296 



-50- 



The general expressions for the strengths of the dlpoles 

 are: 



/I A"^ 3 



at 0^ , there is a radial dipole of strength -n- D ; 



(14) { 



3 

 st Q„,n IT II II II n II II A 3 



The expansion of a unit radial dipole lying outside 

 a sphere at a distance X from the center is 



2 



(15) _ -L - 2R p_^(cog e) - §- P2(cos ©) , 



while a radial dipole inside the sphere has an expansion 



(16) i Pt(cos 9) + ^ Pp(cos ©)+ 



These are obtained from (10), (H) by differentiation with 

 respect to X. See Milne -Thorns on, pp. 442,443. 



For the present purpose we are interested in the 

 expansion of <|>, only up to terms involving P, (cos 9)« 



Using (14), (15), (16), we obtain 



CIO 3 oo 3 



(17) \ = f Z =#- ^ ^^ Z! =f- ^ Pi(°°^ ^) 



.3 ^ , P,(cos 9) 



+ ii- > D — = — s + terms in higher 



^ ^^ ^ R Legendre polynomials. 



It is convenient to evaluate / <i» -i dS and / 4> ^cos 9dS • 

 Using the orthogonality property of Legendre polynomials. 



