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Proceedings of the Royal Society of Edinburgh. [Sess. 
XXX. — The Geometry of Twin Crystals. By Dr John W. Evans, 
of the Imperial Institute and Birkbeck College, London. Communi- 
cated by Professor J. W. Gregory, F.R.S. 
(MS. received January 26, 1912. Read March 4, 1912.) 
In a communication to the Mineralogical Society, published in September 
1910 {Min. Mag., vol. xv. p. 390), I suggested that the relation between the 
component structures of twin crystals was similar to that existing in 
combinations of crystals of different substances with definite relative 
orientation (Barker, Min. Mag., vol. xiv., 1907, p. 235), and was determined 
by equality of molecular distances in the two structures in the plane of 
contact or composition. 
It has been my purpose in the present paper to determine what are the 
possible geometrical relations between crystal structures in which such 
equality exists in all or some of the molecular rows in the plane of contact. 
As a rule, rows with equal molecular distances are equivalent to one 
another. I have, accordingly, investigated in the first place the relations 
between crystal structures in which equivalent lines coincide in orientation. 
The treatment is much simplified by dealing with equivalent lines in 
general, not merely those which are parallel to possible edges and therefore 
to molecular rows ; but special considerations applicable to the latter alone 
are added where necessary. 
I next discuss the coincidence of lines which are not equivalent, but 
in which the molecular distances are either equal, or are related by a 
simple ratio so that equality exists between low multiples of them. 
I am indebted to Mr H. Hilton {Mathematical Crystallography , Oxford, 
1903; and Min. Mag., vol. xiv., 1907, p. 261) for some of the methods em- 
ployed. I have not thought it necessary to give proofs of well-known 
propositions, except where they assist in the development of the argument. 
§ 1. Planes, Lines, and Directions; Operations. 
i. It is convenient in describing the geometrical characters of a crystal 
to consider planes and lines (that is, straight lines) to possess orientation 
but not locality. In other words, parallel planes or lines are treated as 
identical ; for they have the same physical and crystallographic characters, 
and parallel rows of molecules have equal molecular intervals. 
