TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 29 



Also, from (40), for the corresponding stresses at the plane 2 = 0, we have 



(109). 

 J = p. {(2e - V) A + 2i 



Although the above derivation is sufficient for our purpose, it may be worth while 

 to give the direct investigation,* starting from the equations 



3 2 ?.< /, x 3 A ., 3 2 y \ 3 A 



' a? = (x + ^ te + ^ p w = (x + ^ a,, 



" (HO), 



U(/ UfJ 



where 



O.K Si/ 3; 

 In the case of simple-harmonic motion ('/'') these are satisfied by 



U = -|- U 1 , V = - - -f- i./, W = - -- + w' . . . , (112) 



ox ay oz 



provided 



(V- + /,-)c/> = (113), 



and 



(V- + 3 ) ' = 0, (V- + /,'-) v' 0, (V- + /-) ('' = o j 



3 / ? ' ~> <' L ... (1 14). 



+ - + ( ' 1 = " 



where /r, P are defined as before by (28). A particular solution of (114) is 



provided 



(V 3 + F) X = (116). 



On the hypothesis of symmetry about O: we have 



__, 3 2 1 3 o" / , 1 7 \ 



V- = -, + - - ~- + x -., ....... ( l17 ). 



CTS~ -us ons 04 



and the formulae (112), (115) are equivalent to 



* C). ' Proc. Loud, Math, Soc,,' vol. 34, p. 276, for the corresponding statical investigation. 



