132 On tJie Laws of Crystalline 



the equivalence of vibrations is true for metals,* as well as for 

 crystals, which it certainly is not. It is not easy to see why 

 the principle should hold in the one case and not in the other ; 

 but it is probably prevented from holding, in the case of metals, 



* A few days after this Paper was read, I found reason to persuade myself 

 that in metals the vibrations parallel to the surface are equivalent, but not those 

 perpendicular to it ; and that in metals, as well as in crystals, the vis viva is 

 preserved. This persuasion was founded on a system of formulae -which I had 

 invented for expressing the laws of metallic reflexion and refraction ; and which 

 seem to represent very satisfactorily the experiments of Brewster, Phil. Trans. , 

 1830. As metallic and crystalline reflexion are kindred subjects, and will one 

 day be brought under the same theory, however distinct they may now appear, 

 it will not be out of place to insert the formulas for metals here. These formulae 

 are not proposed as true, but as likely to be true ; and they will be found to 

 express, at least with general correctness, all the circumstances that have hitherto 

 been regarded as anomalies in the action of metals upon light. 



I suppose that for every metal there are two constants, M and x> of which 

 the first is a number greater than unity, and the second is an angle included 

 between and 90. The number M I call the modulus, and the angle x tne 

 characteristic of the metal. Both M and x varv witl1 tlie colour of the light, 



and the ratio - is probably the index of refraction. From Brewster' s ex- 



cosx 

 periments it appears that M diminishes from the red to the violet ; and therefore 



I should suppose that cos x diminishes in a greater ratio, in order that the index 

 of refraction may increase as in transparent substances. 



Put t x for the angle of incidence, and i 2 for the angle of refraction, so that 



sin t. M 



- i- = - ; (xn.) 



sin t a cos x 



and let p. be a variable determined by the condition 



cost 



fJL = - -. (XIII.) 



COSi 2 



These two relations combined will give 



which shows that p. is equal to unity at a perpendicular incidence, and that it 

 vanishes at an incidence 90, decreasing always during the internal. 



Now if plane polarized light be incident on the metal, we must distinguish two 

 principal cases, according as the light is polarized in the plane of incidence, or in 

 the perpendicular plane. In the first case, denoting the reflected and refracted 

 transversals by r 3 and r z respectively, let us put As for the change of phase in 



