636 Scientific Proceedings, Royal Dublin Society. 
betray the fact of a dimorphous change having effected the alter- 
ation toa lower symmetry. 
For comparison with the foregoing geometrical conclusions we 
have various instances of the presence of vicinal-faces on crystals, 
in very many of which the fact of the existence of pseudo-symmetry 
is revealed by the optical properties not being in harmony with ~ 
the higher symmetry." 
Among these may be mentioned the pseudo-tetrahedrally-cubie 
pharmacosiderite whose cube faces consist of two planes inclined at 
a re-entrant angle of nearly 180°, the angle edge running along a 
diagonal. Six individuals are discernible in each crystal which 
interlock in the neighbourhood of planes drawn through opposite 
cube edges. ‘They are double-refracting, and sometimes the sepa- 
ration of one of these individuals into two halves by a diagonally 
placed boundary is indicated optically by the different extinction- 
position for one half as compared with the other. There are some 
remarkable optical peculiarities.” 
Brauns notes that among the alums, mixed crystals with wichita 
faces are biaxial,? a fact which suggests that that which produces 
these vicinal faces also produces a lowering of symmetry. 
Tt is, as has been said, conceivable that a dimorphous change 
which lowers symmetry may reveal itself by the production of 
vicinal-faces in the way above indicated, but that the change in the 
arrangement of the ultimate parts which it produces may be so 
slight as to have no perceptible effect on the optical properties. 
And the pure alums which crystallize in the cubic symmetry with 
vicinal-faces but are isotropic,* may exemplify this. 
We have already called attention to the fact that if the 
solid angles at which the twinning-blocks meet are not abso- 
lutely unaffected by the dimorphous change, some strain or 
irregular shifting of parts will be involved in the continuance 
of the blocks in their original relative situations. The nature 
1 Comp. A. Ben Saude, ‘‘ Die wahrscheinlichen Ursachen der anomalen Doppelbre- 
chung der Krystalle.”? Lissabon, 1896. Note 2, p. 14. 
2 Brauns, Joc. cit., p. 349. 
3 Brauns, loc. cit., p. 237. 
4 Brauns says that only those alums which contain isomorphous admixtures exhibit 
optical anomalies. Jbid., p. 228. 
