306 
ON SYMMETRIC FUNCTIONS AND SEMIN VARIANTS. 
[932 
Instead of (a — /3) 8 (a — 7) (ft — 8) which contains a 10 , that is, k, we may take 
(a - /3 ) 8 (7 - 8) 2 , 
that is, 
(г — 8bh + 28eg — 56df+ Зое 2 ) (c — b 2 ): 
this is ciaob 2 e 2 , a blunt form; by subtracting from it ci 00 ce 2 , we could obtain the 
next sharp form dh 00 b 2 e 2 ; but this in passing; it does not appear that there is any 
monomial umbral expression for the last-mentioned form. 
I do not stop to examine the next following forms, but pass on at once to the 
last of them; instead of the expression given, we may take the expression 
(a - /3) 2 (7 - 8) 2 (e - O 2 (v ~ в) 2 0 - к) 2 , 
that is, (c — bj, which is in fact the sharp form c 5 00 b 10 . 
Seminvariants of a given Degree: Generating Functions. Art. Nos. 57 to 59. 
57. We may consider the seminvariants of a given degree, and arrange them 
according to their weights: thus in each case writing down the series of finals, and 
for a reason that will appear also the conjugates of these finals (see Table of 
Conjugates, ante No. 54). 
For degree 2, or quadric seminvariants, we have 
2 3 4 5 6 ... 
G, b 2 - C 2 , c 2 - C\ d 2 
there is here for every even weight (beginning with 2) a single form, and for every 
odd weight no form: the number of forms of the weight w is thus = coeff. of x w 
in x 2 -=r{ 1—x 2 ), or writing for shortness 2 to denote 1 — x 2 (and similarly 3, 4,... to 
denote l—« 3 , 1— oct, ...), say that for degree 2, Generating Function, G.F., is = x 2 + 2. 
For degree 3, or cubic seminvariants, we have 
3 4 5 6 7 ... 
D, b 3 — CD, be 2 D 2 , c 3 G 2 D, be 2 
the counting is most easily effected by means of the conjugate forms; these contain 
all of them the factor D, and omitting this factor we have all the combinations of 
G, D which make up the weight w — 3, viz. for weight w, we have number of ways 
in which w — 3 can be made up with the parts 2, 3: that is, 
for degree 3, G.F. is =x 3 -i- 2.3. 
Similarly for degree 4 or quartic seminvariants, we have terms each containing 
E, and removing this factor, we have all the combinations of G, D, E which make 
up the weight w — 4, viz. 
for degree 4, G. F. is = x 4 2.3.4.