and (from the invariantive property of the determinant) the original determinant is 
equal to the determinant of this new form, viz. we have 
= 9a'b'c'd', 
d, 
3c, 
-3b, 
a 
3c, 
— 6 b + 3d, 
3a — 6c, 
3b 
3b, 
3a— 6c, 
6b — 3d, 
3c 
a, 
3b, 
3c, 
d 
= 9 [(a — 3c) 2 + (3b — d) 2 ] [(a + cf + (b + d) 2 ], 
which is the required theorem. And the theorem is thus exhibited in its true 
connexion, as depending on the transformation 
(a, ...\x, y) n ={a', ...\%{x + iy), \{x-iy)) n . 
Addition, 1th 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 
di Matematica, t. I. (1867) pp. 23—79, 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= 
MB + 
1 
*A 4 I - 1 
A 6 B + 
1 
*A 3 I - 1 
A*B 
+ 
1 
*A 2 I - 1 
A 6 C + 
1 
A 2 BI- 3 
A S C + 
1 
ABI - 3 
A 4 C 
+ 
1 
ABI - 3 
A 3 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 3 C 2 - 
54 
A 2 C 2 - 
42 
AC 2 
— 
30 
A 2 B 2 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.