430 Proceedings of the Royal Society of Edinburgh. [Sess. 
that they may give rise to a contra-directional twin plane, the original 
plane must contain uniterminal lines ; in other words, the normal must not 
he a line of symmetry of the structure, and the structure must not possess 
point symmetry. 
vi. If these conditions are not fulfilled, the operations under consideration 
will be co-directional operations of the plane and give rise to co-directional 
twin planes, provided that in the case of a reversal relatively to a point the 
structure does not possess point symmetry, and in that of a reversal relatively 
to the normal this is not a line of symmetry of the structure ; and they 
must, like all other co-directional twinning operations, be equivalent to 
reversals relatively to the plane ( see § 9, iv., and § 12, i. and ii.). 
§ 10. Modes of Twinning and Types of Twin Axes. 
i. It has now been shown that every twin crystal may be obtained by a 
reversal of a portion of a structure relatively to a plane, a line, or a point, 
and that no other twinning operations need be considered. It has also 
been demonstrated that every such operation will result in the formation 
of a twin plane, provided that it is not a co-directional operation of the 
structure. 
ii. Each of these three reversals may be termed a mode of twinning. 
If the twinning is by reversal relatively to a plane, which may be termed 
plane (or reflexion) twinning , the plane is the twin plane, and its normal 
the twin axis. If it is by reversal relative to a line, line (or rotation) 
twinning, the line is the twin axis, and the plane to which it is normal the 
twin plane. Finally, in twinning by reversal relatively to a point, point 
(or inversion) twinning, every plane has the properties of a twin plane, 
and every line may be regarded as a twin axis. 
iii. In the case of plane twinning the corresponding twin plane is 
always co-directional, but the twin axis may be either co-directional or 
contra-directional; in line twinning the twin plane may be either co- 
directional or contra-directional, but the axis is always co-directional ; 
and in point twinning both twin planes and axes may be either co-directional 
or contra-directional. 
iv. As a twinning operation is not a co-directional operation of the 
structure (§9, ii.), a plane of symmetry cannot be a twin plane with plane 
twinning ; a line of symmetry cannot be a twin axis with line twinning, 
and point symmetry is inconsistent with point twinning. 
v. The different types of twin axes will now be enumerated. In each case 
the type symbol of the line which forms the twin axis is first given, and 
