252 Proceedings of the Royal Society 
If 010, Oil be nearly 45° or 60°, and y — z not very small in tbe 
former case, noi y-zV^ in the latter, while 100, 110 differs con- 
siderably from 45° or 60°, the value of tan ^y'-tan though con- 
siderable, will not be so great as before. 
If 100, 010 and 010, 011 are both nearly equal to 45° or 60°, 
tan (J will differ very little from tan except when x — y and y — z^ 
or X - yV3 and y - z^ differ considerably. 
Hence are explained the variations of the angle between the 
optic axes produced by an increase of temperature in Brookite and 
Ghrysoberyl, as observed by Hes Cloizeaux. 
24. We find that the expression for tan in terms of a, 5, c, con- 
c 
tains the factor For the existence of this factor no d priori phy- 
sical reason can be assigned. Now c = o represents an infinite plane, 
containing the crystallographic axes a and 5, and tan vanishes 
when c = o. The physical interpretation of this factor therefore 
is, that such an infinite plane has only one optic axis which coin- 
cides with the axis c. We may compare this result with the fact, 
that if a ray of light is incident on a crystal of the prismatic system, 
in the direction of one of the principal planes, one of the refracted 
rays, which is polarised in that principal plane, follows the ordi- 
nary law of refraction. 
25. Again, the expression for tan becomes indeterminate if 
a — h=o, and h — cf 3 = o ; 
or, h — c = o, and a — b/d‘3 = o ; 
01 , a — hV3 = o; a.ud h — cV3 = 0 . 
Hence, if the angular element of a crystal of the pyramidal sys- 
tem were equal to 60°, it would not possess double refraction ; the 
same would be the case if one of the angular elements of a crystal 
of the rhombohedral system were equal to 60°. 
26. If a', y , d be the optical constants in descending order of mag- 
nitude, and cd the angle which each optic axis makes with the axis 
of least elasticity, we have, 
tan ‘^o) = 
assuming Fresners expression for the angle between the optic axes 
