1911-12.] The Theory of Circulants from 1861 to 1880. 
147 
Menesson, (1878). 
[Solutions des questions proposees (Question 185). Nouv. Corre- 
spondance Math., iv. pp. 185-187.] 
Not only is Spottiswoode’s result here explicitly obtained by equating 
the two forms of the eliminant of 
a n _ x x n ~ l + a n _ 2 x n ~ 2 + . . . +a 1 £ + a 0 = 0 i 
x n -l = 0) 
which were referred to under Baltzer (1864), but the parallel result is also 
stated, namely, that 
C(a 0 , a x , a 2 , . . . , a n _ x ) = - 1) ... . (#U - 1) 
where /3 1 , /3 2 , . . . , f3 n _ x are the roots of the first equation. 
Minozzi, A. (1878). 
[Sopra un determinante. Giornale di Mat., xvi. pp. 148-151.] 
Minozzi s purpose is to find the final development of a circulant, not 
from the determinant form, but from Spottiswoode’s product. His first 
point is that, multiplication of the n factors having been performed, the 
terms of the resulting expression must be of the form 
A a 0 °a ei a e 2 2 . . . aJ l S x 
where the e’s are positive integers whose sum is n. His next is that any 
a entering into this, say a k , must be accompanied by one of the 0 ’ s raised 
to the power k, and that therefore the form of A is known. His third is 
that A, being thus of necessity a symmetric function of the roots of the 
equation x n — 1 = 0 , can be calculated, and that all the more readily by 
reason of the fact that so many of the simple symmetric functions are 
equal to 0. For example, the coefficient of ala ± a 3 in C(a 0 , a 1 , a 2 , a 3 ) is 
i.e. $101 + $l$\ + 6\e\ + $l$l + (%6\ + $l$\ 
+ $l$l + 6\$l + e\$l + $101 + m + 0101 , 
i.e. ( 0 , + 0 2 + 0 3 + 0 4 )(0f + 01 + 01 + 0!) -(0} + 0l + 01 + 0i) , 
i.e. 0.0 - 4 . 
Minozzi is careful to note that, in accordance with Baltzer’s theorem of 
1870, it is only necessary to calculate A when the sum of the suffixes of 
