160 Proceedings of the Royal Society of Edinburgh. [Sess. 
perfectly general terms. We shall, for the sake of greater clearness, apply 
it only to the case where r*s=15, the elements of the circulant being 
, (^2 ? ^3 j • • • 5 ®15 • 
The fifteenth roots of 1 are the three third roots of each of the five fifth 
roots of 1 , so that if y 1 , y 2 , y 3 be the third roots of 1 and e p e 2 ,. . . , e 5 
the fifth roots of 1 , the fifteenth roots of 1 may be represented by 
71*1* > 72*1* > 73*1* ; 7l*2* > 72*2 i > 73*3* ; • ■ • ; 7 i* 5* > 72*5* » 73*5^ > 
and the first of the fifteen linear factors of the given circulant by 
a i + ^ 2 ( 71 * 1 *) 4 a s(7i*i*) 2 + a 4(7i*i*) 3 + • • • + a i5(7i*d) 14 - 
But this, by attending to the powers of y^* and rearranging accordingly 
is equal to 
(«i + a 4 c i + 4- a 10 e? 4- a 13 el) + (a 2 + « 5 e 2 4- ... 4- a 14 e i)7i*i* 
+ («3 + a 6 *l + • • • + ft 15*9 2 7l*l S 5 
and as the second and third factors differ from it merely in having y 2 , y 3 
respectively for y 1 , the product of the three is the circulant 
a i + a 4*l + • • • ( a 2 + a 5*l + ■ ■ • )* 1 * ( a B + a 6 *l + • • • ) e i* 
(a 3 + a 6 e 4 + . . . )ej$ + a 4 ej 4 - . . . {a 2 4 - a 5 e 1 + . . . )eT 
(a 2 + a 5 e j 4- . . . )ed (a 8 4- a 6 e 1 4- . . . )e 1 ^ cq + aqq + . . . , 
which, by multiplying the rows in order by Cl * , e 4 § , ej , and thereafter 
dividing the columns in order by the same, becomes 
(a 1 + oqq + . . . ol 2 + a 5 €j + . . . cl 3 + « 6 e 4 + . . . 
(a 3 4- a Q e 1 + . . . )e 1 a 1 + a 4 e 4 + . . . a 2 + cqq + . . . 
(a 2 + a 5 e 4 + . . . )e 4 (a 3 4- a 6 e 4 + . . . )e 4 a x 4- o 4 e 4 + . . . . 
Performing the multiplications by e 1 , we see this to be equal to 
a x 4 -a 4 e 1 + • • • + a i3 4 0-2 + a 5*l+ • • ' + a i4 e i a 3 + a 6*l + • ' • + a !5*l 
a i 5 + rt 3 *i 4- . . . + a 12 e\ cq 4- a 4 c r 4- ... 4- a l3 e\ a 2 4- 4 - ... 4- a 14 ef 
a i4"t a 2*l“t • • • + a il*l tt 15 + '' / 3*l+ • • • + a i2*l « 1 4-« 4 e 1 4- • • • 4-a i3 €i , 
and therefore to be expressible as a sum of 5 3 determinants with monomial 
elements which can be grouped so as to take the form 
P 4- Qq 4- Rq 4- Sef 4- Tcf , 
where each of the coefficients of the powers of e 4 is a sum of 5 3-1 such 
determinants. It is next observed that the fourth, fifth, sixth linear 
factors of the given circulant differ from the first, second, third merely 
in having e 2 in place of : their product therefore is 
P 4- Qe 2 4- Rea 4- Se 3 4- Te| : 
