102 
Proceedings of the Royal Society 
The direct transformation may readily be effected by noticing, 
in the first instance, that any two operations of the form 
x -m+iA x m 
dx 
are convertible. We find, in fact, 
d 
dx ' dx 
x 
-m+lfcc . x~ n+1 i X n = x i 
(S 2+(m+n+1) 4 +nm > 
into which m and n enter symmetrically. 
Replacing the operations in the left hand member of the first 
formula by convertible operations, which will be separated by 
points, we find 
d d 
-i 
d 
d 
7 J ^ X 7 • X i ) 
dx dx dx dx 
d d - 1 _2 2 d _i 2 d _i 
~r x~r x — x 7 -a; . x — x 
dx dx dx dx 
and so on. Hence, 
f f\ n _ r ~ n ( r n f r - n + ! 
dx x dx) \ dx 
x «-\ d x -n+2 V 
dx J 
Again, 
n L n - 
\ dx 
x 
X 
-n + 1 . n-lA «-n + 2 
t *J 7 it/ 
dx 
= x 
— n 
n ( d Y ) 2 ( d\n n ( d\ l 
Ms) 
<fa) ( 
<ir/ 
X X = X X ~r X , 
dx dx ’ 
^ o 2 -1^2 
tAs f t b “ tv A d/ 7 JO a 
dx dx ’ 
and so on. Hence 
dx 
x — aS* - x n x r~ n+l — r n r ~ n + 2 — r* -1 
cA/ 7 w/ | w A iA/ -j i/y * lA/ -i <As 
dx dx 
d 
1 ^ o 
— X ' X -r- x . a? — 
6/ 
d ) 2 
X J- 
dx 
r- 
a; 
dx 
dx 
dx 
r - n + 1 — 
W • • • tv ^ IV j 
= x 
d \ n 
dx 
n >» 
