436 
ON THE TRIPLE ^-FUNCTIONS. 
[699 
Here the irrational part vanishes and the rational part is found to be 
= 0- 2 Kbjaob/gh ( ; y - z) 2 
(d — e) CjC 2 de 
+ (e — c) djdoce - 
+ (c — d) eje 2 dc 
+ a^ab^g^ (z — xf 
1 (d — e) Cacdjej ^ 
+ (e — c) d.dcjej 
+ (c —d) e^d^ 
(d — e) ccxd 2 e 2 ' 
+ abajbjfsgahs (x - y) 2 - 
+ (e - c) ddjC 2 e 2 - 
]• 
-1- (c - d) eeid 2 c 2 
The three terms in { } are here =-(c — d)(d — e)(e- c) multiplied by (z — x)(x — y), 
(x — y)(y — z), (y — z) (z — x) respectively ; hence the term in [ ] divides by 6 and the 
result is 
= _(o-d)(d- e) (*-c) [%biaAfgh (y _ z) 
+ a 2 b 2 a b fjgxhi (z — x) 
+ a b a 1 b 1 f 2 g 2 h 2 (x - y)\ 
or finally this is 
= — (c — d) (d — e) (e — c) 
multiplied by 
{(a 2 + ab + b 2 )fgh — (a 2 b + ab 2 ) (fg + fh + gh) + a 2 6 2 (f+g + h)} 
+ (x + y + z){ ~(a + b)fgh + ab (fg + fh + gh) - a 2 b 2 } 
+ (yz + zx + xy) { fgh — ab (f+ g + h) + a 2 b + ab 2 } 
+ X V Z {“ (fd +fh + 9^) + (a + b)(f+g + h)-(a 2 + ab + b 2 ) }, 
that is, we have (d — e) (ABC) 2 + (e — c) (ABD) 2 + (c — d) {ABE) 2 = a sum of four squares, 
viz. we have here a linear relation between 7 squares. 
I have not as yet investigated the forms of the relations between the products 
of pairs of ^-functions. 
Cambridge, 30 September, 1878.