156 MR. GEORGE W. WALKER ON THE INITIAL ACCELERATED 



There is theoretically an additional reaction due to the magnetic force acting on 

 the surface current. It depends on the fourth power of the velocity, and has there- 

 fore been neglected. 



Vibrations then of the first and second order arise, but since these are rapidly 

 damped we may fix attention ou the motion possible when these have become 

 negligible. 



A trial of the assumption f = ^k^ 2 , and determination of the ensuing forms for ^i 

 and X2 fi'om the conditional equations, shows that in the equation of motion we leave 

 unbalanced terms of the third order in t which depend on the time. 



A uniformly accelerated motion is thus impossible with a constant force, but we 

 may destroy the terms of the third order in the equation of motion by assuming for f 

 the form > , ,. ,., , , , 



where k 3 and / - 3 are at least of the third order. For the satisfaction of this condition 

 we require rF J a*F ll m(3mm n -m a -m' 3 ) \ 



1 = ' 5 * ' 5 ' 



, _ 3 mm' 2 

 ~ 5 



In these expressions higher powers of F have been neglected. 



The values of (X', Y', Z') are obtained by adding to the expressions for (X, Y, Z) 

 the vector -(<),//, :) (/'x/' + x/) /r. 



Hence, using condition (-2), the equations that hold at / = are now 



aW'+ZaW+GaXt'+Zx^See/C-^ay+SaW^ 



while the surface density is given by or, where 



4 = ^ + ^('^- 2rt v;)+^ 



The equation of motion is 



m *~ fl ( F -*i") + *( F -t^ 



Proceeding as before, by assuming 



=^ a +-p- 2 i :i +A/, 



we find, up to terms of the third order in F, that k 2 = & 3 = and 



eF 



ro+m'l ' C 2 (m + mj }' 



