bernouilli. 187. If in the integral' fXdx = f(x) =, we make x = x + h, we have (~Xdx)z,+, = f(x + h) = W'; and, by Taylor's formula, du d2J h(' - u = ~ dt A + &Co........(1). dx dx2 1.2 INTEGZRAL CALCULUS. 259 But since du JfXdx = u, Xdx = d, d X d'u dX' d3u d2X dx2 = d' dx3 = dx2-, &c. These values substituted in (1), give dX h' d2X hS - = Xh dx 1.2 +'dx 1.2.3 + &C If in this series we make x = a, h = b - a, and denote dX d2X., by A, A', A"', &c., what X, d-' d'A &c., become under this supposition, it is plain that what u becomes will represent the value of the integral when x = a;. what u' becomes, its value when x = a + b - a - b; then what u' — u becomes, will be the value of the integral between the limits x = a, and x = b; whence b A' A" Xdx = A(b —a) + 1.2(ba)2 123(b-a)'+&c., a series from which the approximate value of a definite integral may be obtained. If b - a is so lalge that the series does not converge, or does not converge rapidly enough, then let it be divided into n equal parts, so that b - a = nuc and take the value, first between the limits a and a + a, then between a + a and a + 2a, &c., and suppose the results to be 260 INTEGRAL CALCULUS. 2 1.2.3 Ca +' + C" + &c... (2), 1.2 1.2.3 a a Da + D' 2 + " 1 2 3 + &c. &then, y article (160), we have; then, by article (160), we have J~Xdx = (B + C + D + &c.)a + (B'+ c + &e).) -2 + &c....(3), and as a is arbitrary, the separate series (2) [and of course the final series (3)] may be made to converge as rapidly as we please. i1TEGRATION OF DIFFERENTIALS CONTAINING TRANSCEDENXTAL QUANTITIES. 188. But few of these differentials admit of exact integrals. We can, however, by the aid of formulas previously deduced, obtain, by series, their approximate integrals. By the examination of a few expressions, we will endeavor, as far as possible, to indicate to the pupil the general method to be pursued, and then leave to his ingenuity and industry its application to the different cases with which he may meet. Take first the expression X (x) "dx, INTEGRAL CALCULUS. 261 in which X is an algebraic function of x. If we divide it into the two factors Xdx = dv, and (x)" - u; whence dx fXdx - v = X', du = n(lx)"-l I-; and then substitute in the formula of Art. (169), we have JX(lx).dx = X'(Lx) - nfX'(Ix)'- L..... (1). x By this the integral of the primitive expression, when the integral of Xdx can be found, is made to depend upon the integral of another similar one, in which the exponent of (lx) is one less than at first. If, then, n be entire and positive, after repeated applications of the formula, the exponent of (lx) will become 0, and the expression upon which the integral depends, algebraic. For a particular case, let Xmq-l X = x, then fxm"dx = X', and this in (1) will give Jxm(x)dx = (1x)n - + fxm(1x) -dx.. (2). If in this we substitute for n, in succession, n - 1, n- 2, n - 3, &c., we have 262 INTEGRAL CALCULUS. fxn(lx)"-Idx m + 1 (lX)' — m + 1fxm(lx)"2dx, xm)+l n - 2 fx l)xsd = + 1 (lx) -m +, These values in (2), will give a general formula, in which, if n be positive and entire, the last term will be.f n (n - 1).....2.1 xIx)(dz) - (0 _ + n + (M + I)" (. + 1)n"+ We shall therefore have Jx(l)nd = m+11 [+ (lx)"-' n (n-)- 1...1 (3). The sign of the last term will be plus when n is even, and minus when n is odd. If m = 1 and n - 1, we have fxlzdxz = if x (-) If m- 0O and n- 1, we have fl xdx: x (l - 1). If m = - 1, the second member of (3) becomes infinite. In this case the differential becomes (x)ndx. x INTEGRAL CALCULUS. 263 dx Making lx = z, we have - dz, and dx Zn+l" (lx)"+' (lx) = fd = d )+1 X n+l n+' which is true for all values of n, except when n — - 1. In this case the expression becomes dx: Making lx = z, we have =- dz, and x dxi - v - Ix = (lx)= 189. Take now the expression Xa dx, in which X is an algebraic function of x. If we divide it into the two factors X and a'dx, and recollect that aelad = da'.........Art. (39), whence da0 as adx = and fadx = we shall have, from the formula for integration by parts, fXa"dx -- Xa f adX. If we take the successive differentials of X, and place 264 INTEGRAL CALCULUS. dX = X'dx, dX' = X"dx, dX" = X"'dx, &c., we obtain jaadX X'a i a d' J la (la)2-aj (la)2dx' &c)-( — (l3dX.X &c. These values, in equation (1), give.a_ X X'::X"I a a~dX"' (ad la (la( ) X2 (1_' padX'.*........ (2). /SXa " x a ( la)2.la(la)+' J(la)~+-. If the function X is 9f such a nature that one of its differential coefficients X', X", &c., is constant, the differential of this will be 0, and the corresponding term ja'dX"' The integral will then be exact. The expression xRafdx, admits of an exact integral when n is entire and positive. If n be fractional or negative, we write for a2 its development, Art. (40), and then integrate as in Art. (185). 190. By article (43), we have d sin nx = ndx cos nx, d cos nx -- ndx sin nx; hence INTEGRAL CALCULUS. 265 cos nx sin nx fdx sin nx - f dx cos nx n n In the expression dx sin x 1 cos 2x we can place for sin2 x, its value, 2- 2 and then have fdx sin 2x dx cos 2 xdx sin Sdsin$ =-2 — 2 2 4s2x and, in general, the integral of similar expressions, containing any power of either the sine or cosine of x, can be obtained by first substituting the value of the power in terms of the functions of the double, triple, &c., arc, as determined in Trigonometry. The expressions dx sin x, dx cos x, when m is entire, may also be integrated as follows: Make dz sinx = z, then x = sin-z, dx -- (1- z- )2 whence fdx sinx = f d (1 -z )2 This expression, by repeated applications of formula A or C, may be made to depend upon or - *} (1 -g2)2. (1 _ z 266 INTEGRAL CALCULUS. In the expression dx tanmx, place tanx = z, hen dz dx + 1-, fdx tanm =j + which is a rational fraction. Exanmples. Integrate dz 1. du = dx sin3x. 2. du - s Cos X dx 3. du =. 4. du= dx tangx. sin x 191. In the general expression dx sin"x cos~x, we may place sin.= z, then cos=-(1-ZI)+ dx dz (1 - z2)' and finally,.fdx sinmx cos"x = jrztdz(l — Z) i, INTEGRAL CALCuLUS. 267 which may be reduced by formulas A, B, C, and D, and in some cases integrated, as in the example du = dx sin4x cos2x; whence u = fz4dz(1 - z2). 192. Take finally the expression Xdx sin-' x. Place Xdx = dv, and sin-x - u, then dx f'Xdx - v = X', and du - (1 - X) Substituting in the formula of Art. (169), we have fXdx sin-1x = X' sin-ix - - X'd (1 -x)2 Thus the integral of the primitive expression is made to depend upon the integral of the algebraic expression X'd (1 -XT Let X = X then fXdx = fx"dx = + X' and we have X"+l 1 x+- + f 11dx sin-' x sin-' x -- -. n:-~- 1'(1 —x') 268 INTEGRAL CALCULUS. By the application of formula A or C, when n is entire, the last term may be reduced, and then integrated; except when n = - 1, in which case the expression becomes dx which can only be integrated by series. In the same way, like expressions may be found for fXdx cos- x,.fXdx tan-' x, &c. INTEGRATION OF DIFFERENTIALS OF THE HIGHER ORDERS. 193. By an application of the rules previously demonstrated, we may readily obtain the primitive function, from which differentials, of a higher order than the first, containing a single variable, may have been derived. Let there be the differential d"u = Xdx", X being any function of x. Dividing by dx1-', we have dnu d -i- = Xdx, and since dx"-l is a constant, this may be written, Art. (26), (d -I U Xdx d kdx1)= Xdx. INTEGRAL CALCULUS. 269 Integrating both members, we have d n-X d' = fXdx X' + C. After multiplying both members of this equation by dx, it may be written d (dxlu\) = X'dx + Cdx; and integrating as before, dxn-2 = X" + Cx + C'; which by another transformation and integration, may be reduced one degree lower, and finally after n integrations, we shall obtain CX n- 1 Ct X x-2 =- F(x) + 1.2.. (n - 1) + 1.2..(n - 2) The above operation may.be indicated thus, u = fnXdxn; the symbol f" indicating that n successive integrations are required. At each integration an arbitrary constant is introduced. The complete integral may therefore be required to fulfil n arbitrary conditions. Examples. 1. Let d'u = ax'dx2. The required operation is indicated thus, 270 INTEGRAL CALCULUJS. u = 2Jax'dx2, and may be read, the double integral of ax'dx2. Let the expression, after dividing by dx, be written d f du\ d - d d x ax'dx; whence, by integration, du ax axa d- =- 3 + C, du -- dx + Cdx. ax 3 3 Integrating again, we obtain ax4 u - + Cx + C'. 12 2. If d3u = bdx', u = f3bdx8, which is called a triple integral. We may write d'u dlu dx d x ] -2- bdx; whence d2u = bx + C; and finally, as in the'last example, bx'3 Cx2 u -f 3bdx3 = + - r C' C". 8~.. 6 dx4 8. d4u - 4. 4 d3u = Vx dxz. x INTEGRAL CALCULUS. 271 INTEGRATION OF PARTIAL DIFrERENTIALS. 194. Hitherto, we have explained the mode of integrating only the differentials of functions of a single variable. It yet remains to extend our rules to the integration of those which contain more than one variable. These differentials are either partial or total, Art. (52). When partial, they belong to one of two classes: I. Those obtained from the primitive function by differentiating with reference to one variable only. II. Those obtained by differentiating first with reference to one variable, and then with reference to another, &c., Art. (48). 195. The differentials of the first class may be expressed generally thus, dne =-f(z, y, z, &c.) dx"; du = f'(x, y, z, &c.)dy"; &c., in which u is a function of x, Y, z, &c., and may evidently be obtained by successive integrations, precisely as in article (193); all the variab.es, except the one with reference to which the differentiation was made, being regarded as constant, and care being taken to add, instead of constants, arbitrary functions of those variables which are regarded as constant during the integration. Examyples. 1. Let d2u = bx2ydxl, which, after dividing by dx, may be written 272 INTEGRAL CALCULUS. d ) bx2ydx; whence du cy bxly + y, d- = fbxlydx = b 3' du= Y dx + Ydx, 3 and u = f2bx2ydx2 by + + _ -- S~12 Y in which Y and Y' are arbitrary functions of y. 2. Let dSu = cxly2z2dy3. 196. The differentials of the second class may be written genex ally thus, d'+"+... = f (x, y, z, &c.) dx dyRdzP......, and the mode of integrating is plainly to integrate first, m times with reference to x, then n times with reference to y, and so on until all the required integrations are made. To illustrate, let 1. dou = 2xzydxdy, which may be written d(w y) = 2 xydy, whence, by integration with reference to y, INTEGRAL CALCULUS. 273 du x= x2Y + X, or du= x2ydx + Xdx, and u = $'2x'ydxdy = 3 +fXdx + Y; there being no necessity of indicating with reference to which variable the integration is first to'be made, Art. (49)..2. Let d3u - ax2ydy2dx. This may be written d 3u d-.u -du = ax' ydx, or d d axydx dy or Integrating with reference to x, d'u ax3y dy2 ~3'' Y, which may now be integrated as in the preceding article. 3. d'u = axz'dxdydz. 4. d4u = (x + y)'dx'dy'. INTEGRATION OF TOTAL DIFFERENTIALS OF THE FIRST ORDER. 197. If u = f(Xy),: we have found, Art. (52), du du du -dx + dy, dx dy 18 274 INTEGRAL CALCULUS. du du in which - dx and d dy are the partial differentials of dx dy f(x, y); and also, Art. (49), d~t d2"z d u d;d d du) d- d dxdy - dydx' or dy dx (1). If, then, an expression of the form Pdx + Qdy.......... - (2), be the total differential of a function of x and y; Pdx and Qdy must be the two partial differentials of the function, and by the integration of either, we shall obtain the function itself. To ascertain, in any given expression of the above form, whether Pdx and Qdy are. such partial differentials, we have simply to see if the condition (1), or dP dQ dy dx' is fulfilled. If so, the given expression is the differential of a function of x and y, and we have U = fPdx + Y........... (3), Y being a function of y, which is to be determined so as to satisfy du the condition - =. dy Since the differential of every term of u waich contains x, when taken with respect to x, must contain dx, the integral of Pdcv will give all that part of u which contains x. The differential of those terms which contain y and do not contain x, will be found only in the expression Qdy. If, then, we integrate those terms of Qdy which do not contain x, we shall have that part of u which con tains y and not x. This will be the required expression for Y INTEGRAL CALCULUS. 