428 
Proceedings of the Royal Society of Edinburgh. [Sess. 
always a possible face in each component structure, such faces being, of 
course, equivalent to each other. In such a plane there is a molecular net 
belonging to each structure, identical in shape and dimensions, and so 
placed that the interspaces of all parallel rows of molecules in the two 
nets are the same, in spite of the fact that the lattices, as a whole, 
of the two structures have in other respects a different disposition 
in space. 
iii. Where a twin plane or other common plane forms a plane of contact, 
it may be termed a contact twin plane or contact common plane. Where, 
on the other hand, the two component planes of a twin plane or common 
plane coincide mathematically but not physically, it may be referred to as 
an abstract twin plane or common plane. In the same manner a common 
line parallel to a plane of contact may be termed a contact common line, 
and one which is not parallel to the plane of contact an abstract common 
line. Abstract common planes and lines have usually no direct structural 
significance. 
iv. A twin plane like other common planes, and a twin axis like other 
common lines, may be co-directional or contra-directional. 
§ 9. Twinning Operations. 
i. We now proceed to ascertain the geometric relations of the component 
structures of a twin crystal as determined by the definition of a twin plane 
given in § 8. We may conceive the two structures as forming in the first 
place portions of one homogeneous crystalline structure and therefore 
co-directional with one another, and suppose that an operation is applied 
to one of them which results in such a change in its disposition in space 
that a contact or abstract twin plane results. Such an operation may be 
termed a twinning operation.* A twinning operation must accordingly 
bring a plane in the portion to which it is applied into coincidence with an 
equivalent plane in the other portion, in such a manner that the two planes 
are co-linear but the structures are no longer co-directional. 
ii. We may restrict ourselves, however, to the case in which the two 
component planes of the twin plane were identical before the application 
of the twinning operation. For if by a twinning operation U a plane q' 
in the portion S' to which U is applied is brought into a co-linear relation 
* It is scarcely necessary to explain that twin crystals are not formed in nature in this 
way, except in the case of twins formed by movements along gliding planes, when a new 
disposition in space is given to a portion of a crystal. Here it is the individual molecules 
that move, and not the structure as a whole, but the change satisfies the definition of an 
operation given in § 1, ii. 
