1900. No. 5. ON PARTIAL SUES GL BENN å ETC. 31 
au 
os 
Ou Ou Ou 
ou ou Ou 
Ar a pr 
Conversely, we may show, in the same manner as the above, that this 
system, supposing that 
AD — HG = BC, 
is relevant to the solution of the given equation. 
3. It remains now to consider the two cases in which £=0, Go, 
F +o, and F =o, G=o, E + o, and when therefore system (II), (p. 16) 
will not lead to the solution of equation (I). 
We must then try to construct a system of partial differential equa- 
tions of the first order, which, in the cases here considered, is equivalent 
. to the proposed equation (I). 
In the same way as before (p. 18), we may show that each first 
integral of the given partial equation (I) will, when considered as a function 
of x, y, z, p, 9g, r, s, å, Satisfy four partial differential equations of 
the form 
of of of 
I aS oe 4 GE 
dy dy or Er 
«B2- EY) 3 + 6 +-4- 
416 e+ 7G) e+ 00 G+) &) 
QL O24- 
We shall here consider two cases, viz. À + 0 and A =o, 
roter 
