178 
MR. A. C. DIXON ON SIMULTANEOUS 
Then 
4 * 1. y) = PA X 1. *2) + hi + gL + h 2 + ^ ft) + g| d(x„ },) + 
gA d (ai , ?2 ), 
and similar expressions may be found for c/(.r.,, ?/), d(. / r ] , 2 ), d(.r 2 , 2 ) in terms of the 
bidifferentials of the pairs of independent variables. 
Let <q, f‘ 2 , c 3 , c 4 be the constants of integration in a new complete primitive 
found by the method of §§ 27-8. Let X be the common value of the ratios 
dp x ,/dq x , dp 2 /dq 2 , dcj)/dxjj. Then, after integrating the equations dpjdq x = 
dpjdq 2 — dcp/dxp (=■ X) by help of an assumed relation connecting, say, p x , q x , p. 2 , q 2 , 
c x , c. 2 we have four relations among 
P\i p z , ^-5 <?!> Co, C 3 Cf, 
and we may therefore 'suppose p x , p 2 , q x , q 2 expressed in terms of X, c x , c 2 , c 3 , c 4 , 
unless X is a constant, and therefore itself a function of c x , c 2 , c 3 , c 4 . Then 
dp | — X clq x , dp 2 — X r/gq, defy — X dip 
will be linear combinations of dc x , dc 2 , dc 3 , dc 4 , and so will some such expression as 
adp x + /3dX, 
where a vanishes if X is one of the constants or a function of them. Conversely, 
dc x , dc 2 , dc 3 , dc 4 will be linear combinations of 
dp] — X d(/j, dp 2 — X fX/ 2 , d<j> — X dxfj, a dp x + (3 d\, 
and the bidifferentials of c x , c 2 , c 3 , c 4 in pairs will be linear combinations of the six 
following expressions :— 
d (p x , p 2 ) — Xd (q x , p>d) — Xd (p x , q 2 ) + X 2 d (q x , q.J, 
d (Pi, </>) ~ Xd ( q x , </>) — Xd ( p x , 1 //) + Xhl (q x , xfj), 
d {Pz > </>) — ^ (?s> </>) — ^d (p 2 , + XH (q 2 , x //), 
/3d (jP l5 X) — aXrZ (q x , p x ) — £Xc/ (gq, X), 
ad(p z ,p x ) + /3d (p 2 , X) — aXd(q 2 , p x ) — (3Xd(q,, X), 
ad (</>, pj) -j- /3d ((f), X) — aXd(xp, p x ) — /3Xd(xp, X). 
These are combinations of the bidifferentials of p x , p 2 , q x , q 2 , in pairs, with the 
expressions 
d(p 1 , X) — Xd (q x , X), 
d ( p 2 , X) — X(7 (f/ 3 , X), 
d (</>, X) — Xd (v//, X). 
