408 Professor MacCullagh on the Law of Double Refraction. 



contains also the squares and products of the alternate deriva- 

 tives Xj, Y,, Z 15 X 3 , Y 3 , Z 3 , &c., then we get, of course, a dif- 

 ferent law. Now I find that there will still be two optic axes 

 for each colour, and that the two directions of vibration in a 

 given wave-plane will have the same relation to them as be- 

 fore ; while the difference of the squares of the two velocities 

 of propagation will continue proportional to the product of 

 the sines of the angles which the wave normal makes with the 

 optic axes; but the sum of the squares of these velocities will 

 be increased or diminished by a quantity proportional to the 

 square of a perpendicular let fall from the centre on the tan- 

 gent plane of a certain very small ellipsoid, this tangent plane 

 being supposed parallel to the wave. Such is the general re- 

 sult for biaxal crystals ; but its bearing will be best perceived 

 by taking the case of a uniaxal crystal, wherein the law of 

 Fresnel reduces itself to that of Huyghens. 



In this case the wave-surface will, instead of the sphere and 

 spheroid of Huyghens, consist of two ellipsoids touching each 

 other at the extremities of a common diameter, which coin- 

 cides with the axis of the crystal; one ellipsoid differing slightly 

 from a sphere, the other slightly from a spheroid. Neither 

 of the rays will be refracted according to the ordinary law, 

 nor will the wave-surface be symmetrical round the axis. As 

 the law of refraction is unsymmetrical, that of reflexion will 

 be so likewise, and thus we may perhaps obtain an explana- 

 tion of the extraordinary phaenomena observed by Sir David 

 Brewster in reflexion at the common surface of oil of cassia 

 and Iceland spar. 



It will no doubt appear strange to call in question the ac- 

 curacy of the Huyghenian law, which is generally considered 

 to be established beyond dispute by the experiments of Wol- 

 laston and Malus. But the fact is that no exact experiments 

 have ever been made on the refraction of the ordinary ray. 

 Neither of those philosophers seems to have entertained any 

 suspicion that the ordinary law might be inapplicable to it ; 

 they both took for granted that it followed the law of Snellius. 

 But their results seem to be quite consistent with the suppo- 

 sition that the ordinary index, for rays passing in different 

 directions through Iceland spar, may vary in the third place 

 of decimals, perhaps even in the second. The experiments 

 of Rudberg throw no light upon the question, for it happens, 

 oddly enough, that though he had two prisms in every other 

 case, he used only one of Iceland spar ; he could not there- 

 fore compare the velocities of rays passing in different direc- 

 tions. On comparing his numbers, however, with those of 

 Wollaston and Malus, there is, as Sir David Brewster has 



