1919-20.] Determinants connected with Circulants. 
23 
III.— Note on the Determinant whose Matrix is the Sum of Two 
Circulant Matrices. By Sir Thomas Muir, LL.D. 
(MS. received October 11, 1919. Head November 3, 1919.) 
(1) Perhaps the most interesting theorem as yet known in regard to 
circulants concerns the determinant whose matrix is the sum of two 
circulant matrices, one of the latter being taken symmetric with respect 
to the primary diagonal and the other with respect to the secondary 
diagonal : for example, the determinant 
<q 4“ b^ 
a 2 4- b 2 
cl 3 4“ b< 
cl 3 4- b 2 
a i + 1) 3 
a 2 4- b 
a 2 ^3 
a 3 + b l 
CL j 4* b. 
whose matrix is the sum of the matrices 
a x a 2 a 3 ^1 ^2 ^3 
a 3 <q a 2 and b 2 b 3 b 1 
a 2 a 3 a l b 3 b x b 2 , 
that is to say, the matrices of 
CK,a 2 ,a 3 ) and ( - l)^-, . C (b x ,b 2 ,b 3 ). 
(2) The property in question is that such a determinant of the n th order 
has the quadratic factor 
(<q + <qo> + ftg(o 2 4- . . . )(oq 1 + $3 to 2 + . . . ) — (b^ + b 2 (x> 4- . . . )(& 1 + b 2 w~^ + . . . ) 
where co is a 'primitive n th root of 1. The proof originally given was of 
that indirect character which consists in the finding of two different forms 
of the resultant of a set of linear equations and thereby arriving at an 
equality. This was first published thirty-eight years ago ( Messenger of 
Math., xi, pp. 105-108), and up to the present no direct proof has been 
arrived at, — no proof, that is to say, resting on mere transformation in 
accordance with the ordinary elementary properties of determinants. 
(3) If we restrict ourselves, merely for ease in writing, to the case 
where n is 5, what we have thus got to show is that, e being a primitive 
5 th root of 1, 
(cq 4- « 2 e 4- ... 4- + tqe -1 + • • • + % e ~ 4 ) ~ (^i + b 2 e 4- . . . 4- b^){b^ 4- b 2 e _1 4- ... 4- 
