138 ON THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS. [656 
In the one case, a, b being conjugate solutions of (H, ©) = (), 
from the equations H = const., a = const., b = const., 
we find p, q, r functions of x, y, z, H, a, b: 
and then p dx + qdy+rdz is an exact differential. 
In the other case, a, b, c, d, e being the solutions of (H, ®) = 0, 
from the equations H = const., a = a 0 , b — b 0 , c = c 0 , d=d 0 , e = e 0 , 
we find p, q, r functions of x 0 , y 0 , z 0 , H: 
and then p dx+ qdy +r dz is an exact differential. 
It may be added that, if from the last mentioned equations we determine also 
Po, <io> I'o as functions of x, y, z, x 0 , y 0 , z 0 , then considering only H as a constant, we 
ought to have p dx + qdy + rdz —p 0 dx 0 — q 0 dy 0 - r 0 dz 0 an exact differential; I have not 
examined the direct proof. 
Cambridge, 28 Nov., 1876.