441 
1911-12.] The Geometry of Twin Crystals. 
iv. If n be odd, all these n lines are equivalent, and so are the plane at 
right angles to them, and as the mode of twinning is the same in each case 
they may be described as equivalent twin axes and twin planes (figures 
3 Uh, 3 UhP, and SUhL, p. 434). 
v. If n be even, there cannot he more than nj 2 distinct equivalent lines 
at right angles to the axis, unless the axis of co-directional symmetry also 
possess co-linear and contra-directional symmetry with cyclic number 2 n, 
in which case there will be n equivalent lines constituting n equivalent 
twin axes (figures 4 Blc, 4tBkPI, p. 437, where n — 2; see also § 17). 
vi. If, on the other hand, n be even and the axis does not possess co- 
linear and contra-directional symmetry with cyclic number 2 n, the n lines, 
which are twin axes with the same mode of twinning, must consist of two 
series such that those of one series bisect the angles between those of the other, 
and that the n/2 twin axes of each series are equivalent to one another, but 
not to those of the other series (see figures 6Uu, QUnP', and OUuL', p. 444). 
vii. The same reasoning may be employed in respect of co-directional 
reversals relatively to lines at right angles to the axis, or planes to which 
they are normal (in other words, planes passing through the axis) ; with 
the result that, if there be one such line or plane of symmetry, there will 
be a number of them equal to the co-directional cyclic number of the axis. 
§ 16. Axes of Co-directional Symmetry which are Twin Axes. 
i. Combining the results of § 14, ii., and § 15, we find that if an axis of 
co-directional symmetry with cyclic number n, and with n lines of symmetry 
at right angles to it, be a twin axis with line or plane twinning, or both, 
there will be n twin axes with the same mode or modes of twinning each 
at right angles to the axis and line of symmetry (see figures 3 Uh , 3 UhP , 
and SUhL, p. 434, and 3 Be and 3 BcPL, p. 436). 
ii. Again, combining the results of § 14, iii., and § 15, if an axis of co- 
directional symmetry with cyclic number n, and with n planes of symmetry 
passing through it, be a twin axis of line or plane twinning, there will be 
n twin axes with plane or line twinning, as the case may be, at right 
angles to the axis, and each will lie in a plane of symmetry (see figures 
3 Bn, 3 BuP, and 3 BuL, p. 433). 
§ 17. Equivalent Twin Axes connected by an Axis of Contra- 
directional Symmetry. 
i. A combination of a rotation through an angle 20 and a reversal 
relatively to a point was shown in § 5, viii., to be equal to a combination 
