148 
Proceedings of the Royal Society of Edinburgh. [Sess. 
the <x’s is 0 or a multiple of n. For example, in the case of C(n 0 , cq , a 2 , af) 
the coefficients would have to be found for 
a 0 a 0 a 0 a 0 ^ cl^cCqCL-^cl^ 
a 0 a 1 a 1 a 2 
a \ a \Q'\ a \ 5 
Lemonnier, H. (1879). 
[Calcul dun determinant. Nonv. Annates de Math., (2) xviii, 
pp. 518-524.] 
The determinant in question is the circulant in which the elements are 
in equidifferent progression. The two modes of procedure followed are 
like Baehr’s of 1860, but neither is so good. 
Glaisher, J. W. L. (1879 January). 
[Theorems in Algebra. Messenger of Math., viii. pp. 140-144.] 
One of the two theorems is connected with determinants, and is to 
the effect that if the 2 n quantities a 1 ,a 2 \\ . . , a 2n be connected by the n 
relations 
(«1> 
n 5 
1 J - • 
. ,-a 2 
$ a l > ’ a 3 > * * 
• ) in) 
= r 
(«3> 
, • • • • 
. ,-a 4 
$ 
) 
= 0 
(<*5> 
-«4» 
a 3 , ... . 
. ,-a 6 
>cx 
) 
= 0 
( a ‘2n-l ? 
^2n— 2 ) ®2n-3 5 • • 
• ) ~ a 2n 
$ 
) 
= 0, 
then 
C(«! ,a 2 ,a 3 , ... , a 2n ) = 1 , 
and the elements of the first row of C are respectively equal to the com- 
plementary minors of the elements of the first column. 
The truth of the first part of the theorem is a direct consequence of the 
same author’s theorem of the preceding year, our illustration of which 
makes clear that if 
2 ac - b 2 - 2 df+ e 2 = 0 and 2 ae - 2bd + c 2 -f 2 = 0 , 
we must have 
C (a ,b,c,d,e,f) = (a 2 - d 2 - 2 bf+ 2 ce) s , 
and with a third condition 
a 0®2%^3 a S Cl 'ii a 3 a '3 
a 1 a 1 a 3 a s 
«l«2 a 2 a 3 
CL 2 2 
C(a,b , c, d , e,/) = 1. 
