264 
Proceedings of Royal Society of Edinburgh. [sEsa. 
x y z 
P = ~p~ = ~ p~ = 
■*- XX -L xy - 1 - xz 
where the denominators are seen to be what we now call certain 
‘ principal minors ’ of S ; or, as Cauchy says, where V uv is 
“ce que devient S, lorsqu’on supprime dans le Tableau 
“ les termes qui appartiennent a la meme colonne horizon- 
“tale que le binome A uu -s, avec ceux qui appartiennent 
“ a la meme colonne verticale que A vv - s , ou bien encore 
“les termes compris dans la meme colonne verticale que 
“ A uu — s , et ceux qui sont renfermes dans la meme colonne 
u horizontal que A vv - s .” 
The ratios x : y : z : . . . . having thus been got, there only 
remains, for the determination of x , y , z , ... , to use the 
equation 
« 2 + ?/ 2 + z 2 + ... - 1 . 
But before doing so it is temporarily convenient to introduce an 
alternative notation, viz., denoting the signed minors 
P - P - P 
xx > yy ) L zz » • • • • 
by 
X , Y , Z , . 
so that the values of these corresponding to x r , y r , z r , .... , and 
therefore to s r , may be denoted by X r , Y r , Z r , . . . We thus 
have from the additional equation 
x y z \ 
X = \ = Z = ' ' ' ‘ = * fX- + Y 2 + Z 2 + . . . 
and therefore 
^ BB ?/i --1 = ..... = + 1 
Xj Yj z, + y 2 2 + z 2 2 + • • • 
*2 _ y%_ _ % _ _ + l 
x 2 y 2 z 2 ~ VX 2 2 +Y 2 2 +Z 2 2 + • • • 
_ .'/« . . _ _ + f 
x. y» z» ' _ \/x„ 2 + y„ 2 z „ 2 + • • • • 
Of course this supposes that the special values of X x , Y 1 , Z x , . . . 
occurring in the denominators do not vanish ; and Cauchy’s con- 
clusion therefore is 
