On Systems of Rays. 141 
within a biaxal crystal. And we easily see that on any wave in a biaxral crystal, whe- 
ther propagated from within or from without, the differential equation (17) deter- 
mines a series of lines of vibration, having the property that at any point of such a 
line the vibration is in the direction of the line itself. ‘To find these lines on Fres- 
NEL’s wave (0°), we may change af y to wy z in the differential equation (17), and 
we then find, by integration, 
ee +ytese, (N”) 
« being an arbitrary constant; and since this integral, when combined with the equa- 
tion (O") of the wave itself, gives 
ate ae bi+e a? +- c + ¢! 2 “e+ +c*) é—a he Cz oO” 
Ly 
we see that the lines of vibration on FRESNEL’s wave, propagated from a point in a 
biawal crystal, are the intersections of two series (N®)(O*) of concentric and co-axal 
ellipsoids. 
By this general integration, extending to the whole wave, or by integrating the 
approximate equations for vibrations near the conoidal cusps and circles of contact, 
obtained from (A”) (J*) by changing the direction-cosines of a ray to the propor- 
tional co-ordinates of the wave, we find that near a cusp the lines of vibration coincide 
nearly with small parabolic arcs on the tangent cone of the wave, in planes perpendi- 
cular to the elliptic normal already mentioned ; and that in crossing a circle of contact 
the course of each line of vibration is directed towards that point of the circle which is 
the end of the corresponding waye-normal of single velocity, that is, towards the foot 
of the perpendicular let fall from the centre of the wave on the plane of circular 
contact. 
In any Uniform Medium, the Curved TVave propagated from a point is connected 
with a certain other surface, which may be called the surface of components, by 
relations discovered by M. Caucuy, and by some new relations connected with a 
General Theorem of Reciprocity. This new Theorem of Reciprocity gives a new 
construction for the WVave, in any Undulatory Theory of Light : and it connects 
the Cusps and Circles of Contact on Fresnev’s WVave, with Circles and Cusps of 
the same kind on the Surface of Components. 
31. The theory of the wave propagated from a point in any uniform medium, may 
be much illustrated by comparing this wave with a certain other surface which appears 
to have been first discovered by M. Caucuy, who has pointed out some of its proper- 
ties in the Livraison already referred to. In that Livraison, M. Caucny has treated 
VOL, XVII. 20 
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