A.—MATHEMATICS AND PHYSICS. 41 
There are therefore four modes in one class: four varieties of internal 
arrangement which all lead to the same external appearance of symmetry. 
Our example is two-dimensional, and the crystal has three dimensions. 
But there are no new ideas to be added: it is only the numbers of sym- 
metries, modes, and classes that are increased. If, for example, we continue 
the study of the modes of arrangement that lead to an external symmetry 
of reflection across two planes at right angles to each other, we find that 
there are four lattices instead of two, and twenty-two modes instead of 
four. The class containing crystals that possesses this particular form of 
symmetry is generally called the ‘hemimorphic class in the orthorhombic 
system.’ Its symbol is C,,: the symbols of the four lattices are 
r, Tj’ Tr,” 1,’’. In every case the content of the unit cell is divisible 
into four parts, corresponding to the ABCD of figs. 2 to 6. The ten 
modes in the Tf, lattice are shown in fig. 7, which will serve to show the 
numerical increase due to the introduction of the third dimension. Under 
each separate figure is given, beside the crystallographic symbol, another 
symbol which describes the shifts: D* means a direct reflection across a 
plane parallel to yz; Ei a reflection across a plane parallel to yz, 
together with a shift parallel to the axis of y equal to half the y edge of 
the cell, and M* a reflection across a plane parallel to yz, together with a 
shift parallel to the diagonal of the yz face and equal to half that 
diagonal. 
Let us now see how the X-ray analysis distinguishes the mode. Let 
us imagine that fig. 2 represented a number of pits in a plane reflecting 
surface. The surface could be used as a grating having many spacings 
instead of one. If, for example, we so placed it that the horizontal lines 
of the figure were parallel to the slit of the spectroscope the spacing would 
be equal to EJ : if the vertical, the spacing would be equalto EF. Again, 
_ if the grating were so placed that EK, for example, were vertical, the spacing 
_ would be the perpendicular distance between EK and FL. If the surface 
is pitted as in fig. 3, the spacing when the horizontal line is parallel to 
the slit is the same as before ; but when the vertical is parallel to the slit 
the effective spacing is only half what it was in fig. 2. This follows from 
the fact that if we divided the surface into a number of vertical narrow 
strips the diffracting effect of each such strip, for this position, depends on 
_ the total amount of reflecting surface contained in the strip, but not on its 
hee 
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distribution along the slip. It does not matter that C and D are upside- 
down as compared to Aand B. The strata consisting of C and D portions 
have interleaved the strata of A and B portions. This halving of a spacing 
of fig. 3 as compared with fig. 2 occurs only when the grating is placed 
so that the slit is parallel to the vertical line of fig. 3, and not when any 
other line is vertical, except by some odd chance connected with the shape of 
the pits. In this way it is possible to distinguish between fig. 2 and fig. 3. 
The mode shown in fig. 4 is distinguished by the halvings of both the hori- 
zontal and vertical spacings, and of no others. In the case of fig. 6, as 
compared with fig. 1, the spacing is halved when the slit is parallel to the 
horizontal or the vertical line of the figure, and also whenever the grating 
is so placed that the parallel to the slit passing through one of the corners 
of a cell does not pass through the centre of that or any other cell, as, for 
example, if EO but not EK is parallel to the slit. It is therefore easy to 
distinguish each of the four modes. 
