IGO UNDULATOEY THEORY OF LIGHT. 



surface ; tliat is, tliey are the angles of incidence and refraction. The law of 

 Snellius is thus very simply deduced from the theory of undulation. It also 

 appears froai the foregoing proportion that the velocity before refraction is to 

 the velocity after refraction as the sine of incidence is to the sine of refraction, 

 or as the index of refraction is to' unity. 



It was not until after Huyghens had perfected his theory of refraction, that 

 ho became acquainted with the remarkable case of Iceland spar. In order to 

 reconcile the phenomena presented by this crystal with his theory, it was 

 necessary to suppose that the incident wave is divided at the refracting surface 

 into ttvo waves having unequal velocities. But, inasmuch as one of these waves 

 must, on this hypothesis, and, in order to meet the phenomena, have a velocity 

 greater in some directions than in others, it occurred to him that the second 

 wave must probably be splieroulal, and not spherical. Following out this in- 

 genious idea, he presently discovered that it contained a perfect explanation of 

 all the apparent anomalies of double refraction ; and by generalizing the 

 method which has just been given for finding the direction of a ray after re- 

 fraction, and extending it to embrace sjihcroidal as Avell as spherical wave 

 surfaces, he contrived a geometrical construction by which the path of the ex- 

 traordinary as well as of the ordinary ray may, in all cases, be exactly deter- 

 mined. 



In order to understand this, let it be observed that the greatest and least 

 axes of the spheroidal wave will evidently be proportioned to the greatest and 

 least velocities of the ray in the crystal ; and these velocities are inversely as 

 the corresponding indexes of refraction. Also, as the least index, in the case 

 of Iceland spar, is that which is found in the plane perpendicular to the optic 

 axis, and as the refraction is there conformable to the law of Snellius, it is 

 manifest that this is the plane of greatest velocity, and is the equatorial plane 

 of the spheroid. The polar axis of the wave corresponds in direction, therefore, 

 with the optic axis of the crystal. Let it also be remembered th.it the least 

 velocity of the (extraordinary ray is the constant velocity of the ordinary ray ; 

 or is, in other words, the radius of the spherical wave. 



If, now, a plane wave of light fall obliquely upon the surf ice of a crystal of 

 Iceland spar, intersecting it in a straight line, any point of this line of inter- 

 section may be assumed as the common centre of a spherical and a spheroidal 

 wave, having one diameter in common, parallel to'the optic axis of the crystal, 

 which common diameter is the axis of revolution of the spheroid. Moreover, 

 as this supposition may be made of every other imaginable point of the line of 

 intersection, it follows that there will l)C an inlinito number of elementary 

 waves, spherical and spheroidal, simultaneously generated. As the incident 

 wave advances, the line of intersection will advance along the surface of the 

 crystal ; and in every position of this line a new set of elementary waves will 

 in like manner originate. By reasoning similar to that which was employed in 

 the illustration of ordinary refraction, it may be shown that, in any position of 

 the incident wave, all the elementary spherical waves Avill be touched by one 

 and the same tangent plane ; and all the elementary spheroidal waves will be 

 touched by one plane likewise. These two tangent planes must intersect each 

 other in the line in which the incident wave intersects, at the moment, the sur- 

 face of the crystal, or the plane of that sm-face ; but they will not coincide ex- 

 cept in the single case in which the velocities of wave progress are equal ; that 

 is to say, when the movement of progress within the crystal is in the direction 

 of the optic axis. 



The geometrical problem of determining the path of the exti*aordinary ray 

 reduces itstdf, therefore, to this. With the point of incidence as a centre 

 describe a sphere. Upon that diameter of the sphere which coincides with the 

 optic axis describe a spheroid of revolution, whoso revolving axis is to its fixed 



