932] 
ON SYMMETRIC FUNCTIONS AND SEMINVARIANTS. 
287 
and conversely any function annihilated by A is a non-unitariant ; comparing here 
with the subsequent theory of seminvariants, this is in fact the theorem that a 
non-unitariant is the same thing as a seminvariant ; or to state this more explicitly : 
for the MacMahon form of equation, a function of the coefficients which is a 
non-unitary symmetric function of the roots is a seminvariant. 
I consider for instance the Table VI (6), but attend only to the non-unitary 
portions thereof, viz. the lines G, CE, D-, C 3 : I convert these into columns, at 
the same time changing the arrangement of the headings g, bf, ce, &c., from CO 
to AO: and then making the foregoing change b, c, d, e, f g into b, |, 
/» 
1^, , but to avoid fractions multiplying the whole by 720, I form the table 
- 720 
II 
1 
9 
6 bf 
15 ce 
20 d- 
30 b 2 e 
60 bed 
90 c 3 
120 V’d 
180 b-c 2 
360 ¥c 
720 b 6 
C 3 
D 2 
CE 
G 
— 2 
+ 3 
+ 6 
- 6 
+ 2 
- 3 
- 6 
+ 6 
- 2 
- 3 
+ 2 
+ 6 
+ 1 
+ 3 
- 3 
+ 3 
+ 3 
+ 2 
- 6 
- 3 
+ 4 
- 12 
+ 1 
— 2 
— 2 
— 2 
+ 6 
+ 1 
+ 9 
- 6 
+ 1 
№ 
[O 
[6V] 
[6 6 ] 
which is to be read according to the columns: and observe that the outside left- 
hand numbers are to be multiplied into the numbers of each column: thus the first 
column is to be read 
C 3 = Sa 2 /3 2 y 2 = 7 ~ (—2(7 + 12 bf - 30ce + 20 d 2 ), 
the second column is to be read 
and so on. 
D- = Sa 3 /3 s = -^ (3g - 18bf... + 90c 3 ), 
By what precedes, the columns are seminvariants,—as afterwards explained, “ blunt ” 
seminvariants; and they are named as such by the outside bottom line of symbols 
with a [ ]; viz. 
[d 2 ] = (- 2g+ Ubf- 30ce + 20d 2 ), [c 3 ] = (3g - 186/... + 90c 3 ), &c.,