154 MR. GEORGE W. WALKER ON THE INITIAL ACCELERATED 



apparent contribution to the initial velocity produced by the first vibrations is exactly 

 destroyed by the second vibrations. In a similar way the apparent contribution to 

 the initial displacement is destroyed. 



F 2 



The contribution 4 7 r- is the displacement due to F acting on (m + m f ) for a 



2 (m+m') 



/T71 , 



time <!. The contribution p-/ -^, on account of the apparent initial velocity, 



could not be expected to disappear. 



p. 

 There is no loss of energy since the velocity established is - ^~f\> an d the energy 



F 2 3 Ft 2 



of the system is i- 7 l .r or FxA-^ ' l and is thus the work done by force F 



J 2 ' 2 (m + iri) J 



acting on (m + m') for a time t^. The dissipation function is now found to vanish. 



The result then shows that the initial motion is attended by the production of a 

 damped harmonic train. On account of the rapidity of damping, a uniformly 

 accelerated motion soon becomes possible. 



The existence of a gyrostatic term in the kinetic energy has been revealed, and also 

 the existence of a dissipation function. 



The production of waves (Ilontgen radiation) by the sudden creation or destruction 

 of a velocity has been already shown by THOMSON (' Conduction of Electricity through 

 Gases,' p. 538). Our investigation shows that the establishment of a constant velocity 

 is really attended by the production of two rapidly damped harmonic trains, which of 

 course combine if the time of action of the force is sufficiently short. The frequency 



/ m >\ 1/2 Q 



of the waves is ( 3 + 4 ) - and the modulus of 'decay C/2a. 



4. Second Order approximation. When the former expressions for the field are 

 carried out to squares and products of g and -^(Ctr), it appears that the motion is 

 modified by the production of damped harmonic waves depending on a second order 

 zonal harmonic. 



We therefore introduce a new function x^(^~ r ) associated with a second order 

 zonal harmonic supposed to be small of the second order, while and ^i (Gt r) are 

 small of the first order. (X', Y', Z') now differs from (X, Y, Z) by terms of the 

 second order. 



We can readily show that this will introduce terms of not less than the third order 

 in the equation of motion of the sphere when condition (1) is used. 



Thus the surface density is given by 47rcr = N, where N is the normal component 

 of electric force. Since the tangential component must vanish at the surface, the 

 force due to radiation reaction is 



! dS. 



= 277 f ( 



