A.—MATHEMATICS AND PHYSICS. 51 
X-ray evidence we learn that the molecule has four intersecting trigonal 
axes. We must place the unique oxygen at the centre of a regular tetra- 
hedron, and the four beryllium atoms at its corners. Each of the six 
acetate groups must be associated with one of the tetrahedron edges, and 
in such a way that the four trigonal axes are maintained. This necessitates, 
_ as crystallographic theory shows, the existence of a dyad axis through the 
_ middle points of each pair of opposite edges of the tetrahedron. The 
C,H,O, groups must be added so as not to interfere with the existence 
of these axes. If they are placed correctly for the trigonal axes, each of 
them has a dyad axis of the kind mentioned. All this agrees with the 
_ chemical evidence as partly stated in the formula, which implies :— 
1. That there is one oxygen differently situated to the rest. 
2. That the four beryllium atoms are all alike. 
8. That the acetate groups are all alike. 
Further, chemists would say that the carbon atoms are not alike; in that 
case, they must both lie on the dyad axis, since if they did not they would 
necessarily be symmetrically placed with respect to that axis and would 
be equivalent. On the other hand, the oxygen atoms in the acetate group 
cannot lie on the axisif, as is probable, they are equivalent to one another. 
They must be placed symmetrically with respect to the dyad axis. As to 
the hydrogens, we must assume either that they do not count, which is 
not at all unlikely, or that they are not all alike. It is impossible to place 
eighteen hydrogen atoms so that the group has four intersecting trigonal 
_ axes and that every hydrogen is like every other. The molecule has no 
_ plane of symmetry, the fault lying with the oxygens. It could not be due 
_ to the hydrogens because there are marked differences in the intensities 
of reflection of pairs of planes, which differences would not exist if there 
_ were planes of symmetry, and would be small if due to dissymmetry in the 
_ positions of hydrogens only. It is by reasoning along such lines as these 
_ that X-ray evidence and chemical evidence can help each other. Many 
3 other instances might be given ; indeed, no complex crystal can be studied 
with success without calling in the assistance of chemical arguments. 
A fourth example of the connection between arrangement and properties 
is to be found in the recent work by W. L. Bragg on the indices of refraction 
_ofcrystals. It has been found possible to calculate the indices of refraction 
of calcite, given the dielectric capacities of calcium, carbon and oxygen 
atoms separately. The difference between the two principal refraction 
indices is almost entirely due to a difference between the dielectric capacities 
of a set of three oxygen atoms, at equal distances from one another, when 
placed :— 
1. So that the plane in which they lie contains the direction of the 
field. 
2. So that this plane is perpendicular to the field. 
If we are able to calculate the refractive indices on these data, then it must 
be possible to find conditions governing the arrangement of the atoms, 
when we know the composition of the crystal and its refractive indices. 
For instance, the near equality of the refractive indices of potassium 
sulphate implies that the dielectric capacity of the SO, group is much the 
same in all directions, and this is in agreement with the hypothesis that the 
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