of Edinburgh^ Session 1863 - 64 . 233 
each of the above forms. The three straight lines perpendicular 
respectively to the faces 101, 121, and 111, are taken as the new 
axes of symmetry; the parameters corresponding to them being 
a, 5, c, respectively, it is shown in a later part of the paper that a,h,c, 
are unequal. 
3. The paper then proceeds to show that the laws of symmetry 
of crystals of the rhombohedral system are the same as those of the 
prismatic system. 
This is done by taking each simple form separately and finding 
what the indices of its faces become when referred to the new 
axes ; it is then found that the new indices for all the simple forms 
follow the laws of symmetry of crystals of the prismatic system. 
The same process is followed with the hemihedral forms. 
4. In order to find the indices of a given face referred to the new 
axes, it is necessary to solve the following problem : — 
“ Griven the indices and parameters of any face of a crystal when 
referred to given axes, to find its indices and also the new parame- 
ters when referred to any other axes originating in the same point.” 
This problem is solved in the paper, but the solution is too long 
to be here given. 
For the purposes of this paper only a particular case of the gene- 
ral problem is required. 
For in the rhombohedral system the axes make equal angles with 
each other, and the parameters are equal. Let o> be the angle be- 
tween the rhombohedral axes, and a the magnitude of each of the 
equal parameters. Let hhl be the symbol of a face of a crystal 
belonging to the rhombohedral system, and KUt the indices of 
this face when referred to axes perpendicular respectively to the 
faces 101, 121, 111, a', c', the new parameters. 
Then the formulas obtained in the general case become, in this 
instance, — 
2h-h-l 
6 
y/ Tl •\-Tc I 
( 1 .) 
These expressions give the new indices. 
