38 Mr. Mac Cullagh on the Laws of 



in ordinary media ; but here it is a variable ratio, and has different values for 

 the same angle of incidence. I have elsewhere* shown how to find the refracted 

 rays and waves when the incident ray is given. 



As we suppose the ethereal molecules to vibrate parallel to the transversals, 

 we may take the lengths of the transversals proportional to the magnitudes or 

 amplitudes of the vibrations ; these lengths being always measured from the 

 common origin O. Then, in virtue of our fourth hypothesis, the transversals 

 will be compounded and resolved exactly by the same rules as if they were forces 

 acting at the point O. 



We must now conceive a wave surface of the crystal, with its centre at O, 

 the point of incidence. As the velocities of rays which traverse the crystal in 

 directions parallel to the radii of its wave surface are represented by those radii, 

 so let a concentric sphere be described with a radius OS, which shall represent, 

 on the same scale, the constant velocity of light in the medium external to the 

 crystal. At any point T on the wave surface apply a tangent plane, 

 on which let fall, from O, a perpendicular OG, meeting the plane in 

 G. On this perpendicular take the length OP from O towards G, so 

 that OP shall be a third proportional to OG and the constant line OS, 

 Then while the point T describes the wave surface, the point P will 

 describe another surface reciprocal! to the wave surface. This other 

 surface may very properly be called the indea: surface,% because its 

 radius OP is the refractive index of the ray whose velocity is OT, 

 or rather of the wave TG, which belongs to that ray; for, if we conceive an 

 incident wave, touching the sphere, to be refracted into the wave TG, touching 

 the wave surface in T, the sine of the angle of incidence will be to the 

 sine of the angle of refraction as OS to OG, or as OP to OS; so that, taking 

 the constant OS for unity, the index of refraction will be represented by OP. 

 The wave surface and the index surface will thus be reciprocal to each other, 



* Irish Acad. Trans, vol. xvii. p. 252. 



f For the general theory of reciprocal surfaces, see Irish Acad. Trans, vol. xvii. p. 241. 



X This is the surface which I formerly called (ibid. p. 252) the surface of refraction; a name not 

 sufficiently descriptive. Sir W. Hamilton has called it the surface of wave slowness, and sometimes 

 the surface of components. But the name index surface seems to recommend itself, as both short 

 and expressive. 



