415 
1907-8.] Dr Muir on the Theory of Hessians. 
transforming equation. His proof is essentially that still followed ; that 
is to say, he recalls that from the multiplication theorem we have 
s±4X>. 
. . u ,a> 
n 
, (1) (2) ( n ) . (1) (2) [n 
^ 12 n 12 n 
if 
(A) 
v> ; 
K. 
(A) (/c) , (A) (k) . . (A) (/c) 
= a\ , w\ , + a\’w\ J + ••• +<V/, 
and 
. . 
n 
9 ■ (1) (2) (?l) 
= r 2 • 2j ± v V . . . v , 
— 12 n 7 
if in 
addition 
K 
= ••• +«^«; 
and he then merely asserts that application to the case where 
■-<*> 
K dx K dx\ ’ A dyx 
P> = 
K 
ay 
accomplishes the desired aim. The result, as stated in later phraseology, 
is that “the Hessian is a covariant.” The case where f—ax 2 -\-2bxy-\-cy 2 
was given by Lagrange in 1773, and the case f=ax 2 -\-by 2 -\-cz 2 -\-2dyz 
-\-2ezx-\-2fxy by Gauss in 1801. 
The ternary cubic is next returned to and shown to be transform- 
able by a linear substitution into the form 
yi+V2*+y* B +fayl$ys 
and to be such that constants c v c 2 are determinable which make 
c i w 33 + c 2 H(w 33 ) 
resolvable into linear factors. 
Cayley, A. (1845, early). 
[Note sur deux formules donnees par MM. Eisenstein et Hesse. Crelles 
Journal , xxix. pp. 54-57 : or Collected Math. Papers, i. pp. 113-116.] 
Cayley, who had, like others, been attracted by Boole’s epoch-making 
paper on Linear Transformations, and was about to publish his own first 
paper on the subject ( Camb . and Dubl. Math. Journ., i. pp. 104-122), was 
naturally interested in that part of Hesse’s paper which concerned the 
“ determinant of /.” He consequently wrote the note we have now 
reached, for the purpose of adding to Hesse’s results and of extending 
an identity of Eisenstein’s not distantly related to the same subject. 
The “ equation remarquable ” of Hesse’s which he starts with he writes 
in the form 
V(U + aVU) = AU + BVU, 
noting that its author had not given the values of the coefficients A, B, 
