Elastic Equilibrium of Semi-Infinite Solid. 27 



P^os^_^^^ = or a 



similarly for the others. 

 Put again 



s = ^(u) 



and denote b}^ (o^ and oj, the real and imaginary half-period 

 respectively and w.^—w^ + oj^, tlien, since s or }^{u) is real and lies 

 between eg and e^, s=e3 and s = 62 correspond to u=(o.^ and w=w2 

 respectively, if we take the sign of }^''(iù) to be positive.'^ Thus 



r " cos/? ,n o f<^-2( 1 . -J 



= «'(-^ei^'^.--'^i)- (64) 



For the evaluation of the otlier integral, write 



2ara 6ara 



then we have 



/'^ a-rcoQd ^jß = J^ ,0, - r-ar r^^^ du 

 { B'^B' + z' 2a ' 4ä^V ^ )^^^)-)^{,u) ' 



etc. 



Now 





(66) 



in which the term mTzi enters l:)ecause of the many-valued property 

 of a logarithm. The actual value of ??^ and ^'(v) will l)e determined 

 by the following consideration. 



1 If we assume 1?'(î() to be negative, then s = 6', and s = e-j correspond to »=(»_, and 

 ii = 2coi+Gûi respectively. But the saoie result will, as a matter of course, be obtained after 

 integration. 



