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Proceedings of the Royal Society of Edinburgh. [Sess. 
ii. In the former case the operation, R d , of reversal relatively to d, 
will be an operation of line twinning * with the line d as twin axis and the 
plane ne at right angles to the cross plane de, and cutting it in the line e 
at right angles to d, as twin plane (see figure 1 UuL' d , p. 448, where L' d 
denotes line twinning with d as twin axis). 
iii. If, on the other hand, de, the co-linear common plane, be a twin 
plane, it may be obtained by applying a twinning operation, T, to one of 
two portions of the same structure. The cross plane will therefore result 
from the application of the operation U=T. R d . 
The twinning operation T must be (1) a reversal relatively to the plane 
de (figure 1 UuP, p. 432), (2) a reversal relatively to the normal n (figure 
1 UuL), or (3) a reversal relatively to a point (figure 1 Uul). The operation 
l J=T.R d will be, in the first case, a reversal relatively to the plane nd at 
right angles to de and passing through the lines n and d (§ 5, iii., group (c)). 
It will accordingly be an operation of plane twinning with e as twin axis 
and nd as twin plane (see figure 1 U uP' e , where P' e denotes plane twinning 
with e as twin axis). In the second case it will be a reversal relatively to 
the line e (group (6)), and therefore an operation of line twinning with the 
same axis and plane of twinning (see figure 1 UuL' e ) as in the first case. In 
the third it will be a reversal relatively to the plane ne, which is at right 
angles to d and to the plane de, and passes through the lines n and e 
(group (a)). It is an operation of plane twinning with d as twin axis, 
and ne as twin plane (see figure 1 UuP' d ). 
In every cross plane, therefore, one common line at least is a twin axis. 
§ 23. Number of Common Lines in Cross Planes. 
i. We shall next determine how many common lines there may be in 
a cross plane. Let d x and d 2 be two common lines in a cross plane K ; then 
a reversal R x of one structure relatively to d v and a reversal R 2 of the 
same structure relatively to d 2 , will each result in a co-linear common 
plane which may be either co-directional or contra-directional. If these 
resulting common planes are both co-directional or both contra -directional, 
R 1 must, as applied to K, be equivalent to R 2 , or, in symbols, [K]R ± ( = ) 
[K]R 2 . If one be co-directional and the other contra-directional, R ± must 
be equivalent to R 2 combined with a reversal relatively to a point, that is, 
[K]R 1 ( = ) [K]R 2 .Ri, where Ri is a reversal relatively to a point. Therefore, 
either R 1 .R 2 or R 1 .R 2 .R i must be a co-directional operation of K (§ 5, ii.). 
* The existence of the cross plane precludes, of course, its being a co-directional 
operation. 
