260 Geometrical Propositions 
In the theory of Fresnex, the wave surface is* a biaxal whose generating ellipsoid 
has its centre at the point O, and its semiaxes parallel to the three principal directions 
of the crystal, the length of each semiaxis being equal to OS divided by one of the 
principal indices of refraction. The surface of refraction is reciprocal to the wave 
surface, aud is (11) theretore another biaxal generated by an ellipsoid reciprocal to 
the former, having its centre at the same point O, and the directions of its semiaxes 
the same as before, the rectangle under each coincident pair of semiaxes being equal 
to k* or OS*.. Hence the semiaxes of the ellipsoid which generates the biaxal surface 
of refraction are equal in length to OS multiplied by each of the three principal in- 
dices. ‘This biaxal surface is of course to be substituted for the surface of refraction 
in the preceding observations. 
55. When the line #S, produced below S, passes through a node WN of the biaxal 
surface of refraction, the points P, M, coincide in the point N, and the interval PM 
vanishes. At the point N there are an infinite number of tangent planes, and the 
perpendiculars from O on these tangent planes give a cone of refracted rays whose 
sections we have already shown how to determine (20). All the rays in this cone, on 
arriving at the second surface of the crystal, emerge parallel to the incident ray OS ; 
‘and if the rays in the emergent cylinder be cut by a plane perpendicular to their com- 
mon direction, they will all arrive at this plane at the same instant, because the inter- 
val PM vanishes. See art. 47. 
56. Suppose fig. 5 to be a section of the wave surface. The right line Od will pass 
through V ; and the circle of contact, described on the diameter di in a plane per- 
pendicular to the right liné OdN, will be a section of the refracted cone. Now it will be 
recollected that, in general, the vibrations of a ray OT, which goes to any point 7’ of 
the wave surface, are parallel to the line which joins the point 7' with the foot of the 
perpendicular let fall from O on the tangent plane at 7’. “In the present case, the per- 
pendicular is the same for all the rays of the refracted cone, and its extremity coin- 
cides with the point d : so that the line d 7; drawn from d to any point T of the circle 
of contact, is parallel to the vibrations of the ray O7' which passes through 7° 
Conceive, therefore, a plane perpendicular to ON at the nodal pomt N. This 
plane will cut the refracted cone in a circle whose circumference will pass through WN ; 
and a line NT", drawn from the node to any other point 7" of the circumference, _ 
will be the direction of the vibrations in aray O7” which crosses the circle at this 
point. The plane of polarisation is perpendicular to the direction of the vibrations. 
57. The transverse section of the emergent cylinder is always a very small ellipse, 
affording a hollow pencil of parallel rays in complete accordance (55). If the crystal 
be thin, this ellipse will be of evanescent magnitude. Hence the line OS will be the 
direction of a line drawn from the eye to the centre of the rings commonly observed 
* Trans. R. I. A. Vol. XVI. p. 76., + Ibid. 
A) 
