431 
1911-12.] The Geometry of Twin Crystals. 
with it, in parentheses, the crystallographic classes (usually indicated by 
their symbols) in which it may occur. These are followed by the type 
symbol or symbols of the twin axis or axes which may he formed by the 
line, with examples in each case where such are known. The type symbol 
of a twin axis is formed by adding to the type symbol of the line the 
capital letter P, L, or I, according as it is an axis of plane, line, or point 
twinning. If it is an axis of two modes of twinning, both the corresponding 
letters are employed.* 
Some of the more important types of twin axis are illustrated by diagrams 
which are stereographic projections on the twin plane of the faces of a general form. 
Two opposite equivalent parallel faces are represented by a single small disc in the 
position of the projection of the upper face. Opposite faces which are not equivalent 
do not occur in the same form. An upper face without an equivalent parallel lower 
face is distinguished by a plus sign with a disc as centre ; a lower face without 
an equivalent parallel upper face by a minus sign and disc in the position which a 
parallel upper face would have occupied .f 
A short radial line drawn inwards from the circumference of the diagram 
indicates the position of a line of symmetry in the plane of projection. Other 
symmetry relations are obvious at once from the projection. For instance, if there 
is point symmetry there are no positive or negative symbols, for every line has two 
equivalent directions. Small marks placed on the circumference divide the diagrams 
into 2, 4, or 6 parts, in order to indicate more clearly the results of the application of 
the twinning operations to the structures. 
The first of each group of diagrams shows the structure before the application of 
a twinning operation, and the other, or others, the same structure after one has been 
applied. 
The position of an axis of line twinning lying in the plane of the paper is 
shown by a radiating line drawn outwards from the circumference of the diagram, 
and that of an axis of plane twinning by a shorter radiating line terminated by 
another longer line at right angles to it, while in the case of a twin axis with 
both line and plane twinning the radial line is continued outwards beyond the 
second line. Point twinning is indicated by a small circle in the centre of 
the diagram. 
vi. Uniterminal lines with odd co-linear and co-directional cyclic 
number yield different twin crystals with the three different modes of 
twinning. With plane twinning the twinning plane is co-directional, 
while with line and point twinning it is contra-directional. With line 
twinning the twin axis is co-directional, while with plane and point 
twinning it is contra-directional. 
1 Uu (in all crystallographic classes without a centre of symmetry) : 
* A line cannot be an axis of all three modes of twinning at the same time (§ 12, i.). 
t The presence of two non-equivalent faces is denoted by (see Min. Mag ., vol. xv. 
p. 401). 
