and which we have represented by a distinct symbol Du, is evidently different from either of the numerators of the fractions du du dy,' dy In order to obviate all possibility of confusion, we might use du du the symbols du, du, to denote the numerators of dy,' dy,' the suffixes serving to point out the origin of the two differentials. Such a notation, however, although remarkably clear, would frequently be very embarrassing, especially in long operations. It will be sufficient for distinctness if we remember to regard and as fractions the denominators of which are insepa dy2 dy, rably attached to the numerators, the symbols dy, and dy,, which express the denominators, thus serving to indicate the true nature of the du in the numerators. If, however, as will be sometimes convenient, we do put du, du, instead of du, in the expressions du du we shall then be at liberty to treat these differential coefficients as ordinary algebraical fractions: thus may be written, multiplying both sides of the equation by dx, The quantity du denotes the differential of u taken with regard to y1, as if y, were constant, du the differential of u taken with regard to y, as if y, were constant, and Du the differential of u due to the simultaneous variations of Y1 and y, dependent upon the variation of x. The quantities du, du, are called the partial differentials of u with regard to y1, y2, respectively, and Du its total differential. The quantities in the equation (1), are called the partial differential coefficients Du of u with regard to y1, y2, respectively. Finally, is called the total differential coefficient of u. dx The equation (3) shews that the total differential of u is equal to the sum of its partial differentials. Differentiation of a Function of any number of Functions of a single Variable. 21. Let u = f(y1, Y2, y), a function of three variables y1, Y 2, Y 3, each of which is a function of x. Then if u', yi, ya, ya, be simultaneous values of u, y1, y2, Y1, we have u = ƒ (Y1, Y2, Y3),_u = ƒ (Y1, Y2', Y3'), u = ƒ(y1', y2', Y¿') — ƒ (Y1, Y2, Y3), x' x case of two functions, du dy + .(1). dy, dx are the partial differential coefficients of u with regard to y1, Y2, Y,, respectively; and the total differential coefficient of u with regard to x. adopt the suffix notation, the equation (1) may be written Du du dy du dy du dy = + + dx dy dx dy, dx dy, dx Multiplying the equation (2) by dx, and putting The same theorem may evidently be extended to any number of functions; so that, if Y1, y2 y3,....y, being any n functions of x, then و dx dy, dx dy, dx dy dx Du du dy du dy2+ du dy = or, replacing du in the expressions du dy, by du, du, du, ყვ Yn equation of fractional forms," Du = du + du + d u + ... + d u. It may therefore be stated as a general proposition, that the total differential of a function of any number of functions of a variable is equal to the sum of its partial differentials taken on the hypothesis of the separate variation of each of the several subordinate functions of the variable. n COR. If any one of the quantities y1, y2 y3,.... Y y, for instance, be equal to x, which is the most simple form of functionality, then from the above demonstration it is plain that we may replace the corresponding term Differentiation of an implicit Function of a single Variable. y being therefore an implicit function of x: then v', x', y', being corresponding values of v, x, y, v = p(x, y) = 0, v' = p (x', y') = 0, v' – v = $ (x', y') -- p (x, y) = 0, p (x', y) − p (x, y) + $(x', y) − p (x, y) y - y = 0..(1). X y' - y y' x Now in the limit, when x' approaches indefinitely near to x, and therefore y' to y, we have and, first replacing x by x, and then y' by y, dx' p(x, y') − $(x, y) _ dp (x, y) hence the equation (1) becomes dy dx dx dy dx (2). This result gives us the expression for dy in terms of the dx partial differential coefficients of v with regard to x and y taken successively as separately varying. If we replace the symbol do in the numerators of the fractions by the expressive forms dv, dv, we have, transforming the equation (2) from differential coefficients to differentials, Dv = d_v + d2v = 0. This result shews that if any function of x and y be always C zero, its total differential or the sum of its partial differentials is always zero. General Theory of the Differentiation of implicit Functions of a single Variable. 23. If v1 = 0, v2 = 0, v2 = 0, ....v = 0, where v1,,,,,...., are n functions of x, y,, Y 2, Y 39 • • • • Yμ> the variables y1, y2, Y ̧‚. . . .ÿ, being therefore implicit functions of x, then V3, together with n - 1 additional equations involving ",, vg, V 4 • • • V μ3 precisely as v, is involved in this. Let x', yi, y', y',....y', ', be corresponding values of X, Y1, Y2, Y3,....,,; then v1 = f(x, Y ̧› Y1⁄2, Y ̧‚• • • •Y„) = 0, v,' = ƒ (x', Y‚'‚ Y2'‚ Ys'‚···Y„') = 0 ; hence 2 0 = v,' — v1 = ƒ(x', Y‚'‚ Y1⁄2'‚ Y ̧'‚• • • •Y„') − f(x, Y1, Y2, Y3,• • • • Yn) ... = f(x′, Y1, Y2, Y3, • • • ·Y„) − f(x, Y1, Y29 Y3,• •• •Y„) + f (x', Y ̧'‚ Y2', Y3 ‚• • • · Y„) − f(x', Y‚'‚ Y2, Yз,• • · ·Y„) whence, 8, 81, d2....d, denoting the partial increments of the functions to which they are prefixed with regard to x, y1, Y2, ....y, respectively, and ▲ the total increment of v1, we have |