423 
1909-10.] The Theory of Persymmetric Determinants. 
power of x in D r _ x is always — of the coefficient of the highest power of x in 
N r _ x , because = + and from (6), by equating coefficients of x n , that 
the coefficient of the highest power of x in f r _ x is the reciprocal of the 
highest power of x in D r _ x : in other words, that 
C r)I = riB rl = nj A r>1 . 
In the next place, denoting the roots of the given equation by 
x 1 , x 2 , . . . , x n he has from the theory of “ partial fractions ” 
^bfr(Xi) _ 0 
S if Ax,) ’ f^Xi) 
i= 1 i= 1 
i=n 
5 /iW 
i— 1 
o, 
0 _ 
A • 
’ Zj /i(*i) 
i = 1 
’ Zj /l(*<) 
i=l 
n r,l ) 
and therefore from (5) 
2 d mW = o, = 
i= 1 i= 1 
o, : 
i=l 
= 0, 
i—n 
....... 2»r 2I) r-l (Xi) 
i= 1 
i=n 
= o, 2^ _1D r-iW 
= Ar, x ; 
and consequently on putting 
B r _ xl x r 1 + 2 + . . 
. 4- B r _ l7 . for 
and s m for aq m + ;r 2 m -f . . 
there results 
B r — i, r s 0 "t B r—1 _ r _j S x + . . 
• +B r _ 1) iS r _i = 0 ' 
B r _ lr s x + B r _ x > r _ x So, + . . 
. + B r _, t ! s r =0 
B r _ lr s r __ 2 + B r _ ltl _ 1 s r _ l + . . 
. + B r _ x t x s 2r _ 3 = 0 
>■ 
■By. 2^ ^7* X By, ^ y, t>y, 4" • 
. + B r _ X) x s. Jr _ 2 = A rjl> 
The solution of this set of equations gives the B’s in terms of A r x and the 
s s , so that the finding of A r>1 in terms of the s’s is the next desideratum. 
This with the help of the relation A r>1 B rl = 1 is easily obtained, for from 
the set of equations it is seen that A r being written for the persymmetric 
determinant of s’s 
and therefore 
T> _ A r> ! A r _! 
l) r-l, 1 ~ 
A r i - 
A, 
A r 1 
Ar— 1 A r _ ljl 
VOL. XXX. 
28 
