844] 
ON A THEOREM RELATING TO SEMINVARIANTS. 
327 
es 
3 bS 
Sum 
-146ЙГ 
Sum 
140 S 
-3 OS 
-=-110 
i 
l 
1 
+ 
7 
+ 7 
+ 
14 
- 21 
+ 7 
0 
bh 
- 
5 
- 
3 
- 
8 
- 
42 
- 14 
- 56 
- 
70 
+ 126 
- 56 
0 
eg 
+ 
2 
+ 
2 
+ 
38 
+ 38 
+ 
28 
- 114 
+ 196 
+ 110 
+1 
df 
+ 
4 
+ 
4 
- 
2 
— 2 
+ 
56 
+ 6 
- 392 
- 330 
- 3 
e 2 
- 
5 
- 
5 
- 
15 
- 15 
- 
70 
+ 45 
+ 245 
+ 220 
+ 2 
b 2 g 
5 
+ 
21 
+ 
26 
+ 
60 
+ 98 
+ 158 
+ 
70 
- 180 
- 110 
- 1 
bef 
3 
- 
27 
- 
24 
- 
96 
- 126 
- 222 
+ 
42 
+ 288 
+ 330 
+ 3 
bde 
+ 
5 
+ 
15 
+ 
20 
+ 
60 
+ 70 
+ 130 
JL 
70 
- 180 
- 110 
- 1 
c 2 e 
- 
30 
- 
30 
- 
30 
- 30 
- 
420 
+ 90 
- 330 
- 3 
cd 2 
+ 
20 
+ 
20 
+ 
20 
+ 20 
+ 280 
- 60 
+ 220 
+ 2 
ьу 
- 
36 
- 
36 
- 168 
- 168 
bee 
+ 
90 
+ 
90 
+ 420 
+ 420 
b\P 
- 
60 
- 
60 
- 280 
- 280 
+ 
40 
+ 
126 
+ 
163 
+ 
185 
+ 588 
± 773 
+ 
560 
+ 555 
+ 448 
+ 880 
± 8 
The columns show ©$ (where observe that, to operate with the bd a of ©, we restore 
the proper power of a, reading S as being = + larh — labg + &c., and putting ultimately 
a = l) and — 3bS, and the sum of these, which is a seminvariant, degree 4, weight 8; 
also QS and —14bS, and the sum of these, which is a seminvariant, degree 4, weight 
8. They also show 14©$ and — 3П$, the sum of which would be a seminvariant, 
degree 3, and weight 8 ; instead of giving this sum, I have added a column equal 
to + 7 (г — 8bit + 28cg — 56df+ 35e 2 ), and given the sum of the three columns which will 
of course be a seminvariant of the same degree and weight ; the coefficients contain all 
of them the factor 110, and, throwing this out, we have in the last column the 
seminvariant eg— %df+... + 2cd 2 of the degree 3 and weight 8, derived by the foregoing 
direct process from the given seminvariant S of the degree 3 and weight 7.