8& On the Laws of Crystalline 



tions, I amused myself, when I first became acquainted with 

 Fresnel's theories, by throwing his algebraical expressions, when- 

 ever I could, into a geometrical form ; and treating in this way 

 the well-known formulae in which he has embodied his solution 

 of the problem just alluded to, I obtained a remarkable result, 

 which gave me the first view of the principle that I have since 

 employed under the name of the principle of the equivalence of 

 vibrations. In order to state this result briefly, I will take leave 

 to introduce a new term for expressing a right line drawn 

 parallel to the plane of polarization of a ray, and perpendicular 

 to the direction of the ray itself. Calling such a right line the 

 transversal of the polarized ray, I found, from the formula 

 of Fresnel, that when polarized light falls upon a singly- 

 refracting medium, the transversals of the incident, of the re- 

 flected, and of the refracted rays are all parallel to the same 

 plane, which is the plane of polarization of the refracted ray ; 

 and that the magnitudes of the vibrations, or the greatest ex- 

 cursions of the ethereal molecules, in the incident and the 

 reflected rays, are to each other^ inversely as the sines of the 

 angles which the respective transversals of those rays make with 

 the transversal of the refracted ray. I was struck by the strong 

 analogy which these relations among the transversals bore to 

 the composition of forces or of small vibrations in mechanics ; 

 but it happened unfortunately that, in the theory of Fresnel, 

 the vibrations of light were supposed to take place, not in the 

 direction of the transversals, but perpendicular to them, so that 

 there was no physical circumstance to support the analogy, there 

 being no motion in the direction of the transversals ; while, on 

 the other hand, no such analogy existed among the vibrations 

 themselves in the directions which Fresnel had assigned to them. 

 It was therefore with some interest that I afterwards learned, 

 upon the publication of the tenth volume of the Memoirs of the 

 Institute, that M. Cauchy* had actually inferred, from mecha- 

 nical principles, that the vibrations of polarized light are in the 



* Memoires de f Institut, tome x. p. 304. 



