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e A—MATHEMATICS AND PHYSICS. 45 
When the determination of structure cannot be carried far enough, 
the X-rays may fail to decide between the presence and absence of polarity. 
For example, resorcinol is an orthorhombic hemihedral crystal: this is 
known by its external form. The X-rays show that, this being so, its 
internal arrangement must be that of M*M” or C;} in fig. 7. If we 
had no help from the study of external form, or from any other source, 
we should not be able to decide between C!° and the more symmetrical 
mode known as Q!: the symmetry of the latter is obtained by adding a 
centre of symmetry to the elements of symmetry possessed by (,): 
that is to say, by removing the polarity of the crystal. As a matter of fact, 
the external form of resorcinol clearly shows polarity: or, if we could be 
sure that the molecule had no symmetry, we could infer that the crystal 
was unsymmetrical about the zy plane, there being only four molecules 
in the cell and all these being wanted to give the symmetry observed by 
X-rays. Thus there are cases where the X-rays cannot decide between 
two modes, one of which can be derived from the other by the addition of 
acentre of symmetry. As, however, the existence of a centre of symmetry 
can generally be decided by other means—for example, by such means as 
I have described above in the case of zinc blende or of resorcinol—this 
incapacity of the X-ray method is of no great consequence. 
The addition of a centre of symmetry moves a structure from one class 
to another—Class 1 to Class 2, Class 31 to Class 32. Consequently, the 
X-ray methods are by themselves sometimes in doubt between two modes 
in different classes when they are rarely in doubt-as to the mode within a 
class. It will readily be understood that the doubt as to class may be of 
far less importance than the doubt as to mode ; though hitherto the former 
kind of difference has been given all the attention because it has been the 
only kind that could be observed. A very slight relative movement of 
the atoms would be sufficient to reduce the symmetry of the crystal from 
one class to another: but the change from one mode to another within 
the same class would mean a complete rearrangement of the molecules. 
There are two cases in which the X-rays cannot distinguish between 
two modes in the same class. These are Q* and Q’ in the enantiomorphous 
class of the orthorhombic system, and T* and T° in the tetartohedral class 
of the cubic system. The ambiguity disappears, however, if there are only 
two molecules in the unit cell, when the former alternative is alone per- 
missible in each case: it would disappear also in any case in which the 
structure could be determined completely by any other means. 
It has been known for many years, thanks to the work of Fedorow, 
Schonflies and Barlow, that the 230 modes of arrangement represent all 
the possible forms of internal crystal structure. In each mode of arrange- 
ment there is a relative disposition of planes, axes and centre of symmetry, 
_ which is characteristic of the mode, and the mode may be described in terms 
of these symmetries. This was the language used in the original work on 
the subject, and the term ‘space group’ was used, instead of the term ‘ mode 
of arrangement,’ in reference to the particular group of symmetry planes, 
axes and centre in space. When the subject is approached from the point 
of view of the X-ray worker, the language of the mode of arrangement has 
its special conveniences. A list of the 230 modes, and of the X-ray tests 
for each mode, has recently been published in the Transactions of the 
Royal Society by Astbury and Yardley. Lists of the same 230 space 
