48 4, M. L. Lorenz on the Reflexion of Light at 
where 
Ji) = -{" du (Jan sea) 
U Uy 
From the last equation we get by differentiation 
f(uy= (“au fea) 
Uy 
and 
flu =f(u) 2 
flu) = cel + eye, 
where the constants ¢ and e¢, are to be determined by the equa- 
tions 
which gives 
Wh f(ua) =1, and f"(u,)=0. 
ence 
U—U ue —U 
4 —_ € a +e€ a 2 
fe) Or eae 
€ tea 8B 
and the value of (7) or the amplitude of the refracted ray is 
oA tan 8 
ee 
ee Vee a 
If now we wish to find the amplitude of the refracted ray 
polarized in the plane of incidence, which we will call B, we 
must in the above expression substitute 
eae PR 
u=—tlogtanz, 
and we shall find 
2) cos aisinye 
Ba2A— eae: aa (9) 
If, on the other hand, in (8) we substitute 
u= + log sin 2a, 
we find for B’, the amplitude of the portion of the refracted ray 
polarized perpendicularly to the plane of incidence, 
eis cosasin 8 
Rae sin («+ 8) cos («—£)’ 
We return, therefore, exactly to Fresnel’s formule, which is a 
remarkable property of those expressions. The calculation only 
assumes the relations indicated by (3) and (4); and these expres- 
sions might have been deduced from many other formule than 
Fresnel’s. 
The amplitude of the reflected ray, which is the sum of the 
(10) 
