Ci)NWAY — On the Motion of an Electrified Sphere. 5 



111.— The Hakmonicoid Functions of the First and Second Kinds. 



If the position of a moving point bo denoted by a, it is necessary to 

 assume that the function a has a diiferential coefficient h. Let us denote tlie 

 values of a at the times t' and ti respectively by a' and ai ; then the function 



e {t - t'y- + {,, - a')- 



has a real positive zero <, between and t if c-f + p- > aud Th < c* 

 These conditions express the facts that p is to be taken inside the sphere 

 of radius d, and having the centre as origin, and that the speed Ti of 

 the point is less than c. If t' is complex, aud if we take a contour integral 

 enclosing only the zero t, the function 



f(t')di' 



ci 



TT 



C\t - tj + (p- (tT 



possesses a pole in real space at the point p = o-„ and satisfies 



9^ 



The integral may be written 



CT 



2^;^ 



fiOdt' 



Tip -^')[c(t-t')-T{p -<,')] 



If now it is possible to draw a contour enclosing <, and no other zero 

 and the point t, and such that on the boundary c\t - t'\>\T (p - (t"j\, then 

 it is possible to expand the integral in inverse powers of c, and we obtain 



ci 

 2^ 



fI^^^^^^i^^ = |'--4F©-K')-<-'"t 



If we change the notation so that p is now the distance of any point 

 from <7, and put Tp = r, we have for the scalar potential P^ of a point- 

 charge £ the series! 



P ^- (-)"c-" /a\" ,.,. 



* Proceedings of the London Miitlieraatical Society, series 2, vol. i. 



t I'rooeediiigs of the Royiil Irish Aciidemy, vol. .\.\vii, Section A, No. viii, G. A. Scholt, AiinaU'ii 

 der rhysik, 2.5, p. 79. 



I Mr. W. R. W. Roberts, f.t.c.d., suggests a compact form for /n nnd similar series ; thus 



Po = .E Exp {d!dt . >■). 



where after expansion the operator is placed at the beginning of each term. 



