[SULLIVAN] PARTIAL DIFFERENTIAL EQUATIONS 127 
(13) & = EEE ae Ue Vi, 
du 
The transformation F parameters which reduces these solutions to 
their simplest form disturbs the coefficients of (A); consequently we 
prefer to obtain the simplest parametric representation of the integral 
surfaces of the Normal System from other considerations. 
Transformation of the Normal System which reduce it to 
the Monge Equation 
1 
2° 
Let (x, y, z) be three linearly independent solutions of the Normal 
System, and let (x, y) be chosen as independent variables. Then z 
will be a certain function of (x, y) satisfying the differential equations 
obtained from (A) and (6) by the transformation which replaces 
(u, v) by (x, y) and 6 by zg. In short, z will satisfy the differential 
equations which result from performing the transformations 
(14) x=x(u, v), y=y(u, v), 0=2 
on equations (A) and (6). 
The relations between the two sets of derivatives are given by 
the equations: 
S—rt = 
00 
(15) ai = pxu+qy, = = px2+ QYo, 
0°0 9 | 2 
rag xPr + 2xyyis + yt + XD + yug, 
940. 0 
Suds #12 + (x19 + X291)S + YiVot + xp + yg, 
fe) 
ov 
where 
20 
Per = Xo2r + 2X2VoS + yort > X22D + Yo0d ; 
pee _ 02 pound eat 02 pos 
ax’? 7 oy’ 9x? Oxo’ dy? 
and the suffix one indicates derivation with respect to #, while 
the suffix two indicates derivation with respect to v If J = 
(eS a be the Jacobian of (x, y) with respect to (uw, v), then J 
cannot vanish identically. Because if J vanished identically, the 
equations 
os oJ 
FC ul 
combined with (7) would necessitate that x and y be solutions of the 
equations 
