SYMMETRICAL AND MORE GENERAL SOLUTION 
Now these equations are of the same form as (32). 
establish the same relations between the functions 
They 
-{X'+pZ'), -(Y' + qZ'), F, Q\ (34), 
as (32) does between the differentials dp, dq, dx, dy. 
It follows that if we give to dx and dy, which are arbitrary, 
the ratio of the last two of the functions (34) then will dp and 
dq have the ratio of the first two, so that the following will be 
a consistent scheme of relations, viz. 
dx dy 
dp 
dq 
X' + pZ' Y' + qZ 
-, (35). 
Now dividing the successive terms of (30) by the successive 
members of (35) we have 
(X + pZ) F + (Y+ qZ) Q' -P(.X' + pZ') 
~ Q [Y' + qZ") — 0 (36). 
This is the relation sought. It might be obtained by direct 
elimination by multiplying the equations of (33) by P and Q 
respectively, and the corresponding equations derived from (30) 
by P' and Q' respectively, and subtracting the sum of the 
former from the sum of the latter. 
It is obvious too, and the remark is important, that we 
might pass directly from (30) to (36) by substituting for dx, 
dy, dp, dq, the functions of (34), and that this substitution 
is justified by the identity of relations established in (32) 
and (33). 
If in (36) we substitute for X, Y, &c. their values, and 
transpose the second and third terms, we have 
dF dF' 
dx ^ dz 
dp 
d+ d+> 
dx dz 
dF 
+ 
dp \dy 
dF dF' 
d<Y 
dq 
dX> dY 
dy +q 
dz 
dF 
dq 
0 (37). 
Such is the relation which connects the functions F and 
When/is given it assumes the form of a linear partial differ-