^{page no 418 note *} Co-directional and contra-directional operations do not correspond to the operations of the first and second sort of Hilton.

^{page no 418 note †} 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.

^{page no 424 note *} See note to § 3, i.

^{page no 428 note *} It is scarcely necessary to explain that twin crystals are not formed in nature in this way, except in the case of twins formed by movements along gliding planes, when a new disposition in space is given to a portion of a crystal. Here it is the individual molecules that move, and not the structure as a whole, but the change satisfies the definition of an operation given in § 1, ii.

^{page no 431 note *} A line cannot be an axis of all three modes of twinning at the same time (§ 12, i.).

^{page no 431 note †} The presence of two non-equivalent faces is denoted by ± (see Min. Mag., vol. xv. p. 401).

^{page no 440 note *} If a line in the plane of symmetry be an axis of both plane and line twinning, there must be a centre of symmetry, and the normal to the plane of symmetry will be a line of symmetry (§ 5, iii. (a) and § 11, ii.), so that the case falls within § 14, ii.

^{page no 442 note *} The same lines cannot in this case be axes both of plane and of line twinning, for if they were, the structure would possess point symmetry (§ 12, i.), which is inconsistent with contra-directional symmetry.

^{page no 450 note *} The existence of the cross plane precludes, of course, its being a co-directional operation.