applied to Statical Electricity, 17 



The normal stress on the surface at any point is 



N =p„ sin 2 9 +p yy cos 2 6 + 2p xx sin 6 cos 



= 2(lJL-}m)(e+g)acos0 + 2macos6Q(e+f)sm 2 d+gcos°-6y,(92) 



or by (87) and (90), 



N=-?wtf(e4-2/)cos<9 .(93) 



The tangential displacement of any point is 



t = £ cos 6 - f sin = - (a 2 /+ d) sin 0. . . . (94) 



The normal displacement is 



w = f-sin0+?co8 0:=(a a (e+/)+rf)cos0. . . (95) 

 If we make 



« 2 (e+/)+d=0, (96) 



there will be no normal displacement, and the displacement will 

 be entirely tangential, and we shall have 



t = a 2 es\xid (97) 



The whole work done by the superficial forces is 



V = iZ(Tt)dS, 

 the summation being extended over the surface of the sphere. 

 The energy of elasticity in the substance of the sphere is 



tt iW^f d V «*? /drj dg\ , M , dP\ 



(!+£>>> 



the summation being extended to the whole contents of the 

 sphere. 



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

 value, namely 



U=-§7ra 5 me(e + 2f) (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. 



We assume the axis of z to be in the direction of maximum 

 variation of the pressure among the particles, and we have to 

 determine the relation between an electromotive force II acting 

 on the particles in that direction, and the electric displacement h 

 which accompanies it. 



Prop. XIII. — To find the relation between electromotive force 

 and electric displacement when a uniform electromotive force It 

 acts parallel to the axis of z. 



Take any element 6S of the surface, covered with a stratum 

 Phil. Mag, S. 4. Vol. 23. No. 151. Jan, 1862. C 



