269 
1909-10.] Dr Muir on the Theory of Orthogonants. 
g m x x 
= affx 1 + affx 2 + . . 
• + affx n ' 
g m x 2 
= a { ^ ) x 1 + affx 2 + . . 
• +aff% n 
g m x n 
= affx x + affx 2 + . . 
■ + aS$«n J 
where 
S\=n S ‘2 = n s m -i—n 
— Q>id ^ = 2 ( ’ ’ ’ * * * ^ s m—i h • 
$1 = 1 s 2 = l $ m _i = l 
In the next place, g 1} g 2 , . . . . g n being the roots of the resultant of the 
initial set of equations, it is readily seen, from the expression for the said 
resultant when arranged according to descending powers of g, that 
#1+02+ * • • + 9n — a il + tt 22 + * • • + a nn» 
Similarly, by considering the resultant of the second set of equations we 
learn that 
gl+gl+ . • • +gl = < ) + <4 ) + • • • + , 
and generally that 
g?+g™+ • • • +g" = < ) + < ) + . • • + afi? . 
Consequently, if we use s m to stand for the sum of the m th powers of the 
g’ s we have 
i= 1 
In the third place the difference-product of the g s being 
^(± f Ag\yl ■ ■ ■ fir 1 ), 
Borchardt has only to use the multiplication-theorem of determinants to 
obtain as an equivalent for the product of the squared differences the 
determinant of the system 
s 0 s i s 2 • • • S n _ i 
& 1 ^2 ^3 • • • S n 
S 2 S 3 5 4 * • • S n + 1 
S n-1 S n S n+1 • • • S 2n-2 * 
It is this last determinant, therefore, which he has to aim at expressing as 
a sum of squares. 
The process devised by him for doing so is very interesting. Returning 
to the original set of equations and the sets derived therefrom, he takes the 
/i th set and multiplies both sides of each equation by g v and then on the 
right-hand side substitutes for g v x ly g v x 2 , . . . , g v x n their equivalents as 
