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Proceedings of the Royal Society of Edinburgh. [Sess. 
former case all co-linear rotations must he co-directional, and there are no 
contra-directional rotations. In the latter, if a rotation through an angle 
a be the smallest co-linear and contra-directional rotation, a rotation 
through an angle 2 a will be the smallest co-directional rotation. The 
co-linear and contra-directional cyclic numbers will in this case be equal 
to one another and double the co-directional cyclic number. The contra- 
directional cyclic number will therefore always be an even integer. 
The co-directional and co-linear cyclic numbers may be 1, 2, 3, 4, or 6 ; 
the contra-directional cyclic numbers 2, 4, or 6. Any others are inconsistent 
with the law of rational indices and the molecular constitution of matter. 
A line with a co-directional, contra-directional, or co-linear cyclic number 
of 2 or more is said to be an axis of co-directional, contra-directional, or co- 
linear symmetry, as the case may be, with the corresponding cyclic number. 
xii. If the co-directional cyclic number of a line be even, say 2m, the 
line will be a line of symmetry of the structure ; for since a rotation 
through a 2mth part of a turn is co-directional, one through a half turn 
which is m times this is also co-directional, and it is equal to a reversal 
relatively to the axis. 
xiii. The relation of a line to the symmetry of a crystal structure may 
be expressed by a symbol consisting of (1) the co-linear cyclic number of 
the line ; (2) a capital letter U (unilateral) or B (bilateral), indicating the 
absence or presence of one or more planes of symmetry passing through 
to the line ; (3) one of the following letters : — k, indicating that the line 
possesses contra-directional symmetry ; u, that it is uniterminal ; c, the 
presence of central or point symmetry ; and h (helical), the presence of 
one or more lines of symmetry at right angles to the original line, and at 
the same time the absence of point symmetry, so that the terminations of 
the line are related in the same manner as the ends of a screw. It is 
necessarily unilateral. 
Thus the symmetry of the principal axis of chalcopyrite is expressed by the 
symbol 4 Bk; that of tourmaline by 3 Bu ; of beryl by 6 Be; of dioptase by 3Uc; of 
quartz (that is to say, ordinary or a quartz) by 3 Uh ; of benitoite by 6Bk. 
This symbol may be referred to as the type symbol of the line. 
xiv. The crystallographic class of a crystal structure may be expressed 
by the symbol of the line with highest co-linear cyclic number, written for 
this purpose as a Roman instead of an Arabic numeral. In the case of the 
classes of the cubic system, however, the symbol expressing the symmetry 
of the four lines with co-linear cyclic number 3 is chosen, but with the 
capital letter C (cubic) substituted for the numeral {Min. Mag., vol. xv., 
1910, p. 398). 
