134 _ Professor Hamitton’s Third Supplement 
so as to pass to a new system of rectangular co-ordinates such that the plane of x, =, 
coincides with the plane of ac, and the positive semiaxis of z, with the line w’ of sin- 
gle normal velocity, we find a new transformed equation of the wave, which may be 
thus written, 
(wit y 2+ 4,2, 07° V@—8 VB—eyY= Q A-—27b-), «L”) 
if we put for abridgment . 
Q=(a° +e?) pe + (at—e*) rr’ —a’e? (1 +2267); (M”) 
and hence it is easy to prove that the plane 
a b, (N*®) 
which is perpendicular to the line w' at its extremity, touches the wave in the whole 
extent of a circle ; the equation of this circle of contact being, in its own plane, 
eity?+ 2,07 vVae—k Ve—Ce=0. (O”) 
It is evident that there are four such circles of plane contact at the ends of the four 
lines +o’, +w’, of single normal-velocity. They are all ae to each other, and the 
common magnitude of their diameters is 0—' Va?—0? Vb?—c%. The same conclu- 
sions may be drawn from FReEsNEL’s equation of the wave in co-ordinates @ y z 
referred to the axes of elasticity: the equations of the fowr planes of circular con- 
tact being, in these co-ordinates, 
eVP—¢ +taVG@—8= tbVa—e, (P”) 
FresNeEL however does not appear himself to have ‘ 3 the existence of these 
circles of contact, nor do they seem to have been since perceived by any other person. 
We shall find that the circles and cusps, pointed out in the present number, conduct 
to some remarkable theoretical conclusions respecting the laws of refraction in biaxal 
crystals. 
New Consequences of Frusnew’s Principles. It follows from those Principles, that 
Crystals of sufficient Biaxal Energy ought to exhibit two kinds of Conical 
Refraction, an External and an Internal: a Cusp-Ray giving an External 
Cone of Rays, and a Normal of Circular Contact being connected with an 
Internal Cone. 
29. The general formule for reflexion or refraction, ordinary or extraordinary, 
which we haye deduced from the nature of the characteristic function VY, become 
simply 
