62 
Proceedings of Royal Society of Edinburgh. [sess. 
Note on the Axes of Symmetry which are Crystal- 
lographically possible. By Hugh Marshall, D.Sc. 
(Read January 17, 1898.) 
In a “ Memoire sur la deduction, d’un seul principe, de tous les 
systemes crystallographiques avec leurs subdivisions,” by Axel 
Gadolin,* there is given a proof that, if we assume the law of 
rational indices, only digonal, trigonal, tetragonal, and hexagonal 
axes are possible with crystals. This proof is adopted by Groth in 
the last edition of his Physikalische Krystallographie , 1895. The 
proof which I shall now give, although somewhat similar in general 
principle, is decidedly simpler than that of Gadolin. It is assumed 
in the former that an axis of symmetry is necessarily a possible 
edge or zone axis, and that there are possible edges perpendicular 
to any axis of symmetry, i.e. t that the plane to which it is normal 
is a possible face. These general propositions are not, so far as I 
am aware, to be found in text-books, although tacitly assumed in 
certain cases, and a proof of them, as also of two similar ones 
concerning planes of symmetry, is therefore indicated here before 
treating the main problem. 
1. Every plane of symmetry is a possible face. — Any face A, 
inclined to the plane of symmetry S, gives by reflection in S a 
similar face A' lying on the other side of S. These two faces 
necessarily intersect in a line lying in S. Similarly, any other 
face B will intersect its image B' along another line lying in 
S. Consequently, S is parallel to two possible edges, and is there- 
fore a possible face. 
2. The normal to every plane of symmetry is a possible zone 
axis. — Let OX be any edge inclined to the plane of symmetry S. 
Its reflection OX ' is a similar edge. The plane XOX' is therefore 
a possible face, and, from the symmetry, it is perpendicular to S. 
* Ada Soe. Scient. Fennicce, ix. p, 1 (1871). A German translation is pub- 
lished as No. 75 of Ostwald's Klassiker der exaden Wissenschaften. 
