342 A TENTH MEMOIR ON QUANTICS. [693 
367. Sylvester obtained an expression for the N.G.F. of the quintic : this is 
a° . 
1 
+ a 3 . 
X? 4 X 3 + X 9 
+ a 4 . 
of + X 6 
+ a 5 . 
X + X? + X 1 — iC n 
+ a 3 . 
X- -f X* 
+ a 7 . 
X 4* x 9 — X 9 
+ a 8 . 
X 2 + X* 
+ a 9 . 
X 3 4 x? — X 7 
+ a 10 . 
x" + X? — X 10 
+ « 11 . 
x + xr i - of 
+ a 12 . 
55, 
r 
1 
it 
o 
+ a 13 . 
55 
1 
H, 
1 
% 
+ a 14 . 
Si 
1 
1 
% 
+ a 15 . 
1 
55 
1 
H, 
+ a 16 . 
rp2 . /v»6 /y>10 
\Aj iAj xAj 
+ a 17 . 
1 
1 
+ a 18 . 
1 — iB 4 — X? — X 10 
+ a 19 . 
1 
it 
1 
55 
4- a-°. 
1 
55, 
r 
% 
1 
4- a 23 . 
— X 11 
1 — axr 5 .1 — a 2 # 2 . 1 - 
- a*x s . 1 — a 4 .1 — a 8 .1 — a 12 ; 
viz. expanding this function in ascending powers of a, x, then, if a term is Na 6 x ,i , this 
means that there are precisely N asyzygetic covariants of the deg-order 6. g. 
368. It is known that the number of the irreducible covariants of the binary 
quintic is = 23; representing these by the letters a, b, c, d, e, f g, h, i, j, k, l, m, 
n, o, p, eg r, s, t, u, v, w, (a the quintic itself), the deg-orders of these, and the 
references'* to the tables which give them are 
[* See also the paper, 143, in the second volume of this collection.]