272 
Transactions of the Royal Society of South Africa. 
the elements taking the simple form here given by reason of the equality 
5 m — e5 m _!+e 2 5 m _ 2 — . . . = 4 m . 
Since the last element 
and its cofactor is our dialytic eliminant of the next lower order, it is clear 
that we shall have finally 
as was expected. 
12. A less pleasing but more quickly effective mode of proof would be 
to attest the existence of any one of the linear factors, say d-\-e, by showing 
that the putting of d-\-e equal to 0 in the determinant causes the latter 
to vanish. Or, again, we might perform the operation 
when we should find that the 4th column as thus altered contains the factor 
d+e* 
RONDEBOSCH, S.A., 
Uh May 1922. 
* Apparently the first to observe this peculiar bi gradient was Mr. A. M. Nesbitt. 
See Educ. Times, lvii (1904), p. 490. 
e 4 +5 2 e 2 +5 4 - e 4 +(e4 1 +4 2 ) e 2 +(e4 3 +4 4 ) 
= e 4 +e 3 4 1 +e 2 4 2 +e4 3 +4 4 
= ( e + a )( e +&)(e+c)(e+eZ), 
^1 ^3 ^5 
1 5 2 5 4 
5 2 5 3 5 5 
. 1 5 2 5 4 
(e+a)(e+b)(e+c)(e+d) 
. (d+a)(d+b)(d+c) 
. (c+a)(c+b) 
. (6+a), 
col 4 — de . col 3 + d 2 e 2 . col 2 — d 3 e 3 col l5 
