86 On the Laws of Crystalline Reflexion. 



It remains to say a word respecting the new principle of 

 equivalent vibrations, the most important, perhaps, of all, as it 

 is certainly the simplest that can be imagned. If we conceive 

 an ethereal molecule situated at the common surface of two 

 media, it would seem that its motion ought to be the same, 

 whether we regard the molecule as belonging to the first me- 

 dium or to the second. Now the incident and reflected vibra- 

 tions are superposed in the first medium, and the refracted 

 vibrations in the second; and therefore we may infer (when 

 the phase is not changed by reflexion or refraction), that if the 

 incident and reflected vibrations be compounded, like forces acting at 

 a point, their resultant will be the same, both in length and direc- 

 tion, as the resultant of the refracted vibrations similarly com- 

 pounded. This is the law of equivalent vibrations, and it 

 gives, at once, three equations. A fourth equation is afforded 

 by Fresnel's law of the vis viva; and thus we have the four 

 conditions necessary for a general solution of the problem. 

 From the principle of equivalent vibrations, as we have 

 stated it, it follows that the vibrations resolved parallel to the 

 separating surface are equivalent in the two media; and, in 

 fact, this part of the general principle was assumed by Fresnel ; 

 but the other part, namely, that the vibrations perpendicular 

 to the separating surface are equivalent, was not assumed by 

 him, nor is it by any means true in his theory. It appears 

 then that three conditions only are afforded by the hypotheses 

 which Fresnel successfully employed in solving the problem of 

 reflexion from ordinary media. These hypotheses, therefore, 

 are not sufficient when applied to crystals; except, indeed, in 

 the case before alluded to, where the azimuth = 0,. which has 

 been solved by M. Seebeck. It should be observed, that though 

 the reasons which I have assigned for the principle of equivalent 

 vibrations are extremely simple, yet it was not by such simple 

 reasoning that I was led to it originally. 



TRINITY COLLEGE, DUBLIN, December 13, 1836. 



