PARTIAL DIFFERENTIAL EQUATIONS. 181 
so that the eight original variables connected by two equations are now expressed in 
terms of six. 
Thus d (3 y — v' d , 4w — v 4 ) can be expressed in terms of d (z — v, 2x — v 2 ), the six 
bifferentials of w, x, y, z and those of 
u, — r>(l + uv), wu — xv(l + uv) + y, yu — zv( 1 + uv) -(- x, 
that is, of 
u, v, wu — xv( 1 + uv) + y, yu — zv( l uv) + x. 
There is no difficulty in finding the relation. It is 
u 2 d(3y — v 3 , 4\v — rd) — 6ir(l -f uv) 2 d(z — v, '2x — v 2 ) 
— 12 u 2 d(y, w) + 12r 2 (l + uv) 2 d(z, x) 
+ 1 2v 2 {(1 + uv) (y — zv 2 ) — u(iv — xv 2 )}d(v, u) 
— 12v 3 (l + uv)d(y , yu — zv — uzv 2 + x) 
4 - 12 uv 2 d(v, wu — xv — uxv 2 + y) = °- 
Here then we have an identical linear relation connecting the bidifferentials of 
seven pairs of functions of six variables. Any one of the seven pairs is accordingly 
by definition a bifunction of the other six. 
Second Application. 
§ 31. Take now a differential equation of the second order, 
f(x, y, z,p, q , r,s,t) - 0 , 
where p, q are the first and r, s, t the second partial derivatives of 2 with respect 
to x, y. 
A complete primitive will consist of a single equation in x, y , 2 involving five 
arbitrary constants, say oq, a. 2 , a 3 , cq, a 5 . If we form the first and second derivatives 
of this equation we shall have, in all, six equations from which cq, a 2 , a 3 , a 4 , a 5 can 
be found in terms of x, y, 2, p, q, r, s, t, and the original differential equation will be 
the result of eliminating cq, a 2 , a 3 , cq, cq. Let tq, u 2 , u 3 , rq, u 5 represent the 
expressions found for cq, cq, o- 3 , a 4 , cq respectively, in terms of x, y , 2, p, q, r, s, t. 
Then from the equations 
7 = 0, u^ — a 1} u 2 — a 2 , 
by differentiating, we can form six equations which will involve the third derivatives 
of 2 ; by eliminating these we deduce the following two differential equations to be 
satisfied by tq, u 2 :— 
