MOTION OF ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. 155 



Thus, if 



cr = c 

 where 



<TI is of the first order, 

 and 



cr 2 is of the second order, 

 the force becomes 



= 4irV f +1 P! (oo + *& + cr 2 P 2 ) 2 dp, 



J-i 



= 477V (10-00-! + -Ay 0-i CT 2 ). 



This result can readily be extended, and it appears that only products of successive 

 cr's occur. 



I had a special reason for wishing to examine the effect of a uniform field of electric 

 force F in this case, so that the new meaning of F must be remembered in comparing 

 results with those of the preceding section. 



The forms to be assumed for the field in accordance with the fundamental equations 

 up to the second order are 



.(X, Y, Z) = y, *) + (*, xy, xz) (rV 



+ (x, 0, 0) (,-V" + 3>V + 6r x / + 6 X2 ) + (-,**'" + 3 



Hence, in order that the tangential component of (X, Y, Z) should vanish at the 

 surface, we must have for r = a 



= 3 F/c, 



Hence we find that the surface density of electricity is given by cr, where 



while the equation of motion of the sphere is 



X 2 



