152 MR. GEORGE W. WALKF.R ON THE INITIAL ACCELERATED 



of its initial value, and thus the vibratory motion may practically become insignificant 

 before the equations become invalid. Since the decay is exponential this can be 

 secured for moderate values of Ct/a while the whole displacement can be made small 

 by making F small. 



In these circumstances the displacement of the sphere is adequately represented by 



f- i F J , 2m'at 



*(nt + m') I (m + m')C (m + m') 2 C 2 J 

 while 



2am /,-,,, -, 



K (C r+a) 



' 



throughout a certain region. 



Within this region the state of the field is given by 



At gi'euter distances the damped harmonic train would have to be retained. 

 Hence a constant surface density is established, given by 



2rF 



(m + m')a (J a 

 We further find that 



, F m' a- 



Hence the sphere arrives at the point as if the equation of motion had been 



with an initial velocity - - and an initial displacement 7 '^ *f . We have 



' 77i+wi'*C m+wC 3 



thus shown tliat a uniformly accelerated motion is possible, and that the reaction of 

 the medium is m'g. 



If we introduce a new variable, <j>, defined by e<f>/C = xeg/G the equations (l) 

 and (2) become the equations of motion of a system in which the kinetic energy is 



T = ir'(M)> + l m p + X(^-<^C/, 

 * In 'Roy. Soc. Proc.,' loc. cif., p. 265, line 5, write m/(m+m') for |- (2 m + m')/( m + ni'). 



