Licathem — On Doublet Distributions in Potential Theory. 37 



10. Derivation of </> from <£. — Assuming <£ to have been determined, we 

 can get an expression for <f> by precisely the same sort of application of 

 Green's Theorem as is employed in § 5, with only this modification that the 

 point P from which r is measured is to be taken in the relevant region. 



To avoid an infinity in the subject of the volume integration an additional 

 boundary must be introduced, namely a sphere a' of radius ij surrounding 

 the point P. This involves the introduction of additional terms in equation 

 (1), namely - JV 1 d<j>/dy da' on the left-hand side, and - J$ di~ l /dri da' on 

 the right-hand side. The limits of these for »; ->- are respectively zero 

 and ±Tr(j> p , where $ P means the value of <p at the point P. 



Consequently we get, instead of formula (2), 



3 (I 



[W ,„ ^m 



dn\ri 



V 



4>--[- dS + iTrA P = -*ff-2-+2-,cosv, (6) 



r r 



with an extra term, if necessary, on the right-hand side for flow at infinity. 

 Tn the potential notation this may be written 



±ir<p = - F, (?) + V 1 (W, - m, - M ). (7) 



11. It is worth remarking that in the application of Green's Theorem in 

 § 5, if for r~ l there were substituted any function ^ which satisfies A</> = 

 at all points of the relevant region, a result very like that of formula (2), but 

 rather more general, would be obtained. 



The integrals on the sphere a surrounding a doublet would require some- 

 what careful treatment ; thus, for \p we should write 



where the values now refer to the centre, and /, m, n are the cosines of e. 



Also, for <p we put 



- (nl<krt?)(Ll + Mm + Nri), 



where L, M, N are the cosines of the doublet. Consequently, 



f . d<b , 2fi fi , fM d^ U\) LI + Mm- + Nn , 



III 



and 



2 f dip „8^ ty\ 



o^\ dos dy dzj 



</> -^- da- _>- + -£- I -^ + m -¥- + n f~)(Ll + Mm + Nn) dw 

 r dn 4!r, 3a; dy dz J 



