386 Proceedings of Royal Society of Edinburgh. [sess. 
would give a lower face, 2 ; and by repeating these operations an 
upper face, 3, would result. Repeating the rotation once more 
would bring the face back to its original position, but the ensuing 
reflection would give a new face, immediately below the first. It 
is therefore evident that in order to return to the original position, 
by repeating the operations characteristic of the symmetry, two 
complete revolutions are necessary, and this produces six faces, as 
shownjin fig. 3 — three above and three below. The diagram now 
exhibits the higher symmetry of an ordinary trigonal axis combined 
with a plane of symmetry at right angles to it ; but this is the 
symmetry of the trigonal bi-pyramidal class which corresponds to 
the tetragonal bi-pyramidal class. There can, therefore, be no tri- 
gonal class corresponding to the tetragonal bi-splienoidal class. 
Similarly, there can be no trigonal class corresponding to the 
di-tetragonal scalenohedral class, as a trigonal axis of compound 
symmetry combined with vertical planes of symmetry leads neces- 
sarily to the symmetry of the di-trigonal bi-pyramidal class. Each 
of these two classes — the trigonal bi-pyramidal and the di-trigonal 
bi-pyramidal — therefore represents, in a sense, two classes of the 
tetragonal system. It is noteworthy that not a single substance is 
known to crystallise in either of them ; they are only ‘ theoretically 
possible.’ 
As hexagonal axes of compound symmetry are possible, there are 
the full number of seven classes possible in the hexagonal system. 
The classes corresponding to the tetragonal bi-sphenoidal and 
scalenohedral are the rhombohedral class and the hexagonal scaleno- 
hedral. Representatives of both are known, especially of the 
f 
\ 
Fig. 2. 
Fig. 3. 