275 which added, with an arbitrary constant, to fPdx, will give the complete integral. If all the terms of the given differential contain x or dx, Y will be 0, and we complete the integral by the addition of an arbitrary constant to the integral of Pdx. Examples. 1. Let du = (2axy - 3bxay)dx + (ax' - bx') dy, which, compared with equation (2), gives P = 2axy - 3bx2y, Q = ax2 - bx, dP dQ = 2ax - 3bx, = Q dy dx This condition being fulfilled, we then have, since all the terms of du contain x or dx, u -= (2axy - 3bx2y)dx = axay - byx3 + C 2. If du= -+ (2y -- dy, U - - + Y. Y Y The term of Qdy which does not contain x, is 2y dy, the integral of which is y; hence Y = ya 276 N1TEG1R:AL'CALCULUS. and the aabove expression:for u:becomes U- - +y C. 3. If du = u tan-'x - C. 4. Let du = (6xy - y) dx + (3x2 - 2xy) dy. 198. The method of obtaining the integral of a differential, containing several variables, is readily dednced "from what precedes. Let du = Pd + Qdy + Rdz = df(, y, z).... ). If for a moment we regard z as a constant, and then, in succession, y and x, it is plain that we shall have the three expressions Pdx + Qdy, Pdx + Rdz, Qdy + Rdz... (2), which, taken separately, are differentials of functions of two variables, if the primitive expression is a differential of a function of three, and the reverse. But the conditions' that these be each an exact differential, are dP dQ dP dR dQ dR dy d -' d d = dx' dz a hence, if we have given an expression of the form Pdx + Qdy + Rdz, and the conditions (3) are fulfilled, it will be the differential of a function of three variables, and we can obtain the function by INTEGRAL CALCULUS, 277 integrating either of the expressions (,2), as in the preceding article, taking care to add to the integral a function of that vari-v able which is regarded as constant. Thus, denoting the integral of Pdx + Qdy.'by v., We have f(Pdx + Qdy + Rdz) = v + Z.....(4), Z being independent of x and y, and a function of z alone; and may be determined by taking, the integral of those terms of Rdz which contain neither x nor y. 199. If a function of two variables, composed of entire terms, is homogeneous with reference to the variables, its differential will also be homogeneous; and such a relation will exist between the function and its partial differential coefficients, as will enable us at once to obtain the function when the differential is given. To explain this relation, let U = f(X, y), and m denote the sum of the exponents of x and y in each term. For x and y, substitute tx and ty respectively; the primitive function then becomes tmu. In this expression, for t put (1 + s); then tu = (1 + S)mU. Under these suppositions, x and y, in the primitive function, have become, respectively, x + sx, and y + sy. Developing this new state of the primitive function, as in article (48), we have J(du du )+ 1 d2u 2 2 dU 2 )+ =(+ SX+ =SY} +m \ dx2 + 2-xdyxy. + &C. - (1 - s)"u = u + mus +- - 1)m &c+. 278 INTEGRAL CALCULUS. Equating the coefficients of the first powers of the indeterminate s, we have du du d + y....... Hence, in the differential du Pdx +'Q dy, if P and Q are homogeneous of the (m - 1)th degree, we shall have, by comparison with equation (1), Px + Qy Px + Qy = mu, u. For- example, let dg = 4xy2dx + aysdx + 4x'ydy + 3axy dy, in which mn - 1 = 3, m = 4, 4-xy + aya = P, 4x2y + 3axy2 = Q; whence Pxr + Qy u = -- 2x'y2 + axy3. 200. If we denote fPdx by v, we have, by passing to the differential coefficient, dv dx Differentiating this with reference to the variable y, we find INTEGRAL CALCULUS. 279 dx dP dyl dy P dy d Art(197); whence. dv \ dy) dx = - dx. dx dr Integrating with reference to the variable x, we have - = y dx, or, since (dP)dx = d (Pdx), dfPdz _ d (rPdx) dy -J dy By which we see, that we may differentiate. with reference to another variable, the indicated integral of a partial differential, by simply differentiating the quantity under the sign. INTEGRATION OF DIFFERENTIAL EQUATIONS. ~201. These equations, when of the first order, and when derived from equations containing but two variables, will appear as paiticular cases of the general form Pdx + Qdy = 0, and may of course be integrated as in article (197), when 280 INTEGRAL CALCULUS. dP dQ dy dx' and give fPdx + Y = C. In practice, however, it will in general be found that, in consequence of the disappearance of a faLtor common to both terms of the differential equation, or when the differential equation has been obtained by the elimination of a constant from the primitive and its immediate differential equation, Art. (58), this condition is iiot fulfilled; hence other means of obtaining the integral must be sought for. In the first place, it is evident that, if by any transformation the equation can be placed under the form Xdx + Ydy = 0, X being a function of a, nd Y of y, the integral can be found by taking the sum of the integrals of the two terms; thus, fXdx - fYdy = C. 202. Among the most simple forms with which we meet, are I. Ydx + Xdy = o. II. XYdx + X'Y'dy = o. The variables may be separated, in I., by dividing by YX; and in II., by dividing by YX'. The results, dz dy o dx + 4= and X Y',dx + y dy = 0, INTEGRAL EALCUrLtS-. 281 are under the proposed form. In general, if the value of d', deduced from the equation, be under the form dy XY, dx we have dy Xd and d = Xdx. Y y d Examples. i. Let yd- xdy = Dividilng by /x,: w; have dz dy -, 0- ly = C; or, making C IC', we have x y 1- = I - C'y 2. Let xy'dx + dy = 0. Dividing by y2, dy xdx t+. d O; integrating, and reducing, x2y - 2: 2- y. 282 INTEGRAL CALCULUS. 3. Let (1 - )2yd (1 + y)x2dy = 0; whence ( - ) dx + Ydy = 0, and 1 21x + - iy-y - C. 4. Let (1 + x2)dy d- Vydx = 0. 5. Let x'ydx - (3y +- 1) v/xdy = 0. 203. III. In all cases where the equation is homogeneous with reference to the variables, they can be separated, and the equation placed under the proposed form. Suppose the general form of the given differential to be Ax"ymdx + BxAykdy = 0, in which n + m = h + k = g. Make y = zx, and substitute; we thus obtain Axgz"dx + Blxzkdy = 0; dividing by xg, and putting for dy its value, zdx + xdz, we have Az"dx + Bzk(zdx + xdz) = 0; dividing by (Az" + Bzk+) x, we have dx Bzkdz x Arm + Bzn+ = 0, which is under the proposed form. INTEGRAL CALCULUS. 283 Examples. 1. Let x'dy - yIdx - xydx = 0. Make y = zx, then dy = zdx + xdz. Substituting in the given equation, we have x2zdx + x'dz - z2x2dx -- xzdx - 0; reducing and integrating, xdz z- 2dx = 0, - - x C. z Putting for z its value, we have finally x ix (C +-c. x~ + yx x 2. If +?/dy = ydx, x - C. x —y 2y x 3. Let xdy -ydx = dx x2 + y. 204. IV. The:equation (a + -bx + cy)dx + (a' + b'x + c'y)dy = 0, may be so transformed, that the variables can be separated and the integral found. For this purpose let us make x = t + 6, y = u + 6'; 284 INTEG-RAL COAGCULUSs. whence dx = dt, dy -= du. These values in the primitive' equation, give (a + b& + c&'+ bt'+ cu)dt + (a' + b' + c'T + b't 1- c')du = o. By placing a + b6 + c' = O, a' + b'6 + c'a'- O, we can determine proper values for the arbitrary quantities 6 and 6', and our equation reduces to (bt + cu)'dt + (b't + c'u)du = 0; which being homogeneous with reference to t and u: may be treated as in the preceding article. This transformation is always, possible, save when the values of a and 6' become infinite, which will be the case only when b c'- cb'=- O0 whence 6b' b' c';' + + c'y (bz + cy). The primitive equation thus becomes adx + a'dy + (bx + cy) (dx + dy) = 0, in which the variables may be separated by making bx + cy = z. INT-EGRA-L CALULUS. 2885 Substituting this, and the resulting value of dy, the equation reduces to ~dx + (a'b +- b'g)dz abc - a'b2 + (bc - bb') z If be - bb' = 0, we have at once the integral 2a'bz + b'z2 x + C, 2 (abc - a'b2) in which the value of z is to be substituted, 205. V. In the equation dy + Pydx = Qdx...... (1),* P and Q being functions of x, the variables may be readily separated by making y = zX~.......... (2), X being a function of x, for whiCh a proper value is to be determined. By differentiating equation (2), we have dy = zdX + Xdz, and by substitution in (1), zdX + X(dz + Pzdx) = Qdx....(a). Suppose X to have such a value that zdX = Qdx...........(4); * Equations of this kind, being of the first degree with reference to y and dy, are sometimes improperly called linear equations. 286 INTEGRAL CALCULUS. equation (3) then becomes x (dz + Pzdx) = o; whence - =- Pdx, = -- fPdx; or, taking the numbers, Z = e-yrP. From equation (4), we have dX - Qdx = Qe/Pd dx; whence X = fQe/Pd~dx. These values of z and X, in equation (2), give y - e-/PdSofQe/Prdx. Example. Let dy + ydx = e,~dx, then P = 1, Q = e-, fPdx = x; hence, by substitution in the above value of y, y = e-afe-V.eddx = e-:(x + C). INTEGRAL CALCULUS. 28'7 206. If we divide the form dy + Pydx = Qymdx...... (1), by y", we have dy + Py-mPy odx = Qdx.... (2), In this make y-m+l = z, whence - (r - 1)y-"dy = dz, and dy ymdz m -- 1 Substituting these values in (2), and reducing, we obtain dz - (m - l)Pzdx = - (m - l) Qdx, the same form as equation (1) of the preceding article. Integrating this, and substituting the value of z, we shall have the primitive equation, from which equation (1) may be derived. 207. Equations of the form aymxndx + bxPdx + cxqdy = 0, may sometimes be rendered homogeneous by making y -= zk k being a constant to be determined. From this, we have dy = }kz- dz, yim = zkrn These values in the primitive equation give 288 INTEGORAL CALCULUS. az x" dx + bxlPdx + ck+xzP-'dz = 0, which will be homogeneous, if knm — 2 n =-p = q + k - 1; that is, when - - 2' 1 - q = k. OF THE FACTORS BY WHICH CERTAIN DIFFERENTIAL EQUATIONS ARE RENDERED INTEGRABLE. 208. It has been remarked, article (201), that differential equations sometimes fail to fulfil the condition of integrability, in consequence of tlh disappearance of a common factor. Whenever this factor can be discovered, by trial or otherwise, the integral can at once be found, as in article (197). Let Pdx + Qdy = 0, be a differential equation in which the condition is not fulfilled, and suppose that z = f(x, y), is the factor by the disappearance of which the given -equation has resulted. The immediate differential equation will then be Pzdx -+ Qzdy = 0, from which we have the condition, Art. (197), dPz dQz dy dx' INTEGRAL CALCULUS. 289 or, performing the differentiation, zdP Pdz zdQ Qdz dy d d' dx' or (dz dz dP d (dy Q dx\ dy dx - This equation expresses a relation between z, x, and y, but its solution in the general case is so difficult, that nothing will be gained by attempting it. 209. If it be possible to find one factor which will render the differential equation integrable, an infinite number of others will at once result. For, suppose an expression for z has been found; then zPdx + zQdy = du is a differential which is integrable. If we multiply both members by any arbitrary function of u, denoted by U, we have UzPdx + UzQdy = ldu. Udu, containing u alone, is a differential; hence the first member is also a differential of some function of x and y, which admits of integration; and zU, or zFf (zPdx + zQdy), is a factor which will render the given differential equation integrable. 