867] 
NOTE ON THE JACOBIAN SEXTIC EQUATION. 
499 
The first three of these equations, or writing A = e 3 — e 2 , /a = e — e 4 , say 
A (A 4- C) 4- fi (B + D) = 0, 
A (B + C) + fi (A — JE) = 0, 
X(A+B) + g(G + F) = 0, 
constitute the entire system of independent linear relations between the square roots 
A, B, C, D, E, F. The coefficients A, /a are such that 
A 2 — /a 2 = A/a (= e — e 2 — e 3 + e 4 , = V5), 
and it is hence easy to verify that the remaining twelve equations can be deduced 
from these by the elimination of one or two of the square roots A, B, G. For 
instance, to eliminate A from the first and second equations, multiplying by — /a, A 
and adding, we obtain 
that is, 
or finally 
(- A/a + A 2 ) G+ (- /a 2 + A 2 ) B - ¡AD - XfjbE = 0, 
/AG + X/iB — ¡AD — A/a E = 0, 
X(B-E) + fi(G-D) = 0, 
which is one of the equations. And so again eliminating A from the first and 
third equations, we find 
X (B — G) + /j, (G + F — B — D) = 0, 
that is, 
(X-ri(B-C) + n(F-I)) = 0, 
or multiplying by A, 
/a 2 (B - G) + A/a (F — D) — 0, 
that is, finally 
A {F — D) + /a (B — G) = 0, 
which is one of the equations. 
Cambridge, 21 March 1887. 
63—2