251 
of Edinburgh^ Session 1863 - 64 . 
for determining the angle between the optic axes, were, from the 
causes mentioned, different from those given by Professor Miller. 
22. But if the angular elements of a mineral vary within certain 
limits, the angle between the optic axes, calculated by the for- 
mulae given in this note, must also vary within corresponding 
limits.* 
The causes, therefore, which produce in a given mineral the 
variations in the angle between the optic axes are the same as 
those which produce the variations in the angular elements. That 
the angle between the optic axes of a crystal varies by isomorphous 
replacement, is established by the experiments of De Senarmont;f 
that it varies with the temperature to which tlie crystals are or 
have been exposed, is established by those of Des Cloizeaux.| 
The existence of this variation, therefore, far from being opposed 
to the existence of a connection between the form and optical 
properties of crystals, as argued by De Senarmont, § is strongly in 
support of it. 
23. Suppose that from either of the causes mentioned, the para- 
meters a, 6, c, become a -i- ic, & -f y, c-i-z, where x, y, z, are small quan- 
tities. Let the optic axes lie in the plane ca, and make an angle 
with the axis c, and let J be the angle corresponding to the new 
parameters. It is easily seen that 
tan a = tan u 
^-y _ y- g _^ a;-yV 3_ y-^V3 
a — h h — c a — hj^o h-CsJ^ 
y--z^ y-z^ ~l 
+ 5-cV3 __ 
neglecting ^ 
Hence, if 100, 110 be nearly 45° or G0°, and x~y not very small 
in the former case, nor x-ys/Z in the latter, while 010, Oil 
differs considerably from 45° or 60°, tan uf - tan cd will be large. 
In this case, therefore, the angle between the optic axes will vary 
considerably. 
Topaz and mica are well known examples of this variation of the angle 
between the optic axes. 
t Annales de Chemie, third series, xxxiii. p. 391, 
J Annales des Mines, sixth series, ii. p. 327, 
§ Ibid., p. 433. 
