419 
1911-12.] The Geometry of Twin Crystals. 
If a reversal relatively to a point be a co-directional operation of the 
structure, the two directions of every line in the structure must be 
equivalent to one another, or, in other words, every line is biterminal, and 
the structure is said to possess 'point (or central) symmetry. If the 
structure contain lines with non-equivalent directions, that is to say, 
uniterminal lines, the reversal will be a contra-directional operation of the 
structure. 
ii. Reversal relatively to a Line. — Every point is transferred to a new 
position such that the straight line joining the old and new positions is 
perpendicularly bisected by the same fixed straight line, the line of 
reversal. If such a reversal is a co-directional operation of the structure 
to which it is applied, the line of reversal is said to be a line of symmetry 
of that structure. 
iii. Reversal relatively to a Plane. — Every point is transferred to a new 
position such that the straight line joining the old and new positions is 
perpendicularly bisected by the same plane, the plane of reversal. If this 
is a co-directional operation, the plane is a plane of symmetry of the 
structure. 
iv. Rotation round a Line. — Every point is rotated round a line, the 
-axis of rotation, through the same angle and in the same cyclic direction. 
If the angle of rotation be an nth. part of a complete turn, n is said to be 
the cyclic number of the rotation. 
The cyclic number of the smallest rotation round a line, which is a 
co-directional operation, is said to be the co-directional cyclic number 
of the line. 
On the other hand, the cyclic number of the smallest rotation (if any) 
round a line, which is a contra-directional operation, is the contra-direc- 
tional cyclic number of the line. 
In the same manner, the co-linear cyclic number of a line is that of 
the smallest rotation round it which is a co-linear operation of the 
structure. 
v. The co-linear cyclic number forms the most convenient basis for the 
classification and nomenclature of the crystallographic systems and classes 
{Min. Mag., vol. xv., 1910, p. 398). 
Thus, in the triclinic system the co-linear cyclic number of every 
line is 1 ; in the monoclinic and orthorhombic systems the highest 
co-linear cyclic number of any line is 2 ; in the rhombohedral 
system it is 3 ; in the tetragonal, 4 ; in the hexagonal, 6 ; while 
in the cubic system there are always four lines with co-linear cyclic 
number 3. 
