160 Proceedings of the Roijal Irish Academy. 



where 



■4/' + 1 4/- + 3 / 4/- + 1 4r + 3 4/- + 1 4/' + 3 - 



/' + 1 4/- + 3 / 4/- + 



9 ^ 



Br, 



^'''' {2r-r2Y ^'' ((2/-+2y- ' (2/'+2)-^ " (2r + 2y^ j 



and 



4r + 1 4r + 3 

 2 ■ ^ 

 ^'■^^= (2. + 2)^ ^'- 



Putting /' - 1 instead of /■, we can get the equation for A,- in the form 



4 7- - 3 4r- l , \ 



Y~ ' ~~2 f 2 2 1\ ( 



4/- - 3 4/- -J. 



2 " 2 

 = — " 92 ,.2 " — {^'-1 - ^C-i^V-ij say : 



4/' - 7 4/' - 5 4/' - 3 4/' — 1 



^ ^ " — {Ar.2 - Kr-.Br.^ - Kr_,Br.^\ 



2}-r,{r-Vf 



1 3 4r_-^ 



2 ' 2 " " 2 



OS'-,. ' >. ' 



{^0-^0(^0 + ^1 + .. + -^,-01 



Now ^0 = 0, and ^^ = 1 ; 



therefore 



13 4r-l 



The complete solution of the differential equation comes out in the 



form 



y = Ay, + B 'jjy log .r - ic), 

 where iji is the series 



13 13 5 7 



and 



2'2 - 2'2'2"2,, 

 X~^ + ^-r 37-2 + ^ "^ " rr-§ 

 2.2 2.2.4.4 



1 3 4/--1 





V 2' 2"" 2 

 v: = x-i 2, Dy.vr-'', 



-1 Qr'r'.r\ 

 where D, = K,, + /^ + . . + Z;._, 



