we might proceed to find the differential coefficients of higher orders by a continuation of precisely the same kind of processes. COR. Suppose that u = f(x, y, z) = 0, there being no other equation connecting x and y: z will thus be a function of x and y. Then, from the equations of Art. (27), dy2 dy dydz dy D2u d'u d'u dz = + + + + dx2 dz dx2 dz2 + dz dy' dy2 d'u dz + du d2z dz' dy' = = 0, d'u dz dz du d'z dxdy dxdy dydz dx dxdz dy dz2 dx dy dz dxdy From these five equations we can determine dz dz d2z d2z d2z dx' dy' dx2' dxdy' dy' = 0. the partial differential coefficients of the implicit function z, in terms of the partial differential coefficients of u, and therefore in terms of the variables x, y, z. Change of Variables. 53. Let it be proposed to change the variables of an equation dy d'y d3y x, Y, dx' dx2' dx3 from x and y into two variables s and t, t being, in the transformed equation, and x in the proposed equation, the independent variable. We suppose s and t to be connected with x and y, which by virtue of the equation (1) are functional of each other, by two equations Differentiating these equations successively n times each, dry supposing to be the differential coefficient of highest order dx" in (1), and regarding x, y, s, as implicit functions of t, we shall get 2n equations which we will denote by From these equations, in conjunction with the equations (2), we may obtain expressions for the 2n + 2 quantities But, by Art. (47), we are enabled to obtain expressions for Hence we are able to obtain expressions for y and its n differential coefficients with regard to x, in terms of s, t, and the n differential coefficients of s with regard to t. The equation (1) may therefore be transformed by substitution into an equivalent 54. Let z be a function of two independent variables x and y. We propose to express the partial differential coef ficients of z, taken with regard to x and y, in terms of those of another function r, taken with regard to two other independent variables and 4. The six variables x, y, z, r, 0, 4, are supposed to be connected together by three equations any four of the six variables, since z is by the supposition some function of x and y, being thus functions of the two remaining ones, which will be entirely arbitrary. It is evident that r may be regarded either as a function of x and y alone, or as a function of 0 and alone, 0 and 4 being in the latter case regarded each of them as a function of x and y alone. Hence we see that, first considering y and next x as constant, coefficients of r, 0, and 4, with regard to x and y, when r, 0, and 4, are expressed entirely in terms of x and y; dr dr being the partial differential coefficients of r, with do' do' regard to and p, when r is expressed entirely in terms of ✔ and . Again, from the equations (1), we have, y being considered constant, DF dF de dF do dF dr = + dx dz dx dF dF dz + + + = 0 dr dx + + dr dx + + dFdF, dz + dx dz dx |