32 Dr. C. E. Weatherburn on Tivo Fundamental 



Each of the expressions [V + ] and [V~] is therefore 

 identically zero. The stratum density p(£) must therefore 

 vanish, because in the degenerate cases referred to the 

 surface traction is zero, and therefore gives no discontinuity 

 at the boundary. Since then p(£) is zero for all values of 

 n>l, it follows that the poles of the resolvent K'(ts) are 

 simple. The same may be proved of the poles \ of the 

 resolvent TL(ts). 



These resolvents may then, in the neighbourhoods of their 

 poles X and X \ be expressed in the forms 



H (ts) = P (ts)/(\ -X) +K (ts) 1 m 



•K'W^Xtsj/iXo'-Xj + K'its)]' 



where K(ts) remains finite at X and K'(ts) at \ '. Since the 

 poles are simple the residues P(te) and P'(te) are given by * 



';■ K • • • • ( 41 > 



P'(**)= WW*) I 

 i=i J 



where n^s) and m^s) are the solutions of the homogeneous 

 equations associated with (34), satisfying the orthogonal 

 relations 



h(«).T/o*i n, if i=n f42 > 



while /j and ¥ are the multiplicities of the roots X and \ ' 

 of D(X) and D'(X) respectively. 



§ 12. Solution in the JS'eiglihourhocd of a Singular Value 

 of X. — The solutions of the boundary problems (25) as 

 expressed by (30) in general become infinite when X is 

 equal to a singular parameter value X ( sa } r ) i° ^ ne first case 

 and X ' in the second. In order that this should not be so 

 it is necessary that the residue of the solution at this pole 

 should vanish, that is 



jf(*).P(*0^ = for v(0l 



]?'(ts).f{s)ds = for u(*)J 



These are equivalent, in virtue of (41) and the linear 



* Cf. Plemelj, Monat. fur Math, und Pliysik, Bd. 15 (1904), S. 127- 

 128. 



