136 
Proceedings of the Royal Society of Edinburgh. [Sess. 
X 2 - 2x 1 . 1 
1 x x . xf 
x 2 - 2 x 2 . 1 
1 x 2 . x 2 2 
xf — 2x s . 1 
X rr 
1 x% . xf 
1 ... 
. . . 1 
where, for the moment, we use x rr to indicate row by row multiplication. 
It should be noted that not more than three of the series are con- 
templated by Cayley, as he viewed the identities mainly on their 
geometrical side, namely, as giving the relation between the distances of 
three points in a straight line, four points in a plane, and five points in 
three-dimensional space.* 
Hesse (1844, Jan.). 
[Ueber die Elimination der Variabeln aus drei algebraischen Gleich- 
ungen vom zweiten Grade mit zwei Variabeln. Crellds Journal , 
xxviii. pp. 68-96.] 
In this paper there appears the first reference to the special form of 
determinant which has for its elements the second differential-quotients of 
a function, and which consequently is axisymmetric. On account of its 
importance this form falls to be dealt with separately : we merely note 
here that Hesse himself viewed it as a special form of Jacobian, namely, 
the Jacobian whose originating functions are the first differential-quotients 
of a single function, and that he called it the determinant of this single 
function, a practice which, when the function is quadratic, is not at 
variance with that introduced by Gauss. 
Cayley (1846). 
• : . * J 
[Probleme de geometrie analytique. Crelles Journal, xxxi. pp. 227- 
230: or Collected Math. Papers, i. pp. 329-331.] 
The problem in question depends on an algebraic identity which, after 
a little examination, is seen to be a property of axisymmetric determinants. 
Cayley writes the identity in the form 
F^U + V^).K(U) - F«,(U).K(U + V») = {F p0 (U)}2 
where 
U = Ax ' 2 + By 2 + C z 2 + 2F yz + 2G zx + 2H xy + 2L xw + TsXyw + 2 'Nzw + P w 2 , 
V = a x + /3y + yz + Stv , 
* To one taking this point of view Sylvester’s paper “ On Staudt’s theorems . . . . ” will 
be of interest. See Philos. Magazine , iv. (1852), pp. 335-345 : or Collected Math. Papers, i. 
pp. 382-391. 
