430 



Mr Basset, On finding the Potentials of [May 10, 



If </> (p) be any given function of p which is finite and con- 

 tinuous for all values of p between the limits infinity and c, it is 

 required to find a function of F (X), such that 



(13). 



<p(p)= F (X) J m {Xp) dX, p>c 



J 



0= X F (X) J m (Xp) dX, p<c 



Jo 



7. In the case of m = 0, consider the integral, 



f) |-CO ,-00 



W = — J dp, I e~* z sin Xv sin fivJ (ftp) dv. 



TT J J e 



By a known formula 



f°° 1 

 e «tJ (tip)dfi = — — 



Jo (£ 2 +p) 2 



Let £" = z — tv, where t = J — 1, then 



r ■ 1 



i I is - '** sin /ryJ" (/xp) dp, — imaginary part of - 



Jo {(z - iv) 2 + p 2 }' 



and if we transform to polar co-ordinates, and put 



R cos 2^ = r 2 — v 2 , 



R sin 2% = 2rv cos 6, 

 as in Art. (5), we obtain 



Jo 



~* z sin pro J (p,p) dp, ■ 



R^/2 



when z = and p > v the integral vanishes ; but when z — and 



p < v, the integral = . . 



H ° Jv 2 -p 2 



Hence when p > c, the limiting value of the integral W is 



sin Xvdv 



P Jv 2 — p 2 



2 r 00 



= — / sin (Xp cosh 6) dd 



by a known formula. 



