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1911-12.] The Geometry of Twin Crystals. 
combination of an even number of contra-directional operations ; whilst 
the combination of an odd number of contra-directional operations, or of 
a co-directional operation and a contra-directional operation, is a contra- 
directional operation. 
As special cases it may be noted that (1) the combination of any 
contra-directional operation with a reversal relatively to a point is a 
co-directional operation, and that (2) every contra-directional operation 
is equivalent to any co-directional operation combined with reversal 
relatively to a point. 
viii. As every co-linear operation is either co-directional or contra- 
directional, and a co-directional or contra-directional operation is always 
co-linear, all combinations of co-linear operations are themselves co-linear. 
ix. The inverse of an operation is an operation in which the action 
of the original or direct operation is inverted. It is denoted by the 
symbol A -1 , where A is the direct operation. Then A.A _1 = 1. If an 
operation A be a co-directional operation of a structure S, the inverse 
operation A -1 will also be a co-directional operation of S, for, since 
[$]A. A -1 — [$]1, and [S]A ( = ) [S]l, [$]A -1 ( = ) [$]1. In the same way 
the inverse of a contra-directional operation B is also a contra-directional 
operation of the original structure; for [8]Ri.B ( = ) [S]l, where R. 
is a reversal relatively to a point (§ 4, vii.). Apply the inverse opera- 
tion B' 1 ; then B.B~ X ( = ) [$]5 _:1 , but B.B~ 1 = 1; accordingly, 
( = ) [$]i? -1 . It follows from the above that the inverse of a 
co-linear operation is always a co-linear operation. 
x. Let C a , a rotation round the line c through an angle a, be the smallest 
rotation round c which is a co-linear operation of a structure S, then all 
rotations of the form C na ( = 0"), where n is any integer, positive or 
negative, are also co-linear operations of S, which is not the case with 
any other rotation round the same line ; for if a rotation G( n+q ) a , where n is 
an integer and q is a fraction less than 1, were a co-linear operation of S, 
C(n +q y.C~^ = C(n +q -.ri) a = C qcL would also be co-linear, which is, by hypothesis, 
impossible, for qa is less than a. 
In the same way it can be shown that, if G p be a co-directional operation 
of S, rotations of the form C n/3 , where n is an integer, and no others round 
the same line are co-directional operations of S. 
As a complete turn is always co-directional and co-linear, the co- 
directional and co-linear cyclic numbers must always be integers. 
xi. The smallest rotation round a particular line which is a co-linear 
operation of a structure must be either the smallest co-directional rotation 
or the smallest contra-directional rotation round the same line. In the 
