﻿Motion of the Lorentz Electron. 5i* 



from the truth, as measured by the number 8, must be ex- 

 ceedingly small. In order to obtain some idea of its amount 

 we must study the general motion of § 9 a little more 



11. The limits of accuracy of the hypothesis. — As we have 

 already remarked in § 9, the theoretically best method of 

 resting the hypothesis in question depends upon a comparison 

 of the kinetic energy, T, acquired by the electron with the 

 work, F.r, done by the external field. We see from (18) 

 and (39) that T differs from Fx by a finite amount, the 

 difference beino- derived from the acceleration energy of the 

 electron. Suppose then that as a result of experiment we 

 find 



T = cosh % -l = (l+/)F.r = (l+/)r[ r smhx^, (41) 



Jo 



where / is a number, which is probably a small fraction with 

 the same sign as 8. We must express 8 in terms of / by 

 means of (38), (39), and (41). Let us substitute for % in 

 (41) its expression in terms of t and 8 given by (38), expand 

 both sides of the equation in ascending powers of F8e T by 

 means of Taylor's theorem and integrate with respect to t. 

 Rearranging the terms according to powers of F8e T we find 



+ Fsi nhF3) 

 r , (l + / -F 2 VsinhF(T-S)-(l + r)F6 r coshF(r-8) + (l + r)F (cosh F8 



4- F cosh 8) 

 _ J _ r ^ (4 + / , F 2 )e LV coshF(T-8)~2(lH-/)Fe^sinhF(T-5)-(l + /) F(2sinhFS 

 I**" 2(4-F») 

 H- ... =/{coshF(T-S)-l}-(l+»{coshF«-l} (42) 



We must combine this equation with (38) so as to eliminate 

 r and determine 8, but the calculation is so difficult that the 

 result will hardly repay the labour expended; hence we shall 

 content ourselves with finding limits for 8. 



We first observe that the series on the left side of: (42), 

 being derived from exponential series by integration, is 

 absolutely convergent for all values of FSe r , and that the 

 coefficients of all powers of F8 increase with r provided 

 that tanh F(r — 8) is greater than fF, a condition which is 

 satisfied in actual experiments on account of the smallness of 

 F. Hence the first term on the left, which tor such values 

 of t has the sign of 8, is less than the right-hand member 

 when 8 is positive, and of course / also positive, but is 



