302 
ON SYMMETRIC FUNCTIONS AND SEMINVARIANTS. 
[932 
as shown in the following table : 
C 3 D 2 CE G 
1 9 
Sx 6 — 
— 2 
+ 3 
+ 6 
- 6 
+ 6 bf 
Sx 5 y — 
+ 2 
- 3 
- 6 
+ 6 
+ 15 ce 
II 
■1* 
— 2 
- 3 
+ 2 
+ 6 
+ 20 d? 
Sody 3 — 
+ 1 
+ 3 
- 3 
+ 3 
+ 30 6 2 e 
II 
js» 
+ 3 
+ 2 
- 6 
+ 60 bed 
II 
to 
- 3 
+ 4 
- 12 
+ 90 c 3 
aSx 2 y 2 z 3 = 
+ 1 
- 2 
— 2 
+ 120 b 3 d 
Sx’yzw = 
- 2 
+ 6 
+ 180 6 2 c 2 
Sx 2 y 2 zw = 
+ 1 
+ 9 
+ 360 Vc 
Sx 2 yzwt = 
- 6 
+ 720 b s 
Sxyzwtu — 
+ 1 
[ci 2 ] [c 3 ] [6V] [5 6 ] 
the numbers whereof are, it will be observed, identical with those of the foregoing 
table No. 33, relating to the MacMahon equation. 
This is to be read 
= C 3 [d 3 ] + D 2 [c 3 ] + CE [6 2 c 2 ] + G [6«], 
viz. il 6 is a linear function of C 3 , D 2 , CE and G, the coefficients of these, being 
given functions of (b, c, d, e, f g), which given functions are the specific blunt 
seminvariants which have been already called [d 2 ], [c 3 ], [b 2 c 2 ] and [b s ], And so in 
general, the developed value of affords a complete definition of these specific blunt 
seminvariants of the weight w. Observe that cl, /3, y, 3, ... are umbrae in nowise 
connected with the roots a, ¡3, y, 8, ... before made use of, and that B, C, D, ... 
are actual quantities in nowise connected with the symbolic capitals B, C, D, ... before 
made use of. 
54. The capital and small letter symbols are conjugate to each other. It will 
be convenient to give here, in reference to subsequent investigations, a table of these 
conjugate forms up to the degree 6 and weight 15.