The Determinant of the Sum of Two Circulant Matrices. 295 
(8) In the foregoing analytic search for a proof the primary minors of 
the determinant 
1 (I3 a.i, 
1 ^2 % 
1 a-^ a2 
1 a- 
1 a.^ a^^ a- ai 
will be found on closer investigation to play an important part. It is naturally 
viewable as the determinant got from the circulant C(aj, a^, a.^, ^4, a-^) by 
removing the factor + «2 + % + + ^5' ^^^^ what our determinants 
of §1 reduce to on making all the h's vanish. 
Two properties of the said minors have been made use of for our main 
purpose, but in their quite general forms they are worth enunciating on 
their own account. The first is that in the determinant C(aj, a^, ^3, a^, a-)/ 
+ 610 + 03 + + ci^) ^^^^ cof actor of the {r,s)^^' element differs only 
in sign from the cofactor of the {r -\- s — 1, 7 — s)^^' element, (r,s) hei7ig 
any place in the secondary diagonal or on the upper side of it hut not in the 
first column. The second is that in the determinant C{ai, a^, a^, a^, a-)/ 
(a^ + a.2 + ffs + a 4, + a^) we have 
cof(r, 2) + cof(r + 1, 3) + cof(r + 2, 4) + co/(r + 3, 5) = 0, 
where r may have any one of the values 1, 2, 3, 4, 5, and, cof {r,s) stands for 
the cofactor of the element in the (r,s)^^'' place. 
(4) Now let M denote the determinant of the sum of the matrices of 
0(^1, ao, a.^, a^, a^), G{h^, h.,, h.., h^, 65), 
the former being taken symmetrical with respect to the secondary diagonal, 
and the latter with respect to the primary diagonal ; and let N denote the 
determinant of the differences of the same two matrices. Then multiplying 
M by N in the ordinary way we find that only three of the twenty-five 
(25) products of pairs of rows are distinct, one occurring 5 times, and each 
of the two others 10 times. Further, we find that the product-determinant 
comes out in the form of a circulant, namely, that 
a-^-\-h^ + 
••• % + ^5 
-ho . 
• 
h 
a. + 63 ai + 63 
... a^-\-h^ 
a- — ho a^ 
-h ■ 
a^+h^ a, + h_^ 
... a^ + h^ 
^4 — ^3 % 
-^^4 ■ 
■ «3- 
ao + &3 
— ^4 «'4- 
ao — 
^2 + ^5 % + 
... % + &4 
a-^ — 
V W W Y 
= C (XJ, Y, W, W, Y), 
Y U Y W W 
W Y U Y W 
W W Y U Y 
Y W W Y U 
