161 
1915-16.] The Theory of Circulants from 1880 to 1900. 
and hence finally the product of the five triads is equal to 
C(P, Q, R, S, T), 
as predicated. Here r is 3 and s is 5. 
Had we written the first linear factor of the given circulant in the 
form 
a i + « 2 ( c i7i*) + ^(w) 2 + • • • + a i 5 ( e i7i i ) 14 > 
we should have changed it at the outset into 
( a i + %yi + a n7i) + («2 + «r7i + a i27i) e i7T + • • • + K + a io7i + a isyl)4y> , 
and so have obtained for the product of the first five linear factors the 
circulant 
C{ (cq + « 6 7 i + ^n7i) ) ( a 2 a 77i -"t a i27i)7i T > • • • > ( a b a io7i a i57i)7i 6 | * 
This by multiplication of rows and division of columns we should have 
changed into 
<%l + ^67l a n7l ^2 ^77l "t ^127 i • • * "t a io7l a i57l 
a i5 a 57i “t tt io7i a i "*■ tt e7i "f" a n7i • • • “** a 9 7i a i47i 
a i 4 + a 47l + a 9 7l ®15 + a 57l + a io7l * * • %+ a 8 7l + tt 137l 
a i3 + a 37l + a 8 7i a i4 + a 47l + a 9 7l * • • a 2 + a 7 7l + a i27l 
a i2 + a 27l + a 1 7i ft 13 a 37l + a 8 7l • • • a i + a 6 7l + a il7l ? 
from which we should have passed on to the form 
X + Yy 1 + Z 7 i. 
Thereupon we should have seen the product of the next five linear factors 
to be 
X + Yy 2 + Z 7 2, 
and the product of the last five to he 
X + Y73 + Z73 ; 
and therefore the product of the whole to be 
C(X, Y, Z). 
Here r is 5 and s is 3. 
From this transformation of a circulant of composite order into one 
of lower order Torelli proceeds to deduce a circulant factor of the original. 
Thus, having reached C(P, Q, R, S, T) as an equivalent of the fifteen-line 
circulant, we know that 
P+Q+R+S+T 
must be a factor of it, and, looking back at the three-line determinant 
whence P, Q , . . . originated, we see that their sum equals 
C(a 1 + a 4 + . . . +.a 18 , a 2 + a 5 + . . . +a 15 , a 3 + a 6 + . . . + a 15 ) , 
YOL. XXXYI. 11 
