182 
MR. A. C. DIXON ON SIMULTANEOUS 
J(x, r, t) 4- pJ(z, r, t ) + rJ(p, r, t ) + sJ(q, r, t) + J (y, s, t ,) -fi qj(z, s, t) 
+ sJ(p, s, t) + tJ(q, s, t) = 0, and 
J (x, s, r) + pJ (z, s, r) + rJ(p, s, r) -f sJ ( q , s, r) -f J(y, t, r) + qJ(z, t, r) + sJ(p, t, r) 
+ tJ(q, t,r) = 0. 
Here J ( ) denotes the Jacobian of f, w l9 % with respect to the variables specified. 
These equations express the conditions which are necessary and sufficient in order 
that 
dz = pdx + qdy, 
dp = rdx + sdy, 
dq = sdx + tdy 
may be integrable without restriction, when r, s, t are given in terms of x , y, z, p, q, 
by the equations 
j — 0, ?q — cq, 2^2 ■— ci 2 , 
the conditions must of' course be satisfied by any three of the six functions u Y , it. 2 , 
m 3 , u-, f We thus have forty equations, of which only eight can be algebraically 
independent. 
§ 32. The conditions to be satisfied by u v u. 2 are linear and homogeneous in their 
Jacobians with respect to the eight variables x, y, z,p>, q, r, s, t; of these, one is 
given in terms of the rest by the equation f — 0, and may, if convenient, be sup¬ 
posed not to occur in tq, u. 2 : hence the auxiliary equations in this case have seven 
independent variables and the dependent variables do not occur explicitly: to 
find a solution we are therefore to form a complete bidifferential, which shall be a 
linear combination of the determinants of the following array :— 
d(r, s), 
0 
- X - pZ - rR 
— sQ 
d(r, t), 
X -j- pZ -fi rP -f- sQ, 
— Y — qZ — sR 
- tQ 
d(s, t), 
Y -fi qZ -fi sP -fi £Q, 
0 
d(p,r ), 
rT, 
- rS - sT 
d(p, s ), 
*T, 
rR, 
d(p, t), 
— rR — sS 
sR 
d(q, r), 
«T, 
- sS - tT 
d(q, s), 
tT, 
sR 
d(q, t ), 
— sR — £S, 
tR 
d(z, r), 
pT, 
- pS - qT 
d(z, s), 
?T, 
pR 
d(z, t), 
1 
hw 
l sJ 
1 
U1 
qR 
d(x, r), 
T, 
- S 
d(x, s), 
0, 
R 
