On Systems of Rays. 139 
and for any plane wave slightly inclined to this plane of wy, the following approx- 
imate relation between the components of normal slowness, 
U,, = ie a Li (s, = jie o,. at 7, )s (G”) 
retaining the meaning (4”™) of r,,; and if we attend to the general connexions, 
established in this Supplement, between the direction-cosines of a ray and the compo- 
nents of normal slowness of a wave, we easily deduce from (G*), by differentiation, 
the following other relations, 
a,, ou, on B,, ou, iia Ty 2 
eee ), Bee 8 Fate pn 
oy, = Ty, By 4 a?) 
This formula contains the theory of the conical polarisation connected with internal 
conical refraction. It shows that the vibrations at the circle of contact on FRESNEL’s 
wave, are in the chords of that circle drawn from the extremity of the normal w of 
single velocity ; and therefore that the corresponding planes of polarisation all pass 
through another parallel normal at the opposite point of the circle. The plane of 
polarisation, therefore, in passing from one position to another, revolves only half as 
rapidly as the revolving radius, so that the angle between any two planes of polarisa- 
tion is only half the angle between the two corresponding radii of this circle on 
Fresnew’s wave. And if we suppose that the direction of the external incident ray 
coincides with the wave-normal w’, and therefore also with the normal to the refracting 
face of the crystal, then the small internal components of normal slowness, o,r,,, 
parallel to this refracting face, are equal (by our general theorem of refraction) to the 
small external direction-cosines a, 3, of the inclinations of a near incident ray to the 
semiaxes of x, and y,; from which it follows, by (J), that the plane of external 
incidence containing this near incident ray revolves twice as rapidly as the corres- 
ponding plane of refraction. 
For the other kind of conical polarisation, connected with the external conical 
refraction, and therefore with the conoidal cusps on FREsNEL’s wave, we find by a 
similar process, 
= > 
8z, o, aE Va? a] ah 
our = By B, (K”) 
and 
oz,= —2br dr, (L”) 
r having the meaning (U"). The formula (A) shows thatthe normal plane to the 
