1900. No.5. ON PARTIAL DIFFERENTIAL EQUATIONS, ETC. 37 
4. The above remarks prove that a first integral, #, of the given 
partial differential equation (I), is, under certain conditions, to be determined 
by the solution of ¢#ree simultaneous linear partial differential equations with 
eight independent variables. We now finally proceed: to construct the 
systems of total differential equations upon which the discovery of the 
integrals of the linear equations obtained depends. 
To reduce the determination of the first integrals of the given partial 
differential equation (I) to the solution of a system of total differential 
equations. 
We must here, as above, consider various cases. We will begin 
with that in which none of the quantities Z, Æ, G vanish. We then 
found, that a first integral of (I) was determined by the linear system. 
Ou ou ou 
EG are ur GE ent 
t 
or 
ou eu ou 
Am + ur AR 
ou ou ou 
ner a J=- 
where À and u satisfied the two quadratics (A) and (u), p. 22 and p. 23, 
and where 
a, — 42 = AG 
SS ER 
To deduce the value of # thus determined, it will obviously suffice 
to multiply the second equation of the given system by an indeterminate 
multiplier #, and the third equation by another indeterminate multiplier 7, 
and add the result to the first equation, so as to form a new equation 
which, like those from which it is formed, will be linear and of the first 
order, and which, on account of the indeterminate characters of mm and x, 
will be equivalent to three equations. From the auxiliary equations which 
we obtain in the process of solution, # and » must be eliminated. 
_If in this way we combine the equations of our system, we have, 
on arranging the resulting equation according to the differential coef- 
ficients of x, 
