368 Mr. B. Williamson on the Solution of 



wliich contains the complete solution whenever \ is of the form 



Or 



2r + l' 



For the practical application of the foregoing solutions, it is 

 only requisite to investigate the expansion of the symbol of ope- 



ration (|^«-)". 



It is readily seen that we may assume this expansion to be of 

 the form 



{D«- ')"= a-"D" + A„«-'^D"- ' + B„«-'"^D"-' + &c. 



where A„, B„ are constants depending on ??, the forms of which 

 we have to determine. 



If we operate on both sides of this equation with ( t-'I'"' )j it 



becomes 



(Da-y+' = rt-'~>D"+'+(A„-?ITl)«-'^D"+(B„-(n + 2)A„)«-^3aD''" 



+ &C. -(2yt + l)P„«-2'^ 



but 



(Da-)"^' = «-'~D"^' + A„+,«-~2D» + B„+,a-^3D"-' + &c, 



Accordingly, equating the coefficients of like powers of D, we get 



A„-(n + l)=A„+, 



B„-(n + 2)A„=B,.+ ,, 

 &c. 

 hence we conclude that 



_ w.(w+l ) -^ _ {7i-l)n. {n+l){n + 2) 

 2 ' 2.4 ' 



and the complete expansion of (D«~')" is 



2 2.4 



±1.3... (2«-l)rt-2'"^(D -«-'). 

 There is no difficulty in proving that this expansion holds for 

 negative as well as for positive values of n ; and hence by merely 

 changing the sign of n, we might have inferred the solution of 

 (3) from that of (1). 



If w€ change the sign of «- and expand the operating symbol 



