The theory of determinants in the historical order of development, by Sir Thomas Muir.

HESSIANS (BRosIOSH, 18,54)39

397

Q u Th1 U2 **.
in in1 i12... infl1
- A- in+ in 'u 2
Ul m11 in12. ln
m2 i21 i22 22.
in Un nm'bl
CAYLEY, A. (1 8 56).
[A second memoir on quantics. A third memoir on quantics.'
Philos. Transac. B. Soc. (London), cxlvi. pp. 101-1216, pp.
627-647; or Collected Math. Papes i pp. 25 0-27, PP.
310-3:32.]
In his tables of invariants and covariants Cayley gives the
Hessian of the binary quartic, binary quintic, binary sextic,
binary octavic and ternary cubic. The results are those numbered 9, 105, 33, 42, 61i'
BELLAYITIS, G. (1857, Juue).
[Sposizione elementare della teorica, dci determinanti. ]Iernorie... Istitinto Veneto.... vii. pp. 67-144.]
Bellavitis (M 79, 80) denotes "lI'Llessiano delle fuuzione 0 " by
I DD, DyDy...,.I
calling it also "ii determinante delle derivate-seconde." He
confines himself to three of the main theorems. Hesse's theorem
of 1851 (March) he amplifies, his enunciation being:-If n, a
homogeneous integral function of the variables xj, X2,. xl,,,be
transformed by means of the subbstitution
x = a~k) + a~k~y + +a ky'
Xk 1 Y, 2 2. +a,
* In the third columnm of this last in the Collected Math. Papers, cij and fid
should he cfj and gil, and in the eighth column ach should be ack.