1899 - 1900 .] Dr Muir on the Theory of Skew Determinants. 1 09 
CAYLEY (1851). 
[“ On the Theory of Permutants.” Camb. and Dub. Math. Journ. 
vii. pp. 40-51 ; or Collected Math. Papers, ii. pp. 16-26.] 
By this time the widened definition of a determinant which 
Cayley had given in his paper of 1847 had been exploited to a 
certain extent, and had been found profitable both by himself 
and his fellow-worker Sylvester. The paper we have now come 
to, however, is the only one of the series that for the present 
concerns us.* In it he implicitly discards his former usage of 
the word “ determinant ” in any wider sense than that employed 
by his predecessors; adopts instead the word “ per mutant” as 
suggested by Sylvester, and in working out the theory of the 
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) 
Permutants 4 
i ( ' 
a) Pfaffians 
'(b) Commutants -] 
f (/^Determinants 
(B) Intermutants *1 
(or hyperdeter- ( 
minants) 
CAYLEY (1854). 
[“ Becherches ulterieures sur les determinants gauches.” Crelle’s 
Journ., 1. pp. 299-313 ; or Collected Math. Papers , ii. pp. 
202-205.] 
The development with which his 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 special- 
ized, the form in which the identity appears being 
* All of them fall to be dealt v r ith when giving the history of the develop- 
ment of the theory of determinants in general. 
