20 ALF GULDBERG. M.-N. Kl. 
[år år Aloe) år" (5) åt OG) et 
+eFel +04] rate 
+7 [By +e +02 | +027 ++ Os] a =? 
Now as this equation is to be satisfied in virtue of the constitution 
of A,B, ... G, H, and the function 7, and independently of 8 and y, 
the coefficients of the various combinations of $ and y must be separately 
equated to zero, when we obtain, just for F, the 4 proposed equations 
of the first order. 
The only condition then, respecting the application of the equations 
obtained above, is that we do not admit any relations which cause cither 
oF oF ; ; : : 5 
are to vanish. It is evident, however, that this may occur, if, 
for instance, in equation (I) we have E +0, F=o, G=o or F $0, 
E =o, G=o. In these cases, the above equations will not lead to the 
determination of a first integral of equation (I), but we must then establish — 
another system of partial differential equations of the first order, upon the 
solution of which a first integral depends. But before treating these 
particular cases, we will occupy ourselves with the general equations 
obtained above. 
2. Proposition. The solution of the system of partial differential 
equations established in the last proposition may, under certain conditions, 
be made to depend upon that of simultaneous linear partial differential 
equations of the first order. | 
In demonstrating this proposition, we shall consider first the case in 
which none of the quantities Æ, Æ#, G vanish, next the cases in which 
one of the quantities 2, Æ, G is equal to zero, and lastly the case in 
which £=0, F=0, G = 0: 
