PARTIAL DIFFERENTIAL EQUATIONS. 
189 
involve only the differentials of oq, a. 2 , a 3 , a 4 , a 5 . One of these five is expressible in 
terms of the other four, since 
3F 0F 0f DF 0F 0F 
da + ^ dh + if db + ~dg + i -df+ f-dc = 0, 
dli db tig * df J ^ 
da 
db 
dc 
while one of the relations connecting X, p, a, b, . . . is 
0F 0F 
da + k dh 
, ar _L >U F j. X 3F , 1 i 
+ f 1 + X M + Vj, + 
db 
■3/ 
0F 
dc 
= 0 . 
Some expression such as vdk — pda will also involve the differentials of oq, a 2 , a 3 , 
a 4 , a- only. Hence the differentials of oq, a 2 , « 3 , a 4 , a 5 will be linear combinations of 
vdk — pda, dh — \da, dg — pda, db — k~da, df — kudu, 2dc — pdda, of which the 
last five satisfy a linear relation. 
Thus the bidifferentials of oq, a 3 , a 3 , a 4 , a 5 in pairs will be linear combinations of 
the bidifferentials of a, h, c, f g, h (only five of the six need be used) in pairs, and 
of the expressions 
d(li, X) — kd(a, X), d(g, X) — pd(a, X), d(b, X) — k~d(a, X), 
d(f X) — kpd(a, X), 2 d(c, X) — p?d(a, X), 
of which last five, only four are independent. 
Now X, p are connected not only by the equation 
aF _nx aF J_ 0F xs 0F _l X 
da + X 0X + ^df +X df + /V 
0F 
S/ 
+ 
1„3 
2 r 1 
0F 
dc 
= o, 
but also by the equation 
* fi" fy d - P- — 
so that they are definite functions of x, y, a, h, j, g, h. 
Again p = ax + hy -f g, 
d(p, x) — hd(y, x ) = xd(a, x) -f yd(h, x) + d{g, x), 
d{p, y) ~ ad(x, y) - xd(a, y) + yd(h, y) -f d(g, y ). 
Thus y{d(h, X) — kd(a, X)} -f- {d(g, X) — pd(a, X)} 
= xd(a, X) + yd(h, X) + d(g, X) 
= ^ L d (p, x) ~ hd(y, x)] + ^ [d(p, y) - ad(x, y)~\, 
+ multiples of bidifferentials of a, b, c, f g, h. 
