417 
1 907-8. J Dr Muir on the Theory of Hessians. 
or in later notation 
u 
x 2 
2 xy 
y 2 
~ A 
B 
C 
Vi 
A' 
B' 
C' 
2x iVi 
A" 
B" 
C" 
2 
3 
then we should have * 
VVU 
= 2 10 
A 
B 
c 
U. 
A' 
B' 
C' 
A" 
B" 
C" 
In connection with the 
first 
of these 
results 
should note that on putting v = 4 we obtain 
V(U + aVU) = ( - 6a J + 54a 2 I)U + (l + 3a 2 J)VU, 
or V(aU + /3VU) = ( - 6aj3J + 54/3 2 I)U + (a 2 + 3/3 2 J)V U , 
and VVU = 54LU + 3J-VU. 
Further, if the particular form of U be 
this gives 
where 
and 
ax 2 + 4 bxhy + 6cx 2 y 2 + 4 dxy 3 + e?/ 4 , 
VVU - 12 3 (432I»U - J-VU) 
I = ace + 2bcd - ad 2 - eb 2 - c 3 , 
J = ae + 3c 2 - 4 bd . 
In his famous first paper (February 1845) “ On the Theory of Linear Trans- 
formations,” of which this is merely an offshoot, Cayley states that the 
invariance of I had been communicated to him by Boole, along with the 
•still more interesting fact that Boole’s invariant (i.e. the discriminant) is 
equal to J 3 — 27 1 2 . 
Cayley, A. (1847). 
[Note sur les hyperdeterminants. Crelle’s Journal , xxxiv. pp. 148-152 : 
or Collected Math. Papers, i. pp. 352-355.] 
As already noted, the second section of this short paper concerns what 
would, a few years later, have been called “ the Hessian of the discriminant 
Qabcd + 35 2 c 2 - a 2 d 2 - 4ac 3 - 4 b*d 
of the binary cubic.” The result, which is rather inelegantly verified, 
is that the said Hessian is a numerical multiple of the square of the 
discriminant. 
* Instead of 2 10 we find in the original 2 5 , and in tfie Collected Math. Papers 2 8 . 
VOL. XXVIII. 27 
