46 Geometrical Propositions applied 



two emergent waves parallel to each other and to the incident 

 wave, moving along the emergent rays with equal velocities V, 



x PM 



and preserving the distance ^ between their planes, the 



minus wave being foremost. If OS, the radius of the sphere, be 

 taken for unity, PM will be a number generally a very small 

 fraction and the interval will be the thickness of the crystal 

 multiplied by this number. 



50. Suppose the right line PMR, remaining always perpen- 

 dicular to the face of the crystal, to describe a cylindrical sur- 

 face, with the condition that the part PM, intercepted between 

 the two sheets of the surface of refraction, shall remain of a con- 

 stant length ; the point R will then describe, on the surface of 

 the crystal, a curve whose radii OR are the sines (to the radius 

 08} 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. Lines 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 trans- 

 mitting polarized 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 refrac- 

 tion. The ring described by supposing pm to remain constant 

 corresponds 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, 

 are proportional (48) to intervals like those in Newton's rings 

 to the intervals, namely, between the reflected ray Os and the 

 rays SPps, SMms, 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 distinguish between the different intercepts, and 

 will therefore include all the rings PM, pm, Pp, Mm, Pm, Mp ; 



