126 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
[655 
We have as a conjugate system a, b, c; also the conjugate systems H, a, b: 
H, a, c; H, b, c; H, b, e; H, c, d; H, d, e\ but the first three of these, considering 
therein H as standing for its value abc — 1, are substantially equivalent to the first- 
mentioned system (a, b, c). 
72. Postponing the consideration of the augmented system, we now consider the 
partial differential equation pqr = 1 + H, where H is a given constant and p, q, r 
denote the differential coefficients of a function V. The most simple solution is that 
given by the conjugate system H, a, b, viz. here p, q, r are determined by the 
1 + H 
equations p = a, q = b, pqr = 1 + H, that is, r = ^ ; or, introducing for symmetry the 
constant c, where abc = 1 + H as before, then r = c, and we have 
V = A + J (cidx + bdy + cdz), = A + ax + by + cz, 
where a, b, c are connected by the just-mentioned equation abc = l+H. This is there 
fore a solution containing say the arbitrary constants X, a, b, and, as such, is a 
complete solution. 
But any other conjugate system gives a complete solution, and a very elegant one 
is obtained from the system H, d, e. Writing for symmetry ¡3 — a, y — a. in place of 
d, e, we have here to find p, q, r from the equations 
H — pqr — 1, qy—px = (3 — a, rz — px = 7 — a ; 
or, if we assume 6 = px — a, then 
H=pqr— 1; px, qy, rz = 6+a, 6 + ¡3, 6 + 7 
respectively, whence 
(1 + H) xyz = (6 + a) (6 + /3) (0 + 7), 
which equation determines 6 as a function of x, y, z (in fact, it is a function of the 
product xyz), and then 
0 + a 0 ¡3 0 + y 
P, q, r = 
X y z 
and we have 
There is no difficulty in effecting the integration directly by introducing 0 as a new 
variable, and we find 
X + 30 - a log - /3 log - 7 log ^t- 7 
Or, starting from this form, we may verify it by differentiation; the value of dV is 
6 + a 0 + /3 0 + 7/ x + y + 2 
a /3 7 \ + a dx + (3 dy + y dz