190 
MR. A. C. DIXON ON SIMULTANEOUS 
In like manner 
y{d(b, X) — A ~d(a, X)} + {d(f X) — Xpd(a, X)} + x{d(h, X) — Xd{a, X)j 
= xd(h, X) -j- yd(b, X) + d(J, X) 
= ^ [<% x ) - *)] + ^ [%> 2 /) - hd i x > y)l 
+ multiples of bidifferentials of a, b, c, f g, h. 
Lastly, 
2xy{d(h, X) — Xd(a, X)} -j- y z {d(b, X) — X z d(a, X)} + 2 x{d(g, X) — yd{o, X)} 
-f 2 y{d(f X) — Xfxd(cL , X)} fi- {%d{c, X) — ydd(a, X)} 
— x 2 d(a, X) -f- 2 xyd(h, X) + y z d(b, X) + 2xd(g, X) -j- 2yd(f X) + 2c/(c, X) 
= 2{d{z, x) — (hx + + f)d(y,x)}dX/dx 
+ 2 {d(z, y) ~ {ax + hy + g)d{x, y)} cX/dy, 
fi- multiples of bidifferentials of a, b, c, f g, h. 
Hence, in all, nine combinations of the ten bidifferentials of oq, a 2 , a 3 , cq, a 5 can 
be expressed in terms of the bidifferentials of x, y, z, p, q and of a, b, c, f g, h ; that 
is, in terms of the bidifferentials of the seventeen known independent pairs of functions 
satisfying the auxiliary equations : thus the new complete primitive adds only one to 
the number of these known bifunctionally independent pairs, and one more must be 
added in order to give the full number. 
This theory enables us again to construct examples of bifunctions of a number of 
known pairs which may reach eighteen. 
§ 42. The foregoing investigation may be modified so as to give singular solutions 
of a pair of differential equations of the form in question, say 
F x (r, 5, t, p, q, z) = 0, 
F 3 (r, s, t, p, q, z) = 0, 
where p = p — rx — sy, 
q = q — sx — ty, 
z = 2 - Up + p) x - £(? + d)y- 
A complete primitive would be given by supposing r, s, t, p, q, z constants con¬ 
nected by the above equations. Another solution would be given by solving the 
total differential equations found hy supposing the relations 
