663] FURTHER INVESTIGATIONS ON THE DOUBLE S-FUNCTIONS. 
167 
that very simply, the actual expressions for the constant factors; and so we can enunciate 
the theorem as follows; the squares of the sixteen double ^--functions are proportional 
to sixteen functions — {a}, + {ab}; where, in a notation about to be explained, 
{a} = Va [a], {a&} = fab [aft]. 
Here in the radical fa, a is to be considered as standing in the first place for the 
pentad bcdef, which is to be interpreted as a product of differences, 
= bc .bd .be. bf. cd .ce . cf. de . df. ef, 
(where be, bd, etc., denote the differences b — c, b — d, etc.). Similarly, in the radical 
Vab, ab is to be considered as standing in the first instance for the double triad abf. ede, 
which is to be interpreted as a product of differences, = ab.af .bf.cd.ce.de, (where ab, af 
etc., denote the differences a — b, a—f, etc.). 
It is convenient to consider a, b, c, d, e, f as denoting real magnitudes taken in 
decreasing order: in all the products bcdef, etc., and in each term abf or ede of a 
product abf. ede, the letters are to be written in alphabetical order; the differences 
be, bd, etc., ab, af etc., which present themselves in the several products, are thus all of 
them positive; and the radicals, being all of them the roots of positive quantities, may 
themselves be taken to be positive. 
We have to consider the values of the functions [a], [crf>], or {a}, [ab], in the case 
where the variables x, x' become equal to any two of the letters a, b, c, d, e, f; it is 
clearly the same thing whether we have for instance x — b, x' = c, or x — c, x = b, etc.: 
we have therefore to consider for x, x the fifteen values ab, ac, ..., af ..., ef\ there is 
besides a sixteenth set of values x, x each infinite, without any relation between the 
infinite values. 
Taking this case first, x, x each infinite, and in [a&], etc., the sign ± to be +, we 
have 
or, attending only to the ratios of these values, 
4#?^ CC ^ 11* 
where rr is infinite, and the values may finally be written 
where 
[a] = 0, [ab] = 1 ; 
whence also, for x, x infinite, 
(a} = 0, {ab}=fab, 
the radical fab being understood as before. 
Suppose next that x, x denote any two of the letters, for instance a, b; then two of 
the functions [a] vanish, viz. these are [a], [6], but the remaining four functions acquire 
determinate values; and moreover four of the functions [a&] vanish, viz. these are 
[ab], [erf], [ce], [de], for each of which the xx' letters a, b occur in the same triad (the