.28 Dr. 0. E. Weatherburn on Two Fundamental 



solution is available. There is no need to eliminate the 



second term of F (s/>) in (7) involving axjnx grad - ). This 



term does not become infinite of the second order as p 

 approaches the boundary point s along the normal. For 



grad- =— r/r 3 ; but n has the same direction as r, so that 



r 1 

 .21 X grad- becomes infinite of order less than two. It is this 

 & r 



fact that makes the integral w 2 (p) of § 6 continuous at the 

 boundary. 



Each of the equations (27) has one and only one solution 

 unless X is a characteristic number of the kernel of that 

 equation. The solutions are expressible in terms of the 

 resolvent dyadics K{ts) and H'(te) of Wits) and x(ts) respec- 

 tively, connected with these by equations of the form 



and a similar set (28') for H'(ta) and x(. ts )- * n terms °^ 

 these resolvents the solutions of (27) are 



T(O=f(0+xJf(*).H(i«)A-i 



u(*)=f(0+xjH'(«*).f(*)^J ' 



and these values substituted in (26) give the displacements 

 that satisfy the boundary problems (25), viz. 



w (p) = jf (*) . \y(s P ) + x Jh (^) . ^r(5rp) <&]<& j 



V(p) = $[®(ps) + XJ<S>(^) . H' ($*)<*$] .f{s)ds] 



§ 9. The resolvents TL(ts) and H'(te), in terms of which 

 the solutions have been expressed, are known meromorphic 

 functions of the parameter X, and either becomes infinite 

 only when X is a root of its denominator D(X) or D\X). 

 These singular values of X depend only on the form of the 

 boundary. The solutions (30) may be written 



W(p) = Sf(s).K(sp)ds | 



v(p)=]r'( ps j.f( s )ds} ' 



if we define the functions H(sp) and T'(ps) by the equations 



HO) =¥(*jt>)+ xjh(^) .¥($p)A 



F(» = <&(ps) +\]&(pd) .H'($*)«Z&. 



