ON A THEOREM RELATING TO COVARIANTS. 
431 
698] 
the number of them which are asyzygetic; for instance, i = 5 as above, j = 6, an 
extract from the table is 
3 
k 
e 
(k + 1) 6 
6 
30 
i 
o' 5 
31 
28 
0 
0 
26 
i 
a3c 
27 
24 
i 
af 
25 
22 
2 
cdb, a 2 c 2 
46 
20 
2 
a?e, acf 
42 
18 
3 
a?d, cdbc, c 3 , f 2 
57 
16 
2 
a?i, abf, ace 
34 
14 
4 
a 2 b 2 , a 2 h, acd, be 2 , ef 
60 
12 
3 
abe, al, ce, df 
39 
10 
4 
a 2 g, abd, b 2 c, ch, e 2 
44 
8 
2 
ak, bi, de 
18 
6 
4 
aj, b 3 , bh, eg, d 2 
28 
4 
1 
n 
5 
2 
2 
bg, m 
6 
0 
0 
0 
462 = 
n(H) 
11(5)11(6)’ 
where, for instance deg-order 6.14, the covariants are a 2 b 2 , a 2 h, acd, be 2 , ef, but the 
number against these in the third column being (not 5 but) 4, the meaning is that 
there exists between these five terms one syzygy, making the number of asyzygetic 
covariants of the deg-order 6.14 to be 4. The second column thus in fact contains 
the several values of k, and the third column the corresponding values of 6; whence, 
forming the several products (k +1) as shown, the sum of these is as it should be 
= 462. 
Cambridge, 13 Jidy, 1878.