439 
1911-12.] The Geometry of Twin Crystals. 
If, therefore, a line of symmetry he a twin axis of plane twinning; an 
indistinguishable twin crystal may be obtained by point twinning ; and 
vice versa, if a twin crystal may be explained by point twinning, every 
line of symmetry is a twin axis of plane twinning (figures 2Bu, 2BuPI, 
p. 437, and figures 3 TJh and 3 Uhl, p. 434). 
iii. The operation of reversal relatively to a plane is equivalent to a 
reversal relatively to its normal combined with reversal relatively to a 
point. Accordingly, if a 'plane of symmetry be a twin plane of line 
twinning, the twin crystal may also be explained by point twinning. Again, 
if point twinning exist, every plane of symmetry is a twin plane of line 
twinning (figures 4 Bk and 4 BkPI, p. 437 : also figures 3 Bu and SBuI, 
p. 433). 
§ 13. Equivalent Twinning Operations connected by Group (6) 
A reversal relatively to a line is equal to a combination of reversals 
relatively to two other lines at right angles to the first and to each other. 
Accordingly, if there be a twin axis of line twinning at right angles to a 
line of symmetry , there will be another twin axis of line twinning at right 
angles to the first and to the line of symmetry (figures 3 TJh and 3 UhL, 
p. 434). 
§ 14. Equivalent Twinning Operations connected by Group (c). 
i. A reversal relatively to a line is equal to a combination of reversals 
relatively to two planes meeting at right angles in that line. It follows 
that if a line of symmetry lie in a twin plane of plane twinning, there 
is another twin plane of plane twinning at right angles to the first and 
intersecting it in the line of symmetry ; or, what comes to the same thing, 
if there is one twin axis of plane twinning at right angles to a line of 
symmetry, there must he another at right angles both to the first and to 
the line of symmetry (figures SUh and 3 UhP, p. 434). 
ii. Combining this result with that of § 13, we see that, if there be a 
twin axis either of line or plane twinning, or both, at right angles to a line 
of symmetry, there will be another twin axis with the same mode or modes of 
twinning at right angles to the first twin axis and to the line of symmetry 
(figures lBc and 1 BcPL, p. 435). This is a special case of § 15 and § 16, i. 
iii. A reversal relatively to a plane is equal to a combination of a reversal 
relatively to any other plane at right angles to the first and a reversal 
relatively to the line of intersection of the two planes ; so that, if a reversal 
relatively to a plane at right angles to a plane of symmetry results in a 
twin crystal, this may also be obtained by a reversal relatively to the line 
