Motion of an Electrified Sphere. 615 



the sphere. The coefficient of the zonal harmonic term in 

 4z7T(T becomes from (3) 



9 r 



3F--x", 

 a *• 7 



and, with a little reduction, it may be shown that the part 

 of £ not evanescent on account of m is 



*F 1*F m ct(m'\i 



, + ,r — e" c/2a COS . -( — I 



m dm a\m / 



and thus by (5) 



3Fa a¥ _. l9a _ rf/»V 



-ct/2a cog# _./_ \ . , # . (12) 



a \m J v 



2c 2c 

 so that the surface density is finally given by 



4«r=J+Fo«itf..-*-o M .f(^)*,. . . .(13) 



and tends very rapidly to the uniform value belonging to a 

 sphere at rest with no applied field, whatever the meaning 

 given to the cosine. This conclusion is in accord with that 

 of Conway. Thus a sphere with no Newtonian mass must 

 move, when placed in an electric field of small intensity, 

 without a change in its electrical distribution, if the usual 

 conditions for a perfect conductor can continue to be valid. 

 The value of o- at £ = 0, before the field has influenced the 

 distribution by setting up vibrations, is of course 



•-&(p + *~*) < u > 



When Newtonian - mass is present, the suriace density 

 soon settles down to the steady value 



~47rVa 2 



w -/Fcos^ • • ( 15 ) 



m + m 



(Walker's first result for this case, given in (19) p. 265, is 

 corrected in a footnote in the second paper), and for a large 

 value of w, gives the ordinary electrostatic formula, as it 

 should. 



