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[609 
609. 
ON THE ANALYTICAL FORMS CALLED FACTIONS. 
[From the Report of the British Association for the Advancement of Science, (1875), p. 10.] 
A faction is a product of differences such that each letter occurs the same 
number of times; thus we have a quadrifaction where each letter occurs twice, a 
cubifaction where each letter occurs three times, and so on. A broken faction is one 
which is a product of factions having no common letter; thus 
(a — b) 2 (c —d)(d — e) (e - c) 
is a broken quadrifaction, the product of the quadrifactions 
(a — b) 2 and (c — d)(d — e) (e — c). 
We have, in regard to quadrifactions, the theorem that every quadrifaction is a sum 
of broken quadrifactions such that each component quadrifaction contains two or else 
three letters. Thus we have the identity 
2 (a — b)(b — c) (c — d)(d — a) = (b — c) 2 . (a — d) 2 — (c — a) 2 . (b — dy + (a — by. (c — d) 2 , 
which verifies the theorem in the case of a quadrifaction of four letters; but the 
verification even in the next following case of a quadrifaction of five letters is a 
matter of some difficulty. 
The theory is connected with that of the invariants of a system of binary quantics.