1901- .] Dr Muir on the Theory of Orthogonants. 
263 
which corresponds to the extreme value s r of s, he has, by a double 
use of each equation of the set, the n pairs of equations 
{-h~xx ~ ®i)^i d" -h-xyl/ 1 d" A xz Z x + ... — 0 ) 
(Aaa — *. 2 ) X 2 d- A xy y 2 + A xz z 2 + ... = 0 j 
A X y x i d- (A yy ~ sfy x +. A y ^ l d- ... = 0 
A xy X 2 + (A yy ~ S g)?/ % + A + ... = 0 I 
A 3a fc 1 + A yz y 1 + (A zz - s 1 )^ 1 + ... = 0 \ 
A xz x 2 + A yz y 2 + (A zz — s 2 )z 2 + ... = 0 i 
From the first pair A xx can be eliminated, from the second pair 
A yy , and so on. Consequently there is in this way obtained the 
n equations 
(s 2 - + A xy (x 2 y 1 - x x y 2 ) + A xz (x 2 z x - x x z 2 ) + • • • = 0 
A X y(y 2 x 1 -yi x 2) + (' q 2- s i)yit/2 + Ay Z (y 2 z 1 -y 1 z 2 ) + ■•• = 0 
AUz 2 x 1 -ZiX 2 ) + A zy (z 2 y 1 -z 1 y 2 ) + (s 2 -s 1 )z 1 z 2 + ••• = 0 
and from these by addition 
(x 1 x 2 + y 1 y 2 + z 1 z 2 + ... )(s 2 -s 1 ) = 0, 
the conclusion being 
“ Done, toutes les fois que les racines s x , s 2 seront inegales 
“entre elles, on aura 
x x x 2 + y x y 2 + z x z 2 + ... = 0 ; 
“ et, si Inequation S = 0 n’offre pas de racines egales, les 
“ valeurs de a?, y , z t . . . correspondantes a ces racines 
“verifieront toutes les formules comprises dans le Tableau 
“ suivant : 
r \ + V \ + ••• = 1 ,x 1 x^ + y 1 y 2 + ... = 0, , X]X n + Vl y n + • = 0 
Vi + yyj j + = 0 , x 2 + y.y 1 + ••• = 1, , x 2 x n + yyUa + - - • = 0 
+ VnVi + ■■■ = 0,x n x 2 +y n y 2 + ... = 0, , x* + y* + ... - 0”. 
This interlude over, the fundamental set of equations is returned 
to, and, the first of them being deleted, there is got from the 
remainder 
