932] ON SYMMETRIC FUNCTIONS AND SEMIN VARIANTS. 
Thus for degrees 
307 
2» 3, 4, 5, 6, 
the G.F.’s are 
= x 2 + 2, x 3 + 2.3, ¿e 4 -7- 2.3.4, a; 5 -r 2.3.4.5, x* + 2.3.4.5.6, ... 
58. We may analyse these results by separating the finals into classes. I use 
the expression b, c, d, ... are discrete letters, meaning thereby that they are distinct 
letters, not of necessity consecutive but with any intervals between them. Thus 
deg. 3, if (b, c) are discrete letters, then the finals are b 3 , and bc 2 \ deg. 4, if b, c, d 
are discrete letters, then the finals are b 4 , be 3 , b 2 e 2 , and bed 2 ; and so on, the number 
of classes being doubled at each step, as will presently appear for the weights 5 and 
6 respectively. 
I notice also a property of the conjugates of these classes; for b 3 and be 2 
themselves the conjugates are D and CD, and these occur as factors, D in the 
conjugate of every form of the class b 3 (for instance conjugates of c 3 , d 3 are D 2 , D 3 ), 
and CD in the conjugate of every form of the class be 2 (for instance, the conjugates of 
bd 2 , ce 2 are C 2 D, C 2 D 2 ); and the like in other cases, viz. for any class whatever the 
conjugate of the first or representative form occurs as a factor in the conjugates of 
the several other forms belonging to the same class. 
59. With these explanations, the expressions for the several G. F.’s are obtained 
without difficulty, and we have 
deg. 2, class C, b 2 G. F. = x 2 -=r 2, 
deg. 3, „ D, b 3 „ =x 3 + 3, 
„ CD, be 2 „ =ic 3 -i-2.3 ; 
we ought here to have 
x 3 -r- 2.3 = x 3 -f- 3 + a? -7- 2.3, viz. in verification 
x 3 = x s . 2 = x 3 — x 3 
+ x 3 + x 5 
—— /JQO • 
deg. 4, class E, b 4 
„ DE, be 3 
„ CE, b 2 c 2 
„ CDE, bed 2 
G. F. = x A -7- 4, 
„ =f-r3.4, 
„ = x s 2.4, 
„ = ic 9 -h 2.3.4 ; 
we ought here to have 
oc 4 t2.3.4= ¿t' 4 -r4 + ic 7 -^3.4 + tf 6 -^2.4 + ic 9 -h2.3.4, viz. in verification 
5 — X 7 + X 9 
39—2