IO2 On the Laws of Crystalline 



versal is such that there is only one refracted ray. It is evi- 

 dent that, in this case, the three transversals must lie in the 

 same plane, since, by the fourth hypothesis, the refracted vibra- 

 tion is the resultant of the other two vibrations ; and, therefore, 

 we have only to prove that the plane of the transversals is the 

 same as the polar plane of the refracted ray. Let TI, r 2 , r 3 be 

 the respective lengths of the incident, refracted, and reflected 

 transversals ; let 0i, 2 , #3 be the angles which they make with 

 the plane of incidence, the angle 9 2 being known from the 

 theory of Fresnel; put ii, t 2 , 3 for the angles made by the 

 respective wave planes with the surface of the crystal, and 

 *i, m z , m 3 for the relative quantities of ether set in motion 

 by each wave. Then our hypotheses will give us the four 

 following equations : 



m l TI = nh T-? + m 3 r 3 2 , (1) 



TI sin & + T 3 sin 3 = r 2 sin 2 , (2) 



TI COS 0i COS (i + T 3 COS 3 COS 3 = T 2 COS 2 COS l z , (3) 



TI cos 0! sin <i + r 3 cos 3 sin 3 = r 2 cos 2 sin j 2 . (4) 



The first equation is manifestly the translation of the law of 



and the refracted transversal is 



sin 2 t 



It has been already observed that our theory differs from that of Fresnel with 

 regard to the magnitude of the refracted transversals. The expressions (n.) and (iv.) 



must, in fact, be multiplied each by in order to produce the corresponding ex- 



SID. i j 



pressions which result from Fresnel' s hypotheses. But the two theories also differ 

 as to the relative directions of the incident and reflected transversals. For, suppos- 

 ing the light to fall upon the denser medium, or i l to be greater than 2 , our con- 

 struction indicates that these transversals, when the angle of incidence is small, 

 point in the same direction ; whereas Fresnel concludes the contrary to be the case. 

 However, the disagreement in this respect ceases as we approach the limiting inci- 

 dence of 90 ; for then, according to both theories, the incident and reflected trans- 

 versals point in opposite directions. This last conclusion is conformable to the 

 inference which Professor Lloyd has drawn from his experiments on the interfer- 

 ence of direct light with light reflected at a very oblique incidence. See Irish 

 Acad. Transactions, VOL. xvn. p. 176. 



