DOUBLE REFRACTION. IGl 



axis as tlie greatest index of refraction is to the least. In the plane of inci- 

 dence lay down the path of the ordinary ray according to the law of Snellius. 

 Through the intersection of this path with the surface of the sphere pass a 

 plane tangent to the sphere ; and through the intersection of this tangent plane 

 with the plane of the refracting surface pass another plane tangent to the 

 spheroid. The radius of the spheroid drawn to the point of contact Avill be the 

 path of the extraordinary ray. 



The following is the construction usually given in this case. It has the 

 advantage of involving only the angle of incidence. Let the velocity before 

 refraction be represented by unity; and after refraction let the velocity of the 

 ordinary ray be v, and that of the extraordinary ray perpendicular to the optic 

 axis be i''. With the point of incidence as a centre let there be described a 

 sphere and a spheroid, as before, the radius of the sphere being = r, and the 

 revolving axis of the spheroid being =«/. On the plane of the refracting sur- 

 face and in the plane of incidence take a distance (in the direction of progress) 

 equal to the cosecant of the angle of incidence, and through the point so de- 

 termined draw a perpendicular to the plane of incidence. The plane passing 

 through this perpendicular and touching the spheroid, determines the direction 

 of the extraordinary ray, which, as before, coincides with the radius to the 

 point of contact. 



To illustrate by a comparatively simple case. In figure 34, let MN be the 

 surfiice of the crystal, and CD the direction of the optic axis. Let also the plane 



of incidence (represented by the plane of the figure) 

 be a principal section. EC being the direction of 

 the ray and C the point of incidence, make RC = 

 unity, draw CG perpendicidar to RC, and RGr par- 

 allel to MX, cutting CCI in G. Draw GQ parallel 

 to RC Then GQ being made radius, QC is the 

 cosecant of QCG = PCR = the angle of incidence, 

 p.^ .,, Then if, with the centre C, a sphere be described 



''' " ' whose radius is CD = v, and also a spheroid whose 



polar radius is CD aloO, and whose equatorial radius is made =: v', the tangents 

 QE, QF, drawm through Q to the sphere and spheroid respectively, will deter- 

 mine the directions of CE, CF, the ordinary and extraordinary rays. 



The hypothesis of Iluyghens comi)letely determines the geometrical law of 

 double refraction, but it leaves the physical cause of the phenomenon unex- 

 plained. It is easy to understand how the disturbance at a single point of the 

 molecules of an ether of uniform density and elasticity shoxild produce a spheri- 

 cal wave; and it is also easy to comprehend how a similar disturbance in an 

 ether of variable density or variable elasticity should produce a wave having 

 a surface not spherical ; but, as undulation was understood in the time of 

 Iluyghens, it was not easy to comprehend how waves of both these descriptions 

 should be goneriited in the same ether simultaneously. We have alreadv seen 

 that Iluyghens himself was very greatly astonished to observe that, when the 

 two rays into which a single incident ray is divided by double refaction in a 

 crystal of Icland spar, fall, after emergence, upon a second similar crystal, they 

 are, each of them, in some, and in fact, in most positions of this second crystal, 

 again divided, so as, of the single original ray, to make four; while in other 

 positions they arc not so subdivided, but remain two only. But, in fact, this 

 new phenomenon presents no really new cause of astonishment. The thing 

 which ought to have surprised him, the point which involved, in truth, all tlnf 

 difficulty of the conception, lay in the actual coexistence of dissimilar waves in 

 the same ethci-. No mode of explaining this fact could well have failed to ex- 

 plain the other at the same time. 



It was only, however, after the entire inadequacy of any theory of light which 

 11 s 



