44 Mr. Mac Cullagh on the Laws of 



that, when there is only one refracted ray, the three transversals lie in the polar 

 plane of that ray, according to the general remark with ivhich we set out. We 

 now proceed to show that the theorem asserted in this remark is a consequence 

 of our hypotheses, and we shall afterwards deduce a few results which may be 

 readily compared with experiments. 



Let us suppose then that the direction of the incident transversal is such that 

 there is only one refracted ray. It is evident that, in this case, the three 

 transversals must lie in the same plane, since, by the fourth hypothesis, the 

 refracted vibration 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 r,, r^, r^ be the respective lengths of the 

 incident, refracted, and reflected transversals ; let 6^, 9^, 0^ be the angles which 

 they make with the plane of incidence, the angle 0^ being known from .the 

 -theory of Fresnel ; put j,, i^, i, for the angles made by the respective wave 

 planes with the surface of the crystal, and m^, m^, tn, for the relative quantities 

 of ethpv set in motion by each wave. Then our hypotheses will give us the four 

 following equations : — 



m T ^=m T '^-{-m t ^ (I) 



, 1122'33 V/ 



.T^sin0_+T^sin6i,=;T^sin0^, (2) 



T,COS0jCOS ij-j-TjCOSOjCOS tj=Tj,COS0^COS i^, (3) 



T.cos^sln t,+T COS0 sin t =tcos0 sin t . (4) 



The first equatidft is manifestly the translation of the law of the preservation 

 V of vis viva ; the other three are obtained from the principle of equivalent vibra- 



each by — — > inorder to produce the corresponding expressions which result from Fresnel's hypotheses. 

 1 



But the two theories also differ as to the relative directions of the incident and reflected transversals. 



For, supposing the light to fall upon the denser medium, or t^ to be greater than t^, our construction 



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 



Iceases as we approach the limiting incidence of 90° ; for then, according to both theories, the inci- 



'dent and reflected transversals point in opposite directions. This last conclusion is conformable to 



the inference which Professor Lloyd has drawn from his experiments on the interference of direct 



light with light reflected at a very obUque incidence. — See Irish Acafl. Trans, vol. xvii. p. 176. 



