414 Proceedings of the Koyal Society of Edinburgh. [Sess. 
— that is to say, a set of n + 1 equations in the n + 1 unknowns 2X x , 2X 1 X 2 , 
. . . . , X^ . . . X n+1 , the solution of which is 
i _ _ Em? _ 
A 0 A, A 2 
where A r is the determinant whose array is got by deleting the (r+l) th 
column from the array 
Q J n + 1 ®'n Q>n — 1 • • • 
^n+2 ^n+1 ... 
a 2n+l a ‘ln a in-l • • • a n* 
Prom this it follows that the X’s are the roots of the equation 
A 0 X n B— A 1 X W + A 2 X m_l - . . . = 0, 
i.e. 
X n+1 
X n 
X”- 1 . . 
. . x° 
<% n+ l 
a n 
a>n - 1 • • 
. . «0 
a n + 2 
«n+l 
a n . . 
. . aq 
= o, 
a -in+\ 
<h n 
Chn - 1 • • 
. . a n 
i.e. 
a n± 1 
a n \ 
a n - 
«»- A • • 
. . flq 
0 
e 
1 
««+2 ~ 
a n + lX 
a n + 1 — 
a n X . . 
. . a 2 
~ a i 
a 2n4-l ~ 
a 2 A 
a 2n - 
®2n-lX • • 
. . d, nJr i d n X 
On substituting in the first n+1 equations of the original set the values of 
X x , X 2 , . • . , X n+1 thus found, the values of 7r x , 7r 2 , . . . , i r n+1 are obtainable 
from a set of linear equations of the type associated with the name of 
Lagrange. 
The latter part of this procedure is not given by Sylvester, who on 
reaching the set of equations in 2X 1? SX x X 2 , .... suddenly draws the 
seemingly irrelevant conclusion “ that 
(x + X$)(x + \y) (x + X n+1 y) 
is a constant multiple of the determinant 
x n+ 1 
— x ll y 
x n+l y 2 ..... 
^71+1 
a n 
«»- 1 
a«+ 2 
C,/ n+ 1 
«« 
^2n+l 
Ct 2 n 
«2n-l 
or A say. 
