Reflexion and Refraction of Light, 425 



X-=(l-jJ,m'i)*; V-Sfl-Jfi*) 1 . . (60); 

 and 



m^tfi? ( 61 )- 



18. When a and « ; are small in comparison with j3, we 

 have, approximately, 



and (58), and (52) with (47) (48), become approximately 



-sin 2 ^ (S ~ ?)2 & r 6 2^> 



and 



4 sece(^-cos?,-l- -^ cos 2 J -g- . . (63); 



which show that the energy carried away by the reflected and 

 refracted condensational-rarefactional waves (62) is very small 

 in comparison with the activity (63) of the incident distor- 

 tional wave, whatever the angle of incidence. It is to be 

 remarked that the wave-length of the condensational-rarefac- 

 tional wave, in the upper medium for example, is a//3 of the 

 wave-length of the distortional wave, while, as we see by 

 (61), (46), (47), (48), their amplitudes of vibration are 

 comparable. Hence if we suppose u//3 infinitely small, we 

 must suppose the ratio of the vibrational amplitude of the 

 incident wave to its wave-length to be infinitely small in 

 comparison with a//3, in order that our formulas may still 

 hold, or, which is the same, in order that the condensations 

 and rarefactions may be infinitely small. 



19. Without further preface, let A = 0; which makes a = 0; 

 and \ = c© } and gives, by (47) and (48), 



jj^piA . . (64) _ 



tan (* + ?,) v fl 



which is Fresnel's " tangent-formula.'' 





6 h 



sin 2 i cot i t 



(i 



sin 2 ^ cot i 



F ~ 



Si + k 



sin* i cot i, 

 sin 2 ij ' cot i 



