Also, using the notation of partial differentials, the above equations become

D2u = d2¿u+d2„u+d22u+2d„dzu+2 d2d ̧u+2dxd ̧u+

D2u =

dz

- a2 sin (ax + by + cz),

− b2 sin (ax + by +cz),

=c2 sin (ax+by+cz),

-be sin (ax+by+cz),

— { a2 dx2 + b2 dy2 + c2 dz2 + 2 bc dy dz

+2ca 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 a be equicrescent; so that d2x=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 + bx3y+cz3 + exy z + 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, ... be written, and suppose the function to become u when these substitutions are made; then, by the definition of homogeneous functions, u = F(tx, ty, tz,... t" F(x, y, z, ...).

=

(108)

.'._u' = F(x', y', z', ..............);

and therefore by (46), Art. 49,

equating this to (110), because they are equal, we have

(110)

(111)

.;

Again, taking the second total differential of u', and dividing

whence, equating (113) and (114), and making t = 1, we have

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 = Ax2 + Bу2 + Cz2 + EYZ + GZX+HXY, which is an homogeneous function of two dimensions and three variables.

= 2u.

2 {Ax2+By2+Cz2 + E y z + G zx + Hxy}

(2), which is a homogeneous function of O

Of F(z) let F'(z) 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 4(x, y, z) = 0 are equivalent equations, it is rc

therefore if we estimate the simultaneous variations of x and z,

let the x- and z-differential of this be taken, and let x be equi

by its

dx2

dz

therefore dividing through by da2, and replacing (dz)

value from (118), we have

2

2

(1) (d) - 2 (dd) (dd) (de) + (d) (de) + (d)" (d) = 0;

dx2 dz

whence (d2

d2% dx2

dx dz

dx

dz

Functions is expressed in terms of the partial derived of

Similarly for the y- and z-partial differential

(db) (d2) + (d21) (do)2 + (d2)3 (12)
(+) (+) (+)=0.

-2 (dy dz dy

Also taking the y- and z-partial differential of (118), and sub

stituting for (d), and

;) (d2)2 - ( 12d.) ((o) (da) – (did.) (dź) (db)

dz

which gives us

of 4 (x, y, z) = 0.

d2z

dz

drdy) in terms of the partial derived-functions

We proceed now to consider the theory of successive derivation of an implicit function in its application to many and important theorems.