250 FRESNEL. 
The intérferences 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?; 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 @ 
mi sin 7 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%—sin2r tan (i— ry 
seHlocted rs sin2i+sin2r  tan(@@+r) (3) 
__-4sin r cost pre tan (i— rr), cos i. 
retracted 5) 2 sin2%+ sin27r ( tan (t+ 7) ) cos r. (4.) 
For vibrations perpendicular to the plane of incidence he found, 
— sin (ti—r) 
Ne sin (t+ 1) (5.) 
__ 2sin r cost (6.) 
sin (@-+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 (+7), (t—r), and 7 —2 7), and their sines being as the opposite 
sides h h! hi we have, considering h for the incident ray as unity, 
: 
| 
