CONTEMPORARY ADVANCES IN PHYSICS 179 



Summarizing: if the corpuscles associated with the waves have the 

 properties of ordinary material bodies — if, let us say, for short, the 

 corpuscles are material particles, their speed is equal to the group-speed of 

 the waves. 



This is a ver\' happy and agreeable result. It compensates very 

 largely for our having been forced to concede that if we want both 

 waves and corpuscles, the wave-speed and the corpuscle-speed must 

 be different. The wave-theory has supplied another velocity which is 

 equal to that of the corpuscles. Moreover it is precisely the \'elocity 

 with which we should expect an isolated segment of a wave train to 

 move from place to place. If someone were to cut a piece out of an 

 electron-jet and measure the time it took to traverse a known distance, 

 the speed which he would deduce from his data would probably agree 

 both with the corpuscle-speed and with the group-speed, and disagree 

 with the wave-speed. It would be interesting to try this out. 



In the equations ^23) I have taken account only of the kinetic 

 energy of the corpuscles; in the equations (25), only of their kinetic 

 energy and of the 'rest" energy associated with their mass. But the 

 explanations of refraction by the two theories will no longer be con- 

 cordant, unless the potential energy also is admitted. Let us denote 

 the potential energy of a corpuscle by U; and, since as yet these theories 

 have been verified only for negative electricity, let us immediately 

 write eV for U, e standing for the charge of an electron and V for the 

 electrostatic potential in the region where it is. For the total energy 

 of the corpuscle, then, we have instead of (25) the relati\'istic 

 expression, 



E = nl,c^i^l - ff' -\- U = WocVVF^^^ + eV, (26) 



which for small values of the corpuscle-speed u (= 3c) reduces to the 

 classical expression, 



E = hnuu- -\- U = hnuii" + eV. (27) 



In an earlier section we interpreted the refraction of an electron beam 

 passing from vacuum into metal by thinking of the metal and the 

 vacuum as being two regions in which different values of electrostatic 

 potential prevail, the potential thus changing sharply from one value 

 to the other at the surface which bounds the solid. Xow when the 

 beam considered as a stream of corpuscular electrons passes across 

 such a surface, the energy of each electron as expressed by (26) or (27) 

 remains the same, though the proportion which is kinetic energy is 

 changed; and therefore the frequency Ejh of the equivalent wave-train 

 remains the same. If then we keep the assumption that the wave- 