210. In the particular case where z is a function of x only, its value can be determined, as we shall then have 19 290 INTEGRAL CALCULUS. dz dy = and equation (1), of Art. (208), will reduce to Qdz dP dQ \ d + T - - z = O' or dz l_ dP dQ\ z Q. dy dx But by hypothesis z is a function of x, therefore I(dP dQ) _,(Z) = q ~dy-~ -d f ) X; then dz Jd = fXdx; z whence. tx = fXdx, z = e~. Let this be illustrated by the example dxr +- 2xydy + 2y2Idx = 0, in which -P= 1 + 2y', Q= 2xy; whence 1 dP dQ\ 1 Q y dx x and z = efXd = e~d = ez = x being the common factor, the immediate differential equation must be INTEGRAL CALCULUS. 291 xdx + 2xIydy + 2xy2dxr = 0, which can be integrated as in article (197). In a similar way, if z = f'(y), its value may be determined. To ascertain whether a proper expression for z has been thus obtained, we multiply by it, and then apply the test as given in Art. (197). 211. If the given differential equation be homogeneous with respect to the variables, a proper expression for z may be found. Let Pdx + Qdy = 0 be homogeneous and of the Vz - 1th degree, and suppose z is of the nth degree; then zPdx + zQdy = 0, will be of the mn - 1 + nzth degree. Hence, by Art. (199), we have zPX ~ _QY f(zPdx + zQdy) + C; m +t n whence C(m 2 1+) 1 Px + Qy. Px + Qy' since, C being arbitrary, we may make C( Jr n) = 1. 212. If the given differential equation can be divided into two parts, and a separate factor can be found which will render each part integrable, a third factor may sometimes be deduced firom these, which will render the given equation integrable. Thus, let 292 InrEGRRAL CALCULUS. Pdx + Qdy = o be divided into the two parts (P'dx + Q'dy) + (P"dx + Q"dy) = 0, and suppose z' to be a factor which will render the first part integrable, and z" a like factor for the second part; then z'P'dx + z'Q'dy = du', z"P"dx + z"Q"dy =- du"; from which u' and u" may be obtained as in Art. (197). Then, from Art. (209), z'U' and z"U" are also factors which will render the respective parts integrable, U' and U" being arbitrary functions of u' and u". If, therefore, we can assign, by trial or otherwise, such values to U' and U" as to make Z'U' = Z"'U". the expression resulting will be a factor which will render the two parts integrable, and of course the given equation. INTEGRATION OF DIFFmRENTIAL EQUAITONS CONTAT/BNG THE IIIGHER POWERS OF dy. 213. Differential equations of the first order, containing the higher powers of dy, may arise, as in the third example of article (58), from the elimination of the higher powers of a constant. Such equations, after division by dx", may be put under the form (dy + (dy.+ U dx + a-U...M INTEGRAL CALCULUS. 293 The determination of the primitive equation will then depend upon the solution of equation (1), or upon the division of the first member into its factors of the first degree. There are n such factors, and it is plain that each, when placed equal to zero and integrated, will give an equation between y and x which may be regarded as a primitive equation.' If, then, the values of dy be denoted by V, V', V", &c., dx equation (1) may be written dy dy dy (dz dz v-v) (d - V")&C. =o0, dx d~ dx which may be satisfied by placing dy dy dy dY _ e _ O &c. (2) and if the corresponding primitive equations be denoted by P =, P' = O, P" = O, &c., respectively, we shall have PP'P"I&c.= 0........... (3), for the most general primitive equation, particular cases of which may be obtained by placing P = O, P' = O, or the product of any of these factors taken two and two, or three and three, &~. It would appear, since a constant is to be added in the integration of each of the equations (2), that (3) ought to contain n arbitrary constants; but equation (1) can only be deduced from its primitive equation by the elimination of the nth power of a constant: [Or by raising (dy -v) to the nth power, in which case the primitive equation must be y f= Vdx]. It is plain, then, that the constants added ought to be equal, or that the same should be added in each integration. 294 INTEGRAL CALCULUS. The n differential equations of the first degree which are factors of (1), are readily accounted for, by supposing the primitive equation to be solved with reference to C, which will have n values, each of which, differentiated, will. give one of the equations referred to. As there will be difficulty in the solution of equation (1), when the degree is higher than the second, it will be well to discuss Some particular cases which admit of integration by other means. 214. I. If the proposed equation does not contain y, and it be dy easier to solve it with reference to x than with reference to dx' which we will denote by p), we (an thetn obtain. = gjl9) aim*. (1). But dy = pdx, and by parts, article (169), y = px - fxdp - px - fq(p)dp + C; whence, if q (p) dp can be integrated, p may be eliminated by the aid of equation (1), and the primitive equation between x, y, and C, deduced. II. If the proposed equation does not contain x, and may be solved with reference to y, we shall have y = f(p)........... (3), dy = df(p), or pdx = df(p);'whence dx: = df (p) j= df (p) C. AdP P INTEGRAL CALCULUS. 295 Combining this with equation (3), and eliminating p, a primitive equation will result between x, y, and C. III. If the proposed equation is homogeneous with respect to the variables of the nth degree, we may make y = t............. (4), and then divide by x", and, if possible, solve the equation with respect to t, and have t = f(p)....... (5). Differentiating (4), we have dy = xdt + tdx, or pdx xdt + tdx, dx dt df(p) - P - p t p f(P)) the integral of which is lx = q(p). By combining this with (4) and (5), a primitive equation between y and x may be obtained. IV. When both variables enter, but,y enters only to the first power, we maytake its value, in terms of p and x, differentiate it, and thus obtain dy = Rdx + Sdp; or, since dy = pdx, (R - p) dx + Sdp = 0. 296 rNTEGRAL CALCULUS. If this can be integrated, the result may be combined with the proposed equation, p eliminated, and a primitive equation between y and x determined. Suppose the deduced value of y to be y = x + P.............(6), in which P f (p). By differentiation, we obtain dP dy = pdx +r xdp + dP or + dp d= 0 which may be satisfied by making dP x + d....(7), or dp = 0....(8). Equation (8) gives p = C; whence, by substitution in (6), y = Cx + C', C' being what P becomes when p = C. Equation (7) expresses a relation between x and p, and if it be combined with (6), and p be eliminated, an equation between x and y will result, which will contain no arbitrary constant. Let there be for a particular example ydx x- dy = ln /dx2 + dy'; whence y =px + nV 71 + p..........(9), INTEGRAL CALCULUS. 297. npdp dy = pdx + xdp - P p 1 + pa dp (+~P+) =0; whence np i + =O dp= 0, or p C. This value of p in (9) gives y = Cx + n /1 + C2. From the other factor we have P = i, -/n - X which, in (9), gives y2 + x2 =- n2 a result containing no arbitrary constant, which will be further explained in the following article. SINGaLAR SOLUTIONS. 215. It has been seen, that many differential equations of the first order result from the elimination of a constant from the primitive equation and its immediate differential. Thus, let f(x, y, C) = 0........(1), be the primitive equation containing the variables x and y, and the constant C; 298 INTEGRAL CALCULUS. Pdx. + Qdy = 0........ (2), Xi:s immediate differential equation; and P'dx + Q'dy = 0.......(3), the result obtained by the elimination of C from (1) and (2). It may now be asked: May not such a function of x and y be substituted for C, that the result of the combination of equation (1), under this supposition, with its immediate differential, shall be the same as before? To answer this, let equation (1) be differentiated, x, y, and C being regarded as variables; we thus obtain Pdx + Qdy + C'dC = 0....(4). Now, if C'dC = O, it is plain that equation (4) will be the same as equation (2), and the result of the elimination of C between it and (1), will then be the same as equation (3). If, then, for C in equation (1), we substitute the variable value deduced from the equation C'dC = o, that equation will contain no arbitrary constant, and yet will be as much a primitive equation as any one containing the arbitrary constant. Stuch results are termed singular solutions, inasmuch as they cannot possibly be obtained from the complete integral, Art. (160), by assigning to the arbitrary constant a particular value; the latter results being called particular integrals. The equation C'dC = 0 clan be satisfied, by making dC - O, or C' 0. The first gives C = a constant, the particular values of which, when substituted in equation (1), give particular integrals. INTEGRAL CALCULUS. 299 The values of C deduced from C' = 0, if variable, will then give the only singular solutions. To illustrate, let us resume the complete integral of equation (9), in the preceding article, y = Cx + n V/l+ C.....(5). Differentiating with reference to C, we have o = xdC + CdC V1I + C2 whence nO + = 0......(6), V + 02 and x x + x'2C = n= C, or C =-; V/n' - X the negative value of C being plainly the only one which will satisfy equation (6). Its substitution in (5) gives v~n~~n - x y = /'2_ x7, or y + x2 = n the singular solution found in the preceding article. INTEGRATION OF DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 216. Of these equations, which, in their most general form, day dy contain d-Xy dy' y' x, and constants, we shall only discuss those particular cases which admit of integration. 300 INTEGRAL CALCULUS. d'y I. The proposed equation may contain only d —x, x, and condxy stants; in which case, solving it with reference to d- Y, we have d y d — f(), which may be integrated as in article (193). 217. II. It may contain only d Y, Y, and constants. Solving the equation as before, we obtain d2y Multiplying by 2 dy, 2dy d'y d 2Ydy, and integrating, dy2 _4dy dg'_.2Ydy +- C, d= dxl =+, or d = 2fYdy + C; whence dy dy C' dx = 4/2' Ydy =-C +. V2fYdz + C V2fYdy + C MNTEGRAL CALCULUS. 301 Examples. 1. If aOd'y + ydx' = O, d2y y 2dy d2y 2ydy dx a2 dx dx a' dy2 _ y dy - - + - C y a2 which may be integrated as in example (5), article (162). 2. Let d'y /V7 = dx'. d2y dy 218. III. The equation may contain only d' dy constants, being expressed generally thus, F 4d"-~ dx) = o.......(1). then dy d Make d then dY- dp and (1) becomes dx(dX' ) -, - 0 302 INTEG1RAL CALCULUS. which is of the first order with reference to dip, and may be solved with reference to dx; whence dx = F'(p)dp....(2), x = fF'(p)dEp + C....(3). Multiplying (2) by p, we have pdx = dy pF'(p)dp, y = fpF'(p)dp + C'....(4). Eliminating p from (3) and (4), we have the primitive equa~ tion between x, y, and the two arbitrary constants C and C'. For an example, let 3 3 (dx2 + dOy2') a or dx( +rP 22 a; dxdy dp whence adp adpdpdy dx = - 3dx = dy = (1 +r p)2 (1 + p)2 Integrating the last two expressions, we have V-+ a = - + - _ y - +1 + P' p/1 +:o and eliminating p, (x - C)2 + (y -')2 a as was to be expected, since the proposed equation expresses constant radius of curvature. IrTEGRAL CALCULUS. 303 219. IV. If the given equation does not contain y, it may be expressed F (dY dy X 0, or F d.p A = 0, kdx2d dx' } O, dx which is of the first order with Wference to dp. Its integral will give an equation of the form f (p, x) = 0, in which, p being replaced by dy- and the result integrated, we shall have f (y, x)= o, with two arbitrary constants. For an example, let d2y dy 1 dTx d= x' or dp p dp dx dz x' p lp = x + C, p = C'x, dy ~C'x' = IZ, and y -+- C" dx 2 220. V. If the given equation does not contain x, it may be expressed (dQ2/ dy a /) = 0........(1). 