452 Proceedings of the Royal Society of Edinburgh. [Sess. 
plane is a common line. It must consequently be either a cross plane or 
a co-linear plane. In the latter case it must be a twin plane, as the two 
structures are not co-directional. 
ii. When the plane of contact of the two structures of a twin crystal 
is not a twin plane, it is found by experience to be at right angles to a 
twin plane, and must therefore be a cross plane. The structural reasons 
why this should be the case have already (§ 8, ii., and § 21, i.) been 
indicated. 
iii. Similarly, a plane at right angles to a cross plane and intersecting 
it in one of its common lines will be a twin plane or a cross plane. 
iv. Where, as in most pericline twins of plagioclase, the plane of contact 
is not a possible face, it appears to be always a cross plane and not a 
twin plane ; one common line (the macro-axis in pericline twins) is a 
possible crystal edge, and the other the line of intersection of the plane of 
contact and a possible crystal face usually a plane of cleavage (the brachy- 
pinakoid in pericline twins) (Min. Mag., vol. xv., 1910, pp. 392-3). This 
is illustrated by figure 1 UcP'L' (p. 449), which shows the relation between 
the two component structures of a pericline twin of plagioclase, when they 
are projected on the cross plane which forms the plane of composition 
known as the rhombic section. 
§ 25. Co-spatial Structures and Operations. 
i. In the preceding pages the coincidence of equivalent lines in the 
constituent structures of compound crystals has been studied because it is 
believed that the determining factor in the joint growth of such structures 
is in most cases the coincidence of molecular rows in which the molecules 
are at an equal distance from one another, and that this equality mainly 
exists where the molecular rows are equivalent to one another. There are, 
however, lines in crystal structures which are not equivalent, but corre- 
spond to molecular rows in which the molecular intervals are either 
equal or stand in a simple ratio to one another. Such lines occur in a 
crystal which does not possess all the symmetry which could be associated 
with a system of axes and parameters to w T hich it might be referred con- 
sistently with the rationality of the indices of its faces and edges. 
ii. If two structures of the same form, which are not co-linear, are so 
related that every line having rational indices (and therefore parallel to 
a possible edge) in the one coincides with a line with rational indices in 
the other, the two structures may be said to be co-spatial. 
iii. An operation which brings a structure into a co-spatial relation to 
