Let x be equicrescent; so that d2x = 0; and from the relation given between x and y in the preceding equation of condition, dy _ x d%y __ d1 and if we express it in terms of x only, 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. Dep. 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 + czs + exyz + gx2z = 0 is a homogeneous function of three dimensions. Let « = r(x,y,z,...) be a homogeneous function of n dimensions and r variables; for x,y,z, ... 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, 2 {Air2 + Bys + C22 + Ey« + G2a? + Ha:y} = 2w. Ex. 2. « = F(^)> which is a homogeneous function of 0 dimensions. Of 9 (z) let r'(z) be the derived-function, that is, be that quantity which multiplied by dz is the xr-differential of r(z); then (*L\-lf(l\. 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 $ {x, y,z) = 0 are equivalent equations, it is re quired to express (g), (^L), (0) in terms of (g),... /<P$\ /d*<p \ \dx»/' "\dydz>'"' |