304 INTEGRAL CALCULUS. Since dy = pdx, dx= dy d2y dp p dp and equation (1) may be written which is of the first order with reference to dp and dy. Its integral will then be expressed F'(p, y) = 0, or F'(d =, y 0, and this may be treated as in case II., Art. (214). 221. VI. If the equation be of the form d2y + X dr X'y = 0.....(1). Make y = ef...........(2); then dy = u e /d, d'y x d= U + ). These values in (1) give (since the common factor emf~' disappears) du d+ u2 + XU + X' = O, dx INTEGRAL CALCULUS. 305 which is of the first order with reference to du. After integration, the value of u being determined and substituted in (2), will give the required primitive equation, y = eff/d,. INTEGRATION OF DIFFERENTIAL EQUATIONS OF HIIGAR, ORDERS THAN THE SECOND. 222. Of these, it will also be sufficient for our purpose to discuss a few of the most simple cases. d y d'y I. Suppose the equation to contain only day' d"-Y and constants; it may then be expressed, F(d', d_,Y) = o......(1). Make d"-iy d"y du _-1 = i; then- These values in (1) give F( du' ) 0, which is of the first order; and its integral will give u in terms of x, or u = X + C, d X + C, and finally, Y = f-'(X + C) dxn-l 20 306 INTEGRAL CALCULUS. 223. II. Suppose the equation expressed thus, F (d, d — Y) - o....... Make d"-ry doy d_ p -dx- u, then dd' and equation (1) will become -(d4.) 0, which may be integrated as in article (217), and the value of u =f(x) determined; we shall then have nd - 2 = f(x), and y = -2f(x)dx2. 224. III. Suppose the equation to be of the form d3y + Ad'ydx + Bdydx'2 + Dydx = O... (1). Make y- e'............ (2),.tbeing an arbitrary function of x; then dy = eudu, d'y = e"(d'u + du2), d3y = e"(d 3u + 3dud'u + &u3). These values in (1) give d3Iu + 3dud'u + du3 + A(d'u + du')dx + Bdudx' + Ddx3 = 0........... (3). INTEGRAL CALCULUS. 307 Since u, in equation (2), is arbitrary, let such a value be assigned to it, that its differential shall be constant; in which case du = nmdx, d=u = 0, d u = O. Equation (3), under this supposition, reduces to m3 + Amn2 + Bm + D = o........(4). From this equation we may determine the value of the constant m. Denoting the three roots by n, ml, a, we have for du the three values du = mdx, du = m'dx, au = m"dx; whence u = mx + C, u = m'x + C', u = nt"x + C", and Y- e+C, Y y emI+c Y et or, calling eC = C, ec = C', ect — C=' y = Cem, y = C'e"% y = C"e" ". But since these values of y each contain but one arbitrary constant, they must be particular cases of the general value of y, which must be of such a form that either of the above particular values can be deduced from it; that is, y = Cem'd + C'em + C"e"'"", 308 INTEGRAL CALCULUS. from which the first particular value is deduced by making C' and C" = 0; and in a similar way, the others. If two of the roots m, Mn', m", are equal, that is, if m = m', we should have the equation y = (C + C')elz + C"em"I = Cemz q_ Creml"', containing but two arbitrary constants, C + C' being denoted by C. It is not then general. But in this case, y = Ce "", being a particular value, y = C'xe........... (5) will be another; for, differentiating it, we have dy C'em(l + mx))dx, d2y = C'e-m(2m + z'x)dx., d3y = C'em-(3m2 + m3x)dx3, and these, substituted in equation (1), give (4+Amn 2 + Bm + D)x + (3m2+2 Am + B) = 0....(6). But the coefficient of x is the same as the first member of equation (4), which has two roots equal to m; and 3m2' + 2Amn + B is its first derived polynomial, which, when placed equal to 0, must have one root equal to m (see Algebra); hence both terms of (6) are 0, and y = C'xemx satisfies the given differential equation, and must therefore be a particular value of the general one, y = Cemz + C'xe` + C"e". If m - m' = m", it may be shown also by trial, as above, that y = C"ix'2em INTEGRAL CALCULUS. 309 is a particular value; whence the general value must be y - e"(C + C'x + C"x2). Two of the roots may be imaginary; but, as the discussion in this case is quite complicated, and of little value to the student, we omit it. To illustrate the above, let d3Sy + 2d'ydx - dydxt - 2ydx = 0O. Comparing this with equation (1), we have A = 2, B =- i, D) - 2; and equation (4) becomes m3 +- 2mI - kn - 2 = 0; whence the three values of m are - 2, 1, and - 1, and the general value of y is y = Ce-2 + C'e' + C"e-~. 225. It is plain that the preceding principles can readily be extended to the general equation d"y +- Ad"-'ydx -- Bd"- 2ydx2.......... Mydx" = 0, and that the general value of y will be y = Cem" + C'em'" + C"em"a + &c. 310 INTEGRAL CALCULUS. 226. If the equation be d3y + Xd'ydx + X'dydx' + X"ydx3 -...(1), in which X, &c., are functions of x, the difficulty of integration is much increased. If, however, we know three particular values of y; Cy', C'y", C"y"', each of which will satisfy the given equation, then the general value of y will equal their sum, that is, y = Cy'+ C'y" + C"y"'......... (2). To verify this, let equation (2) be differentiated three times, and the proper values substituted in (1); we shall thus obtain C(d3y' + Xd2y'dx + X'dy'dx2 + X"y'dx3) + C'(d3y" + Xd2y"drx + X'dy"dx2 +J X"y"dx') =, + C"(d3y"' + Xd'y"'dx + X'dy"'dx + X"y"'d3) J which is satisfied, since each of the three terms is, by hypothesis, equal to 0. 227. The above demonstration can be generalized, and a similar result obtained for the equation d"y + Xdh-'ydx +.........yX('-')'dx" = 0. This, and the equations discussed in the three preceding articles, belong to the class termed linear. See note to article (205). INTEGRAL CALCULUS. 311 INTEGRATION OF PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 228. A partial differential equation of the first order, derived from an equation between the three variables z, y, and x, z being regarded as a function of x and y, contains, in its most general form, the three variables, the two partial differential coefficients, dz dz and and constants. Without attempting to discuss ax dyl the most general, we will confine ourselves to a few of the most simple cases. I. If the equation contains but one partial differential coefficient and the two independent variables, that is, if dz P being a function of x and y; we integrate at once as in article (195). For example, if dz _ x dx= V/x2+y' z = Y. 229. II. Let the equation be'dz dx R being a function of the three variables. Since the partial differential coefficient has been obtained under the supposition that y is constant, the proposed equation may be regarded as a differential equation between z and x, and may be integrated as in article (201), taking care to add an arbitrary function of y. 312 INTEGRAL CALCULUS. Examples. dz vy2 _ z2 1. Let dX. d-x z By the separation of the variables, we have zdz xdx -. z and by integration, 2 Let 2. Let -dx y + x'$ 230. III. Let the equation be dz dz -M + 3 = o, M and N being functions of x and y. Solving the equation with reference to A-, we have dz ~ N dz =- M dx But, since z is a function of x and y, dz dz dz = -dx + Tdy; dx I Y rNTEGRAL CALCULUS. 313 dz or, by the substitution of the value of dy' dz N dz /Mdx-Ndy d (d -Midy) =d M If S be the factor which will make Mdx - Ndy integrable, we may write S(Mdx - Ndy) = du, which, in (1), gives dz 1 du -SM dxd 1 dz to satisfy which, it is only necessary that SM d = F(u); whence dz = F(u)du, ( = (U), the form of this function being arbitrary. Examrples. dz dz 1. If Idf Yd O Mdx - Ndy = xdx + ydy, which is made integrable by the factor 2, and we have x -- y2 = u, and z = q(xz- y2), which is the general equation of a surface of revolution. 314 INTEGRAL CALCULUS. 2. If Ydz + dz 2,.If sl W- WX Mdx - Ndy = ydx - xdy, which may be -integrated by the aid of the factor -; whence x = t, and z= q ). APPLICATION OF THE CALCULUS TO THE DETERMINATION OF CURVES WITH PARTICULAR PROPERTIES. 231. By means of the preceding principles, we are often enabled to deduce the equation of a curve which shall possess a particular property. I. Let it be required to find a curve whose subnormal shall be constant. The constant being denoted by a, we place the general expression for the subnormal, Art. (85), equal to a, and have dy Y dy = a, whence ydy = adx; and integrating, 2 = ax +- C, y2 _ 2ax + 2C, the equation of a parabola. II. Find the equation of a curve whose subtangent is constant. Place dx dy Yd = a, whence a- = dx. ydyy - - a, $y INTEGRAL CALCULUS. 315 Integrating, a ly = x, or x = log y + C, the equation of a logarithmic curve, the modulus of the system being a. III. Let problem I. be generalized, and let it be required to find a curve whose subnormal shall be a given function of the abscissa, denoted by X. Then dy Y = x, ydy = Xdx, y = 2fXdx. As a particular case, let X = -. Then 2 2 x y2 = 2J- dx =- + C. a = ZS~3 a IV. Let problem II. be generalized in like manner. Then dx dx dy dx y = X, X = y As a particular case, let X - 2x. Then ly =/ — = Ix + C, 2 y = lx + 2 C, 1y2 = l + 2C; or, denoting 2C by 1C', ly' = IC'x, y2 = C'x, the equation of a common parabola. 316 INTEGRAL CALCULUS. V. To find a curve whose normal is constant. Place the general expression for the normal, Art. (85), = a, whence y2 + Yy2d = a2 ydy dx; dx 2=a =; (a - y )2 and by integration, -(a2 _ y2) = x + C, or a2 y = (x + C)2, the equation of a circle. VI. The curve whose tangent is constant may also be found by placing dx2 y = _ a, and this problem and the preceding may be generalized as in problems III. and IV. VII. Required to find a curve, such that its normal shall be a mean proportional between a given line and the sum of its abscissa and subnormal. We have at once from the conditions, 2a ( + Y)x y ( + d/) 2a denoting the given line. Solving this with reference to dy Art. (213), we have, for the first value, INTEGRAL CALCULUS. 317 dy _a a2 2ax \2 dx y \+ -y; a'5 = v 7 whence 2adx - 2ydy - dx. 2(a2 + 2a.r- 2 2)The integral of the first member is evidently the radical in the denominator, Art. (25), and we have (a' + 2ax- y2) = - x + C, or a2 + 2ax y = (C - x)2 the equation of a circle. 232. Let it be required to find a curve which shall intersect, at a given angle, a class of curves whose equation contains but one arbitrary constant. Let the general equation of the class of curves be y: = f(a, x').......... (1), a being the only arbitrary constant; by assigning values to which, in succession, the particular curves are determined; and let x and y denote the co-ordinates of the required curve. Then if T denote the tangent of the angle at which this curve intersects each particular curve, we have T. + p'......(2) in which p and p' are the tangents of the angles which the tangents to the curves, at their point of intersection, make with the axis of X. 318 INTEGRAL CALCULUS. If the given equation be differentiated, and the expression for p' be found and substituted in (2), and then the result combined with (1), and a be eliminated, the final equation will belong to no one of the particular curves more than to another. If, in this equation, x' be made equal to x, and y' = y, since for the point of intersection of the curves these variables are equal, we shall have the. differential equation of the required curve. 1. Let the equation of the class be y' = ax'........(3), whence dx a-='; and let the angle be 45~, in which case T = 1. Substituting these in (2), and reducing, we have dy dy - a =1 +- adx dd Eliminating a by substituting its value taken from equation (3), making at the same time x' = x, and y' = y, we have dy y y dy..... 1 -. dx x x dx This, being homogeneous, may be integrated as in Art. (203) or Art. (211), and we shall obtain l/2 + Y = tan-,l C x If in this we put for x, r cos v, and for y, r sin v, by which the reference is changed to the system of polar co-ordinates, w have INTEGRAL CALCULUS. 