136 Proceedings of the Royal Society of Edinburgh. [Sess. 
XII. — The Theory of Circulants from 1861 to 1880. 
By Thomas Muir, LL.D. 
(MS. received September 23, 1911. Read November 20, 1911.) 
Roberts, M. (1861). 
[Question 581. Nouv. Annales de Math., xx. p. 139. Solution by 
E. Beltrami in (2) iii. (1864 February) pp. 64-66.] 
Roberts’ theorem concerns the cireulant C whose elements are the terms 
of the expansion of 
(t+iy-i 
and is to the effect (1) that there are no odd powers of t in the development 
of the cireulant, and (2) that if in the said development £ be put for f 2 , the 
equation in £ 
CUl* - o 
has for its roots the squares of the differences of the roots of the equation 
x n — 1 = 0. 
If od 1 , co 2 , . . . be the n th roots of 1 we have identically 
(t - a h )(t - 0) 2 ) ...(£ — o> B ) = t n - 1 
and therefore also 
| ^ - (cOi - (0 r ) | | t - (a) 2 - (0 r ) | . . . | t - ( (0 n - co r ) | =(t + (Ji r ) n - 1, 
where the r th factor on the left is simply t itself. Hence the expression 
( t + Wj) n — 1 (t + 0> 2 ) n — 1 (t + M n ) n — 1 
t ' t ■ ’ ' t 
consists of n(n— 1) factors which if suitably combined in pairs are 
replaceable by \n{n— 1) factors of the form t 2 — (® r — co s ) 2 - But the said 
expression being equal to C by Spottiswoode’s theorem (which, however, 
Beltrami does not assume *) the desired result at once follows. 
* For bis mode of proof see under Baltzer (1864). 
