1900. No. 5. ON PARTIAL DIFFERENTIAL EQUATIONS, ETC. I 
in 
As, however, in the case of the solution of the partial differential 
equations of the second order, which possess a first integral, the solution 
of the partial differential equation found, ® — 0, may in certain cases be 
avoided. For let & be an integral of the system (13) and w and 9 
integrals of the associated systems obtained by changing A, into A, and 
A, (supposing these quantities to be different), and it is then evident that 
D—0, w=0, 96 = 0 are in involution, and the values of 7, s, ¢ derived 
from these equations, will render the equation 
dp = rdx + sdy 
dg = sdx + tdy 
integrable, and then, also, 
de = pdx + gdy!. 
On the partial differential equation of the third order, 
Aa + BB+ Gy + Då + E(3?—ay) + F(y? — på) 4 G(ad — py) +H =o, 
in which A, B, C, .. G, H are given functions of x, y, 2, p, 
Oh ty 9 Uo 
1. In the previous section we have shewn (page 5) that a partial 
differential equation of the second order, 4 =/(v), in which w and v are any 
function of x, y, 3, p, g, 7, 5, ¢, always leads to a partial differential 
equation of the third order of the above form. 
We shall now shew that when a first integral of the above form 
exists, its discovery depends upon the solution of four simultaneous partial 
differential equations of the first order, resolvable, under certain condi- 
tions, into linear equations. The following propositions will enable us to 
gain this point. 
1 A detailed discussion of the importance of the first integrals for the integrations of a 
given linear partial differential equation of the #th order will be found in the previ- 
ously mentioned interesting memoir by Mr. Zalk, where several examples are discussed, 
