PARTIAL DIFFERENTIAL EQUATIONS. 
165 
The coefficients in 
instance, 
these combinations are partial derivatives of f x or f 2 , thus, for 
= d(Pi, x i) + fj d(Pi> x z) + f* d (Pe V) + f 2 d(p lt z) 
+ 0~' d (Pv 2i) + fy d (Pi’Pz) + d (Pi, <Js), 
and so in other cases. 
The number of these combinations is sixteen, but it is to be lowered by three, 
since d(J x ,f) and d(f 2 ,f 2 ) are identically zero and can be formed by com¬ 
bining the sixteen in two ways, so that three linear combinations of the sixteen 
bidifferentials vanish identically. 
Hence the array contains virtually only fifteen rows (28 — 13) and as there are 
three columns, we have thirteen bidifferential expressions to combine. Any pair of 
the four functions x x , x % , y, z will satisfy the two auxiliary equations, as is clear either 
from the equations themselves or from an examination of the matrix ; of course these 
solutions of the auxiliary equations will not give a complete primitive. 
§17. If a complete primitive has been found it leads, as has been explained, to 
four equations 
u x — oq, u 2 = a :3 , u s — a 3 , tq = cq, 
and any pair of these must satisfy the auxiliary equations. Thus twelve pairs of 
functions satisfying these are known, namely 
Xi and Xj (i,j— 1, 2, 3, 4) 
u, and Uj (i,j = 1, 2, 3, 4). 
These, however, are not all independent, but one pair is a bifunction of the other 
eleven. 
For if <f>(x x , x. 2 , x 3 , aq, a x , a. 2 , cq, cq) = 0 
xfj(x x , x. 2 , x 3 , aq, a x , a,, « 3 , cq) = 0 
are the equations of the complete primitive, they must reduce to identities when 
it x , tq, ti 3 , tq are substituted for cq, a 2 , a 3 , cq respectively. 
Hence 
0(aq, X,, x 3 , x 4 , tq, U,, Uo, u x ) = 0 j 
xjj(x x , x. 2 , Xo, x 4 , tq, tq, tig, q) = 0 I 
identicallv, and 
•v ' 
so that 
2 dx, = — Sy dui, 
i dx i 
% ~ dxi = 
i OXi 
d\Jr ? 
2 dui, 
i OUi 
>.• «</>>+) ;/ \ _ 2 d(fr f ) j, , 
Z 3(*i. 0 a(X ” Xj) Z 3(«» %) 1 ’’ 
( 8 ). 
and the bidifferentials of the twelve pairs of functions are connected by a linear 
relation. 
