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



insufficient, a knowledge of the numbers of chemical 

 molecules to be associated with the unit cube, such as 

 arises immediately from the density and the dimension of 

 the unit, can be of service. For instance suppose that 

 the diffraction data from a certain crystal assigned it to 

 the two indistinguishable space groups T 3 and T 5 , and 

 suppose that the determination of the number of chemical 

 molecules within the unit cube requires but two chemi- 

 cally like atoms within the unit cell, then since only T 3 

 contains as a special case two equivalent positions, the 

 crystal may be assigned to it, rather than to T 5 . In view 

 of the present lack of definite knowledge as to what it is 

 that conditions chemical equivalence in the crystalline 

 state, such information must obviously be used with great 

 caution. 



There naturally arises a question of whether even with 

 atoms in the most general equivalent positions coordinate 

 values may not exist such that the diffraction results may 

 simulate those corresponding to some space group other 

 than the one to which it really belongs. Any such coordi- 

 nates for a space group can readily be found by practi- 

 cally the same procedure which has already been 

 employed in determining the reflection characteristics of 

 planes in different orders. In this process, however, the 

 sets of equations are to be solved for x, y and z rather 

 than for h, k and I. 



The space group T h 6 will again serve as an illustration. 

 The previously established set of equations, [2], must 

 now be solved for x, y and z which can have any values 

 between zero and unity, including the former, instead of 

 for integral values of p, q and r. For the present purpose 

 care must of course be taken to avoid such values of x, y 

 and z as yield special cases with fewer than the maximum 

 number of equivalent positions within the unit cell. By 

 solving these sets of expressions in a manner analogous 

 to that previously used it can be shown for instance that 

 when x = u, y = and z = 0, or when x = -J, y = \ and 

 £ — 0, only all odd planes are to be found in the first 

 order region. "When attempting to ascertain the space 

 group to which a crystal should be assigned, it is impor- 

 tant to take into consideration the possibility of atoms 

 occupying exactly or nearly such positions as these. It 

 must likewise be borne in mind that atoms in special posi- 