319 = -tan v, V/x2 + y = r, $ I -- tan- (tan v), or I a- v, the equation of a logarithmic spiral, Art. (143). 2. Take the parabolas given by the equation dy' m y- mx'......(4), whence d 2 dx- 2 y- = and let it be required to find the curve which cuts these parabolas at right angles. In this case T = ao, and we must have 1 + pp' = o, or l + dy dy' = dx dx' Substituting the above value of p', we have 2yf 1 - +. Eliminating mn by equation (4), and making y' = y, y dy 1 -2x dx 0o, or 2xdx. + ydy = 0. Integrating, x2 + - = C, or 2x2 + y 2 the equation of an ellipse. 320 INTEGRAL CALCULUS. RECTIFICATION OF CURVES. 233. The rectification of a curve is any operation by which the measure of its length is obtained. In article (90), we have shown how to find an expression for the differential of an arc of a plane curve, in terms of either variable and its differential. If this expression can be integrated, we can, by its integration, obtain an expression for the curve itself. From this results the following simple rule for the rectification of any plane curve: Deduce, as in Art. (90), an expression for the differential of the arc, in terms of either variable and its differential, and integrate the result. We shall thus obtain an expression for an indefinite portion of the arc. For the length of a definite portion, take the integral between the limits designated by those values of the variable which correspond to its extremities, Art. (160), and the numerical value of this expression will be the required measure. 234. The curves represented by the general equation in which m and n are entire and positive, are called parabolas. This equation can be written 1m y = pnxn = p r.. By differentiation, we have dy = rp'x'-ldx. By substituting this in the expression INTEGRAL CALCULUS. 321 dz = vdx2x + dy', and indicating the integration, we have z -fv/dxz + dy2 = fdx(l + rp'2x-2t)2. This admits of an exact integral, when either 1 2r 21 o - - 1\. or 2rL2' or - 2r-2 + 2)' is equal to a whole number, Art. (176); and a general expression for the length of the curves may thus be found in terms of x. 3 If, in equation (1), we make r -= 2' we have Y = p'x2, or y2 =p x, which is the equation of a cubic parabola. In this case, 9 i 8 9 r z = fdx(1 +_ 27p,2X(1 -+ -p'x)2 + C. 4 2'7p'" 4 If we wish the length from that point whose abscisga is a, to that whose abscissa is b, we take the integral between the limits a and b. Let us, however, estimate the arc from the vertex, or suppose the origin of the integral to be at the point where x = 0, Art. (160); we then have 8 8 0 -27 + C, or C 2 272311 27 2,; whence, denoting this particular integral by z', = 27 = 2[(1 + p',2) 1], 21 322 INTEGRAL CALCULUS. for the length of any arc from the vertex to the point whose abscissa is x. If r - 2, we have Y= p'x2 or Y p 2 the equation of the common parabola. In this case, 4x + p, z =fdx(1 + p'2x-) 2+ d=, 4 4x which may be made rational, and integrated as in Art. (172). 235. The length of the common parabola may also be determined in terms of y. By differentiating the equation y9 = 2px, v e obtain 1dv 2 ydy = 2pdx, dx - This value in the expression z =.fvdxg + dy2, gives y2 1 + z =fdy I1 + 2 -f dy (p + y)2 which, by formula B, may be reduced to p2 + y p dy 2p +2J 2 Vp + y" But INTEGRAL CALCULUS. 323 JX~p+ y - I (vi + y2 + y)....... Art. (173); 2p2 + y2 hence 2p 2 _ YV~P2+Y) + YPi(p2~y2~y)+C If we estimate the arc from the vertex, where y = O, we have O = P p Jr C, or C P p; 2 2 and finally, denoting the particular integral by z', y Vp2 + y2 pI =.- [Z(~p' + ~ +y + Y) - lp]. 2p 2 236. For the arc of the circle, we have, Art. (90), z = V/R2 -x R sin-' R which can be expressed by a series, and the length of z determined approximately. Differentiating the equation of the ellipse, we deduce b'x dy -, dx; a-y whence bx = dz Va_ -- (a2- ) z= fdx 1+ —-n ase a oey a! eri which can only be expressed by a series. 324 INTEGRAL CALCULUS. 237. The differential equation of the cycloid, Art. (131), is I dy2 /2ry - Y2 By the substitution of this value of dx, we obtain X=- Bf + dy' I = fdy V_ y,= 2-;Jfdy (2 r-y)- 2 whence, article (158), z= -2 V2(,2r( -y)2 + = - 2/2r(2r - y) + C. If we estimate the are from the point D, where y = 2 r, we have:~ " ~ o = 0 - +, or C = O and A 2 z = DM = - 2 r/2 -,( )...(1). From the figure we see that DF = V/DC x- DI = V/2r(2r - y) hence DM = - 2DF, or the are is equal to twice the corresponding chord of the generating circle. If in equation (1) we make y = O, and denote the definite integral by z", we have z" = DMA - 4r = - 2DC, as in article (135). INTEGRAL CALCULUr. 325 238. For the rectification of the spirals, we take the expression in article (138), dz = V/dr2 + rdv 2. By differentiating the general equation r = av", we deduce dr'2 - n2a 2v-2dv2; whence, by substitution, &c, z = fav"'dv v/n' - v+....(1), For the spiral of Archimedes, Art. (140), n = 1, a and the expression becomes z - Sdv I/ +1 27r and the particular integral estimated fiom the pole may be obtained by placing 1 for p, and v for y, in the expression for z', in Art. (235); whence, after multiplying by 2-' z'.(v /1 + vet) + (V1 + v2 + v). For the hyperbolic spiral n = - 1, and expression (1) becomes z = afv-'dv /1 + v. For the logarithmic spiral, when M = 1, we have dr v = lr, dv - =, z =fdri/ = ry/2 + C; 326 INTEGRAL CALCULUS. or, estimating the arc from the pole, where r = 0, we have z' = r /2, or the diagonal of the square upon the radius-vector. QUADRATURE OF CURVES. 239. The quadratulre of a curve is the operation by which the measure of the area limited by it, is determined. To determine the area limited by the curve and either of the co-ordinate axes, we find, as in article (92), an expression for the differential of the area in terms of one variable and its differential, and integrate this. The result will be a general expression for an indefinite portion of the area. For a definite portion, we take the integral between the limits designated by those values of the variable belonging to the extremities of the limiting curve. The numerical value of this will be the required measure. 240. The value of y, taken from the general equation of parabolas, Art. (23'4), is y = p'x............. (1), which, in the formula ds = ydx, gives s = fp'x'd. + C. r -{-1 If we estimate the area from the origin, where x = 0, we have C =.0; whence s __. r + 1 r- - + 1' INTEGRAL CALCULUS. 327 that is, the area of a portion of a parabola, included between the curve, the axis of X, and any assumed ordinate, is equal to the rectangle of the ordinate and corresponding abscissa, divided by r + 1. Hence this portion of any parabola is always commensurable with this rectangle. The same result may be obtained otherwise, thus: The value of from (1) is 1 1 - — l yv ~ ~ ~ Y ydy x - Y,, whence dx d Y p- rp, and this, in the formula, gives - r + - Y r + 1+ C rp'' v as before. For the common parabola, we have r = -; whence s= y y 2,+2 + 3 For the cubic parabola, r -- whence. 2 s' = xy. 241. The value of y taken from the equation of the ellipse referred to its centre and axes, is 328 INTEGRAL CALCULUS. hence -S'= (a'- X) 2dx. a By formula B, we have 1 1 1 1 (a - x) 2dx = -x(a2 - x-)T + -a2fd(a2 -xa)2 2 But dx. fdx (a - x~)-~ = x_ = sin-' + C; fdx'a:i)~ f':/a2 x a whence, finally, b ab x s - xVa2 - x2 + - sin- - + C. 2a 2 a Taking the area between the limits A c B = O, and x = a, we have ab for x = O, s = sin-O + C C; ab. ab r for x a, s =- sin-l C =.+ C; 2 22 and for the difference, or definite integral, ab 1 s"= = CDB - th of the ellipse; hence the entire area is rab. INTEGRAL CALCULUS. 329 If a = b, the ellipse becomes a circle, of which a is the radius; whence the area of the circle is rra' = -r(radius)2. The same- result may be obtained by taking the value y = v/2Rx - x; whence s = fdxf21Rx- x; for the area of an indefinite portion of the circle. 242. In order to find an expression for the area of a portion of the hyperbola, it will be best to take its equation when referred'o the centre and asymptotes, xy = m; and, since the asymptotes are oblique to each other, we must use the formula deduced in article (92), ds - sin 3 ydx, 8 being the angle included by the asymptotes. The value y x being substituted in the formula, gives mdx ds = sin3; whence s = sin $3mlx q- C. 330 INTEGRAL CALCULUS. If we call the distance CB = 1, and estimate the area from the ordinate AB, for which x = I, we have m - 1, and C = 0; whence A S = sin 3 lx; or, since sin 3 may be regarded as the modulus of a new system of log> arithms, we have s' = logx; or, the area between the curve and asymptote, estimated from the ordinate of the vertex, is equal to the logarithm of the abscissa of its extreme point, taken in a system whose modulus is the sine of the anyle made by the asymptotes. 243. The value of dx taken from the differential equation of the cycloid, and substituted in the expression s = fydx, gives A dy which can be reduced by formula E, and finally integrated. A more simple method, however, is to obtain directly the area ALD. If we denote P'M = 2r - y by z, we shall have d ALP'M = ds = zdx, L P- r, - - --- m ds =A (2r-y):d = dy2 - y;. A c whence INTEGRAL CALCULUS. 331 s ='dy V/2ry - y2. But this is evidently the area of a portion of a circle whose radius is r, and abscissa y, Art. (241); that is, the area of the segment CFHI. If we estimate these areas, the first from AL, and the second from the point C, they will both be 0, when y 0= 0; the arbitrary constant to be added in each case will then be 0, and we have ALP'M = CFH, and when y = 2r, ALD) -= CFD 2' But the area of the rectangle ALDC - AC x CD = vr.2r = 2 rr2; hence 3 area AMDC = ALDC - ALD = 7rr 2 double of which, or the area included between one branch of the cycloid and its base, is equal to three times the area of the yenerating circle. From this we see, also, that the area included between one branch of the cycloid and its base, is equal to three-fourths of the rectangle described upon the base and axis. 244. For the logarithmic curve y = logx; 332 INTEGRAL CALCULUS. K or s = zlogx - Mx + C....Art.(169). P X M being the modulus. If we estimate from the point B, wherA x = 1, we have Y IM = -M + C, C = M, and s' = logx - Mx + M. If we take the area included between the curve and axis of Y, s - Ifxdy = fxxM- = Mx + C, x or, estimating from the line AB, for which x = 1, C =- M; whence s' = M(x — 1). If x = 0, we have s8" = - M = area Y'ABM'. If x = 2, " s" = I = area ABMS'. 245. The curve given by the equation ~'i~1 y. = -, c,' to which, as in the figure, the axes of co-ordinates A P X are asymptotes, presents- a similar case. By differentiatioi%, we obtain rNTEGRAL CALCULUS. 333 dx = 2dy. ya; whence 2dy = 2 + C. Estimating the area from the line AY, where y = o, we have o = - i! C, C = 0o,..0= and 2 y By making y = 1 = MP, we have 8"- 2 = APMD; that is, the area APMD is finite, and equal to twice the square APMC, although the curve does not touch the axis of Y at a finite distance. If we take the area between the limits y = 1, and y = 0, we have 2 area FMPX = -- 2 -- oo 246. For the quadrature of spirals, we take ds - rdv Art. (138), or s....(r The value of r' taken from the general equation of spirals, Art. (139), is r' - a2v"'. This, substituted in formula (1), gives a v~, dv Cv,2V2n+ f/a2v'tdV =2+ C. 2- 4itn + 2 334 INTEGRAL CALCULUS. Estimating the area from the pole, where v = 0 when n is positive, and co when n is negative, we have, in all cases except when n is negative and numerically equal to or less than ~ C = O and a 2V2n+1 8t = 4n + 2 For the spiral of Archimedes, n =, and a = 2; whence s = 24 rr If in this wye make v 2 r, we have -/-r which is the area PMA included within the first spire, or that described by one revolution of the radius vector. Since PA = 1, rr represents the o-. area of the circle PA; hence area PMA = of the circle PA'If v = 2(27r),.._'(47T)' ~ we have Is" ( = 3 r, s4 Trr 3 which is the whole area described by the radius vector during two revolutions. But it is plain that, during the second revolution, the part PMA will be described a second time; hence, to obtain the area PAM'B, we must subtract that described during the first revolution; we then have INTEGRAL CALCULUS. 