453 
1911-12.] The Geometry of Twin Crystals. 
itself as it existed before the operation may be described as a co-spatial 
operation. 
iv. A co-spatial operation, though not a co-1 inear operation of the 
structure to which it is applied, will be a co-linear operation of a structure 
with higher symmetry, which may be referred to the same crystallographic 
axes, and is therefore a reversal or a rotation. 
§ 26. Equxspatial and Co-spatial Lines and Planes. 
i. A line with rational indices which is brought by a co-spatial opera- 
tion into the former position of another, non-equivalent, line may be said 
to be equispatial with it ; for the molecular interspaces of such lines, or 
low integral multiples of such interspaces, will be equal to one another. 
This follows from the fact that the structure in its new position can 
obviously be referred to the old axes in their original position with the 
same parameters, and that at least some lines will occupy the former place 
of equivalent lines. 
ii. Similarly, if a plane with rational indices, and therefore parallel 
to a possible face, is brought by a co-spatial operation into coincidence with 
the former position of another, non-equivalent, plane of the same crystal 
structure, the two planes may be described as equispatial, for every line 
with rational indices in one will be equispatial with or equivalent to a 
corresponding line in the other. 
iii. The term equispatial may be extended to include non-equivalent 
lines and planes with rational indices belonging to different structures, 
but capable of being brought into coincidence by an operation which brings 
those structures into a co-spatial relation with each other. 
iv. If a line in one structure coincide with an equispatial line in 
another, the two lines may be described as co-spatial and as forming 
together a co-spatial line. 
v. If a plane with rational indices of one structure coincide with an 
equispatial or equivalent plane of another structure in such a manner that 
every line with rational indices of one plane coincides with an equispatial 
or equivalent line of the other, but the two planes are not co-linear, the 
planes may be said to be co-spatial and to form a co-spatial plane. 
A co-spatial plane may or may not be a common plane or a cross plane. 
vi. If two equispatial planes coincide to form a plane that contains two, 
four, or six co-spatial lines, then that plane may be termed a cross-spatial 
plane. 
vii. Compound crystals may be expected to occur in which the plane of 
