( 482 ) 



the forced- But it does iiot follow from this that also a greater 

 force is required if the retardation exists only in the future. On the 

 contrary when the velocity has decreased to 2}, < 35, the velocity 

 during the active inter\al has been smaller than 25, on an average, and 

 so also the force required will be smaller than that which corresponds 

 to a constant velocity 25,. So with a gradual decrease of the velocity 

 corresponds a gradual decrease of the force. The re\'erse of this 



thesis is not alwavs true: if — is a continuous function of Mhen the 



(It 



(I?) 

 velocity will also vary continuously. Tf on the other hand -y is 



discontinuous though A be continuous then o will vary discontinuously. 



A diminution of the force is therefore accompanied l\v a diminution 

 of the velocity, and inversely. The behaviour of an electron moving 

 with 25 corres|)onds in this res[>ect with that of a body with a posi- 

 tive mass. If the force acting on the electron decreases gradually to 

 zero, the velocity will fall to iv 



Though it seems to me that there is no reason to doubt whethei- 

 the behaviour of an electron has been described here accurately 

 though only in general outlines, and though a complete calculation 

 of the motion is not practicable in consequence of the great intricacy 

 of tiie formulae, 1 will sliow in one simple case that the force 

 required for a given motion agrees with the above description. 1 

 imagine to that purpose an electron which for some time mo\'es 

 with a constant velocity i^ At the instant / the motion is suddenly 

 accelei-ated with a constant acceleration />. In order to render the 

 calculation possible we will assume that we may apply the formulae 

 for quasi stationary motion. We will calculate the force at an instant 

 t in the first interval 'i, so / > t'. The calculation does not present 

 any difticulties, and can be can-ied ont in the way iiulicated by 

 SoMMKRFELD. After introduction of the approximation for the (piasi 

 stationarv motion we may everywhere separate the terms as they 

 would be for a constant velocity 2>, (we call the sum of these 

 terms d,) muI the supplemeiitary terms which depend on the 

 acceleration, and whose sum will be denoted by A,. In this way 

 we Imd : 



1) This rule is given by Sommerfei.d tlioiigh his calculations show that it does 

 not hold good with perfect generality. In most cases and also in Die present one 

 it will give a Irne idea in general outlines of the value which tlie force must 

 assume. 



~) SOMMERFELD III p. '20ij. 



