Vibrations of a Polarized Ray. 105 



ordinary law, and that the character of the ray is not altered in 

 other respects. It will be necessary, first, to determine on this 

 hypothesis the ratio of the condensation of a given portion of a 

 given wave before intromittence, to the condensation of the same 

 portion in the medium, For this purpose let us select a small 

 rectangular portion, such that it has an edge, of length d-^, per- 

 pendicular to the plane of refraction, a second, of length </g, 

 parallel to the direction of propagation, and the third, which 

 will be parallel to the plane front of the wave, of length d^. 

 Let s be the mean condensation of the rectangular parallelo- 

 pipedon before intromittence into the medium, and s' its mean 

 condensation after; and let d\, d'^, d'.^ be the lengths of the 

 edges in the latter case. Then by the law of refraction, (p and 

 <p' being the angles of incidence and refraction, 



^ = 1 and ^3 _ cos (/) ^ 

 d\ ' d'2 cos (f>'' 



Also, since a given number of waves without the medium are 

 compressed after entering the medium within a space which is 

 less than the space they occupied before in proportion to the less 

 velocity of propagation, we have 



dc, sin (f> , ,, d. da do sin <i> cos <i 



jf = ■ ^„ and consequently ,.' ,; •. - = -. — j-. ^. 



a'2 sm^' ^ •' d\d'cid'^ sm0'cos</>' 



But the quantity of condensation remains the same, if we neglect 

 the loss by reflexion, and the increment caused by the occupa- 

 tion of space by the atoms of the medium. Hence we shall 

 have nearly, 



5' _ J, ^2 d^ _ sin (^ cos ^ 



Consequently 



s d\d'^d'^ sin <p' cos ^'' 



, sin 2(6 

 s'=s. ■ J., . 



In the particular case in which s' = s, ^ and ^' are complements 

 of each other, and //, being the index of refraction, 



a= -. — Vi = tau ©. 

 "^ sm 9' ^ 



According as (/> is less or greater than this critical value, s' is 

 greater or less than s. 



Now supposing the wave-ray to be incident on the medium, 

 reflexion necessarily takes place at the surface, on account of the 

 sudden retardation which the vibrations there undergo by the 

 aggregate effect of occupation of space by the atoms of the me- 

 dium, and the obstacles they present to the free motion of the 

 fluid. 'J'hy reflexion, iu this, as iu every instance, is due to aa 



