PROFESSOR CAYLEY’S NINTH MEMOIR ON QUANTICS. 
O o 
-ID 
but here there is not any finite function A (A)' such that this development is 
=A(,-) .-?A(i). 
The numerical coefficients are of course the same as those in the development of the 
untransformed function ; viz. they are the numbers given in the third column of Table 
No. 82 (Eighth Memoir), and also (carried further) in the third column of the following 
Table, No. 87. And we can, from the discussion of these coefficients, deduce the form of 
A(#), viz. this is 
1 — aJ'x 11 
1 — a 3 x' 3 
14 
12 
10 
8 
6 
1 —a 7 x 15 
13 
1 1 
O) 3 
7 
(1-A 13 ) 3 
(10) 3 
( 8f 
( 6/ 
1 —ax 5 1 — a 2 x 6 
2 
1 — aV 
5 
3 
1 — aV' 
4 
0 
1 —a 8 x 7 
3 
1 
1 — qV* 
2 
1 — a 7 x 3 
1 
1 —a 8 x s 
0 
20 
14 
1 -a 18 | . . . 
where, for shortness, I have written 1 — to stand for (1— aV)(l — «V), and so in 
2 
other cases : moreover in the third column of the numerator the (9) 3 shows that the 
factor is (1 — a 7 x 9 ) 3 , and so in other cases: this will be further explained presently. 
Compare herewith the form, Second Memoir, No. 52, viz. the number of asyzygetic 
covariants of the degree 0 is 
= coeffi « 9 in (1 — «) _I (1 — ft 2 ) -2 (l — « 3 ) -3 (l — a 4 )~ 3 (l — « 5 )" 2 (1 — « 6 ) 4 (1 — a 7 f(l — a 8 ) 6 . . . . 
each index being, it will be observed, equal to the number of factors in the numerator, 
less the number of factors in the denominator, in the corresponding column of the 
new formula. 
Article Nos. 337 to 3 46 . —The 23 Fundamental Covariants. 
oo/. Gordan’s result is that the entire number of the irreducible co variants of the 
binary quintic is =23. I represent these by the letters A, B, C, . . ., W, identifying such 
of them as were given in my former Memoirs on Qualities with the Tables of these 
Memoirs, and the new ones, O, P, K, S, T, V, with the Tables Nos. 90, 91, 92, 93, 94, 95 
of the present Memoir. 
