and Brushes " observed through a Spath Hemitrope. 565 

 Let 



• l£-'rmr)-4-n 2 t;=l, 



be the equations of the two waves in unit time af tei passing 

 through a point o of the surface of the crystal. 



[?*! and n 2 are the reciprocals of the intercepts made on 

 the normal (axis f) to the plate through the point o by the 

 refracted waves in unit time after passing through o.~\ 



If 6 be the azimuth of the plane of: incidence with respect 

 to that of £f and v be the velocity of light in air, then 



, sin i . cos 6 sin i . sin 6 



I = . m = 



v v 



where i = angle of incidence. 



Suppose, that the plane ff contains the greatest axis oz 

 of the ellipsoid of polarization, the plate being on the posi- 

 tive side of f, then if ox, oy, oz be the axes of the ellipsoid 

 and zo£=fo we get by transformation from the axes of 

 optical symmetry to the new axes, 



x = % cos cp cos x~V s i n </> + £ cos $ sm %> 

 y = £ sin <£ cos x~V cos </> + £sin <£ sin ^, 

 2 =— f sin^ + ?cos%. 



The equation to the wave surface, referred to the axes of 

 optical symmetry is 



av by' c 2 z 2 



From the condition that the plane Z£ + m?;-f nf=l should 

 touch the wave surface in the new system of coordinates, we 

 find 



5_ V|;2_ a 2 sm 2 i (a 2 — c 2 ) sin ^ cos% cos Osini 

 T a a 2 cos 2 % + c s sin 2 % 



, / v/(a 2 cos 2 % + c 2 sin 2 ^)(w 2 — c 2 sin 2 i) — c 2 (a 2 — c 2 )sin 2 ^ . cos 2 # . sin 2 *, 

 a 2 cos 2 % + c 2 sin 2 ^ 



where S = vT(n 2 — fti). 



Hence, for a crystal cut perpendicular to the axis (^ = 0), 



.-. pp = -j \'v 2 — d 2 sin 2 i — \/v 2 — a 2 sm 2 i\ /a. 



