142 



come to the conclusion that the vihrations of such a I'ay are 

 performed in an ellipse. For if all planes passing through 

 the axis of the crystal be alike in their optical properties, 

 there will be absolutely nothing to determine the position and 

 ratio of the axes of the ellipse ; there will be no reason why its 

 major axis, for example, should lie in one of these planes, 

 rather than in any other. But, whatever may be thought of 

 this case independently of observation, it is manifestly ab- 

 surd to suppose that the vibrations are elliptical in the case 

 of a ray passing through oil of turpentine, or any other liquid 

 possessing the property of rotatory polarization ; for, in a 

 liquid, all planes drawn through the ray itself are circum- 

 stanced alike. From these simple considerations it is evident 

 that the theory of M. Cauchy is unsound; but a closer exa- 

 mination will show that it is entirely without foundation, and 

 that it is directly opposed to the very phenomena which it 

 professes to explain. To make this appear, however, in 

 the easiest way that the abstruseness of the subject will 

 allow, it will be necessary to advert to some former re- 

 searches of my own, which have a direct bearing on the 

 question. 



The same day on which M. Cauchy's letter was read to 

 the French Academy, I had the honour of reading to the 

 Royal Irish Academy, a paper " On the Laws of Double Re- 

 fraction in Quartz" (see Transactions R. I. A., vol. xvii, 

 p. 461), wherein I showed that every thing which we know 

 respecting the action of that crystal upon light is comprised 

 mathematically in the following equations : 



dPl _ J'K , „d\ 



0) 



dt'~''dz''^''dz' 



de-^dz" ^dz^' 



which differ from the common equations of vibratory motion 

 by the two additional terms containing third differential co- 



