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 



mi sin i cos r 



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

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

 unity, 



reflected V = - = tan (i - r> 



sin 2 t + sin 2 r tan (z + r) 



refracted *, - . * in r co. < = ( a _ tan (-r) , cos^ 



sin 2 t + sin 2 r V tan (a-|- r) ) cos r. . 

 For vibrations perpendicular to the plane of incidence he found, 



sra( + r) 

 7 2 sin r cos t 



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 l**4***)i (i r), and TT 2 t), and their sines being as the opposite 

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



