Electromagnetic Theory of Light, 85 



obtained on two distinct suppositions. If ^1=/^; 



and 



sin^-0) 

 ( 18) - sin(0 1 + 0)' 



_ tan^-fl) . 



^ "tan (p^+ey 



but if K 1 = K, then (19) identifies itself with the sine-formula, 

 and (18) with the tangent-formula. Electrical phenomena, 

 however, lead us to prefer the former alternative, and thus to 

 the assumption that the electric displacements are perpendi- 

 cular to the plane of polarization. The formulas for the 

 refracted waves, which follow from those of the reflected waves 

 in virtue of the principle of energy alone, do not call for de- 

 tailed consideration. 



In the problem of perpendicular incidence, we have from 

 (12), if fi be constant and C zero, 



£k + ^ k @=° cw 



For an application of this equation to determine the influence 

 of defective suddenness in the transition between two uniform 

 media, the reader is referred to a paper in the eleventh volume 

 of the Proceedings of the Mathematical Society. 



In order to obtain a theory of metallic reflection, must be 

 considered to have a finite value in the second medium. The 

 symbolical solution is not thereby altered from that applicable 

 to transparent media, the effect of the finiteness of being 

 completely represented in both cases by the substitution of 

 K(l — ?'47rnCK _1 ) for K. Thus, if fi be constant, the formula 

 for the amplitude and phase of the reflected wave in case 1 is 

 to be found by transformation of (18), in which the imaginary 

 angle of refraction 0\ is connected with 6 by the relation 



K^l-^TrnCKr 1 ) : K= sin 2 6 : sin 2 V . . (22) 



In like manner the solution for case 2 is to be found by trans- 

 formation of (19) under the same supposition. 



With regard to the proposed transformations, the reader is 

 referred to a paper 037- Eisenlohr* and to some remarks there- 

 upon by myselff. The results are the formulae published 

 without proof by Cauchy. From the calculations of Eisenlohr 

 it appears that Jamin's observations cannot be reconciled with 

 the formulas without supposing K x : K, i. e. the real part of 

 the square of the complex refractive index, to be negative — a 



* Pogg. Ann, t. civ. p. 368. t Phil. Mag. May 1872. 



