OF THE COVARIANTS OF A BINARY QUANTIC. 
275 
761] 
and so, if 6r(a, b, c,...) is the group of all the positive substitutions, then we have 
the like theorem for the rational and integral two-valued functions of (a, b, c,...), 
viz. the terms P are here the two-valued function (a -b)(a-c)(b - c)and the 
symmetrical functions 
a + b + c + ..., ab + ac + be + ..., abc + ..., &c., 
as before. 
I return to the theorem {A), but instead of the covariants of a root-quantic of 
any order, I consider first the invariants of a root-quantic of any even order. The 
general form is 
(a - /3) m (a - y) n (/3 - ry)p 
where in all the factors which contain a, in all the factors which contain /3, and so 
for each root in succession, the sum of the indices has one and the same value = 9. 
Writing 12 for the index of a — /3, 13 for that of a —7, and so in other cases, then 
assuming always 12 = 21, 13 = 31, &c., the indices, taken each twice, form the square 
0 
12 
13 
21 
0 
23 
31 
32 
0 
the order of which, or number of its rows or columns, is equal to the order of the 
quantic; the terms of the dexter diagonal are each = 0, and the square is sym 
metrical in regard to this dexter diagonal. Moreover, the square is such, that the 
sum of the terms in each row (or column) has one and the same value = 9; and 
conversely, every such square, say R & , represents an invariant. 
Thus, for the quartic (x — a) (x — /3) (x — 7) {x — 8), the square R e is a square of 
four rows (or columns) representing the invariant 
(a - /3) 12 (a - 7) 13 (a - S) 14 , 
(/3-7) 23 (/3-S) 24 , 
in which 
(7 - 8) 34 , 
12 + 13 + 14 = 9, 
21 + 23 + 24 = 9, 
31 +32 + 34 = 9, 
41 + 42 + 43 = 9. 
35—2