296 
ON SYMMETRIC FUNCTIONS AND SEMIN VARI ANTS. 
[932 
and they indicate the sharp seminvariants k oof 2 , ci oo ce 2 , &c.: where observe that 
the power-enders being in AO as before, the non-unitaries are not in GO, but we 
have inversions (c 2 g, f 2 ) and (cdf\ ce 2 ). 
In particular, (1 — D) (/ 2 oo c 2 d 2 ) = 1 indicates the seminvariant f 2 oo c 2 d 2 ; 
(1 — JD) (c 2 g oo b 2 cc¥) = 2, 
means in the first instance that there are 2 seminvariants c 2 g oo b 2 cd 2 , but here the 
set c 2 g oo b 2 cd 2 includes as part of itself the set f 2 oo c 2 d 2 ; so that, if c 2 g oo b 2 cd 2 is 
used to denote any particular form, then the general form is c 2 g oo b 2 cd 2 plus an arbitrary 
multiple of f 2 oo c 2 d 2 , and we have thus virtually a single form c 2 g oo b 2 cd 2 . And 
similarly, the set cdfccb 4 d 2 includes as part of itself the set ce 2 ooc 5 \ and thus the 
general form cdf oo b 4 d 2 is = particular form plus an arbitrary multiple of ce 2 ce c 5 , or 
we have virtually a single form cdf cc b 4 dr. 
I remark that it would be allowable to take as a standard form of c 2 g oo b 2 cd 2 , 
a form not containing any term in f 2 , and similarly for the standard form of 
cdf oo b 4 d 2 a form not containing any term in ce 2 ; but this is not done in the tables. 
46. The diagram for weight 10 is constructed by the following calculation; 
viz. in col. 1 we calculate (1 — D)(kccf 2 ) and for this purpose write down the 
terms of kccf 2 , and D(kccf 2 ) in GO: in col. 2 we calculate (1 — D) (ci oo ce 2 ), and 
for this purpose write down the terms of k oo ce 2 and D(k oo ce 2 ) in GO, the terms 
of ci oo ce 2 and D (ci oo ce 2 ) being thence found by rejecting the terms k, bj and the 
term j at the head of the two halves of the column. So in col. 3 we calculate 
(1 — D) (dh oo b 2 e 2 ), and for this purpose write down the terms of (k oo b 2 e 2 ) and 
]J (k cc Ire 1 ) in GO, and for dhccb 2 e 2 and D(dli — b 2 e 2 ) reject the terms k, bj, ci, b 2 i 
and j, hi at the head of the two halves of the column. And so for the remaining 
columns. It is to be remarked that there is in each successive column a continually 
increasing number of terms to be rejected; by a properly devised variation of the 
algorithm it would have been possible to avoid writing down these terms at all, but 
for greater clearness I have inserted them.