40 
Proceedings of the Royal Society 
Whence, equating coefficients of like factorials, we have 
(rc + l)A ra = A n _i , 
■= B w — i 4 - A n — 2 j 
( n - 1 ) C n = C n _i + B w _2 , 
(n- 2) D n = D w _i + C n _ 2 , Ac., Ac. 
Let 
| n + 1 An = a«, I /Z B n = fin , I ?? - 1 Crc = y ft , Ac. 
then these equations become 
Otra — CLn — i 
1 = fin 4- a w _i 
y«+i = y» + /?n-i 
Sw+1 = 8n 4- y w _i 
Thus we have 
M>n ~ SO. 
/L. = Sa re _i = jya n 
y n — %fin—l — p. ^ ra 
= Syw— i = py»j &c. 
x(x + 1) . . (x + n) 
_ f an fin s n , 1 /2\ 2 „ . ) 
" l [ n + l + x | njf + 4r + l) |H-l(p) a J 
_ / n + 1% n+ln f%\ 2 )#(#4-l) . (# + ? 2 ) 
( X D a?(a?+l)\D/ J 1.2 . . .(n+1) 
= 1 i-2 + 12 -- \d / ) + ••••}“'■ 
W 
Sy + » 
SV-1 0 
.M) 
d) 
v 
for no negative powers of *— are to be retained, as a» is a mere 
constant. 
The trouble of carrying out this process is considerable, depend- 
