Problems in the Theory of Elasticity. 31 



for which the direction of the inward normal is reversed. 

 Since, then, the second member of the second equation (38) 

 is positive, and cannot vanish except in the degenerate cases 

 already mentioned, the coefficient of a cannot vanish, nor 

 therefore that of b. Hence b itself must be zero, making 

 A/ real. Then V(/>) must be real, giving TJ 1 = 0, U = V, and 

 a=y. Thence by (38) 



V={[V-] + [V+]}/{[V-]-[V+]}. . (39) 



Thus the absolute value of \ ' is greater than unity, except 

 in the degenerate cases for which one of the expressions 

 [V] or [V + ] is zero. 



Similarly, starting with the first equation (35), we may 

 prove the same result for the characteristic numbers \ of 



§ 11. To prove next that each singular value \' is a 

 simple pole of the resolvent H ; (7s). We have seen that it is 

 a pole, and therefore also a pole of the density u(t) of the 

 same order. If this order w be > 1, u(t) may in the 

 neighbourhood of \ ' be expressed in the form 



u(t) -p(i)/(X- V)"+pi(«)/(x- V)" -1 +...., 



where p(£) does not vanish identically. Substituting this 

 value in the second of equations (27) written in the form 



and equating to zero the coefficients of (X — X ')~ n and 

 (X— \')~ n+1 , we obtain the relations 



If now we take p(£) and p x (£) as vector densities of simple 

 elastic strata whose potentials are V(j?) and V x (j;>) respec- 

 tively, these equations are equivalent to 



TV" -TV+ -V(TV- + TV + ) = , 



Tvr -TV!+ - V(TVr + tv x + ) = (tv- -tv+)/\ '. 



Multiply scalarly the first of these by Vi and the second 

 by V, subtract and integrate over S; then in virtue of (37) 

 and the continuity of a simple elastic stratum potential we 

 deduce 



[V+] + [V-l=0. 



