141 
1911-12.] The Theory of Circulants from 1861 to 1880. 
where 1 ,0 1 ,0 2 , ... , 0 n are the n th roots of 1, and thence by multiplication 
and the use of Spottiswoode’s theorem 
$0 
$1 
$2 • ' 
• • $ n - 1 
$ 11-1 
$ 0 
$1 • ■ 
■ • $ n — 2 
= e *( i + fli + • . . + e w _ i ) = e ° = 1 
$ 1 
$ 2 
$B * 1 
• • $ 0 
Similarly \js 0 , \p- 1 , \fs 2 , . . . , being the functions got from the (p’s by 
making the even-numbered terms of the latter negative, and therefore 
being solutions of the equation 
it is found that 
$0 
$ 1 
$ 2 
$ n - 1 
~ $ n - 1 
$ 0 
$ 1 
$ n - 2 
~ $ n - 2 
~$ n - 1 
$ 0 • • ■ 
- $ n - 3 
~$1 
“ $ 2 
~ $3 * * ' 
$ 0 
It is properly pointed out that the case of the <p’s where n = 3 had been 
obtained in 1827 by Louis Olivier (see Crelle’s Journ., ii. pp. 243-251). 
Gunther, S. (1875). 
[Lehrbuch der Determinanten - Theorxe, viii + 236 pp., 
Erlangen.] 
Gunther devotes two pages (§ 11, pp. 93-95) to the subject, but they 
contain nothing fresh save the proposal to call the determinant “ doppelt- 
orthosymmetrisch,” a quite unsuitable name which accurately describes a 
very different special form.* 
* A determinant which is doubly orthosymmetric can have only two different elements, 
and must have all its odd-numbered columns identical, and all its even-numbered columns 
identical. This is readily seen on starting with the first two elements and then carrying 
out the requirements of double orthosymmetry. For example 
a 
b 
a 
b 
b 
a 
b 
a 
a 
b 
a 
b 
b 
a 
b 
a 
