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 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 - sin 2 * ~ sin 2 r _ tan &amp;lt;* ~ H (3 .) 



~ 2r ~ tan(t+r) 



4 sin r cos i f tan (t ) \ cos i. 



refracted */ - gin 2 ; + 8in 2 r =( l ~ tan (+ r) ) coTrT (4&amp;lt;) 



For vibrations perpendicular to the plane of incidence he found, 



sin(i r) 

 ^-IfcTi+Tj- 

 2 sin r cos ^ 

 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 + r), (i r), and TT 2 i), and their sines being as the opposite 

 sides h h&amp;gt; hi we have, considering h for the incident ray as unity, 



