1899-1900.] Dr Muir on the Theory of Skew Determinants. 193 
2,3 , ,n, with the exception of (3 '. This determinant of the 
(' n - 2) th order is expected to be seen to be of the same kind as that 
with which we started, and to be temporarily admitted to be 
equal to 
[a' + 1 , . . . , *, 2, . . . , a -1] . [p + 1, . . . , W, 2, . . . , 1]. 
The typical term of the expansion will thus he 
^aa'[a +!>•.., n,2, . . . , a — 1] • +1, . . . , n, 2, . . . , f3' — 1] • 
and the sum of all such terms 
= { ^ 0.2 [31 ...%] + A. a3 [4 . . . n2\ + . . . + A. an [23 . . . (n - 1)] 
‘ {A i 32[34 . . . n\ + A^3[4 . . . %2] + . . . + A^ n [23 . . . (n — 1)] 
and therefore 
==; [a 2 3 . . . n] • [/3 2 3 . . . n\. 
This means, of course, that if the theorem holds for a deter- 
minant of order n- 2 it will hold for the succeeding case. But in 
the simplest case, viz., where n — 2, it is self-evident that the 
theorem holds, for the determinant then 
= \ a ^22 ~ A.2 l sA. a 2 , 
= A. i 32X a 2 , 
= [/32]-[«2]; 
consequently “ Le determinant gauche et symetrique qu’on obtient 
en donnant a r les valeurs a, 2, 3, . . . , n, et a s les valeurs 
/3,2,3, . . . , n (ou n e-st pair) se reduit a 
[a 2 3 . . , n\ . [/3 2 3 . . . n] ; 
et en particulier , en donnant d r, s les valeurs 1,2 , ,n ce deter- 
minant se reduit d [12 3... n] 2 ”. 
Going back now to the expansion of the skew determinant O 
with which the paper opened, and taking for simplicity’s sake * 
\ rr =■ 1 in every case, Cayley readily obtains, 
for n even, 11= [123 . . . n\ 2 
+ [34 . . . rc] 2 + [24 . . . nf + . . . 
+ [56 . . . n] 2 + . . . 
+ 
+ 1, 
* And of course without loss of generality, as Cayley might have said. 
VOL. XXIII. N 
