665] 
A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
213 
As a partial verification, I remark that A A should be symmetrical in regard to 
the constants b, c, d, e, f; this is obviously the case as regards the terms in dudvr 
and (du) 2 , and it must also be so in regard to the term in (dur) 2 . The whole 
coefficient of (dvr) 2 is 
= aa! (ah + ac + bc + de + df + ef) 
— (— 2a + b + c + a + ad (a — d) (a - e) (a — f) — (Vde) 2 , 
and if we interchange for instance b and d, this coefficient becomes 
= aa x (ad + ac + cd + be + bf+ ef) 
— (— 2a + d + c +a+ a l )(a —b) (a —e)(a —f) — (Vbe) 2 . 
These two expressions must be equal; viz. we must have 
(fbe) 2 — (Vde) 2 = — aa 1 (b — d) (a + c — e - f) + (a — e) (a —f) (b — d)(—a + c -f- a + ad 
the left-hand side is 
= ¿,~(bdi — bjd) (efaA — e^ac), 
o 
and we have 
bdj — bjd = (b — d)d-, 
hence, throwing out the factor b — d, the equation to be verified becomes 
(efajCj — ej^ac) = — aaj (a + c — é —f) + (a — e) (a —f) (— a + c + a + a 2 ). 
Writing 
the left-hand side is 
e = e' + a, etc., 6 = aj — a, 
(a + a x ) ef + aaj (e J ) -\- c ej — c aaj, 
and the right-hand side is 
— aaj (o' — e —f) + ef (c' + a 4- ad, 
and these are equal. 
There are of course, in all, six expressions such as A A, each of them being by 
what precedes a sum of squares. And there are besides ten expressions such as 
A AB, =ABd 2 AB-(dAB) 2 , 
each of which should be a sum of squares: but I have not as yet effected the 
calculation of this expression A AB. 
Cambridge, 7th December, 1877.