4 Art. 7. — K. Terazawa : 



which determines the dilatation J. If J is found from this 

 equation, the displacement u can Ije determined by solving the 

 equation 



X-\-[JL 



grad div u — curl curl u = — grad A. (4) 



A* 



The elastic body which we deal with is supposed to be 

 bounded by an infinite (say) horizontal plane in its natural state 

 and to extend without limit both horizontally and downwards. Take 

 the cylindrical coordinates (r, d, z) such that the axis of z coincides 

 with an inward normal to the boundary and the origin lies on the 

 boundar}^ surface of the body in its natural state, then we have, 

 for z^>o, the equations 



i^+J-. -M,+J-ü^+ü^ = o (5) 



to determine J, and 





+ „.^^ + Ji^__±^ ,. -j:-^ +^ 



a<? /^ a?- 



?r'^ /■ ■ Ir Iz-^ r ' 16- i^ ' ^z' 



(6) 



where ur,v^,uz are the components of the A^ector u. The equation 

 (5) follows from (3), and (6) from (4). 



§6. To solve these equations, assume 



^ ^ smj 



where h is a positive constant so that there may be no dilatation at 

 ;2=oo, and m is an integer, positive or negative, or zero so that the 

 solution may be unique round the origin, then the equation (5) gives 



dr- r dr \ r 



of which the solution is 



1) The second solution is rejected for the reason of its singularity at the origin which 

 we exclude from our present investigation. 



