The object of the present paper is to express the conclusions of mathematical crystallography in a form which shall be immediately useful to workers using homogeneous X-rays for the analysis of crystal structures. The results are directly applicable to such methods as the Bragg ionisation method, the powder method, the rotating crystal method, etc., and summarise in as compact a form as possible what inferences may be made from the experimental observations, whichever one of the 230 possible space-groups may happen to be under examination. It is only in certain cases that the spacings of crystal planes as determined by the aid of homogeneous X-rays agree with the values of those spacings which would be expected from ordinary crystallographic calculations. In the majority of cases the relative arrangement of the molecules in the unit cell leads to apparent anomalies in the experimental results, the observed spacings of certain planes or sets of planes being sub-multiples of the calculated spacings. The simplest case (fig. 8) of such an apparent anomaly is found in the space-group C
2
2
of the monoclinic system, where the presence of a two-fold screw-axis, because it interleaves halfway the (010) planes by molecules which are exactly like those lying in the (010) planes, except that they have been rotated through 180°, leads to an observed periodicity which is half the periodicity to be inferred from the dimensions of the unit cell, that is, leads to an observed spacing for (010) which is half the calculated. All screw-axes produce similar results, and, in general, a
p
-fold screw-axis leads to an observed spacing for the plane perpendicular to it which is 1/
p
th that to be inferred from the dimensions of the cell. Besides those produced by the screw-axes, other abnormalities arise out of the presence of glide-planes. The simplest case of this is shown by the space-group C
s
2
(fig. 4) of the monoclinic system, in which the second molecule is obtained from the first by a reflection in a plane parallel to (010) and half a primitive translation parallel to that plane. If we look along a direction perpendicular to this glide-plane, the projections of the two molecules on the (010) plane are indistinguishable except in position, which is equivalent to saying that, for the purposes of X-ray interference, certain planes
perpendicular
to this plane of projection are interleaved by an identical molecular distribution. Furthermore, since the translation associated with the glide-plane must always be
half
a primitive translation parallel to the glide-plane, we know that the interleaving is always a submultiple of the full spacing and the periodicity is again reduced in a corresponding manner. The use of this method for discriminating between the various space-groups of the monoclinic system was described by Sir Wm. Bragg in a lecture to the Chemical Society. In the present paper the method has been extended to the whole of the 230 space-groups possible to crystalline structures. In general, it may be said that if a crystal possesses a certain glide-plane, a certain set of planes lying in the zone whose axis is perpendicular to that glide-plane will have their periodicity reduced by one-half.
One molecule, three crystal structures: conformational trimorphism of N-[(1S)-1-phenylethyl]benzamide
