'30 Dr. C. E. Weatherburn on Two Fundamental 



these define displacements W(/>) and V(p) satisfying the 

 homogeneous problems 



W (*+)- w(r)=\ [ w(«+)+ w(r)]l 

 TV(r)-TV0 + )=x/[TV(r) +tv(*+)] J ' 



The equations (34) are not associated, and the charac- 

 teristic numbers of ^(ts) are in general different from those 

 of xi is ) m We observe, for the following argument, that if 

 V(p) is any regular displacement satisfying the equation (1), 

 and TTJ(£) the corresponding surface traction, the integral 



[U] = -jTJ.TU^ (36) 



represents twice the potential energy of Ihe deformed body, 

 and is therefore a positive quantity, vanishing only for a 

 translation or rotation of the body as a whole. Further, if 

 V is another regular displacement satisfying (1), Beta's 

 theorem gives 



j[U.TV-V.TU]^ = (37) 



To prove now the reality of the characteristic numbers *, 

 consider for example one X ' of %(£s) for which the second 

 equation (35) admits a solution Y\p). If this parameter 

 value is complex (=za + ib), so also is the potential V(p) 

 ( = U + iXTi). Then, on separating real and imaginary parts 

 in (35), we obtain 



(l-a)TU- -(l + a)TU + +&(TUr+TU 1 + ) = 

 (l-a)TUr-(l + a)TU 1 + -6(TU- +TU + ) = 0, 



where TU~ is written for TU(£~),and so on. Multiply these 

 equations scalarly first by Ux and U respectively, subtract 

 and integrate over 2 ; then by U and Ui respectively, sub- 

 tract and integrate as before. Then in virtue of (37) it 

 follows that 



i{[u+] + [u,+]-[u-]-[ur]}=o l 



'«{[u + ] + [u 1 + ]-[u-]-[ur]} = [u + ] + [u 1 +] + [tj-] + [u 1 -]r 



. . . (38) 

 where 



[u-]=j , u.TU(r)^ J 



which by (36) is positive, as it applies to the outer region 



* The proofs in this § and the next follow closely those of Plemelj 

 for the case of the potential theory. Cf., e. g., " Potentialtheoretische 

 Untersuchungen," § 24. 



