32 Proceedings of the Royal Society of Edinburgh. [Sess. 
The value of x n derivable from this is 
2 n— m 
C\5 n A 
B / + m ' 1 ‘ 2,1 
B w 
= 
- » 
from which we have 
x 
C AC 
2 n—in 
n A 2 C m 
B 
+ 
giving 
o 
(i) 
x = 
B 
+ 
m 1 
n+m r 
B 
m 
n 
-m+l 
C 
m 
m 1 ‘ 
B m 
B n 
1 2n - m + 1 A 2 C 
I • — 
2n— 2»i+l 
m 
m 
2 ! 2?l+l 
B rn 
and the next two terms, got in the same way, are 
3n— 3m+l 
1 3 n - m + 1 3 n- 2m + 1 A 3 C 
m m 3 ! 
Ill 
3w+l 
B m 
+ 
1 4 n — m + 1 4 n — 2m +1 4 n — 3 m +1 A 4 Q 
m 
m 
m 
m 
4! 
4m - 4»i+l 
m 
4w+l 
B rn 
I leave the reader to verify these last two terms, and also to verify 
the formula further, generally, by obtaining by direct inversion the root of 
Ax n — Bx + C = 0. 
/ov C ACV 2 n A 2 C 2n -\o /Q lX A 3 C 3 ”- 2 , . /A lx//( ox A 4 C 4 "- 3 
(2) x— + — + — . — ^ ,, + 3 n (?>n— 1) . — —— + in(in - \)(in - 2) — 
\ } x — r g n+1 | 2 ! B 2w+1 v 3 ! B 3n+1 v A 4 ! B 4 '* +1 
etc., got from (1) by putting m = 1. But the latter series is the development 
of the operator Dp . T>c [w-1) • D B ?l , the general type of operator that acts on 
operands with integral indices. The reader may verify further that both 
formula (2) and operator hold when n is negative. 
Turning to formula (1), we shall find that it also is the development of 
an operator D A 4 . D c Vm . D B m similar to the operator D A x .D c (rt 11 .D/, but 
n _/n \ 
the symbols D" 1 D m ~ l can no longer be regarded as those of the Differen- 
tial and Integral Calculus. To investigate the action of the symbol D 
we recur to the formula 
n 
m 
(3) 
D b .B =?.(?- 1) (q -p+ 1) B 
Q-P 
In formula (I) D P m has to act on indices of the form rn + 1 : but whether 
B m 
n be greater or less than m, so long as it is not a multiple of m, we do not 
