250 FRESNEL. 



The interferences of rays have occupied so great a 

 space in this biography that I cannot dispense with 



portional to their refractions or retardations, or inversely as the den- 

 sities, that is, as sin r; sin i ; and drawing parallels to them, the 



breadths of the parallelograms on the same base are easily seen to be 

 in the ratio of cos i; cos r, and thus the ratio of the simultaneously 

 vibrating masses is, 



m sin r cos i 



mi sin i cos 7- 



Hence Fresnel deduced for vibrations parallel to the plane of inci- 

 dence the ratio of the amplitudes, that of the incident ray being 

 unity, 



sin 2 i — sin 2 ?• tan (i — r\ ,„ . 



reflected h' = . „ . , . -„— = , )■, ! (3-) 



sui 2 I -1- sin 2 »• tan {i -f- r) 



4 sin r cos i f^ tan (i — r) v cos J. 



refracted A;/ = -^—^ r— r — =(1— -^ — -r- — ; ) — — - (4-) 



sm 2 I -h sm 2 r ^ tan (i-|- »•) y cos r. 



For vibrations perpendicular to the plane of incidence he found, 



^^^_^n(i^ (5.) 



sm(t+ r) 



, 2 sin r cos i ,„ . 



sm (t+r) 

 As to the mode of deducing these formulas, considerable discussion 

 has arisen, and the question cannot be regarded as yet settled. 



On merely geometrical grounds, the directions of the incident re- 

 flected and refracted rays are seen to form a triangle, whose angles 

 are (i + r), (» — r), and tt — 2 i), and their sines being as the opposite 

 sides h h' hi we have, considering h for the incident ray as unity. 



