1905 - 6 .] Hessians of Certain Invariants of Binary Quantics. 531 
Qabcd + 3 b 2 c 2 - a 2 d 2 - 4ac 3 - 4 b 3 d or I 3 A 
we see that the Hessian of the latter is 
- d' 2 cd 
3cd c 2 - 4 bd 
3bd-Qc 2 ad + 2bc 
3 be - 2 ad ac - 2b 2 
M - 2c 2 
ad + 2 be 
b 2 - 4 ac 
ab 
3 be - 2ad 
3ac - 66 2 
3 ab 
= 144 © say. 
Two facts are then recalled from the early days of the algebra 
of quantics: (1) Cayley’s statement (1847) that the said Hessian* 
contains as a factor the second power of the invariant from which it 
is derived ; (2) Eisenstein’s observation (1844) that by substituting 
3 abc - a 2 d - 2b 3 ' 
r 
a, 
2 ac 2 - abd - b 2 c 
acd - 2 b 2 d + be 2 
- for 
c, 
ad 2 - 3bcd + 2 c 3 ^ 
d , 
respectively, I 3)4 is raised to the third power. These facts have 
long since been absorbed in larger truths ; but it does not appear 
to have been pointed out that there exists a formal operational 
connection between the two. 
In the case of § 1, the invariant was a determinant of the 3rd 
order, while its Hessian was of the 5 th, and a preliminary trans- 
formation was consequently necessary in order to prepare for 
multiplication. Here both determinants are of the same order 
at the outset. Multiplying therefore at once © by I 34 in its 
discriminant form 
we obtain 
| a 2b 
c 
| . a 
2b c 
. b 
2c d 
b 2c 
d . 
I 01 
2 01 
g 01 
01 
2 da 
3 db 
6 0C 
dd 
i 01 
l 01 
~*db 
~ 3 dc 
~hd 
01 
5 01 
2 01 
i 01 
da 
6 db 
30C 
2 dd 
i ai 
l 01 
i 01 
2 da 
3 db 
6 0C 
* Of course not so spoken of by Mm at that date. 
