THE EQUILIBRIUM OF ELASTIC SOLIDS. 



55 



When the elastic sphere is solid, the internal radius a 3 vanishes, and 



, 8F h, 

 A, =p = q, and -== = 



When the case becomes that of a spherical cavity in an infinite solid, the 

 external radius a^ becomes infinite, and 





J. 



m 



SV 



P- 



m 



(44). 



The effect of pressure on the surface of a spherical cavity on any other part 

 of an elastic solid is therefore inversely proportional to the cube of its distance 

 from the centre of the cavity. 



When one of the surfaces of an elastic hollow sphere has its radius rendered 

 invariable by the support of an incompressible sphere, whose radius is Oj, then 



Sr 



therefore 



p=h ti 



3a,V 



= 0, when r = a v 



+ A 2 - 



q = i 



8r 



= ft, 

 r 



3a//i 



a'a* 



m 



r 3 



a/ , afaf 



3 "2 j- ; 



Ain. 8F j, 3a,'- 3d? 



Whenr = a s , -fr=h in T - 



' V ' O/v 3 m _1_ Q/- 8 <i 



(45). 



CASE V. 



On the equilibrium of an elastic beam of rectangular section uniformly 



bent. 



By supposing the bent beam to be produced till it returns into itself, we 

 may treat it as a hollow cylinder. 



