16 ALF GULDBERG. M.-N. Kl. 
Proposition. If u=f(v) be a first integral of the equation 
Aa + BB + Cy + Di + E (8? — ay) + Fy? — på + 
+ G(ad— py)+H=o.., I 
then u and v, considered as functions of x, y, 2, p, q, r, 5, t, will each 
satisfy four partial differential equations of the form 
aut? , poter 
II 
ALT _ BIT of TANA YS _ 
Te EG =) a Ja 
CAE af of VL NY of 
JG Cor at + Dez (DEE ce he 
ae i à SAN KES 
in which GE and (2) stand for = Fn PG ap 0 ag 39 and — 3 + 
+7 2 +52 +t = gee 
To demonstrate this proposition, we shall compare directly the par- 
tial differential equation of the third order, of which =/f(v) is a first 
integral, with the equation (I), and then deduce the conditions for the 
determination of # and v. 
Differentiating «= /(v), first with respect to x, and secondly with 
respect to y, we have 
ou ou ou Ou , av ov ov ov 
re BEST 
(+++ Fer trete ty? | 
Eliminating f’(v), we arrive at the partial differential equation of tlie 
third order, 
