On Systems of Rays. 137 
rigorous: the relations between the components of normal slowness o rv of the cusp- 
ray o are therefore 
. 
(oo'. - Up a)” jr Beir (9'. - U0 a)s 0 4 + v0. =e e (w"’) 
and the equation (ina, RB.) of the external cone of rays corresponding to the one 
internal cusp-ray ois to be found by eliminating these three internal components orv 
between the two relations (IV) and the two equations of refraction (S"). 
For example, if the internal cusp-ray p’ coincide with the inward normal to the 
refracting face F' of the crystal, we may take, for the semiaxes s, ¢ upon that face, the 
projection of a, and the semiaxis 6 of elasticity ; and then the equations of refraction 
(S”) becoming 
aap up yapi=rs (X") 
we have, by (JV "), the following polar equation of the external cone of rays, 
a2+P.2=27, a, 3 Ye) 
or, in rectangular co-ordinates, an equation of the fourth degree, 
(y+ y2) =473%a *(et+y, +2, ) (Z") 
This cone is nearly circular in all the known biaxal crystals, because the coefficient 7, 
is small, by (U'*), when the biaxal energy is weak, that is, when the semiaxes of elas- 
ticity a 6 c are nearly equal to each other : and rigorously the external cone (Z") 
meets the concentric sphere of radius unity in a curve contained on a circular cylin- 
der of radius=r,, one side of this cylinder coinciding with a ray of the cone. 
With respect to the internal conical refraction, the equation of the internal cone 
of rays corresponding to the internal wave-normal w', or normal of circular contact, 
is always, by (V") (0”), 
wit+y?+9r ec 2 =0, if r,=hb?Vae—b Voc, (A®) 
when referred to the rectangular co-ordinates w, yz, by the transformation (™) ; 
and in the simpler rectangular co-ordinates 7 y z which are parallel to the axes of 
elasticity the equation of this cone is 
(tw,—z0 2 +y° +27, (tw, — Zw z) (tw, +2zw'.)=0, (B”) 
in which we may change the co-ordinates x yz to the direction-cosines a (3 y of an 
internal ray of the cone: while the one external ray corresponding is determined by 
the following direction-cosines 
abu, Balu); (Cc) 
or by the ordinary law of proportional sines, since the internal wave-normal of circular 
contact w’, which is one ray of the internal cone, is connected with the external 
VOL. XVII. 2N 
