1911-12.] 
429 
The Geometry of Twin Crystals. 
with a different but equivalent plane p in the undisturbed portion S, so as to 
form a twin plane, it is possible to obtain an indistinguishable result by 
first applying to S' a co-directional operation F, which will bring p', the 
portion of the plane p which lies in S', into the former position of q', and 
then applying the operation U. It will therefore be sufficient to consider 
the twinning operation T= F. U by which the twinning plane is formed of 
two portions of what was originally the same plane (§4, v.). 
It follows that the relation which exists between the component 
structures of any twin crystal may be brought about by applying to a 
portion of the untwinned structure an operation which is a co-linear 
operation of a plane but not a co-directional operation of the structure. 
It is also clear that every operation that satisfies these conditions is a 
twinning operation. Such a co-linear operation of a plane must be either 
a co-directional or a contra-directional operation of the plane. 
iii. A reversal relatively to a plane is, as we have seen, always a co- 
directional operation of the plane itself, and will therefore, if applied to one 
portion of a uniform structure, convert the plane into a twin plane, provided 
of course that it is not a plane of symmetry of the structure, for if it were, 
the tw T o portions of the structure would continue to have a co-directional 
relation with one another and to form a simple crystal. 
iv. Any other operation U which is a co-directional operation of the 
same plane but not of the structure must be equal to the combination of an 
operation F which is a co-directional operation both of the plane and 
structure and a reversal R, relatively to the plane (§ 6, ii.). If, now, the 
operation U be applied to the structure, it will, since F is a co-directional 
operation of the structure, be equivalent to R (§ 4, v). Accordingly, every 
twinning operation which is a co-directional operation of a plane, is equi- 
valent to a reversal relatively to the plane. 
v. A contra-directional operation of a plane cannot be a co-directional 
operation of the structure, as a whole ; it will therefore, if applied to a 
portion of the structure, always give rise to a twin plane. It may be 
resolved into a co-directional operation of the plane and a reversal 
relatively to a point (§4, vii.). But a co-directional operation of a plane 
must be either a co-directional operation of the structure, a reversal 
relatively to the plane, or an operation which when applied to the 
structure is equivalent to the latter (§6, ii.). The successive application 
of a co-directional operation of a plane and a reversal relatively to a point 
must therefore be equivalent to either a reversal relatively to a point or a 
reversal relatively to the normal to the plane (§ 5, iii., group (a)), and no other 
contra-directional twinning operations need be considered. But, in order 
