1896.] Elastic Solid Shells of nearly Spherical form. 67 



Then we easily find 



u/p = G + i(i+ 1) (p - a) (p -p')j(2hE), 



•/S-/Gct38— *-^**h (19) ' 



where G 3 L_ feP^ + >(P-g7| . 



a 3 — o 3 I 3m — w 4w i 



A displacement 



u = Cp, 



be it noticed, would simply change the surface into one similar 

 and similarly situated to itself, and so leave the shape unaltered. 



If pressure be applied over the outer surface only, we have 



u/p = - C\ + i (i + 1 ) (p - a) p/(2hE) (20), 



where Cj is positive and numerically large compared to the other 

 term on the right. The radius vector is thus everywhere reduced, 

 and the proportional reduction is above or below the average 

 according as 



p is less or greater than a. 



Under the same conditions 



eapdp eapj_dp 



4mhd0' 4>nh sin dd<p K h 



The resultant tangential displacement is thus along the 

 direction in which p increases most rapidly ; e.g. in a spheroid the 

 tangential displacement is along the meridian, being directed 

 away from the axis of symmetry or towards it according as the 

 spheroid is oblate or prolate. 



It is thus obvious that pressure on the outer surface tends to 

 exaggerate the departure of the surface from the truly spherical 

 form. 



If, on the other hand, pressure be applied over the inner surface 

 only, we have 



u/p =C 2 -i (t + 1) (p - a)p'/(2hE) (22), 



eapdp _ea£J_dp_ 



bnhdd' 4>nh sin 6 d<p K h 



where G 2 is positive. 



Here every radius vector tends to lengthen, but the lengthen- 

 ing is proportionally least where the radius vector is initially 

 greatest ; also the tangential displacement is along the direction 

 in which the radius vector shortens most rapidly. Pressure on the 

 inner surface thus tends to make the shape more nearly spherical. 



5—2 



