Borchardt's Form of the Eliminant of Two Equations. 
449 
Thus, when ?^ = 3 we have 
0 = <^^(aia2a3).F(ao)ao - ^^(aoa2a3).F(aoai) + ^^(aoaia3).F(aoa2) 
- ^^(aoaia2).F(aoa3), 
from which on division by ;^^(aoaia2a3) there results 
_ F(aoao) , F(aoai) 
+ 
and thence 
(ao - ai)(ao - a^{ao - (a^ - ao)(ai - a^){a^ - a^) 
F(aoa,) _j ^(aod^ 
(a^ - ao)(a2 - a^{a2 - {a^ - ao)(a3 - ai)(a3 - a^)' 
F(aoao) _ F(aoai) F(aoaJ ^(aoag) 
/(ao) 
/(a.) /(«,) f(a,) 
if f(x) be put for {x - a^{x - a-^{x - a,){x - a^). 
4. Another proposition of Borchardt's occurring in the same memoir 
is also readily generalisable, the fundamental theorem being that If an 
array of n — 1 rows and n columns be such that the stem of every one of the 
rows vanishes, the principal minor determinants of the array, lohen taken 
alternately positive and negative are equal to one another. The array being 
a^j 
aj2 
— I, 1 ^n—i, 2 • • • ^n—i, 
and being the minor determinant whose array is obtained from the 
given array by deleting the rth column, we have only to ascertain the 
relation between and M^. To do this we remove the rth column from 
its place in the given array, and attach it by addition to the first column, 
thus forming a square array whose determinant 
a,. 
<Xj2 
^i, r— I 
a 
+ a,r 
^2, r — I 
^2, r+i 
a^n 
^n—j, 
I ~1~ — I , r 
^n— I, 2 
^n— I, r — I 
^n—i,r + i 
a^ — 1, n 
by reason of the data vanishes, and being partitionable into two gives 
M, + (-l)'-^M, = 0 
and .'. 
as desired, 
