63 
1897 - 98 .] Hugh Marshall on Axes of Symmetry. 
Similarly, any other edge OY will, with its image OY', determine 
a possible face YOY r , also normal to S. S is therefore a zone 
plane, being perpendicular to two possible faces, and its normal is 
a possible zone axis. 
3. Every axis of symmetry is normal to a possible face. — (a) Let 
the axis A be of the second (or any even) order. Any face B 
inclined to A will, on rotation round the axis through an angle i r, 
give a similar face B' , and these two faces necessarily intersect 
along a line perpendicular to A. Similarly, any other face G, 
inclined to A, has a corresponding face C' opposite it and equally 
inclined to A, and these two faces also intersect in a line perpen- 
dicular to A. Consequently, a plane parallel to these two edges, and 
therefore perpendicular to A, is a possible face. (b) If A is an 
axis of uneven order it must be at least of the third order, and any 
edge OB inclined to it necessitates at least two others, OB' and 
OB", crystallographically identical with OB and equally inclined 
to A. If these three edges are taken as axes of reference, it follows, 
from the identity of the edges, that the axial ratios must be 
b = b' = b" =1. The face 111 would therefore cut OB, OB', and 
OB" at equal distances from 0 ; but such a plane is evidently the 
base of a right regular rc-sided pyramid, and the axis A is normal 
to it. A is therefore normal to a possible face. 
4. Every axis of symmetry is a possible zone axis. — (a) Let the 
axis A be of the second (or any even) order. Any edge OB inclined 
to A gives, by rotation through tt round the axis, a similar edge 
OB' lying in the plane determined by A and OB. Similarly, any 
other edge OG necessitates one 00' symmetrical to it, lying in the 
plane determined by A and OC. But BOB' and GOG' are possible 
faces, being parallel to pairs of possible edges, and A, their mutual 
intersection, is a possible edge or zone axis. ( b ) If A is of uneven 
order, it must be at least of the third order. Any face B inclined 
to it necessitates at least two others, B' and B", crystallographically 
identical with it. These faces together form the sides of a regular 
n sided pyramid, whose base, perpendicular to A, is also a possible 
face, so that the basal edges are possible zone axes. There is, 
therefore, a whole series of pyramids, having the same base as the 
first, whose sides are also possible faces, such as G, C\ and G". 
The corresponding edges of two such pyramids determine a set of 
