180 
MR. A. C. DIXON ON SIMULTANEOUS 
In like manner 
E {cl (p x , X) - Xd (q lt X)} + F {cl (p. 2 , X) - Xd (q,, X)} 
+ G {d (0jX) — Xd (i (/, X)} = ~ {d (z, aq) — &cZ (a 3 , aq)} + ^{cl ( z,x . 2 ) — qyl (aq, a;,)} 
+ multiples of bidifferentials ofp l5 gq, q 2 . 
Hence the three expressions 
k ) — ^% l5 
d{p 2 > x ) — x <%s> x )> 
d(</>, X) — Xd(\ft, X), 
are all reduced to the same, save for a factor, by adding or subtracting multiples of 
the bidifferentials of aq, aq, aq, aq and of u x , u. 2 , u z , u x ; the same is therefore true of 
the bidifferentials of c x , c 2 , c 3 , cq. Hence all the new complete primitives found by 
the method of §§ 27-8 only add one to the eleven known “ bifunctionally ” indepen¬ 
dent pairs of functions satisfying the auxiliary equations ; one more pair, leading to 
a fresh complete primitive, is yet to be found. 
§ 30. These results may be used to construct examples of bifunctions. For 
instance, the equations 
y = yqaq + p 2 x 2 + q x , 
z = gqaq + q 2 x 2 + p 3 , 
lead to the following case among others :— 
In the equations 
djy _dqq __ dq x _ 
dq x dq 2 dp, ~ 
put q. 2 = X + a, clq. 2 = clX, and integrate. 
Thus 2p. 2 = X 2 -f- b, Sq x = X 3 + c, ±p x = X'* -f- e, 
and the arbitrary constants a, b, c, e in the new solution are respectively equal to 
q, — X, 2 p 2 —X~, 3gq — X 3 , 4 p x — X 4 , where X 3 aq + x, + X = 0. 
Now from § 29 it follows that cl (c, e) can be expressed in terms of d ( a , b), the 
bidifferentials of aq, aq, y , z and those of p lt p. z , gq, q. 2 . 
For convenience, let us write 
u, v, iv , x, y, z for aq, X, p x , p 2 , gq, q 2 respectively ; then 
for x. 2 we must put — v(l + nv), 
for y ,, ,, wu — xv(l + uv) + y, 
for 2 „ „ yu — zv{ 1 + uv) + a 1 , 
