[SULLIVAN] PARTIAL DIFFERENTIAL EQUATIONS 129 
Therefore H is a constant, and equation (17) becomes 
(B) S?— rl = x (where À = const.) 
From this we conclude that the integral surfaces of the Normal 
System are necessarily integral surfaces of the Monge equation (B). 
We now proceed along classical lines to the integration of the Monge 
equation (B). There will be two distinct sets of subsidiary equations 
from which to construct the intermediary integrals of the general 
Monge equation 
Rr+Ss+Ti+ U(rt — S?) = V, 
(WHeLeche G50 Um Voarehunctioncssol 2,9, 24. D,:q) provided inter- 
mediary integrals exist and the equation 
(RT + UV) +0US + U? = 
has distinct roots. If now the values of R, S, rs etc., in equation (B) 
be substituted in this equation, the roots are: 
1 
@é=+ ate 
Hence the two sets of subsidiary equations are: 
Adx — dq = 0, 
(a) hdx + dp = 0, 
dz — pdx — qdy = 0, 
and 
Adx + dq =0 
(b) Ady — dp = 0, 
: dz — pdx — qdy = 0. . 
From these we obtain two intermediary integrals of (B), viz., 
Ax —g = 2ÿ(8), Ay + p = 28, 
Ax +q=2¢(a), Ay—p=2a. 
Ax = (a) + (8), Ay=atB, 
Sor p=—a+B; q=¢(a) — yp (8). 
Substituting these values in the equation 
dz = Xpdx + dAqdy, 
we find 
Az = (—a+ 8) (dp + dy) + (¢—y) (da + dB), 
and therefore (on integration) 
Xe = (6-¥) (a +6) —2| (adé— Bay), 
Now put (for uniformity of notation) 
o(a)=G'(a)=U'(u), ¢(8) =H’ (8) =—V' @); 
then the above equations become (on evaluating the integral) 
(18) eel GE | Ve 
AY =u +, 
A3=2(U + V) — (u—v) (U'— V'). 
A simple transformation of coordinates and parameters establishes 
at once the identity of equations (13) and (18). In short, if in equa- 
tions (13) we make the transformations 
and 
Thus 
