266 
ON SYMMETRIC FUNCTIONS AND SEMIN Y ARI ANTS. 
[932 
I establish an umbral theory of seminvariants which will be presently again 
referred to, and I consider the question of the reduction of seminvariants. The final 
term of a seminvariant may be composite (that is, the product of two or more 
final terms), and that in one way only or in two or more ways, or it may be non 
composite. In the case of a composite final term the seminvariant is reducible, but 
the converse theorem that a seminvariant with a non-composite final term is irreducible 
is in nowise true ; the reason of this is explained. An irreducible seminvariant is a 
perpétuant. In regard to perpétuants, I reproduce and simplify a demonstration 
recently obtained by Dr Stroh as to the perpétuants for any given degree whatever: 
viz. the generating function for perpétuants of degree n is 
—— /y»2 n ^ 1 « 1 /yi2 "I . /yi3 1 /yiW • 
"" lv * JL tAJ « ~I_ it/ 'a t • J- it/ « 
the theorem was previously known, and more or less completely proved, for the 
values n — 4, 5, 6, and 7. Dr Stroh’s investigation is conducted by an umbral 
representation, 
{olx + /3y + yz + ...) B , x + y + z+ ...= 0, 
of the blunt seminvariants of a given weight. 
I consider in regard to seminvariants the theory of the symbols P — 8b and 
Q — 2cob, and the derived symbols Y and Z, each of which operating on a seminvariant 
gives a seminvariant. These are, in fact, connected with the derivatives (/, F) of a 
quantic f and any covariant thereof F; but except to point out this connexion, I do 
not in the present memoir consider the theory of covariants. 
The Coefficients (a, b, c, d, e, ...) or (1, b, c, d, e, ...). Art. Nos. 1 to 9. 
1. I consider the series (a, b, c, d, e, ...), or putting as we most frequently do 
a = 1, say the series (1, b, c, d, e, ...) of coefficients, the several terms whereof are 
taken to be of the weights 0, 1, 2, 3, 4, ... respectively. We form with these sets 
of isobaric terms, or say columns of the weights 0, 1, 2, 3, 4, ... respectively, for 
instance, 
2 
3 
4 
5 
6 
c 
d 
e 
/ 
9 
6 2 
be 
bd 
be 
¥ 
b 3 
c 2 
cd 
ce 
6 2 c 
b*d 
d? 
6 4 
be- 
b~e 
b 3 c 
bed 
b 5 
c 3 
b 3 d 
Fc- 
b*c 
6 6