﻿Motion of the Lorentz Electron. 57 



We now find instead of (33) 



% =F{T + S(er-1)}, R = F{1 + M 2 , • . (38) 

 while we have as before 



v=/3=tanhx, #=1 sinh%^T, *=1 cosher. 



«^o Jo 



Changing the independent variable from t to % we obtain 

 F ^ = f* smh % fo _f* sinh % ^ % , 



Jo i+«jj "~Jo l+a-r+x/F •••••• i • 



F* = f * cosh %^% _ C X cosh % ^ % 



Jo T +^ ? J l + S-T + % /F> • 



E = F 2 {l + ae^} 2 = F 2 {l + 5-r + % /Fp, 



J o R<fc=F^ {l + ^}cosh % ^=F^{l + S- T + % /F}cosh % ^. 



The equations (38) and (39) show that the analytical 

 character of the solution is completely altered by the failure 

 of the hypothesis under consideration ; what change will be 

 produced in the numerical results depends on the magnitudes 

 of^ /3, F, and 8. In estimating this change we must 5 bear in 

 mind that what we measure by experiment is the increase of 

 velocity produced in a measured distance by a field of known 

 strength, and perhaps in certain cases the total energy 

 radiated m the process. Knowing (3 and therefore y we 

 can calculate x and the energy radiated by means of (39): 

 but in order to measure independently of "the hypothesis to 

 be tested we must not use a deflexion method, either with an 

 electric or a magnetic field, because that would again involve 

 the hypothesis and require very troublesome calculations. 

 We must measure the kinetic energy, e. g. by a thermopile, 

 and thence calculate x and by means of (18). 



When the exponential term in (38) for x is negligible in 

 comparison with the first, we have the case already considered 

 in § 8 ; for the sake of brevity we shall speak of it as the 

 Newtonian motion. On the other hand, when the exponential 

 term preponderates we have another extreme case, which w 

 shall call the exponential motion and shall now examine. 



10. The exponential motion. — We retain only the expo 

 nential term in (38), and accordingly only the term yfF in 

 the expression l + S—T+y/F, which occurs in (39). Then 

 the denominators in the integrals for x and / vanish ai the 



(39) 



\\ e 



in 



