PY 
applied to the Wave Theory of Light. 259 
52. If the equation of the surface of refraction be transformed, so that the plane of 
xy may coincide with the face of the crystal, and the axis of z be perpendicular to it, 
the origin of coordinates being at the centre O, no change will be produced in & or in 
y by the motion of the surface, because P.M, the direction of the motion, is now pa- 
rallel to the axis of z ; but z will be diminished or increased by 7; and accordingly, 
if U'=0 be the equation of the surface in its first position, when the centre is at O, 
and if U’ become U' + V"’ when z becomes 2 +2,—the equation of the surface in its 
second position, when the centre has moved through a distance equal to J along the 
axis of z, will be U'+ V'=0; and these two equations combined will give G=0) 
V’=0, for the equations of one of the ring-edges. ‘The equations of the other ring- 
edge are deduced from these by changing the sign of J. 
The projection of each of the ring-edges on the plane of «y is the curve traced by 
the point # on the surface of the crystal (50). This curve may be called a ring- 
trace. Its equation is obtained by eliminating z between the equations of a ring-edge ; 
and as the result must be the same whether J be taken positive or negative, the equa- 
tion of the ring-trace, when found by this general method, will contain only even 
powers of J. The radii drawn from O to the points # of the ring-trace, are (50) 
the sines (to the radius OS,) of the angles of incidence or emergence of the rays that, 
form an optical ring ; the rays that come from this ring to the eye being parallel to 
the sides of the cone described by the right line S'OS while the point # describes the 
ring-trace. 
53. It is evident that tangents to the ring-edges, at the points P and M, are paral- 
lel to each other, and therefore parallel to the intersection of two planes touching 
the surface of refraction at P and MM, because these tangent planes pass through the 
tangents. But the directions OP’, OM’, are perpendicular to the tangent planes, 
and therefore the plane P OM, containing the two rays, is perpendicular to the inter- 
section of the tangent planes, and of course perpendicular to the parallel tangents. 
Hence the plane P’'OM intersects the face of the crystal in a right line perpendicular 
to the projection of the parallel tangents on the face of the crystal. As this projec- 
tion is a tangent to the curve described by R, it follows that the normal to the ring- | 
trace at the point # is parallel to the line joining the points in which the two refracted 
rays cut the second surface of the crystal. 
In like manner, taking any two consecutive rays (P and m) having a common extre- 
mity on one surface of the crystal, the line joining the points where these rays cut the 
other surface, is parallel to the normal at the point # of the ring-trace which is de- 
scribed when the intercept (Pm) between the letters that mark the rays is supposed to 
remain constant. 
54. In all that precedes we have made no supposition about the surface of refrac- 
tion except that it isa surface of two sheets ; andif we supposed it to have three sheets, 
the conclusions would be easily extended to this hypothesis. 
VOL. XVII. 3.6 
