Elastic Equilibrium of Semi-Infinite Solid. 



13 



Öd 



À + fxJ 



(31) 



U: 



+ 



'o *0 



^? = 0, 



T'y = z'fz{l-)e-''J,{kr)Mk, 







7d = 0; 



where the auxiliary function Z is connected with the prescribed 

 condition by 



Z(k) = k ff[a)J,(ka)ada. (82) 







We shall now apply the general theorem to a few special 

 examples. 



Example I. 



§13. The first example, which is discussed by Prof. Lamb, is 

 to find the effect of a given normal pressure concentrated at a 

 jDoint on the boundary, on the supposition that/(r) is zero for all 

 values of r except those in the immediate neighbourhood of the 

 origin, where it becomes infinitely great in such a manner that 



/f(r).27rr.dr= -II, 



(83) 



in which // is the total amount of the pressure applied and is 

 finite. The function Z{k) now is 



Z(k) = 



n 



Putting this in (30) and (31), and then integrating we get 



