1911-12.] The Theory of Circulants from 1861 to 1880. 139 
the sum of the suffixes differs from a multiple of n by 
— 1 — 2 — 3— .... + r + s + ^ + . . . . , 
that is by 0. 
Stern, M. A. (1871). 
[Einige Bemerkungen liber eine Determinante. Crelles Journ., lxxiii. 
pp. 374-380.] 
Stern opens with what he means to be merely a fresh proof of 
Spottiswoode’s result : what he actually obtains, however, is something 
more important, namely, not merely the establishment of the fact that 
is a factor of 
0(dq , $2 j • • • j ^ti) ) 
but the further fact that the cofactor is 
Ajfl" + A 2 0; i_1 + A 3 0;?~ 2 + . . . +A J r , 
where A 1 , A 2 , . . . are the signed complementary minors of the elements of 
the first row of C. By way of proof he notes that if the multiplication 
of these two factors be performed, the coefficient of d” in the product is 
Ai«i + A 2 a 2 + ... + A n a n , i.e. C \ 
and the coefficient of any other power of 0 r being equal to the product of 
A x , A 2 , . . . , A„ by a row of C other than the first must vanish. The proof 
is thus seen to be based on the identities 
cqA^ 4- a 2 A 9 -f- . . 
. . -f- CL n A. n 
4 c 
CC n Aj + Cq A 2 + . . 
• + a n _ 1 A n 
= 0 
® 2 A 1 + a 3 A 2 4- . 
. . + a x A n 
= 0 
which Stern also uses to obtain by means of addition the special case 
(cq + # 2 + . . . + a n )(A 1 + A 2 + . . . +A n ) = C, 
or, say, . 2A = C . 
Differentiating, he next obtains from this, with the help of a result 
of Cremona’s, 
2A + 2a.^ = «A», 
oa h 
'V a "V 92A . 
2jA + 2,a . — — = nA K , 
oa„ 
A v 92A\ 
and/ - ^ a \-wruJ = ■ re ^ A ' ) > 
