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, 



in sin r cos 



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, 



sin 2 i sin 2 r tan (i r\ 



reflected Jet = . - : --- - = - - - ' (3.) 



sin 2 i + sin 2 r tan (i + r) 



f 4 sin r cos i r tan (a* r) \ cos i. 



refracted &/ = - : - - - =( 1 -- : - '- \ -- (4.) 

 sin 2 i -f- sin 2 r \ tan (-{- ?*) ) cos r. 



For vibrations perpendicular to the plane of incidence he found, 



sin (i r) 



h> = - --r.- '- (5.) 



sm(i + ?) 



2 sin r cos i 



(6.) 



sin (i -\-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-t-r), (i r), and TT 2 i), and their sines being as the opposite 

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



