1911-12.] 
The Geometry of Twin Crystals. 
443 
§ 19. Combinations of Independent Twinning Operations with 
Parallel or Perpendicular Twin Axes. 
i. If a twin axis of one twinning operation coincide with, or be at right 
angles to, a twin axis of an independent twinning operation, the result of 
a combination of these two twinning operations will not depend on the 
order in which they are applied ; for a twinning operation does not alter 
the position of a line (though it may reverse its directions) which is 
parallel or at right angles to a twin axis produced by that twinning opera- 
tion, and every combination of twinning operations the twin axes of which 
are parallel or at right angles to each other falls within one of the groups 
of three reversals (§5, iii.) in which the order of application of the opera- 
tions makes no difference. If one of the twinning operations involve point 
twinning, it must necessarily have a twin axis parallel to a twin axis of 
the other twinning operation (§10, ii.). 
ii. Let T 1 and T 2 be two independent twinning operations which satisfy 
these conditions, so that the result of their combination is independent of 
the order in which they are taken. These may be applied to a structure 
in such a manner that one portion is affected by neither, a second by T v 
a third by T 2 , and the fourth by a combination of both, the nature of 
which we now proceed to determine in particular cases. 
iii. If one of the twinning operations consist of point twinning, and the 
other of plane or line twinning, the combination will be a twinning opera- 
tion with the same twin axis and twin plane as the plane or line twinning, 
but with the mode changed from plane to line twinning or vice versa (§ 5, 
iii., group (n)). There cannot be a combination of point twinning with a 
twinning operation resulting in plane and line twinning with the same twin 
axis, for such a twin axis implies symmetry relatively to a point, so that a 
reversal relatively to a point would be a co-directional operation. 
iv. A combination of plane and line twinning having the same twin 
axis will obviously result in point twinning. 
v. If twin axes of the two different twinning operations be at right 
angles, three cases may be distinguished. If both be twin axes with line 
twinning or both twin axes with plane line, the combination will consist of 
line twinning with a twin axis at right angles to both (groups ( b ) and (c)). 
If one be a twin axis with plane twinning and the other a twin axis with 
line twinning, the combination will be a twin axis at right angles to both 
with plane twinning (group (c)). 
vi. In all these cases the combination of two twinning operations is 
itself a twinning operation, and as the three twinning operations form one 
