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

SKEW DETERMINANTS (CAYLEY, 18,51)

267

general functions under this name assigns to determinants and
Pfaffians their proper niches in the new structure, the scheme of
classification being

'A) (no name)

{(a) Pfaffians

f b)
1). Intermutantsl
(or hyperdeterm11inants)

Commautants

f(() Determinants.

CAYLEY, A. (1854).
[Recherches ulterieures sur les determinants gauches. Crelle's
Jommn., 1. pp. 299-313; or Collected Math. Papers, ii. pp.
202-205.]
The development with which this paper of 1847 closes is here
recalled and repeated for the case where the skew determinant
is of the 5th order and the elements of the diagonal are specialised, the form in which the identity appears being

12345 1 13343 =

11.22. 33. 44 55
~ 11.~22.33.(45 )2
~ 11~22. 44-.(35)2
+ 11.22.55.(34)2
+ 11. 33 44. (25 )2
~ 11 33.55.-(24
+ 11.44.55. (23)2
~ 22. 33. 44.-(15)2
+ 22.- 33 3.55.(14)2
+ 22.~44.~55.(13)2
+ 3-3. 44.55.-(12)2
+ 11.(2345)2
+ 2.2.(1345)2
* 33.(1245)2
+ 44.(1235)2
+ 55.(1234)2,