288 ON SYMMETRIC FUNCTIONS AND SEMIN VARIANTS. [932 
where it will be observed that the symbol within the [ ] is, in fact, the power- 
ender which is in AO the lowest term of the column; and further that this is also 
the conjugate of the capital letter symbol at the head of the column. 
The (b) Tables I to X, with only the change b, c, d, e, ... into b, ~^, ¿rr,... 
At O Zl! 
are given in my paper, “ Tables of the Symmetric Functions of the Roots to the 
degree 10, for the form 
1 + bx + + ... = (1 — ax) (1 — fix) (1 — yx) ...” 
American Mathematical Journal, t. vn. (1885), pp. 47—56, [829]. 
34. By what precedes, it appears that P — b8 operating on a seminvariant gives 
a seminvariant, and that Q — 2bco operating on a seminvariant gives a seminvariant; 
these operators will be further considered in the development of the theory of semin- 
variants. We see further that £A, =bd c + 3cda+6dd e +..., operating on a seminvariant 
gives sometimes but not always a seminvariant, e.g. 
(bd c + 3cdd + 6dd e ) (e — 4bd — 3c 2 + 12& 2 c — 6b 4 ) = 6 (d — 3be + 2b 3 ). 
Seminvariants—the I-and-F Problem, and Solution by Square Diagrams. 
Art. Nos. 35 to 47. 
35. Writing 
1 =1, 
b 1 = b + 0, 
Cj = c 4- 2bd + 6-, 
d, = d+ 3 cd + 3 be 2 + e s , 
ei = e + 4 dd + 6c# 2 + 4 bd 3 + 6 4 , 
&c., 
then there are functions of the unsuffixed letters which remain unaltered if for these 
we substitute the 
have for instance 
suffixed letters: any 
such function is termed a seminvariant. 
<h = 
c + 2 b6 + # 2 , 
i.e., c 1 - bi 2 = c - b 2 , 
-b? = - 
b 2 — 2b6 - 0 2 , 
d i = 
d + 3c# + 3 bd 2 , 
d 1 — 3b 1 c 1 + 2 b x 3 = d — 3 be + 2b 3 , 
— 3b 1 c l = — 
3be - 6b 2 6 - 3bd 2 , 
-3c# -6bd 2 -3d 3 , 
+ 2 b? = 
2 b 3 + 6b 2 d + 6## 2 + 2d 3 ,