140 Professor Hamitton’s Third Supplement 
wave, containing any vibration near the cusp, contains either the cusp-ray itself, or a 
line parallel to this ray ; so that the direction of any near vibration coincides with or 
is parallel to the projection of the cusp-ray on the corresponding tangent plane of the 
wave, or of the cone which touches it at the cusp: and the formula (L”) shows that 
all these near vibrations are parallel to one common plane, which is easily seen to be 
perpendicular to the plane of ac, and to contain the tangent at the cusp to the 
elliptic section (I') of the wave, made by this latter plane ; so that all the planes of 
polarisation near the cusp, contain, or are parallel to, the normal of this elliptic 
section. And the direction of any near vibration on the wave, or on its tangent cone, 
may be obtained by cutting the corresponding tangent plane of this wave or cone by 
a plane perpendicular to this elliptic normal. 
If the cusp-ray be incident perpendicularly on a refracting face of the crystal, then 
the internal components o, 7, are equal to the direction-cosines a, 8, of the corres- 
ponding ray of the emerging external cone ; and therefore, by (A), the plane of 
refraction of this external ray contains the internal vibration, and therefore also, by 
FresNnev’s principles, the external vibration corresponding: so that, i the external 
conical polarisation, produced by the perpendicular internal incidence of a cusp-ray, 
the plane of polarisation of an external ray is perpendicular to its plane of refrac- 
tion ; and therefore revolves about half as rapidly as the plane containing, this 
emergent ray and passing through the approximate axis of the nearly circular 
emergent cone, when the biaxal energy is small. We see also, by (AX™), that the 
plane containing the cusp-ray and containing or parallel to a near internal ray, 
revolves with double the rapidity of the plane containing the cusp-ray and parallel to 
the near waye-normal ; and therefore, in the case of perpendicular incidence of the 
cusp-ray, the plane of incidence of a near internal ray revolves with double the 
rapidity of the plane of external refraction, which, as we have seen, contains here the 
external vibrations. 
In general, the equations of polarisation (#""*), which we have deduced from Fres- 
NEL’s principles, conduct, by (J) (L"*), to the following simple formula 
aradx + b°Pdy + c’ydz =0, (M”) 
dx, dy, z being still the components of displacement parallel to the semiaxis a, 6, ¢, 
and a, (3, y being still the cosines of the inclinations of the ray to the same semiaxes of 
elasticity : and this formula (J17*°), when combined with the equation of transversal 
vibrations, 
8V=0, or, od4+ roy + +vdz=0, (A*®) 
determines easily the direction of vibration for any given direction and velocity of a 
ray, that is, for any point of Fresnev’s curved wave propagated from a luminous origin 
