i i i ii te Me ell 
A.—MATHEMATICS AND PHYSICS. 39 
goniometer. The interior arrangements of the crystal, of which the outer 
form is one consequence, are so varied as to furnish 230 different modes. 
With very few exceptions the X-rays now allow us to carry the classification 
to this higher degree. If the modes are grouped according to the external 
features of the crystals that follow them, we come to the well-known 
thirty-two classes, there being several modes in every class. I may be 
permitted to illustrate this important point by examples, although it is 
familiar to those who have studied crystallography. Let us consider 
first a two-dimensional example, which is much easier to describe than 
the three-dimensional actuality, and contains all the essential ideas. 
Consider an arrangement of figures in a plane which displays symmetry 
across two planes at right angles to one another. Such arrangement may 
be exhibited diagrammatically, as in fig. 2. The unit cell may be drawn 
in various ways, EFKJ, EFLK, RSUT, and so on. The cell contains, 
however it is drawn, either a whole diamond or enough parts to make up a 
whole diamond. Hach diamond can be divided into four parts: B and D 
are the reflections of A and C across a plane ; C and D are the reflection of 
A and B across a plane at right angles to the first plane. Unless the dia- 
mond, the content of one cell, could be divided in this way there could not 
be the double symmetry. But, granted this division into four portions, 
it is not necessary that the four should be arranged as in the figure in order 
that the double symmetry may be obtained. There are two alternatives 
(figs. 3 and 4). 
In fig.3 the lower half of each diamond—that is to say, the portions Cand 
D—are shifted, whether to right or to left is immaterial, by an amount equal 
to one-half of one side of the cell EFKJ. The symmetry about a vertical 
line in the plane of the paper is obviously retained. It is not so obvious 
that there is still any symmetry about the horizontal line until we realise 
that we mean only ‘ observable symmetry ’: that which is to be seen in the 
outer form of the indefinitely extended figure, corresponding to the crystal. 
Clearly, the whole figure will present the same appearance from below as 
from above. In fact, we can see that as a whole the lower part of the 
figure is symmetrical with the upper part by imagining the upper and the 
lower to be further shifted relatively as in fig. 3a: the two parts sliding on 
one another along the line SS. The two parts are then the image of 
each other across SS in the full sense of the word. 
From fig. 2 we may also realise that the amount of the original shift 
must be equal to one-half of EF: no other shift will give the symmetry 
which fig. 3a shows. In figs. 5 and 5a a different shift has been given, and 
the failure is clear. 
In fig. 4 not only are C and D shifted parallel to the horizontal line, 
but also B and D are shifted parallel to the vertical ; this time the amount 
of shift is one-half of the side KJ. 
The three modes of figs. 2, 3 and 4 all lead to the same external sym- 
metry. There is one more which is based, as we should say, on a different 
lattice and is symmetrical, like the others, about two lines at right angles 
to each other. It is shown in fig. 6. There are no variations of fig. 6, as of 
fig. 2, to be obtained by the introduction of shifts. If in fig. 6 we shift 
C and D relatively to A and B, as we did in fig. 3a, we find that they 
can now be described as the direct reflection of A’B’ into CD and of 
A’C into B’D, and the mode of fig. 6a is the same as that of fig. 6. 
