932] 
ON SYMMETRIC FUNCTIONS AND SEMIN VARIANTS. 
295 
The subsequent squares w = ll to 16 are, for convenience, given at the end of 
the present memoir (pp. 331 et seq.). 
44. It is to be observed that in each square the outside left-hand terms are 
the non-unitaries in CO and the outside bottom terms are the power-enders in AO. 
I have inside each square written down only the significant numbers, but we might 
fill up the whole square. For instance, when w = 7, the filled-up square would be 
h 
1 
2 
3 
4 
c f 
0 
1 
2 
3 
de 
- 1 
0 
1 
2 
c 2 d 
0 
0 
0 
1 
bd 2 
be 3 
b 3 c 2 
b 7 
where in the first column the numbers relate to the sets hoobd 2 , cfoobd 2 , deoobd 2 
and c 2 d<x>bd 2 (this last set c 2 dccbd 2 is non-existent since c 2 d is in AO inferior to 
bd 2 , i.e. as well for the set as for the diminished set, number of terms is = 0, and 
we have for the compartment 0 — 0, = 0). And similarly for the remaining three 
columns. The process of thus filling up the whole square is a direct and non- 
tentative one, and the conclusions to which the numbers lead are as follows: col. 1, 
the final being bd 2 , the initial cannot be c 2 d, de or cf, but taking it to be h, we 
have the seminvariant h oo bd 2 . Col. 2, the final being be 3 the initial cannot be c 2 d 
or de, but taking it to be cf we have the seminvariant efcobe 3 : it may be added 
that the top number 2 shows that there are two seminvariants h oo be 3 , these are of 
course the foregoing ones hcobd 2 and cf <x> be 3 . Similarly, col. 3, the final being b 3 c 2 , 
the initial cannot be c 2 d, but taking it to be de, we have the seminvariant de oo b 3 c 2 , 
and col. 4, we have the seminvariant c 2 d oo b 7 . 
For the several weights up to 9, we have simply units in the dexter diagonal 
of each square, viz. the non-unitaries in CO correspond to the power-enders in AO, 
or the sharp seminvariants are c<x>b 2 , dec b 3 , &c. See post, Table of Reductions, 
No. 62, which exhibits these correspondences. 
45. For the weight 10, we have deviations: the figures 1 and 2 denote as follows: 
1-1) k oof 2 = 1 
ci „ ce 2 „ 1 
dh „ b 2 e 2 „ 1 
eg „ bd 3 „ 1 
f 2 „ c 2 d 2 „ 1 
c 2 g „ b 2 cd 2 „ 2 
ce 2 „ & „ 1 
cdf „ b 4 d 2 „ 2 
d 2 e „ b 2 c 4 „ 1 
&e „ b 4 c 3 „ 1 
c 2 d 2 „ b 6 c 2 „ 1 
c 5 „ b w „ 1,