172 
ON A SYSTEM OF ALGEBRAIC EQUATIONS. 
[256 
where a = v — c 2 . The equation in c 2 is thus of the fourth order; and in like manner, 
if instead of c 2 we take <r as the unknown quantity, and substitute therefore for c 2 
its value v — cr, the equation in a will be also of the fourth order: and effecting the 
reduction, this equation is 
8a- 4 — 12ikt 3 + (6v 2 — 2A./a) a 2 + (A, 3 + f — v 3 — Xpv) a + (vX — /a 2 ) (v/u, — X 2 ) = 0. 
It may be remarked that if a = 0, then a or b vanishes; and therefore, from the 
original equations, vX — /a 2 = 0, or vg — A 2 = 0, which agrees with the result afforded by 
the foregoing equation in cr. Again, if a = v, then c = 0; and therefore, from the 
original equations, v' 2 — X/a = 0. The left-hand side of the equation in a, writing therein 
a = v, should therefore contain the factor v 2 — A/a ; its value in fact is v 4 — 2A/ai> 2 -I- A,'ta 2 , 
or (v 2 — Ayu,) 2 . 
vC 'll 
If in the original equations we write a = -, b = -, 
the equations become 
a? + cyz — A.z 2 = 0, 
if + czx — fjbz 2 = 0, 
(c 2 — v)z 2 + xy = 0, 
which are three homogeneous equations of the second order; from which, if the 
variables x, y, z are eliminated, we have the required equation in c. And it would 
not, I think, be difficult, from the known formula for the general case, to deduce the 
foregoing result corresponding to the very particular case which is here in question. 
2, Stone Buildings, W.C., September 25, 1860.