418 Proceedings of the Royal Society of Edinburgh. [Sess. 
vi. If the two directions of a line are equivalent, the line is said to be 
biterminal. If they are not equivalent, it is uniterminal. 
If two structures are co-linear, the directions of every biterminal line 
in one must coincide with equivalent directions in the other, but the 
directions of a uniterminal line in one of two co-linear structures may 
either coincide with or be opposed to the equivalent directions of the corre- 
sponding line in the other. If the former be the case with all uniterminal 
lines, the structures are, as we have seen, co-directional ; if the latter, they 
may be said to be contra-directional. 
An operation which brings a structure into a contra-directional relation 
with itself, as it existed before, is said to be a contra-directional operation .* 
vii. All co-linear structures must either be co-directional or contra- 
directional, for if the directions of some uniterminal lines coincided with 
equivalent directions, and those of others were opposed to equivalent 
directions, other coincident lines which are intermediate in position to 
these could not be equivalent, for they would have different crystallo- 
graphic relations in the two structures. Every co-linear operation must 
in like manner be either co-directional or contra-directional. 
viii. The co-directional, contra-directional, and co-linear operations of 
a structure express the symmetry it possesses. 
ix. If one structure can be brought into a co-directional relation with 
another, by an operation such as may be imitated by the movement of a 
rigid body, the two are said to be congruent. If, on the other hand, one 
structure can be brought by such an operation or movement into a contra - 
directional, but not into a co-directional, relation with another, they are 
said to be enantiomorphic. Two enantiomorphic structures are considered 
to possess the same form (§ 2, ii.) although their disposition in space 
prevents them from being brought into co-directional coincidence by such 
movements as can be carried out with a rigid body. 
§ 3. Different Kinds of Operations. 
i. Reversal relatively to a Point. — Every point in the structure is 
transferred to a new position such that the straight line joining the old 
and new positions is bisected by the same fixed point, the point of reversal .f 
* Co-directional and contra-directional operations do not correspond to the operations 
of the first and second sort of Hilton. 
t In describing these operations, the point, line, or plane of reversal or axis of rotation is 
supposed to have a definite position, but the result is independent of that position. The 
difficulty is avoided if it be supposed that all planes and lines of reversal and axes of 
rotation pass through the same point, and that this is also the point of reversal. 
