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Proceedings of the Royal Society of Edinburgh. [Sess. 
§ 5. Combinations of Reversals. 
i. A second application of the same reversal (whether relatively to a 
point, line, or plane) restores a structure to its original disposition in space. 
In other words, a reversal is the inverse of itself. Thus, if R be a reversal, 
R 2 = 1 and R = R~ x . 
ii. The combination of two equivalent reversals (or of any two operations 
equivalent to the same reversal) will be a co-directional operation ; for if 
[S]R ( = ) [$]!£', where R and R' are reversals, [ S]R .R'( = ) [$]hf 2 = [S] 1. 
Inversely, if the combination of two reversals be a co-directional 
operation, one will be equivalent to the other, for, if [S]R . R' ( = ) [$]1, 
[^] J R( = )[ > Sf]^ , - 1 = [>Sf]E / . 
iii. There are three groups, each of three reversals, such that the 
combination of any two reversals in the same group is equal to the 
third. These groups are (<x) reversals relatively to a plane, its normal 
and a point, (6) reversals relatively to three lines at right angles, ( c ) 
reversals relatively to two planes at right angles and their line of inter- 
section. 
These relations are easily proved. For instance, in group (a), let any point in 
the structure have an angular distance </> from one direction of the normal considered 
as pole, the point of intersection of plane and normal being taken as centre and 
point of reversal,* and azimuth ifr from any meridian. Then, after a reversal 
relatively to the plane, the co-ordinates will be tt - cf> and if/, and after a subsequent 
reversal relatively to its normal, tv - cf> and 7 r + if/, which are those that would result 
from a simple reversal relatively to the centre. The distance from the centre 
obviously remains unchanged throughout. 
In the case of group (6), a direction of one line may be taken as pole, and one of a 
second line to mark the meridian. Then after a reversal relatively to the latter the 
co-ordinates will be 7 r - cf> and — if/, and, after a subsequent reversal relatively to the 
third line, <£ and -jr + if/, which would also result from a simple reversal relatively 
to the first line. 
The results of these combinations are independent of the order of appli- 
cation of the reversals, for let R 1 . R 2 = R s , where R v R 2 , and R s are reversals 
belonging to the same group, then R 1 . R 2 = R 3 = R^ 1 = R^Ri 1 = R 2 . R r 
iv. It follows from group (a) that a rotation with cyclic number 2 round 
an axis (§ 4, ii.), followed by a reversal relatively to a point,* is equal to a 
reversal relatively to the plane at right angles to the axis. 
v. If the cyclic number of an axis of contra-directional symmetry be 
