1896-97.] Dr Muir on Bordered Skew Determinant. 
343 
and lie explains that the expressions 12 , 1234 , etc., are Pfaffians, 
whose law of formation is — 
12 = 12 , 
1234 = 12- 34 + 13- 42 + 14 -23, 
123456 = 12 • 34 • 56 + 13 • 45 • 62 + 14 • 56 • 23 + 15 . 62 • 34 + 16 • 23 • 45 
+ 12 • 35 • 64 + 13 • 46 ■ 25 + 14 • 52 ■ 36 + 15 • 63 • 42 + 16 • 24 • 53 
+ 12 • 36 • 45 + 13 ■ 42 • 56 + 14 • 53 • 62 + 15 • 64 • 23 + 16 • 25 • 34 • 
No proof is given, and the law of formation of the development 
itself is not explained. 
Rather more than three years afterwards (Nov. 1857)* he 
returned to the theorem, stating it then by means of two instances, 
as follows : — 
al23 
93 — a 0 • 11 . 22 . 33 
P " + a/3. 12 . 12. 33 
+ a/3. 13 . 13 . 22 
+ a/3 . 23 . 23 . 11 
+ al .01.22.33 
+ a2.02.11.33 
+ a3 . 03.11 . 22 
a!234 
+ al23 . 0123 ; 
01234 
= a/3 . 11 . 22 . 33. 44 
+ a 0. 12. 12.33.44 
+ 
+ a01234. 1234 
+ ...... 
Manifestly the two statements regarding the determinant of the 5th 
order do not agree ; and, as again no proof is offered, there is no 
immediate means of ascertaining the correct statement. On turn- 
ing to the Collected Mathematical Papers (vol. ii. p. 203), the 
confusion is worse confounded, for there neither of the original 
statements is followed, the theorem being given in a third form, viz., 
«1234 
01234 
= a/3 . 11 . 22 . 33 . 44 
-fa/3.12.12.33.44 
+ . 
+ a0. 1234. 1234 
+ ........... 
"Cayley, A., “ Theoreme sur les determinants gaudies,” Crelle’s Journ. 
lv. pp. 277, 278 ; or Collected Math. Papers , iv. pp. 72, 73. 
