1897 - 98 .] Hugh Marshall on Axes of Symmetry . 
65 
on the two equivalent axes, and gives rise, by the symmetry, 
to a regular w-sided pyramid. The plane passing through two 
alternate pyramidal edges, as CB and CB", will also he a possible 
face, and must therefore have rational indices on the axes of 
reference. As it passes through the two points C and B it can 
have a rational index on the remaining axis only if the ratio 
OD/OB' is rational, OD being the parameter of BOB" on the axis 
OB'. But OB = OB', therefore OD\OB must be rational; since 
27 r 
BB" is perpendicular to OB\ ODjOB is cos BOB', i.e ., cos — . 
The law of rational indices, therefore, limits the value of n to 
27 r 
those cases where cos — is rational. Further, from the nature of 
n 5 
an axis of symmetry, n must be a whole number. It is shown by 
N. Boudaief, in an appendix to Gadolin’s paper, that the only values 
of n which satisfy these two conditions are 2, 3, 4, and 6. Con- 
sequently, only digonal, trigonal, tetragonal, and hexagonal axes of 
symmetry are possible with crystals. 
The construction employed above is possible only when n is not 
less than 3. That digonal axes are possible follows at once, how- 
ever, from the possibility of tetragonal and hexagonal axes, so that 
the case of n = 2 does not require to be specially considered. 
VOL. XXII. 
7 / 3/98 
E 
