Problems in the Theory of Elasticity. 29"' 



The dyadic H(s/>) so defined is an extension of the 

 resolvent K(st) obtained from it replacing t by a point p 

 not on the boundary. From the preceding relation it is 

 easily verified that 



\V(s$) . H(fy>&= j* H(s3) . ¥(■&/>) A 



so that TL(tp) is defined by the alternative relations 



H(«p)— ^(^>)=xjH(**) .^(«p)d*=xJ^(**>.H(«p)rf*. (32) 



An exactly similar pair of relations (32') define the dyadic 

 H'(/y>) in terms of %(£/>)• Then the dyadic r'(/?s) defined 

 as above may be extended, replacing s by another point q 

 not on the boundary, the new function F'(pq) being- 

 specified by 



r(pq) = ®(pq)+\$®{p8) . K'(sq)ds. 



It is then easily verified that 



and thus the dyadic T'(pq) is defined by the alternative 

 relations 



r(pq)-<S>(pq) = \$<P(ps) .KXsq)ds = \$Ti(ps) . X (sq)ds, . (33) 



in which q may be replaced by a boundary point t. 

 Similarly, if we define a dyadic Y(pq) by the equation 



T(pq) = ®(pq)+\§<f>(ps) .K(sq)i 



it satisfies the alternative relations 



T(pq)-®{pq)^\§®(ps).IL(sq)ds = \§r{ps).y(sq)ds. . (33') 



§ 10. Singular Parameter Values. — We may now prove 

 that the characteristic numbers X. and \'. of the kernels 

 Wits) and x(ts) respectively are real, and in absolute magni- 

 tude not less than unity ; also that each is only a simple 

 pole of the resolvent involved. For these values the homo- 

 geneous equations 



admit each one or more non-zero solutions. Regarded as 

 moment and density of double and simple strata respectively,. 