335 8 1 7 PAM'B - - 7 - 7r; 3 3 3 and in general it will be seen, that by each revolution of the radius vector, the area before described will be increased by the area from the pole out to the last spire; hence, to obtain the area from the pole out to the rnth spire; from the whole area described during nz revolutions, take the area described during in - 1 revolutions; or take the integral between the limits v = (mn- 1)2r, and v = m2r, which gives (m2r))3 [(nz - 1)2r']3 m3 - (m- 1)3 247r' 24r2 - 3 The area terminated by the (m + 1) th. spire is then (m + 1)3 - nZS 3 7 and the difference between the two expressions gives the area included between the mth and (mn + 1) th spires, thus (in + 1)3 - 2m3 + (m - 1) = 3- — 2 - 27r - rM. 2 7r. If m = 1 in this expression, we have the area included between the first and second spire equal to 2 7r; hence, in general, the area between the mth and (m + 1) th spires is equal to fin times that included between the first and second. If the area PAC be required, AC being a portion of the second 2spirecorrepondig to the arc AD, we should have, f spire corresponding to the are AD I,, we should have, for 336 INTEGRAL CALCULUS. the whole area generated'when the generating point has arrived 27r at C, since v _ 27 +,, (2r + 2'.24' from which subtracting the area PMA, we have (2Tr + ) (27r) I+ APC — 247r 24ir= n' + + ) or, if we call AP (which has been regarded as unity), R, APC — = (7+- = L7( 1 + + 2) If AC = circumference = -, then n'= 4, and:APC = I( + 4- R+. For the hyperbolic spiral n = - 1, and the general value of s' becomes 2 2 v' which is infinite when v = 0. For the integral between the limits v = b and v = c, we have q A INTEGRAL CALCULUS. 337 In the logarithmic spiral, when M = 1, dr v. r, d -- - - rp2dv jrdr 7+2 s =J 2 =-2-'~ 4+ C; or, estimating from the pole, where r = 0 and C O, we have r2 S'= -; that is, equal to one-fourth the square described upon the radius vector of the extreme point of the curve. AREA OF CURVED SURFACES. 247. I. Of surfaces of revolution. In article (93), we have found, for the differential of the area of a surface of revolution, du = 2 7ry V/dx' + dy2; whence, for the indefinitearea, we have u = f27ry Vdx2 + dy2..........(1), the axis of X being the axis of revolution, and V/dx' + dy the differential of the arc of the generating curve. The indefinite area of any particular surface will then be obtained by deducing, as in Art. (93), the expression for the differential of the surface, in terms of one variable and its differential, and integrating the result. 22 338 INTEGRAL CALCULUS. 248. Let the line AC, by its revolution about AB, generate the surface of a right cone. The origin of co-ordinates being at A, the equation of AC is y = ax; whence dy = ad.r, and u = f27raxdx Via2 + 1 = 7rax~ /a + 1 + C. Estimating the area from the vertex, where x = 0, we have C = 0, and = rax2V/a2 + 1. Making x = AB = h, we have the area of the cone whose altitude is I, and the radius of the base BC = b,," = 7rah2V /a' + 1; b or, since a =' 2rbV/b'2 +q h2 AC 2 2 that is, the circumference of the base into half the side. 249. From the equation of the circle, we have y= v2R - 2x dy = (R- x)dx The surface of the sphere is then INTEGRAL CALCULUS. 339 =2.,my/ + (R - x)'dx2.U f /d += f2 7rRdx, or u = 2 rRx + C. Taking the area between the limits x = 0, and x = 2R, we have " = 4 TR2 = four great circles. 250. From the equation of the ellipse, we have b bOx y - - - x_,2 dy =- 2 dx; a a2y whence, for the area of the ellipsoid of revolution, u =j-~-dxa - (a — 6)x a2 _6 - x2 2 7/b a2 rrb a4 or, placing 2 — a2 - C, and R', a2 _ bu = C'fd dxR'2 _- x'~. But fJdx V/R' - x = area of a circular segment whose radius is R', and abscissa x, Art. (92). Integrating this between the limits x =0, and x = CB-a, ra and calling the segment CBFG = D, we have 1,,._ _ _' u"-= C'D -- area of ellipsoid. E A "-C2~~~~~ 340 INTEGRAL CALCULUS. If a = b in the primitive value of u, we shall have u = -f27adx = 2Trax + C, for the surface of the circumscribing sphere. Let the area of a paraboloid of revolution be determined. 251. By the substitution of the value of dx, Art. (237), in the general expression for u, we have for the surface generated by the revolution of a cycloid about its base, u = 2 ri/~fydy (2r - y)-. Placing 2r - y - z, and integrating as in Art. (159), we have U = 12 r - 4.(2 r-y)2 + C.(2r- )) + C. Taking the area between the limits y = 0, and y = 2 r, we have 32 3 for one-half the surface. The whole is -~4 the area of the generating circle. 252. II. Of curved surfaces generally. In article. (150), we have found, for a partial differential of the second order of a surface, the expression d2u = dxdy 1+ + (-. ) + (d ) () INTEGRAL CALCULUS. 341 If we differentiate the equation of any surface, first with reference to one independent variable, and then with reference to the other, and find expressions for the partial differential coefficients and d in terms of x and y, and substitute in (1), and then integrate between proper limits, we shall obtain an expression for a definite portion of the surface. For the sphere, we have xI + y2 + Z = 2; whence dz x -X dz z vR/ _ x_ 2 y dz y -- y ddy zRy - (dz)/ dy) VR2g ___ y' _'y and /r2= Rgdxdy Making 1/R _- y2 = R', and integrating with reference to x, we have u = SRdyj -= fRdy sin-l R _ SRdy (sin-' + Y). VR 2 -_ y 2 342 INTEGRAL CALCULUS. Taking the integral between the limits xZ 0, and x = ce' = / - y we have fihX ~ z~u = JRdy2' |'A 0 Integrating again with reference to y, we have u — 2Y -+ C, and between the limits y = 0, y = R, 2' for one-eighth of the surface. The entire surface is then 4 7rR2. CUIBATURE OF VOLUMES. 253. The cubature of a volume is any operation by which the measure of its contents is determined. I. Of volumes of revolution. For the differential of a volume of revolution, we have found, Art. (94), dv =- rryIdx; whence v = f-ry'dx. For the cubature of any particular volume, we find, as in Arlt. (94), an expression for its differential, in terms of one variable and its diferential, and then integrate; the result of the integration will be an expression for an indefinite portion of the volume. INTEGRAL CALCULUS. 343 254. Let the rectangle ABCD revolve about AB/and generate a right cylinder. The origin of co-ordinates being at A, the equation of DC will be y = AD = b, then - = frydx = f'rbd = rrb'x + C.. Taking this between the limits x = 0, and x -= AB -, we have v" - 7rbh = the base into the altitude. 255. The equation of the ellipse gives Y = 4( - xZ); A c a whence, for the ellipsoid of revolution, v =/ Sr- (aS - x)dx X + C. Estimating the volume from the plane through the centre perpendicular to the transverse axis, we have x = 0O C = 0, and v= b_ (a2 -3_ Making x = a, we obtain, for one-half the volume, V b2 ( _ a ) = -rb'a; and for the whole, 344 INTEGRAL CALCULUS. 4 2,-rbba = rrbI X 2a; 3 3 or, equal to two-thirds of the circumscribing cylinder. If the same ellipse revolves about its conjugate axis, we have v = Jfrx'dy = fJr-(b2 - y2)dy, which, between the limits y =- b, and y = b, gives 4 2 v - rab - 7ra X 2b. 3 3 The latter volume is called the oblate spheroid, and the former the prolate spheroid; and we have the proportion the prolate: the oblate:: rbla: ra2b:b: a. 3 8 If in either expression a = b, we have 43 a3 = volume of a sphere. Let the origin be now taken at A, when ba y2 = b(2a-. x), and the volume be determined. Give also the cubature of a sphere directly, by using the equation y2 + x2 = R2. INTEGRAL CALCULUS. 345 256. Give also the cubatures of the following volumes of revolution: The right cone, v" base x'- of altitude. 2. The paraboloid, v"- 1 circumscribing cylinder. 3. The volume generated by a given portion of the common parabola revolving about the tangent at its vertex, v" = cylinder with same base and altitude. 4. The volume, the bounding surface of which is generated by the curve whose equation is y2 _ 5. The volume, the bounding surface of which is generated by one branch of the cycloid revolving about its base. 257. II. Of volumes bounded by any surface. We have found in article (151), for the partial differential of a volume limited by a surface and the co-ordinate planes, the expression d2v = zdxdy..........(1). To obtain an expression for the volume, we have simply to deduce from the equation of the bounding surface the value of z in terms of z and y, substitute it in (1), and then take the integral between proper limits. The result will be an expression for a definite portion of the volume. To indicate the process, we place equation (1) under the' form d d = zdy. Integrating with respect to y, 346 INTEGRAL CALCULUS. dv dx = fzdy + X. From this, dv = dxfz dy + Xdx. Integrating with reference to x, v = fdxfzdy + fXdx; or, Art. (196), v = lfzdxdy +f Xdx + Y. The integral fzdy + X is evidently the area of one of the parallel sections dMd', Art. (239). To obtain the whole volume represented in z the figure, we must first take the integral between the limits y = 0, and y = bd', this value of y being that - deduced in terms of x from the equation -.- 4 i x,.,< of the curve Yd'X, and then the second X 1F ~"' integral between the limits x = 0, and x = AX. To illustrate, let us determine the volume of the pyramid ABD-C; the equation of the plane BDC, being x -- 2y + 3z - 2 = 0; I;n whence 2 - 2y - 3 xJa The equation of DC is x + 2y = 2, or y 1 - INTEGRAL CALCULUS. 347 2 AD = 1, AC = 2, AB -, v= fdxdy -- (2- 2y - x) v = f zdxdy = SdxfSdy Integrating with respect to y, 2y- y12- xy; v = fdxz +X; or, taking the integral between the limits y = O, and y= bd' 1 - 2' X2 x2 XS 1 —x x- x + 4 2 12 v=fdx = - +- C. Taking this between the limits x = 0, and x = AC = 2, we obtain for the volume, 4 1 2 2 1 1 v - x - x 1 x - = -AB X AD x -AC 18 2 3 3 2 3 = BAD x AC. 258. As the first integral with respect to y will often be complicated, it will be better, if possible, to obtain directly an expression for the area of the parallel section as dMd', in terms of x, multiply this by dx, and then integrate between the proper limits x = 0, and x = AX. Thus, for the elliptical paraboloid (see Analyt. Geometry) whose equation is 348 INTEGRAL CALCULUS. ny'2 + piz' m x, by making first y = 0, and then z - 0, we have, for any section at a distance x from the plane YZ, I __ z m = = bd, n / Y.~ and for the area of the entire section, 7rbd X bd' = rr. Vnp Multiplying this by dx, we have MI;xdx 7ITnn? IX 2 dv = r, and v - C. Vnp 2 -Vnp Taking the integral between the limits x = O, and x = AB = h, we have 7rmt"h2 rrn?" h V= -- X -. 2 V/np 1np 2 The first factor of this is the area of the entire ellipse DBD'; hence, the volume of the paraboloid is equal to half that of the circumscribing cylinder. In like manner, it may be shown that the volume of an ellipsoid whose equation is b2c X2 + ac 2y2' a-2b2z = a2b2c', is equal to two-thirds that of the circumscribing cylinder. 'PART III. CALCULUS OF VARIATIONS. FIRST PRINCIPLES. 