Also, since z is a function of x and y alone, and since the expression D (dz dx dx dz dx' denotes the total differential coefficient of a function of x dz and y, only so far as the variation of is affected by the varia dx tion of x when y remains constant, it is plain that d2z where is the second partial differential coefficient of z with dx2 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), X we shall obtain, by the simple repetition of the preceding reasonings, D3u d'u d2u dz d2u dz2 du d2z +2 + dx2 dx2 dxdz dx dz2 dx2 + 0, dz dx2 + d'u d'u dz dxdy dxdy dydz dx From these five equations we can determine d'u dz d'u dz dz du dz 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 ƒ x, y, dx' dx2' dx3 )= = 0. .(1) 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, 0, 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 4 alone, 0 and 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 o, 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 regard to and 4, when r is expressed entirely in terms of e and p. Again, from the equations (1), we have, y being considered From the former of the equations (2) combined with (3), we dr dy dy dz dy dF dr dF dF dz 2 2 2 = 0 dr dy dy dz dy in terms of de' do and, from the latter of equations (2) combined with (4), we may find de do dr dz Proceeding to the second order of partial differentiation we shall have, from (2), expressions for |