159 
1915-16.] The Theory of Circulants from 1880 to 1900. 
if for shortness’ sake we write 
a ) P J T f° r Aj ” A 2 J A 2 — A 3 
Multiplying this new determinant by 
1111 
L 3 ’ ^3 ^-4 
1 1 
.1111 
. . 11 . 
. 1 . 
. . . . 1 
• 1 , 
1 
1 1 1 
1 
in succession we obtain 
( C r )M2a) 2 
a 
P 
y 
. 
P 
a + /l + y 
P+y 
. 
y 
P + y 
CL + (3 
y 
P 
« p 
y 
-y 
y 
P CL + P + y 
P+y 
-P 
-y 
y P+y 
CL + P 
and thence 
C 7 = 
= 2 a . 
A 1 — A 2 
A 2 ~ A 3 
A 3 — A 4 
a 2 -a 3 
Aj — A 4 
A 2 — A 4 
A 3 ~ ^4 
A 2 — A 4 
A 1 — A 3 
5 
as desired. 
The latter part of the proof is seen to turn on the transformability of a 
certain six-line persymmetric determinant into the square of a three-line 
axisymmetric determinant, and this property is stated to hold for a per- 
symmetric determinant of the (2 n) th order whose distinct elements are 
q>2 , . • . j — ct> n j 0 , cl h , . . . j cq , — oq , . . . , <i n , 0 , a n , . . . , a 3 , 
the elements of the Ti-line axisymmetric determinant being of the form 
a h + a h+ 1 + • . . +a h+k _ 1% 
Torelli, G. (1882). 
[Sui determinanti circolanti. Rendic. . . . Accad. delle Sci. Fis. e 
Mat. (Napoli), pp. 3-11.] 
This paper, following avowedly on those of Glaisher and Scott and 
Muir, contains important generalisations. 
The first theorem is to the effect that any circulant of the (rs)** order 
is expressible as a circulant of the s th order , and each of the elements of 
the latter is an aggregate of r-line minors, r s_1 in number, of the former. 
Torelli’s proof, which is the natural extension of Glaisher’s, is stated in 
