Crystalline Reflexion and Refraction. 177 



the plane of incidence, the transversals are all perpendicular 

 to that plane. Taking 2 sin *\ to represent the length of the 

 incident or reflected ray, the proportional length of the re- 

 fracted ray is 2 sin fl' 2 , and the projections of these lengths on 

 the plane of y So are 2 sin e\ cos ^ and 2 sin i* cos * 2 , or sin 2^ 

 and sin 2 i 2 . The transversals applied at the extremities of the 

 rays are not altered by being projected on the plane of y s ; 

 therefore the moments of the incident, reflected, and refracted 

 transversals, projected on this plane, are represented by the 

 quantities TI sin 2^ - T\ sin 2&i, and r z sin 2i 2 respectively. 

 Equating the last moment to the sum of the other two, and 

 the refracted transversal to the sum of the other two trans- 

 versals, we get 



(TI - T'I) sin 2ii = T Z sin 2/ 2 , TI + r\ = r 2 ; 



and thence 



tan (! - 4) sin 2\ 



i I . -/-, . Z i . . 



tan fyi + 4) sm (*i + h) cos (^ - 4) 



This case has been considered by JPresnel. The relative 

 magnitudes of the incident and reflected transversals, as given 

 by him, are in accordance* with the formulae (32) and (33) ; 

 but with respect to the refracted transversals, his results do 

 not agree with the formulae. 



SECT. VI. PRESERVATION OF Vis VIVA THEOREM OF THE 

 POLAR PLANE CONCLUSION. 



Eeturning to the general question, if we resolve the trans- 

 versals parallel to the axes of # , 2/ 0> *o, and equate the sums 

 of the parallel components in one medium to the corresponding 



* There is, however, a difference as to the relative directions of the incident 

 and reflected transversals. "When the second medium is the denser, and the inci- 

 dence is perpendicular, these transversals, according to the present theory, have 

 the same direction, but according to Fresnel they have opposite directions. 



TV 



