166 G. E. Weatherhurn: 



' ' V yiP'j) = log ^01' t-lie logarithmic potential 



y{p'l) = ^' ^or the Newtonian potential 

 which are solutions of Laplace's equation; and 

 ^^'^ I il(.P<]) ^ -/U''-) fo>" tlie plane 



( !?(y"7) = - •«"*'■/*■ ^O'' •'^P^c^ 



■when the potential is generalised correspduding to the equation 

 (2). In this /• is the radius vector joining the i)(>ints /; and q, 

 and f{z) has the same meaning as in my paper already referi-ed 

 to. The functions H(</j) and G(^0 '^'^'^ equal to tlie correspond- 

 ing expressions of (3') in st|uare brackets. The foiiuer is an ex- 

 tension of the solving function in which any ])oint y rejilaces the 

 boundary point .*;. The latter may be defined more generally for 

 any two points pq by — 



G(m) = 'An) + Vi/(p^)H(%)f^« 



This function is the Green's functicni^ for the lioundary problems 

 (1). It will be seen that H(#/;) can be expressed in terms of it by 

 normal differentiation, so that both solutions (•)) can be given 

 in terms of it by a representation of Green's type. It is easily 

 verified that 



j'g{qt)VL{tp)dt = /'G {q()h{tp)dt 

 .so that the equations defining and connecting these tinutions aie — 



(5) j liitp) - hilp) = XJ'h{l6)\l{ep)dd=\i H{/0)h{dp)d& 



\ G{qp)~g{qp)^\/cj{qd}K{dp)d6 = k/\;(qO)h{6p)d6 



Now when X. is ecjual to a characteristic number (singular value) 

 Ao, each of the functions 'il{tp) and G{qp) has a simple pole."^ The 

 solutions expressed by (3) are therefore infinite, and cease to have 

 .a meaning. Since the pole is simple we mav write — 



where IHjp) and <'(/j[') are functions of A, which depend on A,, 

 and remain tinite when A— A,, ; the residues !'(//>) and A„ Qiqp) 

 -do not involve A but depend on A„. It iiow v.e siiiistitute from 

 .(6) in (5), multiply l)y (Ay-A) and pr<icecd to the limit A=A,„ 



we obtain tlie following relations : — 



(7) ( ¥{tp) = \jTite)h{0p)de = Xj7,{tt^)V{Up)dO 



) Q(qp)=/y{q6)'P{dp)de = XJ-q(q(J)hidp)de 



1 Of. VVeatheibuni. "Green's Functions for the equntioii A "-^ (/-/.-'.!(( =0, etc." (^Hiarterly 

 Journal, vol. 4G. The remaining references are to my earlier paper. 

 •2 Weatherburii. Loc. cit. § 3. 



