PARTIAL DIFFERENTIAL EQUATIONS. 
177 
§ 29. Let us now consider the new solutions of the auxiliary equations, given by 
the new complete primitive. The old solutions are the six pairs of the form x,, Xj and 
the six of the form u p u p where u x — p x , u. 2 =■ p 2 , u 3 = q x , u v = q, 2 . The bi- 
differentials of these twelve satisfy the relation 
d(y, z) — p x d(x x , z) -~p i d(x 2 , z) — q x d(y, x x ) — q,cl(y, x 9 ) + (p x q 2 — p z q x )d{x x ,x 2 ) 
«1 + 
d<f> 
dpi 
x 2 + 
8cj) 
8p* 
d (Pv 
Pn) 
+ 
x x + 
8(f> 
¥i 
8<f> 
8q i 
d{ P i 
> 7i) 
8\Jv 
dpi 
8-yfr 
dp* 
8\]r 
¥’ 
x x + 
8p 
8q x 
x x + 
8<f) 
¥’ 
8(f) 
8q * 
d (Pi, 
<k) 
+ 
x x + 
8(f> 
8pP 
8(f) 
8q x 
d (P» 
<h) 
¥i 
X 2 -4- 
8\fr 
8q*_ 
8y]r 
¥' 
x x + 
8-fr 
8q\ 
+ 
dcf) 
dp* 
8cf) 
8q* 
d(p%} 
<h) 
+ 
8(f> 
¥ 
8(f) 
8q 2 
%i> 
<!■>) 
8 \fr 
¥: 
x . 2 -f- 
8yfr 
8q 2 
aT + 
8)fr 
8q x 
X . 2 -f- 
8ifr 
¥ 
In the auxiliary equations we may take x x ,x 2 , p x , p 2 , q x , q„ as independent variables, 
since y, z are given explicitly in terms of these six. 
From (12) follow three more relations connecting the six differentials du, dv, dp h dp 2 , dqi, dq*, so that 
their ratios are determinate, and therefore u, v,p x , qi,P-h can on ty functions of one variable. The two 
equations last written will then, generally, give x x , x* in terms of this variable, which may not be. Hence 
we must have 
du = A dv, dpi = Xdq x , dp 2 = A dq 2 , 
and since dp 
equation: 
0, df 2 = 0, dp 0, and du, dv, dp h dq h dp*, dq 2 do not vanish, A must satisfy the 
df 1 + dp 
du dv ’ 
x d L + d A, 
dpi dq x ’ 
dp* 
dp dp 
du dv ’ 
A dp + 
dpi dqi ’ 
x cip 
dp* 
dp + dp ^ 
du dv ’ 
A dp_ , dp 
dpi dqi ’ 
A d A 
dp* 
+ 
+ 
+ 
= 0 . 
If A satisfies this equation the differential relations du = A dv, dp x = A dqi, dp- 2 = A dq- 2 reduce to two only, 
since u, v, p x , q x , p< 2 , q 2 are connected by the equations 
fi = 0, p = 0, p — 0. 
By integrating these two we find two more relations involving two arbitrary constants. Hence, we. may 
suppose v, pi, P'2, qi, q* expressed in terms of u, and find a solution by eliminating u from the following :— 
u = px Xx + P'2 x 2 - y, 
v = q x x x + q 2 Xo - z, 
1 = x x dpi/du + x<i dp 2 jdu. 
2 A 
VOL. CXCV.-A. 
