554 PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
- d, 
3c, 
-35, 
a 
= 9a'b'dd' 
Sc, 
—65 -{-Sd, 
Sa— 6c, 
3 b 
-3 b, 
Sa— 6c, 
65 — 355, 
3c 
a. 
3 5, 
Sc, 
d 
= 9[(«~3 C ) 2 -f-(35-^][(«+c) 2 +(5+^], 
which is the required theorem. And the theorem is thus exhibited in its true connexion, 
as depending on the transformation 
(a, ... x®, y)"=(« f » • • • XMx+iy), ¥p—w)Y- 
Addition, 7th October, 1867. 
Since the present Memoir was written, there has appeared the valuable paper by 
MM. Clebsch and Gordan “Sulla rappresentazione tipica delle forme binarie,” Annali 
de Matematica, t. i. (1867) pp. 23-27, relating to the binary quintic and sextic. On 
reducing to the notation of the present memoir the formula 95 for the representation of 
the quintic in terms of the covariants a, /3, which should give for (a, b, c, d, e, f) the 
values obtained ante , No. 312, I find a somewhat different system of values ; viz. these 
are 
36a = 
36b= 
36c 
= 
36d= 
36e: 
35f= 
A 7 B + 
1 
*A 4 I - 1 
A 6 B 
+ 1 
*A 3 I - 1 
A 5 B 
+ 1 
*A 2 I - 1 
A 6 C + 
1 
A 2 BI - 3 
A 5 C 
+ 1 
ABI - 3 
A 4 C 
+ 1 
ABI -3 
A 5 B 2 + 
6 
*ACI +24 
A 4 B 2 
+ 6 
*CI +12 
A 3 B 2 
+ 6 
A 4 BC - 
39 
A 3 BC 
- 27 
A 2 BC 
-15 
A 3 B 3 + 
9 
A 2 B 3 
+ 9 
AB 3 
+ 9 
A S C 2 - 
54 
A 2 C 2 
- 42 
AC 2 
-30 
A*B*C - 
126 
*AB 2 C 
- 90 
*B 2 C 
-54 
ABC 2 + 
288 
BC 2 
+ 144 
C 3 +1152 
where I have distinguished with an asterisk the terms which have different coefficients 
in the two formulae. I cannot at present explain this discrepancy. 
