404 NEUMANN ON A METHOD OF ESTIMATING THE 
P= + DF 
a cn beam 
cos (4 + ¢) R,’ + cos? ( — ¢’) D,*. 
From these we obtain— 
R,? = P? tan (¢—4/)\7,, 2 sin 2 $ sin 2 4! p 
tan(¢+¢')/ ~” ~~ cos? (¢— ¢!) sin? ( + ¢') 
J ee saler@)\'p o_ Sin? dsin2 ¢ gy 
Rh =§S ees —— sin? ($ + ¢') 
Fresnel’s formulz are thus proved completely and alone from 
observation. The same considerations apply to the posterior 
surface of a transparent medium; moreover, Brewster* from his 
experiments has deduced the same expression for z and = as 
for the anterior surface, whence R,?, R,”, D,*, D,” have the same 
value as at the anterior surface. 
M. Cauchy has deduced formulz for the intensity of the re- 
fracted and reflected light at the surfaces of two media, from his 
new principlest. In his second letter, addressed to M. Librif, 
he has given the following formule for the refracted light, using 
the same signs: 
D?2= 4 sin? ¢! cos? 9__ 4sin? ¢! cos? > S? (a.) 
Pp ™~ sin? (¢ + ¢') cos? (¢— > rhe . = "sin? (¢ + ’) 
M. Cauchy draws a conclusion from these formule which must 
have greatly astonished philosophers, 7. e. that at the moment at 
which total reflexion occurs, and when, consequently, the reflected 
light possesses the same intensity as the incident, the trans- 
mitted ray, instead of disappearing, becomes much increased in 
intensity. In fact this result is at once evident. If sin ¢'=1, 
: 1 : : 
and consequently sin ¢ = ay? Mm expressing the coefficient of re- 
fraction, we obtain from (a.), 
D4 Pe ae 
whence the transmitted ray would be 4 m?, or four times as great 
as the incident, according as it has become polarized perpen- 
dicular, or parallel to the plane of incidence; for ordinary 
light, the increase would be 2 (m? + 1), ¢. e. when glass is used 
* Phil. Trans., Part I., 1830, p. 145. 
+ Mémoire sur la Dispersion, &c. p. 203. 
t Comptes Rendus, 1836, lier Sem. p. 427. 
