NOTE ON THE STANDARD SOLUTIONS 
20 
[800 
where the * denotes a value which is not in general = 0, but which may in any 
particular case happen to be so. 
For instance, let it be required for the binary quartic (a, b, c, d, e§cc, yY, to find 
the asyzygetic seminvariants of the degree 4 and weight 6. Assuming for the 
seminvariant the value in the left-hand column of the diagram, the unknown coefficients 
being A, B, C, jD, E, F, G, this must be reduced to zero by the operation 
ctdi•) + 2 bd c + 3cdd + 4dd e 5 
and we thus obtain as many equations as there are terms of the degree 4 and 
weight 5, as appearing by the second column 
cidb + 2b?) c + 3 cdd -f- 4 dd e 
A arc e 
a 2 b e 
2(7 + 2A 
B „ d 2 
„cd 
D + 6B + 4A 
C a b 2 e 
a b' 2 d 
SF -f 2D + 4(7 
D „ bed 
»be 2 
2G+6E + SD 
E „ c 3 
a°b 3 c 
4<G + SF 
F a°b 3 d 
G „ b 2 c 2 
viz. the equations are 
A 
B 
G 
D E 
F 
G 
2 
+ 2 
= 0, 
4 + 
6 
+ 1 
= 0, 
4 
+ 2 
+ 3 
= 0, 
3 + 6 
+ 2 
= 0, 
3 
+ 4 
= 0. 
We have first a solution beginning B= 1, and secondly a solution beginning A — 1, with 
B = 0: the resulting two seminvariants, say P and Q, are 
F = 
Q = 
I 
II 
A 
a 2 ce 
0 
1 
1 
1 
B 
,,d 2 
1 
o 
-1 
o 
G 
a b 2 e 
0 
-1 
-1 
- 1 
D 
„ bed 
- 6 
- 4 
+ 2 
- 4 
E 
,,c 3 
+ 4 
3 ! 
- 1 
+ 3 
F 
a°b 3 d 
+ 4 
4 ! 
+ 4 
G 
„ b 2 c 2 
- 3 
- 31 
+ 3