530 Proceedings of Royal Society of Edinburgh. [sess. 
Jc-H(J)= -8 
d(bc - ad) - J - 2 d(bd - c 2 ) d(cd - be) 
2 c{bc - ad) 4:c(bd - c 2 ) — J - 2 c(cd — be) 
b{bc — ad) -2b(bd-c 2 ) b(cd -be) - J 
— -8[- J 3 + J 2 {b(cd - be) + 4:c(bd - c* 2 ) + d(bc - ad )}] , 
b{cd - be) - 4:c(bd - c 2 ) - d(bc - ad) 
*= 8J 2 
a 
b 
c 
b 
c 
d 
c 
d 
e 
= 8J 2 
- 
a 
b 
c 
. 
c 
• 
c 
d 
e 
- 4 c(bd - c 2 ) 
= 8 J 2 -c-(ae + 3c 2 - 4 bd) , 
whence we have 
H( J) = 8JI ; 
so that, in words, our result is — The Hessian of the cubinvariant 
of a binary quartic is 8 times the product of the said invariant by 
the quadrinvariant. 
(2) The next invariant of a similar kind belongs to the binary 
sextic, its Hessian being a seven-line determinant with quadratic 
elements. Unfortunately the process just followed does not 
continue to be useful. Other considerations, however, seem to 
show that in this case also the invariant is a factor of its own 
Hessian ; and that very probably there exists the theorem that 
the Hessian of a persymmetric determinant of 2n - 1 independent 
elements contains as a factor the (n - '2) th power of the said deter- 
minant. As for the cofactor, its nature is unknown ; save that 
when n = 4, and when therefore the cofactor is of the 6th degree in 
the elements of the determinant 
abed 
b c d e 
c d e f 
defy , 
w'e can easily show that it is neither the sextinvariant of the 
sextic nor the third power of the quadrinvariant. 
(3) Turning now to the binary cubic and to its unique 
invariant 
