180 R. W. G. Wyckoff — Determination of the 



expressions for any possible integral valnes of p, q, and r. 

 Thus, 



2px + 2qy + (2r+l)z= + [2pz + 2qx+(2r+\)y + i+s~], 

 2px+2qy + (2r+l)z = -^[2py+2qz+{2r + l)x+^ + s], 



etc. 

 2px + 2qy + (2r+l)z = + [2px—2qy-(2r+l)z + $+s], 



etc. 



It is readily shown that all of the solutions to any of 

 these equations are comprehended by making p = and 

 letting q and r have any values. A similar set of equa- 

 tions can be set up for the second and for each succeeding 

 term of A and values of p, q and r which will make A = 

 can be selected from those solutions which are common to 

 all of these sets of equations. Because of the simplicity 

 of these expressions this detailed procedure can be mater- 

 ially shortened in actual practice. 



If the B term were not invariably equal to zero, a simi- 

 lar procedure would have to be followed and solutions 

 common to it and to the A term chosen. Since 



sin 27r(a) = — sin 2tt(/?) 



both when a = — (3 and when a— (/?±-|), two sets of 

 expressions somewhat different from those of the A 

 terms must be established. 



The carrying out of this procedure for each of the three 

 groups of planes shows that 



(1) When the indices are two even and one odd, A = 

 0if2p = h,2q = k = 0, (2r + 1) = I; 



(2) "When the indices are two odd and one even, A = 

 if 2p = h = 0, (2q + 1) = k, (2r + 1) = I; ' 



(3) When the indices are all odd, A is never equal to 

 zero. This absence in the first order of planes of the 

 class hol, where h is even, and of the class Okl, where both 

 k and I are odd, will then be a universal characteristic of 

 the def raction effects from all crystals having the symme- 

 try of the space group T h 6 . Since a further study of all 

 of the space groups shows that there is no other one for 

 which these classes of planes and no others are absent, a 

 unique method is thus provided for determining from a 

 study of its Laue photographs whether or not a crystal 

 has the symmetry of the space group T h 6 . 



By extending the typical treatment applied here to 

 each of the cubic space groups, a series of criteria can be 



