412 BELL SYSTEM TECHNICAL JOURNAL 



can be reached from the one first chosen by a sequence of three dis- 

 placements, one along each of the three directions and each an integer 

 multiple of the minimum distance, or "spacing," or "grating-constant," 

 or "edge of the unit cube." In other words, any atom-group can be 

 brought into coincidence with any other by a sequence of three such 

 shifts. This way of putting it brings out the point that all the groups 

 in the lattice are oriented alike. 



Denote by a the magnitude of the spacing. Install a system of 

 rectangular coordinates with its origin at some one atom-group, say A 

 of our previous picture, and its axes along the three directions stated. 

 Among the six neighbors of A we pick out three to serve as B, C, D 

 of our previous picture; say the three groups shifted from A through 

 the interval a in the positive senses of the three axes, so that the 

 coordinates of the four shall be: 



^(0,0,0); 5(a, 0, 0); C(0, a, 0); Z) = (0, 0, a). 



The directions of the dififraction-spots are to be deduced from the 

 general equations (3), (4) and (5), with all the simplifications from 

 which we benefit thanks to the lattice being cubic. The three spacings 

 are now all equal; but the greatest advantage is, that the various 

 cosines which figure in the equations are now direction-cosines, and 

 we can avail ourselves of the theorems to which direction-cosines 

 conform. There are two of these which we shall use: the theorems 

 that the sum of the squares of the direction-cosines of any line is 

 unity, and that the cosine of the angle between any two lines is the 

 sum of the three products formed by multiplying together corre- 

 sponding direction-cosines of the two. Denote by ai, a<>, a^ the 

 direction-cosines of the primary, by /3i, (82, fiz those of the diffracted 

 beam. The first three are the quantities cos 0, cos </>', cos 4>" of 

 equations (3), (4) and (5); the second three are cos d, cos 6', cos Q" . 

 To bring the notation fully into harmony with usage I further write 

 hi, ^2, hz for the integers n, n' , n" . 



Then by translating equations (3), (4) and (5) into the new notation 

 and adding two more supplied by the first of the foregoing theorems, 

 we form a family of five equations: 



a(((3i — ai) = }ii\, (6a) 



a{^2 - a.) = /ioX, (6b) 



a(/33 — as) = hz\, (6c) 



ai2 + a.^ -\- as- = 1, (6^) 



i3r + /32- + ^3- = 1. (6^) 



