200 
Proceedings of Royal Society of Edinburgh. 
12345 | 13345 = 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
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)2 
11 
44 
.55. 
(23)2 
22 
33 
.44. 
(15)2 
22 
33 
•55- 
(14)2 
22 
44 
•55* 
(13)2 
33 
44 
.55. 
(12)2 
11 
(2345) 2 
22 
(1345) 2 
33 
(1245)2 
44 
(1235)2 
55 
(1234)2, 
where the symbol on the left stands for the determinant whose 
elements are 11, 12, . . . , 21, 22, . . . and the peculiarity of 
skewness is understood but not expressed. Had the specialization 
of the elements of the diagonal been as before, the development 
would clearly have been 
1 
+ (45) 2 + (35) 2 -f (34)2 + (25)2 + (24) 2 + (23) 2 + (15) 2 + ( 14) 2 + ( 1 3) 2 + (12) 
+ (2345)2 + (1345) 2 + (1245) 2 + (1235) 2 + (1234) 2 , 
which, if the order be reversed, agrees exactly with the result of 
putting n — 5 in the identity towards the end of the paper of 1846. 
By way of explanation Cayley adds the sentence “ Les expressions 
12, 1234, etc, k droite sont ici des Pfaffiens — which is noteworthy 
as being the first intimation that he desired “ les fonctions de M. 
Jacobi/’ as he had formerly called them, to be known by the name 
of the mathematician whose integration-method had led Jacobi to 
the discovery of them. The change is easily accounted for by the 
fact that it was more appropriate to attach Jacobi’s name to 
another class of determinants which were of greater importance 
and to which Jacobi had given far more attention. 
