Linear Differential Equations: 189 



may frequently be put under the required form more easily than 

 by the rules given in the former paper. 

 Making •sy = D# + \(D), we have 



A=D- 2 ot 2 m +D- 2 (1-\(D))^+D- 2 (\(D) 2 -\(1>) + DV(D))w 



and changing u into D m u, 



a? 2 D m M=D-V 2 D m «+D" 2 (l-X(D))OTD m M + D- 2 



(\(D) a -X(D) + DX'(D))D m M . 



Substituting these values, and operating with D 2 on every 

 term of the result, we find 



w 2 (D m + k)u + (1 -X(D))w(D m + k)u + (\(D) 8 -\(D)+D\'(D)) 



(D m + k)u-p{p-l)iy m u=D*0. 

 Here make X(D) = 0, and the last is reduced to 



■ur*(D m + k)u + v{T) m + k)u-p{p-l)I) m u=T>% 

 or 



(V + w -p( p _ l))D w « + k{zr* + «r)u = D 2 0, 

 or 



(CT- j o + l)( OT + ja )D m « + te( OT 4-l)M=D 2 0. 

 To solve this, assume 

 u _ {z:—p + Y){zT—p + m + \) (jsr— j» + (■/•— l)m + l) 



ct(ct + ?«) . . . . ('ra , + (?* — 1)?/A 



Then by (C) of the former paper we have 



(g-/> + OT + l) .... (g-j> + r»t + l) 



-U w= 7 ; \ t ; ^ JJ «,. 



(ct + ?«) . . . . (ot + m) ' 



If rm=p, substituting these values, and operating with the 

 inverse of all the factors common to the first member of the re- 

 sulting equation, we have 



TV" -i_i v (^ + m)....(^+p-m) 

 11 * («r— p + 1) . . . .(ot + 1) 

 Therefore 



Ml=S (D' B +*r%, 



whence w may he found. 



In this and the former example m is supposed to be an integer. 

 Again, make 



(■us — m)(vs — 2m) .... (■us — r?n) 



Jl — 1 ' ' 1 2 1 7/ 



(■us+p— m)(vs+p — 2m) . . . .{zj+p — rm) 



Thru 



n«„_ CT •••• ("-rm + m) 



u u ~ (vr+p) {vr+p—rm + m) U M » - 



