497 
1887 .] Dr T. Muir on the Theory of Determinants. 
we know (xxxn.) that 
32|& 2 f 8 | 2 = SX^ 2 + SY, 2 ^ 2 + SZ^ 2 
+ ^ + 2^Z x X x . SfA 
+ 22X 1 Y 1 . ; 
whence, by using the set of six results just obtained, we have 
3S|f 1% &| 2 
= SIa . .2 f I 
1 1 2 ' 3 1 + 22 v^v^ih + 2S^. 2%*! + 22^.2^ ) 
and therefore, again by (xxxn.) 
SI4%&P = {S^AI 2 } 2 - (xxxvi.) 
It is finally pointed out that from the third triad of rows there 
might, in like manner, be formed a fourth triad, and analogous 
identities obtained ; also that, instead of starting with three rows, 
we might start with four , 
1 1» 
^2> 
^3» * 
5 t, n 
X l> 
«2» 
x. 3 , . 
. . . , X n 
Vi> 
V 2 3 
Vs ’ • 
• * * } Vn 
z v 
^2’ 
%3 • 
• • * ) } 
form from them other four 
tWsI* 
> 
|^1 f 2*^3 1 j 
V\ x 2y^ 3 
thence in the same way a third four, and in connection therewith 
establish the identity 
+ = 0 (xxxi. 2) 
and other analogues. (xxxn. 2 + xxxv. 2.) 
The rest of the memoir, 52 pages, consists of geometrical appli- 
cations of the series of theorems thus obtained. 
CAUCHY (1812). 
[Memoire sur les fonctions qui ne peuvent obtenir que deux 
valeurs egales et de signes contraires par suite des transpositions 
