IN THE DIFFERENTIAL CALCULUS. 45 


D (SG 2 al D 
o(2 +1) Hay" apt 0(7) 






= ee p+ 
| 
D jaeee 
Hence p (=) .; Sees 
[Pes 
[ane 
ee 
feo! A ee ie 
[D 
[or 
=i. 
evel Fs A B —(u—1)r 
/-—D 
i= 
Hence ae (= i) & u=(—1)* aa (er nm) eed oe 
ae 
a relation between differentials with positive and negative values of 7. 
Section II. APPLICATION OF THE PRECEDING THEORY TO THE SOLUTION OF 
DIFFERENTIAL EQUATIONS. 
12. The first example which I propose to give is the solution of equation (B), 
Art. 9. 


dred ir 2 
Ex. 1. 7aok (2 N =) =a’ xy, where w may be anything whatever. 
Let bt, en 
aa} r d =f r 
area pa baci Gao 
Le aeraalre taitcd e -1 
or 2 Te ie Dae — aE ae 2 
bo 
or D(D-1)... . D—r+2)x (D—rptlje*z=a' ee *2 
Let = or e~’z= (- >) »; then 
x r 
D D 
WA... O-r+2)x Dorp Ly (- =) °= 
ef (- >)» = cee ey. 
