258 Geometrical Propositions 
cepted between the two sheets of the surface of refraction, shall remain of a constant 
length ; the point £ will then describe, on the surface of the crystal, a curve whose 
radii O& are the sines’ (to the radius OS) of the angles of incidence of a cone of 
rays; and every ray S’O of this cone, when refracted by the crystal, will afford two 
emergent rays, or two waves, having the same given interval between them. Panes 
drawn from the eye parallel to the sides of this cone are the emergent rays belonging 
to a ring, when rings are made to appear, in any of the usual ways, on tenance 
polarised light through the plate of crystal. In nominal conformity to this, we see 
that the line PM describes a ring of constant breadth between the two sheets of the 
surface of refraction. The ring described by supposing pm to remain constant cor- 
responds to the interval between two rays p and m reflected at the same point of the 
second surface of the crystal, and then emerging at the first. The other intercepts 
Pp, Mm, Pm, Mp, ave proportional (48) to intervals like those in Newton’s-rings ; 
to the intervals, namely, between the reflected ray Os and the rays SPps, SJMdms, 
SPms, SMps, emerging at the first surface after one reflection within the crystal ; or 
to the intervals between rays that are twice reflected in the crystal and the rays trans- 
mitted without reflection. 
51. The general investigation of the figure of a geometrical ring does not distin- 
guish between the different intercepts, and will therefore include all the rings PW, 
pm, Pp, Mm, Pm, Mp ; so that it will be sufficient to contemplate any one of them, 
as PM, of which the breadth P.M is equal to a given line J. 
The points P and M describe, in general, similar and equal curves of double cur- 
vature, which may be called ring-edges, as being the edges of the ring ; and if we 
imagine the surface of refraction, carrying these curves along with it, to be shifted 
either way, in a direction parallel to PM, through a distance equal to J, it is clear 
that the new position of one of the ring-edges will exactly coincide with the first po- 
sition of the other, and that therefore the curve of the latter ring-edge will be given 
by the intersection of the two equal surfaces in these two positions. Let U=0,— 
where 0 is a function of x, y, z, and given quantities—be the equation of the sur- 
face of refraction in its original position ; and, the axes of coordinates being fixed, 
suppose that by the shifting of the surface the coordinates of a point assumed on it are 
diminished by the given lines f, g, h, which are the projections of the given line J 
on the axes of x, y, z, respectively. Then the equation of the surface in its new po- 
sition will be had by substituting +f, y+g, 2+h, for x, y, z, in the equation U=0, 
which will thus become U + V=0, where V is the increment of U produced by the sub- 
stitution. These two equations combined are equivalent to the equations U=0, 
V=0, which are therefore the equations of one of the ring-edges. If the surface 
had been shifted the opposite way, in a direction parallel to PM, the intersection 
would have been the other ring-edge, whose equations are therefore deducible from 
those already found, by changing the signs of f, g, h. 
