AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 23 



If possible, let polarized light consist of vibrations parallel to the plane of polarization ; 

 then taking the equations of connection in their most general form, viz. (D), (E), (F), 

 Section ii., and proceeding as in Articles (14) and (15), we find, for light polarized perpen- 

 dicularly to the plane of incidence, the following formulae : 



- ' -■— s + e/xs 



C + D' 



C+D 



Now, by the law of tangents, F, ought to become zero when the tangent of the angle of 

 incidence is /x, i- e. when s = fxs. Therefore we have 



and this gives us 



€u?s = 0, and therefore e = — , 



r, = iii^F, F'= '"' 



flS + S /IS + s 



Now if U, Ui, U', U2, U", be the velocities, when the light is polarized in the plane of 

 incidence, we have, by the laws of the rotation of the plane of polarization, 



C/i s — fjis' /xs + s U U' fxs + s U 

 Fi s + fxs' ' fxs - s' V ' V' s + fxs' V 

 Hence, by the above expressions for F, and V, we have 



s + fxs s + fxs 



and from these equations we find 



M(f/+ tr,) = U' (1), s(U- U,)=s'U' (2). 



Now from the equations of connection (D), we have, as in Article 19, equations (8) and (9), 

 s(U-U,)- U,p, = s'U'- U"p", 

 p{U+U,) + U,s,^p'U'+ U"s", 

 which, by (l) and (2), and since p = ixp', become 



U2P2 = U"p", U^si = U"s". 



Now since -77 = — , and i>", v.^ are essentially positive, p" and />2 have the same sign ; 



therefore U2 and U", and therefore s^ and s" have the same sign. Now by Article (19), 

 equations (10) and (u), s^ and s" have opposite signs, which is absurd. The only way to 

 get over this, is to suppose that U^ and U" are zero, but then it will be impossible to satisfy 

 the equations of connection for all values of p. (See Article 19). 



Hence it follows that, if we adopt the hypothesis which supposes polarized light to consist 

 of vibrations parallel to the plane of polarization, and take into account the experimental 

 laws above stated, we arrive at an absurd result. 



We may therefore conclude, that Fresnel's hypothesis is true. 



