1916-17.] Operators applied to Solution of Equations. 
39 
The (r+l) th term here is 
1 rm — n + 1 inn — 2w + 1 rm — 3 n + 1 
n n n 
The (r + 2) th term is 
1 (r + \)m - n + 1 (r + l)m — 2n + 1 
n 
rm—rn + 1 
rm - r — 1 . n + 1 B' C n 
r\ 
n 
rm 4-1 
A~™~ 
( r + 1 )m — (r - 1 )n + 1 (r + 1 )m — rn + 1 
n 
n 
n 
71 
B 
r +1 
X 
C 
n 
{r+l)(m—n )-\- 1 
The ratio of the latter to the former is 
(r + l)??i — n + 1 (r + l)m-2r+l (r + l)m - 3?i + 1 
+ \ \ ! ( r+i)m+l 
v ’ A n 
rm — 7i + 1 
rm - + 1 
rm - 3n + 1 
(r + l)??i - (r - 1 )n + 1 
rm - (r - l)rc + 1 
(r + l)m-r?z + l BC 
m—n 
n 
(r + l)7i 
The expression within the square brackets, which we call L, is 
1 + m i t Yi + ? 9 ) • • • to (r — 1) terms, 
rm-7i + 1/\ 7in — 2n + lJ 7 
and is seen to be akin to T, ( 1 + M . 
£+» \ VJ 
m 
A Tb 
To evaluate L when r= oo, raise it to the power of 
Thus 
n 
m 
1 + 
m 
1+-? 
m 
+ n [n_ l \ 1 
rm — n + 1 
which, when r becomes infinite, tends to 
rm + 1 
terms 
7m 
(rm — 7i + 1 ) ? 
77i V7n — 71 + 1 m \7ii ) 2 ! (rm — r. + 1 ) 2 
rm + 1 
rm -7i+ 1 
r??i — n + 1 
Similarly, the second factor, ( r + l) m . — by raising it to the 
J rm — 2w + 1 17 & 
( n ) power, becomes — f t-, an( q gQ q or qp e 0 q} ierg> 
\m/ rm — 27 i + l 
Hence 
- (rm + 1 ) (rm - n + 1 ) rm — 2?z + 1 
T m _ ' • ' 7 - 
rm - (r — 2 )?i + 1 
1 r??z —7i+l rm - 2 tz + 1 rm - (r - 2)?z + 1 rm — (r — \)n + 1 
rm _L q _ 
, which, when r = oo, becomes 
rm — (r — Ipr + l 
m — n 
Hence when r becomes indefinitely great, L m tends to the limit 
m 
m — n 
and L itself to 
m 
m \ n 
m — n 
