144 Professor Hamiiton’s Third Supplement on Systems of Rays. 
surface B, and reciprocally ; and, accordingly the cusps and circles on FREsNEL’s wave 
are connected with circles and cusps on the corresponding surface of components, which 
latter surface is indeed deducible from the former by merely changing the semiaxes of 
elasticity abc to their reciprocals. And it was in fact by this general theorem that I 
was led to discover the four circles of contact on FrrsNnew’s wave, by concluding that 
this wave must touch four planes in curves instead of points of contact, as soon as I 
had perceived the existence of four conoidal cusps on the surface of components, by 
obtaining (in some investigations respecting the aberrations of biaxal lenses) the 
formula (G@*), which is the approximate equation of such a cusp. I easily found also 
that there were only four such cusps on each of the two reciprocal surfaces, and there- 
fore concluded that there were only four curves of plane contact on each. I may 
mention that though I have taken care to attribute to M. Caucuy the discovery of 
the surface of components, yet before I met the Hwercices de Mathématiques, I was 
familiar, in my own investigations, with the existence and with the foregoing properties 
of this surface: it is indeed immediately suggested by the first principles of my view 
of optics, since it constructs the fundamental partial differential equation 
8V OV OV 
(Seay? & )=° 
which my characteristic function V must satisfy in a final uniform medium. 
The surface of components possesses many other interesting properties, for exam- 
ple the following, that in a final uniform medium any two conjugate planes of ver- 
gency (") are perpendicular to two conjugate tangents on it: which is analogous 
to the less simple relations considered in the twenty-first number. But the length to 
which this Supplement has extended, confines me here to remarking, that the general 
equations of reflexion or refraction, 
Nise 10, Ac =O; (Q”) 
may be thus enunciated; the corresponding points (c, 7, v, and « + Ac, 7 + Ar, v + Av) 
upon the surface or surfaces of components (0 = 2, 0 = 2+AQ,) before and 
after any reflexion or refraction ordinary or extraordinary, are situated on one 
common perpendicular to the plane which touches the reflecting or refracting 
surface at the point of reflexion or refraction ; a new geometrical relation, which 
gives a new and general construction to determine a reflected or refracted ray, simpler 
in many cases than the construction proposed by Huycuens. 
