278 LAMBERT— ON THE SOLUTION OF [April 20, 



gives the terms of the first series in the vaUie of y multipHed by a 

 constant. This new series is combined with the first series in the 

 value of y. 



The first term in the bracket gives 



^'n+l 2^"~ 



^«!(;/- I)! [~ 2Xn + i) + ¥{;r+~T) \ ^ "^ «T~i )\ ' 



_ - B r ,1-"+-' log X 



J'n+2 = 2^'^'n\{n- I)! [ 2 ! 2\n + i)(;7T^) 



\ ^ "*" 2 "*" ;/ + I ;74- 2 /J 



2 ! 2\;i + i)(;/ + 2) 



The solution of Bessel's differential equation when // is a positive 

 integer is therefore 



J' = ^,f" I , + 7 ry- 



4 



4- 2) 2*.2l 



I x' 



{11 + i)(;/ + 2)(« + 3) 2''.3!^ 



r I A'2 I x^ 



y + -^-^;^j -2 + (;^ _ i)(;^ _ 2) 2^^721 



i>lr" log x r I ,1-^ I .1'* 



~ 2^"-Vr!(;/ -^lyi L^ ~ n + I 2- "*" (7; + i)(;rT2) 2*.2 ! 



] 





] 



(„+ !)(;,+ 2)(;/ + 3) 2^.3! 

 ~ 2^"-'n\{n- i)! [ ;7T~i V ^ "^ ^'^ I / 2" 



- (« + i)(n^2)y "^ 2 "^ «> I "^ «+ 2 j 2*. 2 ! "^ ■ ■ ■ J ■ 



This is also the solution of the ditferential equation when 11 is a 

 negative integer. 



