449 
19X1-12.] The Geometry of Twin Crystals. 
viii. If l x be a line of symmetry in a co-linear common plane K , a 
rotation of one component structure S' round the normal n of K through 
any angle 20 will convert K into a cross plane, provided that the rotation 
is not a co-linear operation of K (figure 1 TJhL'). Let C 20 he a rotation 
through an angle 20 round the normal n, R 1 the operation of reversal 
relatively to l v and R 2 that of reversal relatively to l 2 , making an angle 
0 with l v then [$](7 20 = [$]jR 1 ..R 2 ( = ) [S]R 2 , f° r a co-directional 
operation of the structure. Then, since the operation R 2 is equivalent to 
the rotation C 2e , it cannot, by hypothesis, be a co-linear operation. It 
must therefore result in a cross plane, and C 20 must do so likewise. 
(In these figures hollow discs and interrupted lines correspond to the 
original disposition of the structure.) 
Again, if p 1 be a plane of symmetry at right angles to a co-linear 
common plane K , a rotation through an angle 20 round the normal n of K, 
if it be not a co-linear operation of K , will in the same manner convert it 
into a cross plane ; for if R 1 be a reversal relatively to p v and R 2 be one 
relatively to a plane p 2 also at right angles to K and making an angle 0 
with p v we have as before [S]C 2 e — [>S]I^ 1 I^ 2 = [>S^]i^ 2 . 
§ 22. Relations between Structures possessing a Cross 
Plane in Common. 
i. We have seen that every cross plane can be obtained by the reversal 
of one of two structures which possess a co-linear common plane, de, 
relatively to a line d in that plane. This co-linear plane must be either 
(1) a plane common to two co-directional structures or, what comes to the 
same thing, two portions of the same structure, or (2) a twin plane. 
vol xxxn. 29 
