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Proceedings of the Koyal Society of Edinburgh. [Sess. 
of a reversal relatively to a line at right angles to the axis of rotation and 
a reversal relatively to a plane whose normal is at right angles to the axis 
and makes an angle Q with the line of reversal. 
ii. If a structure has an axis of contra-directional symmetry with 
cyclic number n, a rotation through an ^th part of a complete turn is a 
contra-directional operation of the structure, and a combination of the 
same rotation with a reversal relatively to a point will be a co-directional 
operation of the structure (§4, vii.); consequently, if a line at right angles 
to an axis of contra-directional symmetry with cyclic number n be a twin 
axis of plane twinning or line twinning, another line at right angles to the 
same axis and making an angle equal to a 2^th part of a whole turn with 
the first will be a twin axis of line twinning or plane twinning. 
iii. There will therefore, as in § 15, be n distinct twin axes, and, as n is 
always even (§ 4, xi.) in the case of an axis with contra-directional 
symmetry, there will be w/2 twin axes of line twinning and ti/2 twin axes 
of plane twinning alternating with each other (figures 4 Bh and 4 BkPI, 
p. 437).* 
§ 18. Combinations of Twinning Operations. 
i. As all twinning operations are reversals, or equivalent to reversals, 
a repetition of the same twinning operation will result in identity (§5, i.), 
so that a portion of a structure to which it has been applied twice or 
any even number of times will have the same disposition in space as a 
portion to which it has not been applied at all ; and all portions to which 
it has been applied an odd number of times will have the same disposition 
in space as those to which it has been applied once only. These relations 
are well illustrated in the lamellar twinning of plagioclase. 
ii. A combination of any number of equivalent twinning operations 
will obviously have the same result as the application of any one of them 
the same number of times. 
iii. If, however, two twinning operations, which are independent, that is 
to say, which are not equivalent to one another, are applied to a portion 
of a structure, its disposition in space will differ from that of a portion to 
which neither has been applied and may depend on the order of applica- 
tion. We now proceed to consider the results of such combinations of 
twinning operations. 
* The same lines cannot in this case be axes both of plane and of line twinning, for if 
they were, the structure would possess point symmetry (§ 12, i.), which is inconsistent with 
contra-directional symmetry. 
