( 273 ) 
NOTE ON THE PRODUCT OF A SPECIAL ^-LINE DETER- 
MINANT BY ITS CENTRAL MINOR OF THE (?^^-4)TH 
ORDER. 
By Thomas Muir, LL.D. 
(Read August 19, 1914.) 
1. The main attraction of Mr. Roseveare's paper centres in the very 
interesting equivalent given by him for the product of a special n-line 
determinant by its central minor of the (w-4)th order, this equivalent 
taking the form of an aggregate of products of pairs of minors of the 
(w-2)th order. 
For such a product an outwardly similar expression, perfectly general 
in character, has been known since 1879 ; and, not unnaturally, one 
has some expectation that on putting into this general result the very 
special elements occurring in Mr. Roseveare's paper there would be 
evolved by degeneration the new theorem under consideration. This, 
however, is far from being the case. For example, when n is 7, the 
general theorem gives 
11234567; |345l 
1 123451 
345671 1 
[123451 
345671 
1234567 i 345 1 
~ 12345 
134567 ~ 1 
134561* 
234571 
12345] 
34567 
123451 
i 34567 
13457 
' 23456 
23456 1 
13457 
j 12345 
1345671 1 
123451 
1 345671 
~ 23457 
'13456 ! 
34567 ' 
'12345 ; 
whereas the new theorem 
gives 
! 12345671 3451 1 
234561 2 
1 23456 P 
|23456| 
1 234561 
12345671 345 ~ 
23456 
234571 ^ 
1234561 
1234671 
* Trans. R. Soc. of Edinburgh, xxix., pp. 47-51. 
