166 
[663 
663. 
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. 
[From the Journal fur die reine und angewandte Mathematik (Crelle), t. lxxxiii. (1877), 
pp. '220—233.] 
I CONSIDER six letters 
a, b, c, d, e, /; 
a duad ab not containing f may be completed into the triad abf and then into the 
double triad abf.cde; there are in all ten double triads, represented by the duads 
ab, ac, ad, ae, be, bd, be, cd, ce, de, 
and the whole number of letters and of double triads is =16. 
Taking x, x' as variables, I form sixteen functions; viz. these are 
[a] = a — x . a — x', 
r n _ 1 { fa — x .b — x ./— x ja — x'.b — x x'\ 2 
{x — x') 2 {V c — x' .d — x'. e — x ~ \ c — x.d — x.e — x\ ’ 
where the function under each radical sign is the product of six factors, the arrangement 
in two lines being for convenience only: the sign + has the same value in all the 
functions, and it will be observed that the irrational part is 
2 fa —x.b —x.c —x.d —x.e —x.f—x 
~ (x — x') 2 \ a — x'. b — x'. c — x . d — x'. e — x' . f — x' ’ 
viz. this has the same value in all the functions. 
The general property of the double ^-functions is that the squares of the sixteen 
functions are proportional to constant multiples of the sixteen functions [a], [a&]; but 
this theorem may be presented in a much more definite form, viz. we can determine, and 
V