
EQUILIBRIUM OF ELASTIC SOLIDS. 107 
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 a,, then 




0 
—=0, when 7=a, 
ec 3 a,> evan 2m 
Phra im+3a,ept re 2a>m+3a,° wb 
i 3a,> Lg G,* a,® ™ 
Ae 22a,>m+3a,°p 7 rr 2a m+3a 3p 
(45.) 
or a,® h aras 1 
r *2aim+d3a,p 7? PF 2a 3m+3a,3 ph 
OV 3a,2—3 a,° 
When =a, v oo Siar cera 

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. 
Let a rectangular elastic beam, whose length is 2 7c, be bent into a circular 
form, so as to be a section of a hollow cylinder, those parts of the beam which lie 
towards the centre of the circle will be longitudinally compressed, while the op- 
posite parts will be extended. 
The expression for the tangential compression is therefore 
Or _ec—r 
r ae 

Comparing this value of or with that of Equation (20.) 
e—r_/1 1 gq 
8 Ganga) ore + 
d 
and by (21.) g=ptroe. 
By substituting for g its value, and dividing by r Ga + sal , the equation 
becomes 
dp 2m+3y Pp _Imp—(m—3 p) 0 9m 


dr' m+6m@ r  (m+6p)r (mt+6p)e 
a linear differential equation, which gives 
2m+3 wo 23 
aC ete Om Foe a8 eas 6) 
mM+3 "hc 2m+3 ph 
VOL, XX. PART I. 2F 
