426 
Proceedings of the Royal Society of Edinburgh. [Sess. 
of the remaining symmetry of the structure. It will include so much of 
the symmetry of the structure as a whole as is consistent with the fact 
that a plane is without thickness ; that is to say, it may include a plane, 
line, or axis of symmetry at right angles to the plane, a line of symmetry 
lying in it, and point-symmetry ; but not a plane, line, or axis of symmetry 
oblique to the plane, or an axis of symmetry lying in it with cyclic number 
greater than 2. 
As there is, on the assumption which has been made, no difference 
between the opposite faces of a plane, a reversal of a plane relatively to 
the plane itself — resulting as it does merely in an interchange of the faces 
— will always be a co-directional operation of the plane, whether it be a 
co-directional operation of the structure as a whole or not. 
If a reversal relatively to a plane be combined with an operation which 
is a co-directional operation of both the plane and the structure of which 
it forms part, the combination will be likewise a co-directional operation 
of the plane, but will or will not be a co-directional operation of the 
structure according as the reversal relatively to the plane is or is not a 
co-directional operation of the structure. 
For instance, if a structure possess point symmetry, the normal to a plane will 
be a line of symmetry of the plane (§ 5, iii., group (a)), but will only be a line of 
symmetry of the structure if the plane he a plane of symmetry of the structure. 
ii. Accordingly, the only operations which can be co-directional opera- 
tions of a plane and not of the structure as a whole are (1) a reversal 
relatively to the plane, and (2) combinations of such a reversal with an 
operation which is a co-directional operation common to the plane and to 
the structure of which the plane forms part. 
iii. A plane which is a plane of symmetry of the structure can possess 
no symmetry which is not shared by the latter. 
iv. A rotation about the normal to a plane through a half turn is 
always a co-linear operation of the plane. The co-linear cyclic number of 
the normal in respect of the plane must therefore always be even, and if 
its co-linear cyclic number in respect of the structure as a whole be odd, 
the former must be twice the latter. 
§ 7. Common Lines and Planes. 
i. The coincidence -of lines and of planes may be described in terms 
similar to those employed for the coincidence of structures (§ 2). 
If equivalent lines of two structures coincide, the two lines constitute 
a common line, whether their equivalent directions also coincide or 
