of Edinburgh^ Session 1863 - 64 . 245 
supposing that these factors enter to a degree higher than the first, 
and we are therefore led to assume that they enter to the first degree 
only. Hence the expression for tan to, becomes 
tan 0 ,, = C_( «-6)(a-&Vj ) 
A (6-c)(5-cV3) ^ 
Similarly, if the optic axes lie in the plane containing the axes a 
and 6, and if to« be the angle which each makes with the axis a, 
£(c-a) (c-aV3) 
( 2 .) 
Also, if the optic axes lie in the plane containing the axes h and 
c, and if to,, be the angle which each makes with the axis &, 
tan „^^ S(o-a)(e-»V3) 
G{a-h) (a-bf3) 
(3.) 
We shall show, by the comparison of these formulm with obser- 
A B C 
vation, that — = — . 
a 0 ~ c 
10. These formulm can be easily expressed in terms of the angular 
elements a^. 
h 
For tan a, = 
tan a,^ = - 
tan a,=- 
^ a 
Hence, putting the formula (2.) may be .written in the 
form, 
, _ (cot a., - cot 45°) (tan a., - tan 60°) 
(cot «3 - cot 45°) (tan «3 - tan 60°) 
and so for the others. 
11. The angular elements of a crystal are to a certain extent arbi- 
trary; thus the parameters may be changed from a, h, c, to pa, qh, 
rc — where p, q, r are positive integers none of which are zero — 
provided the symbols of the simple forms are altered accordingly. 
With the new parameters formula (2.) becomes 
pa (qh — rc) (qb - rcs/3) 
qb (rc — poi) (^rc — paV 3) 
We conclude, therefore, that finite and integral values of p>, g', r 
may be found such that, with the angular elements given in 
tan 
