250 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 
f (=): é ux" f (s+ r).u (B) 
Let x=e and suppose w expanded in the form vw=3a, «~": also write D for 



d 
aie then 
aye 1 n+ —n 
iy (oy ee 8 a (ate | Bb 
a (=) a” ( ) In z 
d\& —n 
soe | 
=(—1)" 3a, se 
/—D+p —né 
=(-1f 30, 6 by (A) 
=(-1)" asked 0h Pe 
=p : 
[-D+e 
(1) C 
(“I tae: a0) 
As a particular case of formula (B) we have 
pe eDea EDs, 
iy Epemiee ee: 
These four theorems will be found of the utmost importance in reducing dif- 
ferential equations. Formule somewhat analogous have been applied to the so- 
lution of common differential equations by M. Caucuy, Evercices, vol. i., p. 163, 
and Lxercices d Analyse, ii., 343; by Mr Grecory, Cambridge Mathematical Jow- 
nal, 1., 22, &c.: and by Mr Boots, Philosophical Transactions, 1844, 225. Under 
the different heads in which we shall arrange differential equations, we shall 
solve only the most simple examples, our object being to illustrate the method of 
proceeding rather than to exhibit its power. 
Cuiass I. Equations which are capable of solution without transformation. 
2 
o- Exo id —ay=0. 
dx? 
ae d : : 
By writing d for ae this equation becomes 
a 
(fey oe0 ory =e 8G 