259. A FUNCTION may be regarded as given, when the form of the algebraic expression, which determines the relation between it and the variable or variables, is given, and the constants which enter this expression are known. In this case, the only change which the function can be made to undergo, is that which arises from a change in the variables. When these variables receive infinitely small increments, the corresponding infinitely small increment or change of the function is taken for the differential of the function, Art. (88). All our previous applications of the Calculus have been made to functions of the kind above referred to, and the term differential can, with propriety, be applied to no other change. It will at once be seen, that if a function be not given as above described, but merely subjected to certain conditions, it may be made to undergo a change by altering the relation which exists between it and the variables; and this may be done by changing either the form of the expression for the function, or the constants which enter it, in any way consistent with the given conditions. Now, if such a change be made as to give another function consecutive with. the first, the infinitely small change which the first undergoes is 350 CALCULUS OF VARPIATIONS. called its variation, and the corresponding changes of the variables are their variations. The difference between the terms "differential" and "variation," will be made more plain by geometrical illustration. Let BC be any curve, a function of x, Art. (90), of which M antid M' are any two consecutive points, the co-ordinates of AI being x and y. Now, if the constants --.... — which determine the curve be clanged,,I' in anv way so as to give a ditffilent."': i curve 13'C', inrfinitely near to BC, and jno:M>.is qso that the points AM and M' shall take the positions mt and mn', Pp. will be the A 1 J' P'p X variation of x, and TmS the variation of y, while PP' is the differential of x, andt M'Q the differential of y, Art. (88). The conditions under which the variation is made, may be such that one of the variables will have no variation; and when this is the case, the operations to be performed Yn',,.__will be much simplified. Em Ih, _ Thus, if it be required that the points f/ Mt LJ M and M' shall be found in lines parallel / -I Q to the axis of Y at m and in', Mm will be the variation of y, while x has no A i 1 --- variation; the differentials of x and y being PP' and M'Q, as before. As the differential is denoted by the symbol d, the Greek character 8 is used to denote the variation; and from the illustrations just given, it appears that while the former symbol denotes the changes which take place in passing from one point to another of the samne curve, the latter is used for a very different purpose, to denote the changes in passing from points of one curve to the corresponding points of another infinitely near to it. 260. From the nature of the term as above explained, we see that to obtain the variation of any function of x, y, z, &c., we CALCULUS OF VARIATIONS. 351 have only to put for x, y, z, &c., x + Jx, y t- Jy, &c.; ax, Jy, &c., being, not arbitrary, but the infinitely small changes which take place in'x, y, z, &c., in consequence of that change in the function which gives its variation; and then take, as in the Differential Calculus, Art. (52), those terms of the development which are of the first degree with reference to the variations of the variables. Or, since the development may be made precisely as in Art. (51), by substituting Ax, Jy, &c., for A, k, &c., it is plain that we shall have du du du au d= x + du y + d Sz + &c. dX dy d It is also plain that the principles contained in articles (15) and (17), as also the particular rules demonstrated in articles (20) (26), are equally applicable to variations. 261. In the function u f(x)...........(1), let us substitute x + 6x for x, and denote the new function by f'(x); then, by the definition, Art. (259), s6 = f'(x) - f(z)......(2); and since, from the relation expressed in equation (1), x is a function of u, the second member of equation (2) will be a function of u, and we may write u = (u).......... (3). If, in this equation, we put for ue, u + du = u', we shall have J' = p (u'), 352 CALCrULS'OF VARIATIONS. and, subtracting equation (3), au' - = Q) (u') - pq () = dQ (u) = du. Taking the variation of the expression Ut - U = du, we have du'- Ju = 6du; hence -du = dmu...........(4). That is, the variation of the dif'erential of a function of a single variable is equal to the differential of its variation. Or, when both of the symbols d and J are prefixed to a function, the order in which they are written, or in which the operations indicated, are performed, can be changed at pleasure without affecting the result. The principle above enunciated is true for any order of the differential; for if, in equation (4), we put du for u, we have Cd (du) = ddu,'or 6Cd2u = dd6u = d26u. If, in the last equation, we put du for u, we have Md2(du) = daJdu, or 6d3u = d'6u, and so on; hence we may conclude that 6d"u = dutu. 262. Let v be any differential of a function of x, and place,fv = v', then dv' = v, odv'= v, or ddv' = 6v, CALCULUS OF VARIATIONS. 353 and by integration, Jv' = Jfv, or f J -= jfv. The principles demonstrated in this and the preceding article, are evidently true for functions of any number of variables; since the variation of the differential of such a function is but the sumn of the partial variations, and the converse. 263. In order to consider the subject of variations in its most general sense, when applied to differential expressions, we must regard the differentials of all the variables as variable, as well as the variables themselves. In this sense, if u be a function containing x, y, and their successive differentials, we shall have, Art. (260), u- = MWx + M'6dx + M"d2'x + &c. ) + NMy + N'Jdy + N"8d2y + &c. ) in which M, M', M", &c., are the partial differential coefficients of u taken with respect to x, dx, d2x, &c.; and N, N', N", &c., the corresponding ones taken with respect to y, dy, d2y, &c. This expression may be extended to any number of variables, by adding for each, an expression of the form M6x + M'6dx + M"6d'x + &c.; and may then be made to give every particular case which can arise, by making the particular suppositions upon dx, d'x, dy, d2y, &c., which the case requires. 264. If the differential expression contains only the variables x, dy d'y Y' = P' d x2 - q &c., we may denote it by v, and shall have, as in Art. (260), 23 354 CALCULUS OF VARIATIONS. Sv = MJx + NMy + N'Sp + N"Sq +- &c.... (2). And if this expression be taken in its most general sense, dxr must be regarded as variable; in which case, we put for Sp, 6q, &c., their values obtained as in Art. (26), viz.: -p _ dy _dxSdy - dySdx dSy - pd6x dx - dx2 - dx dp -dxdp - dpCdX d~p qddx dx dx'" dx If dz be regarded as constant, equation (2) is under its most simple form. 265. If we indicate the integration of both.members of equation (1), Art. (263), we have fdu = (Mlx + M'6dx tJ M"Wd2X + &c.).... + f(NMy + N'Sdy + N"Jd2y + &c.) By the application of the rule for integrating by parts, we find fM'6dx - fM'dsx = M'6x - fdM'S&; jM"6d2 - f M"d 62 = M"d6x - fdM"d6x = M"dSx - dM"6x + fd2M"6x; fM,"'.d3x -= JM"'d1dx3R M"'d'6nx - fdM"'d26x = M"'d2Sx - dM"'dSx + fd2M"'ddx = M"'d2x. -dM"'dJx +- d'M"'x - fd3M"'6x. Also, CALCULUS OF VARIATIONS. 355 fJN'Jdy = N'Jy - fdN'6y; f N"6d2y = N"d6y - dN"Jy + fd2N"6y; fN"'ad3y - N"'d26y - dN"'dly + d2N"'Jy - fd3N"'6y, Observing that the second member of equation (1) is equal to the sum of the integrals of the terms taken separately, and substituting the above values, we obtain &-u= (M'-dM"+d2M"' —&c.)6x + (M"-dM"'+ &c.)dZx + (M"'-&c.....)d'6x + &e. + (N'- dN"+ d2N"'- &c.) Jy + (N"- dN"'+ &c.) d6y + (N"'-i &c.....) d2y + &c. + f(M - dM'+ d2M"-d3M"'+ &c.) Jx ).....X (2). + fJ(N - dN'+ dN"-dN"'+ &c.)Jy ) By examining the above expression, it will be seen that there is no term under the sign / which contains the symbols d and a applied the one to the other; and also that the parts. containing Sx are exactly similar to those containing Jy. The formula may therefore be extended to any number of variables, by adding, for each new variable, similar parts containing its variation. 266. It should be remarked, that if the multipliers of 6.r and 6y following the sign J; in equation (2) of the preceding article, are both equal to zero, fSu will be complete, or Ju will be the differential of some function. But in the expression J6- = V6u, 356 CALCULUS OF VARIATIONS. it is evident that if fu contain any terms which cannot be freed from the sign f, 6f u must contain the variations of these terms still under the sign, and.f6u cannot be complete. Hence, if &ue is a differential, u itself must be so. And conversely; for if fau is entirely freed from the sign f, then 6fu cannot contain this sign, and its equal f6u must be complete, or 6u be a differential. lfence, if the conditions M - dM' + d2M" — &c. = 0, N - dN' + d2N" - &c. - 0, are satisfied, u will be the differential of some function, which may be obtained by integration. If the above conditions are not satisfied, u cannot be an exact differential, and f u cannot be obtained, 267. If we take the variation of the expression fvd., in which v, as in Art. (264), is a function of x, y, p, q, &c., we have, Arts. (21) and (155), 6Svdx = f (vdx) = vSdx + f dx6v. But, Art. (169), Sv6dx -f vd6x v= rx - fdvLx; hence 6Svdx = vA + f (dx6v - dvx).......(l). Substituting in that part of the second member which follows the sign Jf, the values of dv and 6v, Arts. (52) and (264), dv = Mdx + Ndy + N'dp + N"dq + &c., 6v = MWx +.N6y + N'Jp + N"6q + &c., we have CALCULUS OF VARIATIONS. 357 dx&v - dvx = N(dx&y - dyJx) + N'(dx&p - dp.v) + N" (dx$q - dqpx) + &c............ (2). Since dy = pdx, we have dx6y - dyx = dx(y - p6x) = dx, by making 6 - pSx = Wo. Also, if for Sp, we put its value, Art. (264), we have dxdp - dp6x = dly - pdzJx - dpyx = d (y - px) = do. If, in this last expression, we put p for y, and q for p, and recollect that q - dp we have dxp - dpx- dw' dxq -dqx = d (Sp - qx) = d ( dx =d kdxJ Substituting these values in equation (2), and prefixing the sign. we have J'(dx-v - dvx) = fNwcdx +./N'dc + JN"d (d) + &c... (3). Again, by Art. (169), fN'dc = N'w, -- dNwx dx, JfN"dd = N" d d dr d,:x dx d- o dN" (dN"' dx dx dx dxr 358 CALCULUS OF VARIATIONS. Now, substituting these expressions in (3), and the result in (1), we obtain dN" dw 6fvd = vax + (N- dx + &c.) + (N" - &c.)d + &c. dN' 1 dN" + f(N d- +' d d _ &e,) wdx....(4). If we now put for w, its value Jy - pdx, the part affected with the sign f will become dN' dtN' f (N - + &c.) dxAy - (N - d-+ &c.) pdxx. From which we see that, in this case, the coefficients. of 6y and Sx have such a relation that if one becomes equal to zero the other will. M~[AXIbMA AND hMINIMA OF INDETERMINATE INTEGRALS.'268. The principal, and far the most important application of variations, is to the determination of the maxinma and minima, of indeterminate integrals, that is, of integral expressions of the form /v-dx-+ dy2, frryIdx, &c, containing x, y, &c., and their differentials; in which the relation between the variables is entirely unknown. Thus, if it be required to determine the relation between x and y, in order that f7ry'dx taken under certain conditions, shall be a maximum or minimum, the problem is one not capable of solution by the ordinary method of article (69), since the principles there developed require the form of the function to which they are to be applied, and the constants which enter it, to be given, and the search is for particular CALCULUS OF VARIATIONS. 359 values of the variables, which will make one or more values of the function a maximum or minimum; whereas the object now proposed, is to ascertain what this form and these constants must be, in order that the function, when subjected to the given conditions, shall be a maximum or minimum, the variables being entirely indeterminate. Questions of this kind are readily solved by the aid of variations. 