EQUATION AND ITS TRANSFORMATIONS. 
809 
Now, identically, <£ denoting any function, 
4£i<K<*)=v(£iw>). 
yd a) ' \db 
and therefore (34) may be transformed into 
a i+ H I db ) cT»= -b i+ H V +1 e- 
■ 2 V(ab) 
which, putting a= 1, becomes 
d \ i+1 
[ db\e~^' b =-b i+ 'l-\ "~ ZVb 
Therefore 
db 
d\ i [ d V+i 
iHi ; 
replacing e 3Vi by e' 74 , this formula becomes 
2 K+ 1 (l) v+! (IJ +1 ^ s = e ’' s - 
whence, taking b=a 2 x 2 , 
i f\>,Ayv 
x ax] \x dx 
+l 
e"-=o 2 i+ V' r 
(35). 
As was shown in art. 32, this is a particular case of the more general formula 
1 _f?Y 
x dx) 
1 rfY +1 
x dx, 
- — 1 X 2i+ H - —) <Kx )=(£) ; 
^\ 2 i+l 
dx) 
and this formula itself admits of generalisation, as it can be proved that 
b £j xiH1 (b £TM=(jLT'*to- 
and generally, r being any positive integer, 
JJ_ _d 
.z ’' -1 dx 
X 
ri+1 
~V 1 d \ i+1 
-7 — <f>( X ) = 
x r 1 dx' 7 
d \ ri+1 
dx 
These formulae are obtained in a paper “ On Certain Identical Differential Relations,” 
published in the ‘ Proceedings of the London Mathematical Society,’vol. viii. (1876), 
pp. 47-51. 
MDCCCLXXXI. 
0 M 
