420 Proceedings of the Royal Society of Edinburgh. [Sess. 
If the tetragonal system is taken as an example, we find that in the IV Bk 
(chalcopyrite) and IY Uk classes (see § 4, xiv.) the co-directional cyclic number of 
the principal axis is 2, and the contra-directional and co-linear cyclic numbers 4, 
while in the remaining five classes the co-directional and co-linear cyclic numbers are 
4 : similar relations hold in the other systems. 
§ 4. Combination, Equality and Equivalence of Operations. 
i. The successive application of the operations A and B constitutes a 
third operation which may be expressed in the form of a product A.B , 
and the repeated application of the same operation A is, on the same 
principle, denoted by the corresponding power of A. 
ii. If two operations A and B result in the same final change, they are 
said to be equal, and we may write A=B. For instance, a rotation with 
cyclic number 1 is equal to the absence of change, and one with the cyclic 
number 2 to a reversal relatively to the line which forms the axis. 
iii. If, when applied to a structure S (that is to say, to a particular 
structure with a particular disposition in space), the results of two operations 
A and B are indistinguishable on account of the distribution of equivalent 
directions, the operations are said to be equivalent to one another, and we 
may write [$].4 ( = ) [S]B ; where [S] indicates that the operation which 
follows is applied to the structure S, and ( = ) expresses equivalence. 
iv. The equality of two operations is independent of the structure 
to which they are applied, but it is convenient to consider equality as a 
special case of equivalence. 
v. If an operation A is equal to an operation B followed by an 
operation C, and B is a co-directional operation of a structure S, the 
operation A, when applied to S, will be equivalent to the operation G, for 
[S]A = [S]B.C( = )[S]C. 
vi. It is obvious that all co-directional operations of the same structure 
are equivalent to one another and to the absence of change, which may be 
denoted by 1. Thus, if A and B are co-directional operations of S, we 
may write [S]A ( = ) [S]B ( = ) [$]1. In the same manner, all contra- 
directional operations of the same structure are equivalent, so that, if A 
and B are contra-directional operations of S, ( = ) [S]B ( = ) 
where R t is a reversal relatively to a point ; for the existence of a 
contra-directional operation implies the presence of uniterminal lines, so 
that a reversal relatively to a point will necessarily be contra- directional 
(§ 3, L). 
vii. It is easily seen that the combination of any number of co- 
directional operations is itself a co-directional operation, and so is a 
