DOUBLE REFRACTION. 123 



forming a species of cone. External conical refraction corresponds 

 to the cusp on the wave-surface ; and takes place without, when a 

 single internal ray coincides with either of the lines of single ray- 

 velocity. Internal conical refraction, on the other hand, takes 

 place within the crystal, when a single ray is incident externally 

 at an angle corresponding to the line of single wave-velocity 

 within. In this latter case, if the crystal be hounded by parallel 

 planes, all the rays of the cone will emerge at the second surface 

 parallel to the ray incident on the first, so as to form a small 

 elliptic cylinder, whose magnitude will depend upon the angle of 

 the cone and the thickness of the crystal. All these remarkable 

 conclusions have been verified in the fullest manner by experi- 

 ment.* 



I shall now proceed to give a brief account of the labours of 

 M. Cauchy in this interesting department of analysis. The 

 researches of this eminent mathematician, on the propagation of 

 motion in elastic media, are scattered through various livraisons of 

 the Exercices de Mathematiques ; and he has given a valuable sum- 

 mary of the results of these investigations, as applied to the wave- 

 theory of light, in a memoir read to the French Academy in the 

 year 1830.f 



Having assigned the general equations of motion of a system 

 of molecules, acting on one another by attracting or repelling 

 forces which vary according to any function of the distance, M. 

 Cauchy observes that it is not necessary to have recourse to their 

 general integrals in order to determine the laws of undulatory 

 propagation. It is sufficient, in fact, to determine the law of 

 propagation of a plane wave. For if we consider a great number 

 of plane waves inclined to one another at small angles, and which 

 are at first superposed in the neighbourhood of the point which is 

 considered as the origin of the disturbance, the vibrations in the 

 elementary waves, to which each of these gives rise, may be sup- 

 posed too small to affect the sense separately, and these waves 

 become efficacious only by superposition. Consequently the general 

 wave-surface will be the locus of all the points in which the 

 elementary plane waves are superposed ; and will therefore be the 



* " On the Phenomena presented by Light in its passage along the axes of biaxal 

 Crystals," Transactions of the Royal Irish Academy, vol. xvii. 



t " Mcmoire sur la Theorie de la Lumiere," Mem. List., torn. x. 



