138 Proceedings of the Royal Society of Edinburgh. [Sess. 
“ H (1) ” (1846, Nov.). 
[Mathematical notes, ii. Camb. and Dublin Math. Journ., i. p. 286.] 
A correspondent signing himself as above puts on record without proof 
two identities which, when the notation of determinants is used, may be 
written in the form 
ad - bb' - cc ah' + ba' ca' + ac' 
ah' -f ba 
ca + ac 
bb' — cc — ad be + cb' 
be + cb’ cc - ad - bb’ 
= (a 2 + 6 2 + c 2 )(ad + bb’ + cc’)(d 2 + b"> + c' 2 ) 
and 
2 ad 
ah' 4- ba 
ca + ac' 
ab' + ba 
2 bb’ 
be' + cb' 
ca' + ac 
be' + cb' 
2 cc 
= 0 . 
It is seen that both determinants are axisymmetric, that the second is 
expressible as the product of two vanishing determinants, and that the first 
is formable from the second by subtracting aa' -{-bb' -{-cc' from each 
diagonal element, — a fact which, taken along with the vanishing of the 
second, shows that aa' -{-bb' + cc' is a factor of the first. 
Cayley (1847). 
[Note sur les hyperdeterminants. Crelle’s Journal , xxxiv. pp. 148- 
152 : or Collected Math. Papers, i. pp. 352-355.] 
The second paragraph of this note concerns the expression 
Gabcd + 3 b 2 c 2 - a 2 d 2 - 4 ac 3 - 4 b z d , or y sa y 5 
soon afterwards (1851) to be called the “ discriminant” of the binary cubic 
ax 3 ■+• 3 bx 2 tj + 3 cxy 1 + y 3 ; 
and Cayley’s proposition is that the determinant whose elements are the 
second differential-quotients of y with respect to a, b, c, d, namely, the 
axisymmetric determinant 
- 2d 2 6cd 6bd - 12c 2 6bc - iad 
Qcd 6c 2 - 24 bd bad+Wbc 6ac~ 12& 2 
6bd - 1 2c* 2 6ad + 12bc 66 2 - 24 ac Gab 
Gbc-iad Gac - 12& 2 Gab -2 a 2 
is a numerical multiple of y 2 - As a matter of fact he says the multiplier 
is 3 ; but this is because, instead of writing the determinant as here, he 
