885] 
ON THE DIOPHANTINE RELATION, y" + yb* — SQUARE. 
599 
Hence 
{a — If (a — pf + (a — mf (a — qf = 0, 
(b — If (b — pf + (b — mf (b — qf = 0, 
(c — If (c — pf + (c — mf (c — qf = 0, 
(d — If (d — pf + (cL — mf (d — qf = 0; 
we cannot from these obtain three equations 
(a — to) (a — q)— i (a — l)(a— p) — 0, 
(b — m)(b — q) — i (b — l) (b — p) = 0, 
(c — to) (c — q) — i (c —l)(c —p) = 0, 
with the same sign for i; in fact these would give 
(1 + i) (b — c) (c — a)(a—b) = 0, 
but 1 + i is not = 0, and the a, b, c are essentially unequal. Hence we must have 
equations such as 
(a — to) (a — q) — i(a — l)(a — p) = 0 ; (c — to) (c — q) + i (c — l) (c — p) = 0, 
(b — to) (b — q) — i (b — l) (b — p) = 0; (d — to) (d — q) + i (d —l){d — p) = 0, 
two of them with — i, and two of them with +i; viz. the a, b, c, d divide themselves 
into pairs which are taken to be a, b and c, d. 
We hence easily obtain 
a + b — m — q — i (a+ b — l— p) = 0, ab — mq — i (ab — Ip) = 0, 
c + d — to — q — i(c + d — l — p) = 0, cd — mq — i (cd — Ip) = 0, 
and thence 
a + b — c — d = i (a + b + c + d) — 2i (l + y>), 
ab — cd — i (ab + cd) — 2ilp. 
Forming from these values of l+p, Ip the expression for 2i(a—l)(a — p), we find 
2% {a — l) (a — p) = (i + l)(a — c)(a — d); and we have thus the set of equations 
2 i (a — l) (a— p) = (i + 1) (a — c) (a — d), 
2i (b — l) (b —p) = (i + l)(b — c) (b — d), 
2i (c — l) (c — p) — (i — 1) (c — a) (c — b), 
2i (d — l) (d—p) = (i — 1) (d — a) (d — b). 
Hence also 
2 (a — If (a — pf = — i(a— cf (a — df, 
2 (b — If (b — pf = — i(b — cf (b — df, 
2 (c — If (c — pf = i (c — af (c — bf, 
2 (d — If (d — pf = i(d— af (d — bf; 
and, substituting these values in a former set of equations, we obtain 
2a (b — a) = — (a — c) (a — d), 
2/3 (a — b) = — (b — c) (b — d), 
2y (d— c) = (c — a) (c — b), 
2S (c — d) = (d — a) (d—b);