188 BELL SYSTEM TECHNICAL JOCRXAL 



infinite sequence of groups at distances a', 2a', 3a', . . . and (— a'), 

 (— 2a'), etc.; and along a third or 2-direction through A, there is an 

 infinite sequence of atom-groups spaced at intervals a". 



Now think of the atom-groups as hard particles, and the corpuscle 

 of light or of electricity (the "X-ray quantum" or the electron) as a 

 hard particle which rushes into the lattice, hits one of the atom- 

 groups — A, say — and bounces oft". Denote by 0, </>', 0" the angles 

 which its original direction of motion makes with the x, y, z directions 

 respectively; Ijy 0, C, 6" the angles which its final direction of motion 

 makes with these three. Before the defiection, the corpuscle has a 

 momentum of magnitude p, parallel to its original direction of flight; 

 afterward it has a momentum of the same magnitude, but parallel to 

 its final direction of flight. At the deflection, then, it loses — that is, 

 it communicates to the lattice — a momentum of which the three 

 components along .v, y, z have the values: 



/?(cos d - COS (p); picos d' - cos 0'); picos d" - cos (/>"). 



Now if, following the foregoing procedure, we equate the first of these 

 to some integer multiple of h/a, the second to some integer multiple 

 of h/a', and the third to some integer multiple of h/a", and then 

 translate momentum of corpuscles into wave-length of waves by the 

 now-familiar formula p = h/X, we get: 



a(cos d — cos (f)} = ii\, 

 a'(cos d' - cos 4>') = ii'X, (50) 



a"(cos d" - cos 0") = ;/"X, 



where ;/, ;/', ii" stand for any three integers. Now these are the 

 equations (numbered 3, 4, 5 in the eighteenth article) to which conform 

 the " Laue beams," which is to say, the directions in which electrons 

 and light are actually diff^racted by crystals. 



Perhaps I should close with two or three admonitions. To make the 

 wave-theory and the corpuscle-theory equivalent for a few simple cases 

 is of course not at all the same as making them equivalent universally. 

 Also, the examples in this article are not always so elementary as they 

 may seem. The first involved two distinct media with a sharp bound- 

 ary between; and discontinuity is always less agreeable than continuity 

 to the mathematician. The last but one involved a non-sinusoidal 

 vibration, which is much more complex than a sinusoidal one. More- 

 over, the concepts of light-waves and quanta are not nearly so beauti- 

 fully welded together as those of electricity-waves and electrons. 

 Nevertheless these illustrations may help to weaken the idea that there 

 is no way out of the present situation but to abandon either waves or 

 corpuscles; for decidedly, there is a way. 



