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 less 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 (p) at (ij 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 i = 0, (3 = a, or the plane of 

 polarization is unchanged. 



When (i + r) 90, p = 0, or at the angle of complete polarization 

 p coincides with i. 



When i =. 90, /3 = a again, or p has its original position. 



If a = 0, h sin a = 0, and if at the same time (i + r) = 90, then 

 M = 0, or we also see that at the polarizing angle an incident ray polar 

 ized in i 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 observation originally made by Malus.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 known if i =. 90, sin i = . 



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 h and M, sin 

 (i r) = cos i, sin (i + r) = sin i and tan (i r) = cot i tan (i + r) 

 = tan tV; whence h = 1 and k = 1, which accords with total re 

 flexion. 



At incidences greater than this the values become imaginary ; and 

 by introducing into them empirically certain terms i multiplied by 

 V-^Fresnel obtained in such cases an expression of the form, 



2?r 



(cos + +/ I sin 0) sin (vt x) 



A 



i See Airv s Tract, Art. 153. 



