1900. No. Ke ON PARTIAL DIFFERENTIAL EQUATIONS, ETC. 19 
Qu 2 Qu Ou Ou Ou GUN ou Ou\ du 
EN TAN Se D = 
af) Ga ot ac os ot 2) dé cf) ot 
QUN ou 
le 
Hence, # considered as a function of x, y, 2, , g, 7, 5, å, satisfies the 
proposed four partial differential equations of the first order. As x and 
v enter symmetrically into the system (a), (0)... (g), (A), v will also 
satisfy the given four partial differential equations. 
Remark. We may also in a more direct manner prove that each 
first integral, 
F(z, J; CA D; 2 7,5, 0, 
of the given partial differential equation (I) satisfies the four proposed 
equations of the first order, 
For differentiating the proposed first integral with respect to x, and 
with respect to y, we have 
OF) oF | oF. , OF 
ler mer 
am oF, 3F | aF 
2 Fødte has 
If we then determine algebraically two of the quantities a, ß, y, 0, 
(we select & and 0) from this system, and substitute their values in the 
given equation (I), that equation ought to be satisfied independently of 
the value of the remaining quantities 8 and y. Now supposing 7 and Z 
Å : ; oF oF Å 
to be both contained in Æ, so that neither ag Dor vanish, we have, 
from the last system, 
Kerr) 
dx 
he oF 
or 
oF | OF oF 
“UE AE JE 
oF 
ot 
which, when substituted in equation (I), give the result 
