614 Dr. J. W. Nicholson on the Accelerated 



very small. It may be shown without difficulty that the 

 various constants take the forms 



A/ 3«F, ■ 3 7?i a 2 F w 771 a 3 F 



4 c 2 m c Am c 



. . m a 3 F . 5 /m\*« 8 F 



A sin €= -=—-, . , A cos e= — - . ( — I , 



2m' c ' 4 \?tt 7 c ' 



so that on reduction, 



C 2 7m' L a \ m / ^ \7?i / CI \77l / J 



1 <>F <?F a£ 1 <?F a 2 



+ ^- / i 2 --,-+rl- / - 2 (10) 



2 ?n 7?i c* 3 m' c 2, y 



and, except at £ = 0, f tends to involve the sine and cosine o£ 

 an infinite angle as the Newtonian mass decreases to zero. 

 But even in the immediate neighbourhood of the limit m = 0, 

 it may be shown that this expression continues, like the 

 corresponding value of ^, to satisfy all the conditions of the 

 problem, and moreover, that no other forms can do so. 

 Whatever the interpretation to be put upon the sine and 

 cosine when m is zero, they cannot exceed unity, so that the 

 vibrational term of £ will very rapidly disappear on account 

 of the damping. A slight departure from the usual condi- 

 tion of perfect conductivity in the sphere may perhaps 

 remove the indeterminate character of the limit, by preventing 

 the argument of the sine and cosine from becoming infinite, 

 so that when £ = 0, this argument vanishes, and the initial 

 conditions continue to be satisfied w T ith no Newtonian mass 

 present. On this supposition, f and f vanish with t, and the 

 initial conditions are satisfied, although the initial acceleration 

 of the sphere would be practically infinite. 



The displacement of the sphere may be regarded as a 

 superposition of a periodic part upon a part corresponding to 

 uniformly accelerated motion, and the damping factor is 

 such that the periodic portion is evanescent after an ex- 

 tremely small time. The displacement thus tends to the 

 form 



*-~2w»' 77i' c + 3m' c 2 ' • ' * { } 



In the formula as given by Walker (p. 268) the sign of 

 the second term is positive, and the factor J has been dropped 

 in the last term. 



We proceed to a determination of the surface density on 



