914 Proceedings of Royal Society of Edinburgh. [sess. 
had in view necessitated reflection on the definition of such 
determinants, and the outcome was a suggestion which it would 
be a serious mistake to pass over. In explaining the character of 
the functions known afterwards as Pfaffians, and which he was 
about to show were closely connected with skew determinants, it 
was natural that he should be struck with certain points of 
resemblance between them and general determinants, and that in 
consequence he should seek a general definition which would 
include both. The new definition given is 
. . . en exprimant par (1 2 . . . b) une fonction quel- 
conque dans laquelle entrent les nombres symboliques 1,2, 
. . . , n , et par ± le signe correspondant a une permutation 
quelconque de ces nombres, la fonction 
Z±(l 2 . . . n) 
(ou 2 designe la somme de tous les termes qu’on obtient en 
permutant ces nombres d’une maniere quelconque) est ce 
qu’on nomme Determinant .” 
It is readily seen that this is much more general than any defini- 
tion in use up to that time, and that it agrees with the ordinary 
definition only when the function (12 ... n) takes the particular 
form \ al Xjs 2 . . . Kn or \ la A . 2j8 . . . X nK . Further, it is not the same 
generalisation as we are familiar with from Cauchy’s great memoir 
of 1812, where determinants are viewed as a special class of 
alternating symmetric functions. This is shown quite clearly by 
the only other case brought forward by Cayley, viz., the case 
where the function (1 2 ... n) is given the form X 12 X 34 . . . X n _ l n . 
Other examples, not given by Cayley, are — 
^ — a 1 23 
i.e. 
a i23 
a l32 
— cz 213 + a 231 + a 312 
a 32l » 
Si 
e 1 
+ 
i.e. 
a !2 _ 
a !3 
_ a -2l _j_ a 23 a 31 
_ a 32 
a 23 
a 23 
ft 32 
a \3 a 31 a i2 
a 21 
There is also in the same paper a more direct contribution 
to the theory of general determinants, viz., the theorem after- 
wards associated with Cayley’s name, and which, — to use later 
