Also, using the notation of partial differentials, the above equations become D2u = d22u + d2u+d22u+2d,du+2d2d ̧u+2dxd ̧u+ ... = c cos (ax+by+cz), (d) =c2 sin (ax+by+cz), be sin (ax+by+cz), D2u = — {a2 dx2 + b2 dy2 + c2 dz2 + 2 bc dy dz +2 ca dz dx + 2 ab dx dy} sin (ax+by+cz). In the preceding inquiry all the subject-variables of the function have been assumed to be independent, and thus the results are general. It however frequently happens that some of them are dependent on others; and although it is unnecessary to consider generally the modifications which the results undergo in such cases, yet it is expedient to give an example, so that the student may perceive the kind of results which such problems present. Let x be equicrescent; so that d'x=0; and from the relation given between x and y in the preceding equation of condition, 82.] As one of the first and most useful applications of the general results of the last Article, let us prove certain properties of homogeneous functions, which are due to Euler, and are generally known by the name of Euler's Theorems of Homogeneous Functions. DEF. A homogeneous function of many variables is one which has the sum of the indices of the variables in every term the same; and if the sum of the indices in each term = n, the function is said to be homogeneous of n dimensions. Thus ax3 + bx2y + cz3 + exyz + gx2z = 0 is a homogeneous function of three dimensions. Let u = F(x, y, z, ...) be a homogeneous function of n dimensions and r variables; for x, y, z,... let tx, ty, tz, let tx, ty, tz, ... be written, and suppose the function to become u when these substitutions are made; then, by the definition of homogeneous functions, ... u' = v(x', y', z', ......); and therefore by (46), Art. 49, equating this to (110), because they are equal, we have (111) let t1, then xx, y=y, z=2, ...; .'. n F(x, y, z,...) = NU=X dx Again, taking the second total differential of u', and dividing whence, equating (113) and (114), and making t = 1, we have ;) + y2 (d2u) n (n-1) u = x2 (du) + 2xy (dx dy d2u d3u dy2 + .. (115) dx2 Similarly it may be shewn that d3u n (n − 1) (n − 2) u = x3 (d/34 ) + 3x3y (dx3 dy and similar theorems are true for every other order of differentials. One or two examples are subjoined, in which the theorems are shewn to be true. Ex. 1. F(x, y, z) = u = x2+By2+cz2+EYZ+GZX+HXу, which is an homogeneous function of two dimensions and three variables. Ex. 2. u = F dimensions. (du) + 2y z ( d2u d2u dy dz (d2) + 2xy dz dx d2u ( dx dy) 2 {Ax2 + By2+cz2 + Eyz+Gzx+Hxy} = = 2u. (2), which is a homogeneous function of O Of F(2) let F'(2) be the derived-function, that is, be that quantity which multiplied by dz is the z-differential of F(2); then Article 53, Ex. 4, is another case in which the preceding theorem is exhibited. 83.] As another application of the results of Article 81, let us investigate the equivalents of the second derived-functions of a function of three variables, when one, say z, is an explicit function of the other two, in terms of the derived-functions, when all are implicitly involved in an equation. That is, if z = f(x, y) and p(x, y, z) = 0 are equivalent equations, it is red2z quired to express (4), (dry), (d) in terms of (),. therefore if we estimate the simultaneous variations of x and z, let the x- and z-differential of this be taken, and let a be equicrescent; then we have therefore dividing through by de2, and replacing (dz) by its dx dz is expressed in terms of the partial derived of Similarly for the y- and z-partial differential 2 (172) (dt) (12) + (124) (dg)2 + (de)' (d) dz dy dz dy dz αφ dz2 dy dz = 0. Also taking the y- and z-partial differential of (118), and sub dy), as they are given in Art. 50, we dz dz stituting for dx ), and We proceed now to consider the theory of successive derivation of an implicit function in its application to many and important theorems. |