1905 - 6 .] Dr Muir on the Theory of Circulants. 
397 
Painvin (1858), Roberts (1859). 
[Questions 432, 465. Nouv. Annates de Math., xvii. p. 185 
xviii. p. 117; xix. pp. 151-153, 170-174.] 
Here it is special circulants that are set for consideration, 
namely, by Painvin the circulant whose elements are the first n 
integers, and by Michael Roberts the circulant whose elements are 
a , a + d , a-\-Zd , . . . . , the result in regard to the former 
circulant being 
C'(l ,2 ,...,«) = ( - 1 Y n{n ~v . \n n ~\n + 1) , 
and in regard to the latter 
C \a, a + d,...,a + n- l‘d) = ( - l) iw(n_1) • ( nd) n ~ l • (f + • 
The first to offer a proof was Cremona, who, after repeating 
(xix. pp. 151-153) Brioschi’s demonstration regarding the re- 
solvability of a circulant, says that in Roberts’ case 0 r being 
= a 
1 “ a r 
■P d 
{- 
1 — a r 
(n^2 "r 
na 
nd 
a r — 1 
and Q n = na + \n{n - 1 )d , 
and that consequently 
6 1 6 2 ... 6 n = 
f or r = 1 , 2 , . . . , n - 1 
( nd) n 1 
/ 1V ,, , — ,-Ana + ln(n-\)d}-, 
(«i - l)( a 2 -!)••• ( a n-l - 1) 
whence the desired result readily follows, because the denominator 
is equal to 
( - 1)" -1 (1 - + - Soqagag-P . . . ) 
where Scq = — 1 , '2a l a 2 = 1 , = — 1 , . . . . 
A proof was also given by G. F. Baehr of Groningen (xix. pp. 
170-173), who changes 
C'K , a 2 , . . . , a n ) into ( - lf n{n - 1] C(a n , a n _ x , . . . , of) , 
performs on the latter determinant the operations 
co^ - col 2 , col 2 - col 3 , . . . . 
row 1 + row 2 + . . . +row n , 
removal of factors d n and ( - l) n_1 • \n\fla + (n - l)eZ} , 
