﻿54 Prof. G. A. Schott on the 



and using (8) and (18), we find 



x — \ sinh^rfr, (24) 



f=| r cosli^ ( /r (25) 



o 



We have now fully utilized the initial conditions so far 

 as they relate to the initial values of the coordinate and the 

 velocity of the electron, but there still remains an arbitrary 

 element — the arbitrary constant B in (23) to be determined. 

 Here we are brought face to face with one point of difference 

 between the ordinary mechanics of JSewton and the electron 

 mechanics founded on the electron theory. Very slight con- 

 sideration shows that the presence of the third arbitrary 

 constant is due to the fact that the equation of motion of 

 the electron, (1), or (14), or (17), when regarded as a dif- 

 ferential equation for the coordinate, is of the third order r 

 and that the differential coefficient of the third order arises- 

 from the radiation terms. It is important to bear in mind 

 that these terms must be present whether we adopt the- 

 Theory of Relativity for accelerated motions, or base our 

 mechanics on the hypothesis of the extended electron ; only 

 in the latter case every additional term of higher order which 

 we introduce into our equation of motion brings with it 

 another arbitrary constant. These additional arbitrary 

 elements, in so far as they must be determined by the initial 

 conditions, represent the effect on the motion of the electron 

 of its past history, a point which I have emphasized on 

 previous occasions *. Unfortunately, the past history is un- 

 known in many problems, and therefore we are compelled to 

 make some additional hypothesis to overcome the difficulty. 

 We must choose it so as to preserve the continuity of the 

 electron mechanics with the ordinary mechanics, which we 

 know suffices in all cases where the velocity of the electron 

 is infinitely small compared with that of light : thus the 

 proper hypothesis suggests itself, namely, that in these cases 

 Newton's Laws of Motion hold without alteration. Hence 

 we assume provisionally : 



When the velocity of the electron is zero, its acceleration is 

 equal to the external mechanical force per unit mass. 



This hypothesis has the advantage, as we shall see later, 

 that it leads to simple results which can be controlled by 

 experiment. 



* Schott, Annalen der Physik, 1908, p. 63; E. R. p. 155. 



