244 
Proceedings of the Royal Society 
But crystals belonging to the rhombohedral system have only 
one optic axis, and this, as is shown in ihe paper referred to, coin- 
cides with the axis c if the angular element 100, 110 = 60° ; i.e.^ if 
Hence when a - h V3, the angle which each optic axis makes 
with the axis c vanishes. 
Hence, and tan w,, vanishes when a = h V3. 
tan cDc is divisible by a -h Vd. 
But we have before shown that it is divisible hy a — h. 
tan coc is divisible by (a -h) (a -b V3). 
But all uniaxal crystals belong either to the pyramidal or rhom- 
bohedral systems. 
Hence these are the only factors containing a and 5, by which 
tan (Oc is divisible. 
7. Again, if Wc = ?j tlie optic axes coincide with the axis a. 
But when this is the case, by the same reasoning as before, 
— 7T 
either 6 = c, or h = cVS ; must =^, or tan co^ must become in- 
finite when b’-c = o, or h — c V3 = o. 
Hence the expression for tan w,, in terms of a, b, c, must be a 
fraction, the denominator of which contains the factors b — c and 
b-cV3; and, for the same reason as before, these are the only 
factors in the denominator containing b and c. 
Hence, since tan co^ is of no dimensions, the expression for tan w,. 
in terms of a, b, c, must be of the form 
A(b-cyXb-cs/3f 
where C is independent of a and b, and A of 5 and c. 
8. Crystals belonging to the cubic system do not possess double re- 
fraction ; in other words, they have an infinite number of optic axes. 
Hence, tan ought to become indeterminate, v/hen a — b = o and 
b^c = o^ which is the case^with the above expression. 
9. We have shown from physical considerations, that the factors 
a — b, andci-W3, must enter into the expression, for tan at 
least, to the first degree ; there is, however, no physical reason for 
