932] 
ON SYMMETRIC FUNCTIONS AND SEMIN VARIANTS. 
289 
and thus c — 6 2 , (6-36c + 26 3 are seminvariants; they are, in fact, the first and second 
terms of the series 
c — b 2 , 
d — 36c + 26 s , 
e — 46(6 + 66 2 c — 36 4 , 
f — 56c + 106 2 (6 — 106 3 c + 46 5 , 
g — 6bf + 156 2 c — 20b 3 d + 156 4 c — 5b 6 , 
where the law is obvious; the numbers in each line are binomial coefficients except 
the last number, which is the next binomial coefficient diminished by unity. The 
successive terms are, in fact, what c x , d x , e x , f x , g x , ... become upon writing therein 
6=-b. 
36. Any rational and integral function of these forms is a seminvariant, and it 
is to be observed that we can form functions for which (by the destruction of terms 
of a higher degree) there is a diminution of degree; for instance, 
(e — 4<bd + 66 2 c — 36 4 ) + 3 (c — 6 2 ) 2 
gives a seminvariant e — 46(6 + 3c 2 . 
It is important to remark that a seminvariant is completely determined by its 
non-unitary terms; thus for e — 4>bd + 3c 2 , the non-unitary terms are e + 3c 2 , and for 
this writing e x + 3cx 2 , and for e x , c x substituting their above values for 6= — b, we 
reproduce the original value e — 46(6 + 3c 2 . 
37. It is at once seen that a seminvariant is reduced to zero by the operation 
A = + 269 c + 3cda + ..., or say that A is an annihilator of a seminvariant; in fact, 
if in any function of 6, c, d, ... we write for these the suffixed letters b x , c x , d x , ... 
then the coefficient of 6 herein is at once found by operating on the function of 
(6, c, (6, ...) with A, and therefore in the case of a seminvariant the result of this 
operation must be =0. And conversely, every function of (6, c, (6, ...) which is 
reduced to zero by the operation A is a seminvariant. 
38. For a given weight, the number of seminvariants is equal to the excess of 
the number of terms of that weight above the number of terms of the next 
preceding weight, or what is the same thing, it is equal to the number of power- 
enders of the given weight. More definitely, considering the terms of a semin 
variant as arranged in A 0, we have seminvariants the finals whereof are the several 
power-enders of the given weight: and we arrange the seminvariants inter se by 
taking these power-enders in AO: thus for the weight 6, we have seminvariants 
[° 7 ']> [° 3 ]> [6 2 c 2 ], [6 (i ] ending in these terms respectively. We may, if we please, consider 
all these seminvariants as beginning with g, or say the forms may be taken to be 
C. XIII. 37