155 
1906-7.] Dr Muir on Axisymmetric Determinants, 
et pour n pair 
X,, I X n (l) I . I X w (l , 2) I . . . . | X n( 1 , 2 , . . . , \{n - 2) I X n (T, 2, 3, , %n) | = A 
ou le symbole 
X n (T , 2 , 3, . . . 
denote que dans ce produit Telement x x entre toujours parmi les elements 
auxquels on a change de signe. Le determinant A resultera de la multi- 
plication successive de l’equation X n = 0 par 1’ unite, et par chacune des com- 
binaisons deux a deux, quatre a quatre, . . . , (n— 1) a (n — 1) si n est 
impair : na>nsin est pair, des elements x 1 ,x % , . . . , x n 
In addition, it is shown under this head that the number o£ equations 
in the set which originates the determinant is 2 n_1 , and, a little un- 
necessarily, that the number o£ linear factors in the product is the same. 
It is also noted that if x 1 , x t , . . . , x n be quadratic radicals, the product of 
the linear factors is rational. 
Following Cayley’s paper still further, Brioschi similarly makes clear 
that one of the nine-line determinants there obtained, namely 
Ill 
. . 1 . . 1 . z 3 . 
. . 1 . 1 . z 3 . . 
. 1 . . . 1 . . >/ 
.1.1. . y* . . 
1 ... 1 ... a 3 
1 . . 1 . . . z 3 . 
?? y‘ 6 a? 3 
111 , 
may be viewed as originating in any one of the nine equations 
x + y +z =0, x + y +az = 0 , x + y +/3z = 0, 
x + ay + /3z = 0 , x + ay + z =0, x + f3y + z =0, 
x + /3y + az = 0 , a x + y +z =0, /3x + y +z —0, 
where a , (3 are the imaginary cube roots of unity, and could thus be shown 
to be equal to the product of the nine left-hand members of those equations. 
Spottiswoode (1851, 1853). 
[Elementary theorems relating to determinants. Second edition, 
rewritten and much enlarged by the author. Grelles Journal , 
li. pp. 209-271, 328-381.] 
The information given by Spottiswoode regarding axisymmetric 
determinants appears under a variety of headings. What little the first 
