67 
“ ™ = U (3) 
ay ax 
M) 
dy dx "" ' ^ 
and if we put 
M = P + p and N = Q + g' 
then, developing and remembering (4) and putting u=0, 
dy dx ^ ^ 
3. Let ^ {x, y, 0 , c)=0 be the complete solution of (2) 
and suppose that some particular solution, say ^ {x, y, 0 , a)=0 
is also a solution of (1). Then ^ (x, y, 0 , a,) =0 gives M=jp 
and N=g, and therefore 
DP DP 
P = 0, ^ =0and ^ = 0 
^dx dy 
Q = 0,^=0and^ = 0 
and consequently, by (5), it gives U=:0. Hence if (1), 
not being integrable, has a single solution, that solution is 
contained in TJ—0.^ 
4. If we develope and then eliminate p between 
0 and 0 and q between ^ = 0 and 0, we 
dx dx dy dy 
find 
dV c?Q c?Q dP _ ^ T c?Pc?Q c?Q cZP _ 
dx dz dx dz dy dz dy dz 
which, combined, give 
dx dy dx dy 
Hence P and Q are functions one of the other. 
* I call sucli solutions “ discriminoidal.” For an example of tliis dis- 
criminoidal solution see Art. 26 of my paper On Particular Integrals” 
in the Quarterly Journal of Pure and Applied Mathematics (vol. XIII. 
p. 239). In the present communication to the Society it will he seen 
that I do not have any recourse whatever to the general dual solution. 
“ Oakwal” near Brisbane, Queensland, Australia. 
October 10 th, 1876* 
