in the equation (2), or their equivalents du du in the equation (1), are called the partial differential coefficients Du of u with regard to y,, y2, respectively. Finally, is called the total differential coefficient of u. 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. 29 21. Let u = f(y1, Y2, y1), a function of three variables y1, Y2, Yз, each of which is a function of x. Then if u', y, y2', y', be simultaneous values of u, y1, Y2, Y,, we have u = ƒ (Y1, Y2, Y3), u' = ƒ (y1, Y2, Y3'), Du coefficients of u with regard to y1, y2, Yз, respectively; and dx the total differential coefficient of u with regard to x. If we adopt the suffix notation, the equation (1) may be written Du du dy du dy2 du dy + 2 = + dx dy, dx 2 dy, dx dy, dx .(2). Multiplying the equation (2) by dx, and putting we have The same theorem may evidently be extended to any number Y1, Y2 Y3,....Y,, being any n functions of x, then Du = n du dy+ dx dy dx dy, dx dy, dx or, replacing du in the expressions du , du du dy, dy' dys by du, du, du,.... du respectively, and clearing the equation of fractional forms," 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. COR. If any one of the quantities y1, Y, Y,,.... y y1 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', ', y', being corresponding values of v, x, y, v = p(x, y) = 0, v' = 4 (x', y') = 0, v' – v = $(x', y') -- p (x, y) = 0, p(x', y) − p (x, y) ̧ $ (x', y') − p (x', y) y' - y + x' y' - y = 0..(1). Now in the limit, when ' approaches indefinitely near to x, and, first replacing x' by x, and then y' by y, Þ (x', y') — $ (x', y) _ p (x, y') − p (x, y) _ dp (x, y) _ dv dy (2). This result gives us the expression for 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 dv in the numerators of the fractions dv dv dx' dy by the expressive forms dv, dy, we have, transforming the equation (2) from differential coefficients to differentials, Dv=d_v + d2 = 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. 23. If v1 where v12, V3 32 the variables y1, Y2, Y.,... .Y,, being therefore implicit functions of x, then V2, V3, V4•••Vn' together with n - 1 additional equations involving v2, 3, ... precisely as v, is involved in this. Let x', y', y', Y.,....y, v, be corresponding values of х, Уг, X, Y, Y2, Y3,....y, v,; then + f(x', Y1, Y2, Y3, ....Yn) - f(x', Y1, Y2) Y... Yn) whence, 8, 1, 2,....,, 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 v,, we have Proceeding to the limit, we have, x, y1, Y2, Yз,. . . . Y., being the ultimate values of x', y', Y, Ya',....Y be The analogous equations in regard to v2, v, ... .vn, may established in the same way. Multiplying the equation (1) by dx, we have or, if we express the different partial differentials of v by From the equation (1) together with the (n - 1) analogous equations, making in all n linear equations, we may determine the n differential coefficients in terms of the n (n + 1) partial differential coefficients of V1, V2, V39.. .v. Total Differentiation of a Function of Functions of independent Variables. 24. We have now fully considered the principle of differentiating a function of functions, the subordinate functions being dependent each of them upon one and the same variable. Suppose, however, that u = f(Y2, Y2, Y3,• • • •Yn), and that y1, y2, Y.,....y, are not all of them dependent upon a single variable, but that they are functions of several independent variables. Let u', yi, ya, ya,....y, be corresponding values of u, y1, Y2, Y3, . Yn : then |