424 
Proceedings of the Royal Society of Edinburgh. [Sess. 
The notation used for the Hessian of <p is H (<p): by “second Hessian” he 
says he means “Hessian of the Hessian”; by “post-Hessian ” the determinant 
of the function got by taking r = 3 ; and similarly for “ prseter-post- 
Hessian ” ! 
We may at once remark that much of this nomenclature had a 
very short Bife, being supplanted by other coinages made by Sylvester 
himself. The functions 
(i + % + ••• ) 
he soon named the emanants of <p : and thus the Hessian, post-Hessian, 
prseter-post-Hessian, and other determinants forming the “ Hessian (or 
Boolian) Scale ” became known as the discriminants of the emanants 
of <p. To the first member of the scale, however, the word “ Hessian ” 
became permanently attached, although Sylvesters mode of defining it 
as the “ discriminant of the quadratic (or second) emanant ” * did 
not spread. It was introduced by Salmon into the first edition of 
his Higher Plane Curves (see p. 72) about a year after Sylvester’s 
first use of it, and met with rapid acceptance. 
Salmon, G. (1852). 
[A Treatise on the Higher Plane Curves, .... By the Rev. 
George Salmon, M.A xii + 316 pp. ; Dublin, 1852.] 
In sect. ix. (pp. 181-195) Salmon deals with the “General Equation of 
the Third Degree ” on the lines of Aronhold’s paper of 1849, and the 
Hessian naturally comes in for attention. The ternary cubic U which 
constitutes the non-zero side of the equation he writes in the form 
cqcc 3 + b 2 y 3 + c z z 3 + 3 a 2 xhj 4- 3 b s y 2 z + 3 c-pPx 
+ 3 a d x 2 z + Zbgpx + 3 c^y + 6 dxyz , 
and gives its Hessian f as 
& Y x 3 + b 2 y 3 + c 3 z 3 + 3a 2 x 2 y + • • • • , 
* See Philos. Magazine , v. p. 122 : or Collected Math. Papers , i. p. 591. 
t It should be noted that this is 216 of the Hessian as defined, that aj is expressible as 
a three-line determinant, and that the performance of the circular substitutions a 1 , b 2 , c 3 
= b 2) c 3 , a lt a 2 , b s , ^ = 63, c lt a 2 , a 3 , b lt c 2 = 6 1} c 2 , a 3 , on the expressions for a l? 3a 2 , 3a s 
gives us six other of the expressions. 
