1904 - 5 .] Dr Hugh Marshall : Crystallograjphical Notes. 385 
the inclusion of them in any other group. We may therefore take 
the tetragonal system as a standard, and compare the trigonal and 
the hexagonal with it. The seven tetragonal classes and their 
characteristic symmetry are as follows : — 
1. Bi-sphenoidal class. — One axis of compound tetragonal 
symmetry. (Representatives of this class are not actually known, 
however.) 
2. Pyramidal class. — One axis of tetragonal symmetry. 
3. Trapezohedral class. — One axis of tetragonal symmetry ; two 
pairs of lateral axes of digonal symmetry. 
4. Scalenohedral class. — One axis of compound tetragonal 
symmetry ; one pair of lateral axes of digonal symmetry ; one pair 
of planes of symmetry intersecting each other, normally, along the 
principal axis. 
5. Di-tetragonal 'pyramidal class. — One axis of tetragonal 
symmetry; two pairs of planes of symmetry intersecting, all at 
equal angles, along the axis of symmetry. 
6. Tetragonal hi-pyramidal class. — One axis of tetragonal 
symmetry ; one plane of symmetry normal to the axis. 
7. Di-tetragonal bi-pyramidal class. — One axis of tetragonal 
symmetry ; two pairs of lateral axes of digonal symmetry, all 
equally inclined to one another ; one principal plane of symmetry 
and two pairs of planes of symmetry, each plane normal to an axis 
of symmetry. 
At first sight it might he expected that, corresponding to these, 
there would be possible seven classes in each of the other two 
systems, the only differences being those due to the lower or higher 
order of the principal axis of symmetry. So far as regards the 
classes in which the axis is not one of compound symmetry, this 
is the case ; but not so when the symmetry is compound. Axes 
of compound symmetry of even order are possible, but axes of com- 
pound symmetry of odd order are not possible merely as such, 
therefore two classes must be lacking in the trigonal system. This 
is easily seen by a reference to the usual symmetry diagrams re- 
presenting projections on a plane at right angles to the principal 
axis of symmetry. 
Fig. 2 represents the case in which the only symmetry assumed 
is that of a trigonal axis of compound symmetry. An upper face, 
PROG. ROY. SOC. ED1N. — VOL. XXV. 25 
