451 
1911-12.] The Geometry of Twin Crystals. 
ii. Now if R x and R 2 make an angle 0 with each other, R V R 2 will be 
equal to a rotation through 20 round the normal to K, and R 1 .R 2 .R i to such 
a rotation combined with a reversal relatively to a point. But a reversal 
relatively to a point as applied to a plane is equivalent to a reversal 
relatively to its normal (for their combination is equal to a reversal 
relatively to the plane, which is a co-directional operation of the plane ; see 
§ 5, ii. and iii., group (a), and § 6, i.), and this is equal to a rotation round 
the normal through a half turn, 7 r. The operation of rotation round the 
normal through either 20 or 20±ir must accordingly be a co-directional 
and therefore, also, a co-linear operation of the plane. But a rotation 
round the normal through a half turn, 71-, is always a co-linear operation of 
a plane. It follows that, if two common lines in a cross plane meet at any 
angle 0, rotations round the normal through both the angle 20 and the 
angle 20 ± 7 r will be co-linear operations of the plane. 
Let such an angle of rotation be denoted by (p . Then, inversely, 
if (p be any co-linear operation of the component planes K' and K", and 
if d x be a common line, there will be other common lines making angles 
ir- and with d x . 
Z z z 
The following table may now be constructed : — 
Co-linear cyclic number of 
normal, in structure .1,2 4 
Co-linear cyclic number of 
normal, in plane (§ 6, iv.) 2 4 
3, 6 
6 
</> = any co-linear rotation 
round normal in plane . 7 r 
0 = angle between any com- 
mon line and a given 
common line = i- + — . 0, -— 
2 + 2 ’2 
7T 7T 7T 27 T 57T 
6~’ t* y ’ y 5 it 
Total number of common 
lines .... 2 
4 
6 
(See figures QUu, 6UuP', QUuL', p.444 ; 1 UuP' d , 1 UuL' cl , 1 UuP' e , 1 UuL ' e , 
p. 448; 1 UhL' and lTJcP'L', p. 449.) 
§ 24. Planes at Right Angles to Twin Planes and Cross Planes. 
i. As the normal to a twin plane is a common line, a plane at right 
angles to a twin plane contains two common lines, for every line in a twin 
