170 C. E. Weatherhurn 



In virtue of (7) and (8) the second member reduces to the first two- 

 terms; so tliat V(p) satisfies the boundary problem. 



All the functions involved are regular for the singular value X = Ao, 

 so that F(/j) is the solution of the problem (19b) regular even when 

 X is equal to this singular value. The problem (lb) does not 

 admit a solution by simple stratum only, Avhen X=Ap, unless the 

 condition 



/F{te)i{e)d$=:0 



is satisfied, in which case the required solution is obviously V(p), 

 Tlie problems (19), derived from (1) by altering the second member, 

 we shall speak of as the modified problem for the singular value Ao- 

 The functions H(t-p) and G{2)t) bear the same relation to the solution 

 of the modified problems that 'H.(fp) and G{pf) bear to the original 

 problems (1). 



iil. — Expansions. — From the formulae (8) and (18) we may ob- 

 tain, by the method of successive approximations, expansions for 

 the various functions in ascending powers of X. These are cer- 

 tainly true for | X | < 1, and in particular cases even for jXl^l. 

 For the present we shall assume that the absolute value of X is less 

 than unity. 



Thus from (8) in virute of (7) we find 



(20) [H{ts)=^[]i{ts) - IP(^«)] +x[a,(<s) - \-;P{ts)] 



G{ps)=[g{ps) -q{ps)-\ + x]^l,{ps)- ^ q{ps)\ 



+ >^'[0.{ps) -\.^{Vs]+ •••• 

 where the suflixes denote functions formed by successive operations 



h,{ts) = /h{t6)h{es)de, 

 h.lts)=/K{te)h{es)dd, etc. 

 and 



g,{ps)=/c,(j^e)h{e.)d6, 



r,.Xps)=/g,{p$)Hes)de, etc. 

 If Ave extend the notation and repbue •>-• by any point p we may 

 write 



h^(t/>)=/ h(te)h(ep)d6, 

 h,,{fp)=/hn^,{fe)/i{ep)de, etc. 



