425 
1911-12.] The Geometry of Twin Crystals. 
2m, where m is odd, there will be a plane of symmetry at right angles to 
the axis. 
For, let C a be the smallest rotation round the axis which is contra- 
directional ; then G ma ( = (7 m a ) is also contra-directional, since m is odd. 
Let R t be a reversal relatively to the axis, R p a reversal relatively to a 
plane at right angles to it, and R t a reversal relatively to a point. Then 
[S]R P = [S]R t . R i = [S]C r . R< = [£]G ma . R, ( = ) [>Sf]l (§ 4, vii.). 
vi. A combination of two reversals R and R! relatively to lines l and V 
making an angle 6 with each other is equal to a rotation round an axis at 
right angles to both through an angle 2d in the cyclic direction from l to 
V , or V to l, according as R or R' is applied first. Thus R . R' = C 2e , and 
R . R — C _2e. 
For, let any point on the structure have polar distance cf> from one direction of the 
axis considered as pole, with the point of intersection of the two lines and axis as 
centre ; assume its azimuth about the axis to be if/, when the azimuth of l is 0, and 
that of V, 0. Then after the reversal relatively to l the polar distance and azimuth 
of the point will be tt - cf> and — if/ respectively, and after the reversal relatively to 
l , <f> and 0 0 ifr = 20 -\- if/. 
vii. In like manner a combination of two reversals R and R' relatively 
to planes making an angle 6 with each other is equal to a rotation round 
the line of intersection through an angle 26 ; or, what comes to the same 
thing, a combination of reversals relatively to two planes whose normals 
n, n' make an angle 6 with each other is equal to a rotation round an axis 
at right angles to both normals through an angle 26 from n to n' or from 
n' to n, according as R or R' is first applied. 
viii. The only combination of two reversals still to be considered is 
that in which a reversal relatively to a line is combined with reversal 
relatively to a plane whose normal makes an angle 6 with the line. If 
the reversal relatively to a plane be resolved into a reversal relatively to 
its normal and a reversal relatively to a point, it follows that a combina- 
tion of a reversal relatively to a line with a reversal relatively to a plane 
whose normal makes an angle 6 with the line is equal to a rotation round 
an axis at right angles to the line and normal through an angle 2d in a 
cyclic direction corresponding with the order of application of the reversals, 
combined with a reversal relatively to a point. 
§ 6. The Symmetry of a Plane. 
i. In the present communication the expression, the symmetry of a 
plane, is restricted to its symmetry in two dimensions. This is determined 
by the distribution of equivalent directions in the plane, and is independent 
