﻿Motion of the Lorentz Electron. 61 



experiments he measured the magnetic force required to 

 produce a prescribed radius of curvature in the path of 

 the electron, and compared their product with the ratio 

 /3/v (1 — /3 2 ) to which it should be proportional for the 

 Lorentz electron. This proportionality was found to hold 

 throughout the whole ran ore of the measurements to within 

 about 1 in 4000. It is obvious that this constancy of the 

 ratio of the two quantities to be compared could only be 

 possible either if the hypothesis were nearly true, or if in 

 the event of its failure the errors compensated each other 

 exactly. Of course it is extremely improbable that the effect 

 of radiation on the kinetic energy should balance its effect 

 on the magnetic deflexion so as to produce exact compensa- 

 tion, but in the absence of a complete theory of the magnetic 

 deflexion absolute certainty is impossible. We may, however, 

 draw the conclusion that the number /, which measures the 

 difference between the kinetic energy and the work done by 

 the external field, is of the same order of magnitude as the 

 errors in Hupka's experiments. By far the greatest error 

 is that in the determination of the fall of potential, given 

 above as 1 in 400; hence we conclude that /is about 1/400. 

 From six experiments with about equal falls of potential 

 we find that the fall of potential used by Hupka for a velocity 

 /3 = 0'5 is nearly 20,000 volt/cm., which corresponds to 

 F = 7*2 . 10" 15 . The corresponding upper limit for / given 

 by (43) is 0*47, which is far beyond the error possible in 

 the experiments ; hence we may apply (43). On account of 

 the very small value of F, the last factor of the right-hand 

 member of the first equation is alone effective in determining 

 the order of 8. Taking logarithms of both sides we find 



Log 10 (l/S) > 10 13 (44) 



13. Let us now consider the case where 8 is negative. 

 From (38) we see that F(t — 8) is greater than ^, so that 

 the whole investigation of § 12 applies provided that the 

 sign "less than " be replaced by " greater than." Thus (44) 

 gives a lower limit for —8. 



We may, however, obtain an upper limit for — 8 by a 

 different line of argument, based on the fact that according 

 to (38) % increases to a maximum as t increases, and 

 thereafter diminishes again. The maximum is given by 

 t= log (—1/8) and is equal to F{log(-l/S)-l-8}, and 

 there is a corresponding maximum value of /3* which is 

 tanhFjlog ( — 1/8) — 1 — 8}. Experiment shows no trace of 

 the existence of such a maximum, so that wo may bo sure 



