ON SEMIN VARIANTS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xix. (1883), 
pp. 131—138.] 
The present paper is a somewhat fragmentary one, but it contains some results 
which seem to me to be worth putting on record. 
I consider here not any binary quantic in particular, but the whole series (a, h, cfx, y)-, 
(a, h, c, d][x, y) 3 , &c.; or in a somewhat different point of view, I consider the indefinite 
series of coefficients (a, h, c, d, e,...); here, instead of covariants and invariants, we 
have only seminvariants; viz. a seminvariant is a function reduced to zero by the 
operator 
A = adi + 2 bd c + 3 cd^ -f- ... ; 
for instance, seminvariants are 
a, ac — h-, a-d — 3 abc + 2b 3 , a-d 2 + 4ac 3 + 4 b 3 d — Qabcd — 3 b-c-, 
ae — 4<bd + 3c 2 , ace — ad- — b-e + 2bed — c 3 , &c. 
A seminvariant is of a certain degree 0 in the coefficients, and of a certain 
weight w (viz. the coefficients a, b, c, d,... are reckoned as being of the weights 
0, 1, 2, 3,... respectively); it is, moreover, of a certain rank p; viz. according as the 
highest letter therein is a, c, d, e,... (it is never b), the rank is taken to be 0, 2, 3, 4, ..., 
and we have w= or <^p9. The seminvariant may be regarded as belonging to a 
quantic (cl, ..fix, y) n , the order of which, n, is equal to or greater than p; viz. in 
regard to such quantic the seminvariant, say A, is the leading coefficient of a covariant 
(A, B,..., K\x, yY, 
where the weights of the successive coefficients are w, w +1,... up to u9 — w; hence 
number of terms less unity, that is, p, is = n9 — 2w; the least value of p is thus 
— p9 — 2w, which is either zero, or positive; in the former case, w = ^p9, the semin 
variant is an invariant of the quantic («,...\x, y) p , the order of which is equal to 
the rank of the seminvariant; but if w < ^p9, then it is the leading coefficient of a 
covariant (A, B,..., KJfx, y)p 0 - 2w of the same quantic («,...]£x, y) p ; and in every case, 
taking n > p, the seminvariant is the leading coefficient A of a covariant 
(A, B,..., K\x, y)M-™ 
of a quantic (a, ..fix, y) n .