A.—_MATHEMATICS AND PHYSICS. 37 
been defined. This we are able to do because the carbon atom is tetra- 
hedral ; a tetrahedron has a centre, and we can state the distance between 
the centres of two tetrahedra, no matter how the tetrahedra are oriented. 
We know that the carbon atom, as built into the crystal, is tetrahedral, 
because the X-ray observations show that four trigonal axes meet in it. 
The two atoms in the cell are oriented differently ; one may be said to be 
the image of the other, if translation shifts are ignored, in each of the faces 
of the cube. Considerations of symmetry or X-ray observations show 
that the centre of an atom of the one orientation lies at the centre of a 
tetrahedron formed by four atoms of the other orientation. The edge of 
this tetrahedron is the edge of the unit cell, and its length is 2.52 A.U. It 
may then be calculated that the distance between the one atom and the 
others, its nearest neighbours, is 1.54A.U. Wemay call this distance the 
diameter of the carbon atom, but we must remember our original definition 
of the meaning of the term. Thus the 2.52 A.U. is the result of a direct 
unaided X-ray measurement, but the 1.54 A.U.is not, and has no meaning 
except after special definition. 
Only such distances between atoms as can be calculated from the 
dimensions of the unit cell can be measured directly and without 
qualification. The determination of these distances may be looked on as 
the result of the first stage of the analysis by X-rays. 
We now come toa second stage. tis possible to make other statements 
of the relative positions of atoms and molecules which, though less complete 
and informative than those of distances, and their orientations, are necessary 
to the solution of the crystal structure problem. These also are deduced 
by means of the X-ray methods. 
It often occurs that the atoms or molecules in one cell can be divided 
into two portions which are the reflections of one another across some 
plane, or can be brought to be the reflection of each other by a shift 
parallel to the plane. In that case the orientation of the plane and the 
amount of the shift can be stated definitely, the former by inspection of 
the crystal or by X-ray observations, the latter by X-ray observations 
alone. So also it may happen that the atoms or molecules in the same 
cell may be divided into portions which can be made to coincide with each 
other by a rotation round some axis with or without a shift parallel to that 
axis. The direction of the axis can be found by inspection of the crystal 
or by X-ray observations; the amount of the shift can be found by X-ray 
observations alone. 
In these cases the distances that are found by the X-ray method are 
all that can be stated without special definition. It is not possible to state 
the distance between an object and its image in a mirror, if the object has 
any extension in space ; but it is possible to state the magnitude of a shift. 
Measurements of this sort constitute a characteristic feature of the 
X-ray analysis, for which reason I would like to discuss them briefly. 
We know that it is possible to separate crystals into thirty-two classes, 
~ according to the kind of external symmetry which they display. As we 
have hitherto been unable to look into the interior of the crystal, we have 
_ been obliged to be content with this imperfect classification by outer 
appearance. It has been shown, however, that there is a classification by 
_ inner arrangement which is perfect and includes the other. It is beyond 
the limits of ordinary vision: out of the range of the lens and the 
