[Proc. Koy. Soc. Victoria, 27 (N.S.), Pt. II., 1914]. 



Art XIII. — Singular Parameter Values in the Boundary 

 Problems of the Potential Theory. 



By C. E. weather burn, M.A., B.Sc. 



(Ormond College, University of Melbourne). 



[Read 9tli July, 1914]. 



The method of integral equations has been successfully applied 

 to the boundary problems requiring the determination of poten- 

 tials, Vs{p) and \(p) satisfying the boundary relations^ — 



i^&>-'-£V)]-KS<'-»-£<->]='(" 



respectively, whether the potentials are ordinary2 corresponding to 

 Laplace's equation, or " generalised " corresponding to the equa- 

 tion — 



(•2) v-'U-FU = 



The latter potential I have considered in a paper^ recently com- 

 municated to the Quartei-ly Journal. It is shewn that solutions to 

 the problems can be uniijuely determined, except for certain singu- 

 lar values of the parameter, in the form of potentials of double and 

 simple strata respectively, given by'* — 



(3) i W{r>)^/i{t)n{tp)dt 



\ Y{'p)=/(A{pt)t{t)dt. 

 At a singular i^a'-ivnietei' value A,,, however, the solutions become 

 intinite, since each of thy functions ii(fp) and (^{pt) has a simple 

 pole, unless certain conditions are satished. It will be sliewn that 

 the parts H{tp) and (T{pt) of these functions remaining finite at the 

 pole Ay, form tlie corresponding functions for the solutions at this 

 pole of the problems (1), which, however, must be modified"' in their 

 second members. The residues P(^//) and <.}(pf) of B.{t p) and G:{pt) 

 respectively, also play an important part in the following argu- 

 ment. 



1 Poineare. " Sur les Equations fie la Physique." Rendicoiiti, Paleinio, ISiU. 



2 Plemelj. Moiiatsfefte \\\v Math, umi Physik, Bd. 1.5, .S. ;«7-4U (1!»U4): Brt. 18, 8. 180-211 

 (1907). 



3 " Boundary ]irol)leiiis for tlic <;eiK'iaHsed potential coiiesponfliny to tlie equation 

 V2u-A;-U = 0." Quarterly Journal, vol. 4G, \^\^. «(J-S2. 



4 The integ-ration throujrhout is e.xtended over the boundary of tlie resiou considered, uidess 

 otherwise stated. The notation of my previous payier is adhered to. 



5 Weatherburn, loc. eit. § ; also Plemelj, loc. cit., S. iO\-U. 



