﻿Motion of the Lorentz Electron. 53 



Rectilinear Motion, 



5. As an example of the use of the equations we have 

 obtained, we shall now consider the case where the electron 

 moves in a straight line under the action of an electrostatic 

 field in the same direction. We shall take the straight line 

 as the axis of #, so that w' = iv'=w'. Then we find from 

 (13) and (14) respectively j 



id' 2 



R"=.r^3' (16) 



l + w 



ww' 2 



w'-w"+ f^=W{l+w 2 ) . (17) 



l + w 2 v ' v ' 



In order to reduce these equations to a simpler form we 

 write 

 w= sinh%, whence v=/3= tanh^, and T= cosh% — 1, (18) 

 (16) and (17) now give 



R=x' 2 , • • (19) 



X'-X"=F (20) 



When F is known as a function of t, (20) may be solved at 

 once in the form 



x =i T FdT-eTC¥e-TdT + A + Be T , . . (21) 



where A and B are arbitrary constants to be determined 

 from the initial conditions. A third arbitrary constant will 

 be introduced when we determine x from the differential 

 equation # = io=sinh ^, but we may make this constant zero 

 bv choosing the origin of coordinates so that os vanishes 

 when t and t vanish. 



6. Determination of the arbitrary constants A and B. — One 

 relation can be obtained at once between A and B, for we 

 are at liberty to choose the origin of time so that r, and 

 therefore also ^, vanishes when t=0. This condition with 

 (21) gives 



A + B = , (22) 



Substituting for A in (21) we obtain 



% =( T F(?T-e^ r Fe- T ^T-fB(e T -l). . . (23) 

 Jo Jo 



Bearing in mind our choice of the origins of space and time 



