454 Proceedings of the Royal Society of Edinburgh. [Sess. 
contact is a co-spatial plane or cross-spatial plane ; for these planes contain 
co-spatial lines, in which the molecular intervals, or low integral multiples 
of them, are equal to one another in the two structures. 
viii. A co-spatial plane may be formed by applying to a part of a uniform 
structure either a co-spatial operation alone, or such an operation followed 
by an operation which, if the co-spatial operation had not been previously 
applied, would have been a twinning operation. 
§ 27. Classification of Co-spatial Operations. 
i. As we have seen, a co-spatial operation is either a reversal or a 
rotation. It cannot be a reversal relatively to a point, because that is 
always a co-linear operation. Nor can it be a reversal relatively to a 
plane at right angles to a line of symmetry, or a reversal relatively to 
a line at right angles to a plane of symmetry, because these opera- 
tions are equivalent to a reversal relatively to a point (compare § 12, 
ii. and iii.). 
ii. We shall first consider co-spatial operations which would be co-linear 
operations of crystals belonging to classes with higher symmetry in the 
same system. These must be reversals relatively to a line or plane, and 
therefore twinning operations, or a rotation with cyclic number four ; for 
a rotation with cyclic number two is equal to a reversal relatively to the 
axis of rotation ; and if a line is an axis of symmetry with co-linear cyclic 
number three or six in one class of a system, it will be so in all, so that if 
a rotation through a third or a sixth of a complete turn be co-linear in 
one class, it will be co-linear in all the classes of the same system (see § 3, 
v., and table, § 4, xiv.). 
iii. Twinning operations which are also co-spatial operations give rise 
to supplementary twins, that is to say, twins of crystals in which the faces 
of the two component crystals together supply the faces required by the 
higher symmetry of another class of the same system. 
In the supplementary twins formed by such a co-spatial operation 
there are only a limited number of twin planes, but all other planes with 
rational indices are co-spatial planes. For instance, in the supplementary 
twins of pyrites the planes parallel to the rhombic dodecahedron faces are 
twin planes, and those parallel to other possible faces are co-spatial planes ; 
while the cubic faces are also cross planes. 
Other analogous supplementary twins are formed by co-linear (here 
contra -directional) operations equivalent to a reversal relatively to a 
