1902 - 3 .] 
Dr Muir on a Special Circulant. 
549 
and, if in this determinant we add to the first column all the 
others, it is possible to remove the factor n and arrive at the 
result 
(6 - 4)-2 5 “6 
1 
- 1 . 1 . 
. -1 . 1 
. -1 
. -1 
1 
in which the determinant of lower order vanishes because the order 
is odd and the determinant zero-axial skew. The true result is 
thus seen to be 
(6 - 4) • 2 5 • 6 • 0 
instead of 
(6 - 4) • 2 5 • 6 • ( - l) 12 - 2 . 
(4) Were we to take the case where p = 2 and n — 7, and treat 
Catalan’s subsidiary determinant as in the preceding paragraph, 
we should find ourselves left, after removing the factors ( n - 2 p), 
2 n ~ x , n , with a determinant co-factor similar to that above but of 
even order, and therefore with the value 1. In this case the 
circulant, viz., C( — 1 , - 1 , 1 , 1 , 1 , 1 , 1 ) would be equal to 
3-2 6 * * * * * as asserted by Catalan. 
A criterion is thus wanted to distinguish the cases p = 2 , n = 6 
and p — 2 i n = 7. 
(5) Theorem. When p and n are not 'prime to one another 
Catalan’s circulant C( - 1 , - 1 , ... , 1,1 ) n , in which p is the 
number of consecutive negative units is equal to 0. 
Looking at the circulant in question, not as the eliminant of a 
set of linear homogeneous equations, but as the eliminant of the 
two equations 
— 1 -x- • ■ • -aP- 1 + z p + • • • + x”- 1 = 0 ) 
l-aj" = 0 J 
we see that, if the common factor of p and n be h, the first of these 
expressions has for a factor 
1 + x + • • • +x ll ~ x 
