384 Proceedings of Royal Society of Edinburgh. [sess. 
of order n such that n is divisible by 2 but not by 4), opposite 
directions are equivalent to one another. So far as external shape 
is concerned, this involves, essentially, the occurrence of parallel 
faces on every form. These are the characters of centro- 
symmetrical bodies, however, so that at first sight it appears as if 
the symmetry might he referred indifferently either to the com- 
pound axis or to the centre. There is, unfortunately^ one grave 
objection to the former method which seems to he generally over- 
looked. In all other cases an axis of symmetry is some perfectly 
definite direction in the crystal, and the number of axes is never 
large — not exceeding six of any one order, even in the most 
symmetrical classes. An axis of compound symmetry of the 
second * order, however, is not a definite direction in the crystal, 
and every centro-symmetrical crystal possesses not one such axis, 
hut an infinitude of them, because any direction whatsoever may 
he chosen as the axis without affecting the final result. It is 
therefore much better to avoid this lack of definitiveness in the 
expression ‘ axis of symmetry ’ by giving up the use of the * axis 
of compound symmetry of the second order,’ and restoring the 
‘ centre of symmetry ’ to its former position. 
II. The Glassification of Trigonal and Hexagonal Crystals. 
For teaching and ordinary crystallographical purposes, the 
classification of crystals is largely a matter of practical convenience ; 
questions of structure or arrangement of crystal molecules may be 
entirely overlooked in this connection. Bearing this in mind, it 
is a matter of some importance that the crystal systems which 
resemble one another in possessing one principal axis of symmetry 
(the trigonal, tetragonal, and hexagonal systems) should he so 
arranged as to accentuate their similarities ; by doing so it becomes, 
for students beginning the subject, a much easier matter to 
appreciate and remember the various classes (nineteen out of 
the total of thirty-two) included in these three systems. 
The tetragonal system is defined quite sharply, and the seven 
classes belonging to it present no characters which would lead to 
* This does not apply to compound axes of higher order, because an axis of 
compound symmetry of order n is necessarily an axis of ordinary symmetry of 
order w/2. 
