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1911-12.] The Geometry of Twin Crystals. 
not. If each direction of one line coincide with an equivalent direction of 
the other, the lines are co -directional. If the component lines are biter- 
minal or, in other words, if the two directions of each are equivalent to 
one another, the two lines will necessarily be co-directional. If, on the 
other hand, they are uniterminal, so that each has two non-equivalent 
directions, they will either be co-directional or they will have their 
equivalent directions opposed to each other, and will be contra-directional. 
The terms co-directional and contra- directional may also be applied to the 
common line formed by two co-directional or contra-directional lines. 
ii. If two equivalent planes forming part of different structures coincide, 
they form a common plane, whether equivalent lines in the two planes 
coincide or not. If every line in one component plane coincide with an 
equivalent line in the other, so that every line in the common plane is 
a common line, the component planes and also the common plane are said 
to be co-linear (cf. § 2, v.). If in a co-linear common plane every direction 
in one component plane coincide with an equivalent direction in the other, 
so that every common line in the common plane is co-directional, the 
planes are also co-directional (§2, iii.). If every line in the component 
planes of a co-linear common plane is biterminal, the planes are necessarily 
co-directional. This will be the case if the structure possesses point 
symmetry or if the normal is a line of symmetry of the structure. If, 
however, there are lines with non-equivalent directions in the component 
planes of a co-linear common plane, the corresponding common lines must 
be either co-directional or contra-directional, and the planes will be co- 
directional or contra-directional accordingly. A contra-directional common 
plane will, however, contain co-directional common lines if the component 
planes contain biterminal lines (cf. § 2, vi.). 
iii. If two equivalent planes coincide to form a common plane, the 
normals will coincide to form a common line. Inversely, a plane at right 
angles to a common line will be a common plane (§2, iv.). 
§ 8. Twin Planes and Axes. 
i. If in a compound crystal made up of two structures of the same 
form there is a co-linear common plane but the structures are not co- 
directional, the common plane is termed a twin plane, and the common 
line which is its normal a twin axis ; and the two structures together 
constitute a twin crystal. 
ii. A twin plane forms the most frequent plane of contact or “ com- 
position ” between the component structures of a twin crystal, and is then 
