The Hon. J. W. Strutt oti BessePs Functions, 343 



Again, by the formula of reduction, 



putting n = 0, m = 2, we find 



{>J„{,-)dr=2r%{r) -r%(r)-4r"rr/rJ„(r); 

 Jo Jo 



or, taking p as the upper limit of integration, 



r>Jo(r)f;*-=2/Jo(p). 

 Thus 



JVyrJo(pr) = i . 3/Jo(;^) = ^ Jo(p) • 



Accordingly, to determine a^, we have 



-i'^[Jo(;')]^=J^Jo(;')> 



whence 



4fi 1 6^' 



««= Trr^T—z; «? = 



The complete value of (f> is thus 



If we put r=l, we find 





l+^ = l + i/^ + 4yaS-2-; 



SO that 5 Jl __ 1 



in accordance with what we have proved already. 

 The energy of the motion 



-^S^H^P'"'''^' 



Jo 



— when z=l=<xi , 

 dz 



— h\ 27rrd?'d>-f- when 2=0, 

 ^Jo ^- 



by Green's theorem. 



