417 
1911-12.] The Geometry of Twin Crystals. 
Every line in a structure has two opposite directions, which may have 
either the same or different physical and crystallographic characters. 
ii. It is often desirable to suppose that the disposition in space of the 
whole or part of a structure is changed subject to the condition that the 
distance between any two points is the same after the operation as it was 
before. Such a change is referred to as an operation. 
§ 2. Coincidence and Equivalence. 
i. If two structures are so related that not only every plane and line 
but every direction of one structure coincides with a plane, line, and 
direction of the other, possessing the same physical and crystallographic 
characters, they may be said to be co- directional (or parallel) to one 
another. 
ii. If by an operation, as above defined, one structure can be brought 
into a co-directional relation to another, they are said to possess the same 
form ; and a plane, line, or direction in one structure is said to be 
equivalent to any plane, line, or direction of another structure, if they can 
be brought into coincidence by an operation which brings the former 
structure into a co-directional relation to the latter. 
iii. An operation which, when applied to a structure, leaves it in a 
co-directional relation to itself, as it existed before the operation, may be 
termed a co-directional operation ( “ symmetry operation ” of Hilton), and 
every plane, line, or direction which can by means of a co-directional 
operation be brought into coincidence with another plane, line, or direction 
of the same structure, as it was before the operation, is likewise said 
to be equivalent to it. It is also often convenient to refer to a plane, 
line, or direction as equivalent to itself. 
iv. It is clear that if two planes, lines, or directions are equivalent to 
the same plane, line, or direction, they are equivalent to one another ; that 
if two planes are equivalent, the lines they contain are respectively 
equivalent ; and that if two lines are equivalent, their directions are 
respectively equivalent ; also that if two planes are equivalent, their 
normals are equivalent, and vice versa. Molecular rows parallel to 
equivalent lines must obviously have equal molecular distances. 
v. If two structures are so related that every line in one coincides 
with an equivalent line in the other, they are said to be co-linear, 
and an operation that brings a structure into a co-linear relation with 
itself, as it existed before the operation, is said to be a co-linear 
operation. 
vol. xxxii. 27 
