494 ON PHYSICAL UXES Of FORCE. 



In order that T may be proportional to sin0, the first term must vanish, and 

 therefore 



?-+/ ............................................. (90), 



T=-~(e + 2f)Bm0 ........................... (91). 



The normal stress on the surface at any point is 



N=p<a sin 1 + p m cos' + 2p a sin 6 cos 6 

 = 2 (/i - ?n) (e +#) a cos + 2ma cos {(e +/) sin 5 Q +g cos 1 0} ...... (92) ; 



or by (87) and (90), N= -ma(e + 2f) cos ........................... (93). 



The tangential displacement of any point is 



= cos0-sin0 = - (a ! /+c/)sin0 ..................... (94). 



The normal displacement is 



7i = sin0 + cos0 = {a'(e+/) + d}cos0 .................. (95). 



If we make a'(e+f) + d = ................................. (96), 



there will be no normal displacement, and the displacement will be entirely 

 tangential, and we shall have 



t = a'esm0 .................................. (97). 



The whole work done by the superficial forces is 



the summation being extended over the surface of the sphere. 

 The energy of elasticity in the substance of the sphere is 



the summation beuig extended to the whole contents of the sphere. 



We find, as we ought, that these quantities have the same value, namely 



Z7= -fwo i -me(e + 2/) .............................. (98). 



We may now suppose that the tangential action on the surface arises from a 

 layer of particles in contact with it, the particles being acted on by their own 

 mutual pressure, and acting on the surfaces of the two cells with which they 

 are in contact. 





