PARTIAL DIFFERENTIAL EQUATIONS. 
155 
§ 5. The functions w, v are not uniquely determined. They may be replaced by 
W, V, where W, V are functions of w, v, one of which, say W, is arbitrary, while V is 
only restricted by the condition 
3(W,V) = 1 
8p, v ) 
The transformations of w, v which are allowable will thus form a group. For a 
single integral the operations of the corresponding group consist in the addition of 
different constants, that is, in varying the constant of integration ; the theory of 
periodic functions is connected with discontinuous sub-groups of this. It is possible 
that an investigation of the discontinuous sub-groups of the group of transformations 
of two variables which leaves their bidifferential unchanged may lead to an extended 
theory of periodic functions of the two variables. 
§ 6. The finding of the functions w, v may be considered as the indefinite integration 
of the bidifferential expression. It is simplified by Jacobi’s theory of the last 
multiplier, which is here a constant. 
Since 
we have 
Y _ dp, v ) 
Xdy — Yclx = 
Y = 
0p, V) 
8 (z, x) 
and thus, on the supposition that v is constant, 
dw = 
X dy — Yclx 
dv 
8 z 
Y clz — Z dy 
dv 
dx 
Zdx — X dz 
dv 
xy 
pZ — vY)dx + (vX — A. Z) dy + (\ Y — /^X) dz 
dv dv 
+ % 
+ v 
dz 
Hence iv may be found, if v is known, by integrating this last expression on the 
supposition that v is constant; X, (±. v may have any values and the constant of 
integration is to be replaced by an arbitrary function of v. Thus, when one of the 
functions w, v is known, the other is found by ordinary integration. The only 
restriction on the one found first is the equation 
x|+y|+z|=o 
ox cy dz 
§ 7. Let us now suppose a greater number of independent variables. Let u be a 
function of 
We have the relation 
