where 



of Light by thin Layers of Metal. 455 



(q,q 3 -l)sm(e-w) 



tani/r'= 



ll + ls+Wlfc+l) C0S ( e - U ) 

 frt.rt_ — "H av 



tan^)' 



j_ (qiq3~l)sin(6-^) 



q J +q 3 -(qiq3 + 1 )cos(e-w) 



In order to exclude all uncertainty in respect of the quadrants 

 in which the phases are to be taken, the tangent-formula? are so 

 written that their numerators and denominators always agree in 

 sign with the sines and cosines. The amplitude of the incident 

 ray is put equal to 1 ; and the remaining amplitudes, as well as 

 the square roots occurring in the expressions for d u „ b JU , are 

 always to be taken according to their absolute positive values. 



The formulae-systems I. and II. differ only in this, that in the 

 latter (\ u and — u take the place of q v and u ; consequently also 

 U„ and 2?„ take the pace of U„ and Y v ; while L and D preserve 

 their values unchanged. Moreover the expression for a lu 2 con- 



2 



tains in addition the factor -I. 



c 2 • - 



3 

 For perpendicular incidence, 



and the two systems become identical. It is true that, for re- 

 flected rays, a difference of phase d' — b'= ±7r is indicated in the 

 formula? ; but this proceeds solely from the reversal of direction 

 of the ray. That is to say, if the a?ther-vibrations take place 

 perpendicular to the plane of incidence (chosen as plane of coor- 

 dinates), the same direction of vibration corresponds to the same 

 phase of the incident and reflected ray; but if the vibrations 

 take place in the plane of incidence, to the same phase corre- 

 sponds the opposite direction of vibration of the normally re- 

 flected ray. 



When 6i>0 3 , the formula? remain valid for all values of the 

 angle of incidence ; then ^ x =go , ^ = 0, and constantly 



If, on the contrary, 6 3 > d v the formula? hold good only as far 

 as the limiting angle of total reflection at the third medium, for 



which angle SJ = 7p, s 3 =l, q = co , q 3 =0. The cases 3 >lre- 



. ■ . 3 

 quires a special discussion. 



For a vanishing thickness of the metal or for A = 0, L = 0, 



and D = l, the expressions for the intensity become FresnePs 



formula? of intensity ; while the tangents of the phases become 



0, as was to be expected, since in the derivation of system II. 



the coefficient of ellipticity was taken as = 0. 