269. Let u be a function of the nature discussed in Art. (263), and suppose x, dx, y, dy, &c., to be increased by their variations; and let the difference between the corresponding function u' and u be developed, which is done at once by putting (rx, by, dx, &c., for h, k, 1, &c., in the development of Art. (51); we shall thus obtain u' — u = M6x + N6y + M'Jdx + N'6dy + &c., plus a term of the second degree with respect to 6x, 6y, &c.; plus other terms. By the same course of reasoning as that contained in Art. (77),. we see that u can be neither greater nor less than u', for all values of 6x, by, &c., unless the term, of the first degree with reference to these variations, is equal to zero. But this term, Art. (263), in the variation of u': Hence, in order that u-be a maximum or minimum., ue inust be equal to zero. If the conditions which. make the variation of u equal to zero, make the termi of the second degree, in the above development, positive, for all values of' x, by, &c., u will be a minimum; if negative, u will be a maximum. The discussion of the various circumstances in which this term will not change its sign, is of too complicated a nature, and likely to lead too far, for an elementary treatise. Neither is it necessary in general, as we shall be able, from the nature of nearly every case, to determine, without a reference to this second term, whether we have a maximum or minimum. 360 CALCULUS OF VARIATIONS. 270. In the application of the foregoing principles to the indeterminate integrals referred to in Art. (268), it may at first be remarked, that if the integral be indefinite, Art. (160), from its nature it can have no maximum nor minimum. The application can then only be made to definite integrals, or those which are taken between some well-defined limits. If, then, it be required that fu be a maximum or minimum, we may write the variation of fu, Art. (265), thus: Sfu = f6u = m6x + n6y + m'Jdx + n'6dy + &c., +r f(klx + k'6y)..........(1); and this, when taken between the prescribed limits, must be equal to zero. We have seen, Art. (266), that this expression cannot be integrated unless the quantity which follows the sign f, in the second member, is equal to zero; that is, there can be no integral to be taken between limits, and of course no maximum nor minimum. We must then have, for the first condition, kNx + k'6y = 0........... (2). If, in the particular case under discussion, the variations of x and y are entirely independent of each other, we must also have k = 0, and k' - 0; or, Art. (265), M - dM' + dM" — &c. = 0 (3). N - dN' + d'N" - &c. = 0) Again, if we denote by I and 1' the results obtained by substituting'the limits in succession in the remaining part of equation (1), we must have, for a second condition, 1'- I = 0.............(4). CALCULUS OF VARIATIONS. 361 Should there be more than two variables in the function u, the quantity following the sign f, in equation (1), will consist of as many terms as there are variables, each of which, if the variations are independent of each other, must be placed equal to zero, and will thus give an equation expressing a relation between these variables and their differentials. If, however, the conditions under which the variations are made are such as to render these variations in any way dependent, we shall be able, by means of the equations which express these conditions, to eliminate from equation (1) one or more of these variations; then, by placing the coefficients of those which remain under the sign f, equal to zero, we shall have a system of equations from which we may determine the nature and extent of the required finction. The system of equations (3) will, in every case, express the relation which must exist between the variables and their differentials, in order that the function shall be a maximum or minimum; but they must be subjected to the conditions deduced from the equation 1' - 1 = 0, which can, of course, contain no variables except those which belong exclusively to the limits.,Where u is under the form vdx, it has been seen, Art. (267), that the two equations (3) will both be satisfied, if one is. They will therefore give but one independent equation, viz.: dN' 1 dN" N- d + - d - &c. =- (4); d~: dr dx and the condition 1'-I- = O must be deduced by substituting the limits in that part of equation (4), Art. (267), which is independent of the sign f. The solution and discussion of the following problems will serve to illustrate and more fully develop the preceding principles. 362 CALCULUS OF VARIATIONS. 271. Problem 1.-Required the nature of the shortest line joining two given points in a plane. Let x', y', and x", y", be the co-ordinates of the points. The g neral expression for the length of the line, Art. (234), is z = f /_dxI + dy2. Taking the variation of this, we have Jfu f] ( dx + d Jy;) which, upon comparison-with equation (1), Art. (263), gives dr d y M = O N, = O, M'- N'= dz' dz' and all the other terms equal to zero. In this case, since Sr and (iy are independent of each other, we use equations (3) of the preceding article, and have dx dy d — O, and d dz dz whence, by integration, dx dy d = c, d c'. Eliminating dz, and integrating again, we have C, dy ='-dx = adx, y = ax + b....(1) which gives the required relation between y and x, and indicates that thle linell must be straight. CALCULUS OF VARIATIONS. 363 The first part of equation (2), Art. (265), becomes M'Jx + N'My. Since, in this case, the limits x', y', and x", y", are absolutely fixed, we must have Jx', Jy', &c., equal to zero, which, being substituted in the above expression, give M'6x' + N'6y' 0, M'6x" + N'Jy" = 0; whence results the fulfilment of the second condition, 1' - 1= ~, and it remains only to determine the constants a and b, in equation (1), on condition that the line shall pass through the two. given points. 272. Problem 2.-Re-quired the shortest line that can be drawnfrom one given curve to another, in the same plane. Let y = f.(x), and y = f'(x), be the equations of the curves; their differential equations being dy = p'dx, dy = p"dx....... (). As in the preceding problem, we have Z w hVdi 2 + dayeis 6lf U - bo dt + e y from which is deduced, precisely as before, the equation of tile required line, y = ax + b............ (2). 364 CALCULUS OF VARIATIONS. But since the ends of this line must be in the given curves, the variations of x and y, at the limits, must be confined to these curves, that is, &y', Wx', Wy", &X" must be the same as dy and dx in eqhations (1); whence y' = p'Jx', jy" = p"Sx". Substituting these, in succession, in the first part of equation (2), Art. (265), and subtracting the results, we must have dx' dyl d" d,,\ I'_ = (i-~ + -d')x'- (- + dp")& " - o; and since this contains two independent variations, it can only be satisfied by making the coefficients separately equal to zero; hence dx' + dyp' = 0, dx"+ dy"p"= O; whence dy' 1 dy"' 1 dx' - p dx' &"' But these are the equations of condition that the required line shall be normal to both curves at the points (x', y'), (x", y"), respectively, Art. (84). In order to determine the constants a and b, in equation (2), we must first find the values of x', y', x", y", on condition that the normal to the first curve at the point. (x', y') shall also be normal to the second at the point (.'", y"), and then cause the line to pass through these points. This problem and the preceding may also be solved by placing z = fV'dx2 + dy2 + dy2) dx = fvdx, dy in which v is a function of d p. In this case, we should use equations (4), Arts. (267) and (270). equations (4), Arts. (267) and (270). CALCULUS OF VARALTIONS. 365 273. Problem 3.-Required the shortest line, on the surface of a sphere, joining two given points of the surface. Let the equation of the sphere be x +- y2 + z = R'.............(1). The general expression for the length of a line joining the two points will be, Art. (91), s= Vdx2 + dy2 + dz2 the variation of which is f3~ =/( dx d + d ydy+ d4; whence, by adding an expression containing 6z to the second member of equation (2), Art. (263), and comparing, we find M = 0, N 0, P =0, dx dy dz M'~ N' d' I_ ds' ds' = ds' and all the other terms equal to 0. The first condition required in Art. (270), is then (d \x +,Id y ++ I(d 6\ = 0.... (2). ds I ds kds In this case the variations are not independent, but must be confined to the surface of the sphere; that is, taking the variation of equation (1), we must have2zxx + 2y6y + 2z6z = 0. '366 CALCULUS OF VARIATIONS. Combining this with equation (2), and eliminating 6z, we obtain (d d _ d d>x + (d d jy ) which, containing two independent variations, gives dx dz d dz zd xd - 0 zd yd O ds d8 ds Now, if we regard ds as constant, these equations become d2x d2-z d2y d2z z -— x -O -Zd - y- 0, ds ds ds ds froom which we deduce d2y dcx x — y - O. ds ds Integrating the last three equations, we have dx dz dy dz dy dx z - a, z - y -- b, X Y =c. zdsxdsa, dT ds ds ds Multiplying the first by y, the second by - x, the third by z, and adding, we obtain ay - bx + cz = o.......(3), which is the equation of a plane passing through the centre of the spllere. The required curve must lie in this plane, and therefore is the arc of a great circle. The limits in this case, as in problem 1, being absolutely fixed, \we have at once, as in that problem, the fulfilmlent of the second condition, L' -- = O. CALCULUS OF VARIATIONS. 367 Equation (3) may be put under the form a b c Y c - + z _ 0~ or z c'x + d'y, and the constants c' and d' determined, by causing the plane to pass through the given points. 274. In many cases where there are conditions confining the variations, whether at the limits or not, the method of reducing the number of independent variations, explained in Art. (270), and pursued in Arts. (272, 273), will be found of very difficult application. In all these cases, the following less direct, but very elegant method may be used. Let r = S - 0, &c., be the eqqations between x, y, &c., expressing the conditions to which the variations are subject; then, at the same time that we have fu - 0,O we must also have =r 0-, s -= 0, &c.; or, denoting by c, c', &c., arbitrary constants, we must have the equation Vfu 4- car + c's +- &e. = 0......(1), for all values of the variations of x, y, &c. Placing the coefficients of these variations separately equal to zero, we obtain equations from which we can eliminate the constants c, c', &c., and 368 CALCULUS OF VARIATIONS. thus deduce an equation or equations which will express the proper relation between x, y, &c. As an illustration, let us take Problem 4.-Required the nature of the line, of a given length, joining two points, which, with the ordinates of the points and axis of X, will inclose the greatest area. In this case we have, Art. (239), 6f u = Jfydx; and since the length of the are between the limits is to be constant, the variations must be subject to the condition fdz = f /dx' + dy2 = a; hence Sf/Vdx~ + dy2 0. Equation (1) will then become f ydx + c6f A'dx 2 + dy = 0; or, putting for the variations their values, we have /(yx + dxy ~ cydx dx + e6dy) =O. Comparing this with equation (1), Art. (263), we see that dx dy M = 0, M' = y + CT, N= dx, N = c and these, being substituted in equations (3), of Art. (270), give -- d y + - d cd d( ) =; and by integrating, CALCULUS OF VARIATIONS. 369 dx dy y + C x - v.. Eliminating c from these two equations, we obtain dy x- a' which is evidently the differential equation of a circle whose equation is, Art. (98), (Y - f)2 + (x - ), f3 a, and R being arbitrary constants, which must be determinedon condition that the circle pass through the two given points, and that the included are be of the given length. 24