146 
Proceedings of Royal Society of Edinburgh. [sess. 
it is readily seen that the result will be an aggregate of expressions 
like 
A„ ,H. 
+ M w>s - -J 0 2 - (n - 2s 
.H. 
+ (n - 2 )(n - 3)(x - n + 3)(x - n + 4)A„_ 4iS H k _ 4 _ 2s . 
Now if we hear in mind that by definition 
I If }h„_ 2 _ 2s = H, 
the second of the three terms of this 
M m> A,_ 2 i 5 H w — 2 _ 2s A n _ 2 )S H n _ 2s , 
or, if we write s - 1 for s in one case, 
== — M njS _ 1 A n _ 2|S _ 1 H w _ 2s — A n _ 2>s H n _ 2s 
— 25 'I lA7i_2,s— 1 "b A n _ 2>s I* , 
and the third, by writing s - 2 for s, 
= {n - 2 ){n - 3)(x -n + 3){x -n + 4=)A n _ 4iS _ 2 H n _ 2s . 
Consequently the sum of the three will vanish if 
A n>s - + A n _ 2>s ) + (n - 2 )(n - 3){x -n + 3)(x - u + 4)A n _f_, = 0 
and therefore if 
B n>s (* - n + 1 ) - B n _ 2 s (x - n + 2s + 1 ) 
- B n _ 2)S _ 1 M niS _ 1 + B n _ 4>s _ 2 (n - 2)(n - 3){x -n + 4) = 0, 
that is, if 
(x — n) j^B n s - B w _ 2jS — (2 n — 3)B n _ 2 s _ 1 + (n — 2)(% — 3)B n _ 4s _ 2 
+ [\, s ~ ( 2s + !) B »-v - { 6» - 8 - (re - 2s + 1) 2 ]■ B„_ 2 s _ t + 4 (n - 2 )(» - 3)B„_ 4 s _ 2 = 0 
But this is the case ; for, as Cayley shows, both the cofactor of x-n 
and the other similar expression following it vanish identically. 
The verification aimed at is thus attained. 
