254 FRESNEL. 
ference appear wholly inexplicable. I will add besides, 
that none of the partisans of the system of emission 
This formula exhibits remarkable changes at successive incidences: 
at incidences Jess than that of complete polarization, the new plane of 
polarization (as indicated by the sign of the tangent) deviates on the 
side of the plane opposite to that of polarization (rp) — at (1,) inci- 
dences greater, it deviates on the same side as P; results which agree 
exactly with numerous and accurate observations of Fresnel, Arago, and 
Brewster. 
We have also the following results of this last formula: 
While a has any finite value, when ¢ = 0, 8 = a, or the plane of 
polarization is unchanged. 
When (+7) = 90°, 8=0, or at the angle of complete polarization 
P coincides with I. 
‘When ¢ = 90°, 8 = a again, or P has its original position. 
If a=0,/ sina=0, and if at the same time (¢+ 7) = 90°, then 
k! = 0, or we also see that at the polarizing angle an incident ray polar- 
ized in 1 will cease to give any reflected ray; which agrees with the 
observation originally made by Malus. 
From the same formulas another more curious inference was made 
by Fresnel as follows: In passing out of a denser into a rarer medium, 
in general it is well knewn if 4 = 90°, sin i = =z 
Consequently a ray making this incidence internally on the bound- 
ing surface will not be refracted out; and at incidences more oblique 
is experimentally found to be totally reflected internally: theoretically, 
the conversation of vis viva would require that the whole vibratory 
force, since none of it is expended on refraction, must be occupied in 
communicating vibrations internally, which can only produce internal 
waves or internal total reflexion. 
Now at the critical incidence, in the formulas for 2’ and #, sin 
(i—r) = cos i, sin (¢-+ r) = sini and tan (i— rr) = cot? tan (i++ 7) 
= tan i; whence = 1 and k = 1, which accords with total re- 
flexion. 
At incidences greater than this the values redding imaginary; and 
by introducing into them empirically certain terms! multiplied by 
»/—1 Fresnel obtained in such cases ari expression of the form, 
(cos @ + ./—i sin @) sin = (vt— x) 
1 See Airy’s Tract, Art. 153. 
