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of intersection of the two planes, and vice versa. Here the twin axes both 
of the plane twinning and the line twinning lie in the plane of symmetry 
and are at right angles to each other. Accordingly, if a twin axis of plane 
twinning lie in a plane of symmetry, there is a twin axis of line twinning 
in the same plane of symmetry and at right angles to the twin axis of plane 
twinning (figures 3 Bn and 3 BuP, p. 433) ; and if a twin axis of line twin- 
ning be in a plane of symmetry, there is a twin axis of plane twinning 
at right angles to it in the same plane of symmetry (figures 3 Bn and 
SBuL, p. 433).* These are special cases of § 16, ii., and § 17. 
§ 15. Equivalent Twinning Operations connected by an Axis 
of Co-directional Symmetry. 
i. It has been shown that a rotation through an angle 20 is equal, in the 
first place, to a combination of reversals relatively to any two lines at right 
angles to the axis, making an angle 0 with each other, and, secondly, to a 
combination of reversals relatively to two planes at right angles to two such 
lines (§5, vi., vii.). If the rotation be a co-directional operation of the struc- 
ture, each of these pairs of reversals will represent either two twinning 
operations with the same mode of twinning, or two co-directional operations. 
ii. It follows that, if a structure possess an axis of co-directional 
symmetry with cyclic number n, so that a rotation through an nth part 
of a complete turn is a co-directional operation of the structure, and if a 
line at right angles to the axis be a twin axis of plane or line twinning or 
both, a second line at right angles to the axis of co-directional rotatory 
symmetry and making an angle with the first line equal to the 2nth part 
of a whole turn will also be a twin axis with the same mode or modes of 
twinning. There will accordingly be a succession of lines making angles 
of a 2nth part of a complete turn with each other, which will all be twin 
axes with the same mode or modes of twinning. The n-\- 1th line will, 
however, coincide with the 1st, for n 2nt\\ parts of a complete turn are equal 
to a half turn. In the same way the ^ + 2th line will coincide with the 2nd, 
and so on, and the number of distinct lines and twin axes will be only n. 
iii. If, therefore, a twin axis be at right angles to an axis of co-directional 
symmetry with cyclic number n, it will be one of n equivalent twin axes 
with the same mode or modes of twinning, at right angles to the same 
axis and making equal angles with each other. 
* If a line in the plane of symmetry be an axis of both plane and line twinning, there 
must be a centre of symmetry, and the normal to the plane of symmetry will be a line of 
symmetry (§ 5, iii. (a) and § 11, ii.), so that the case falls within § 14, ii. 
