16 Prof. Powell on the Demonstration of Fresnel's Formulas 



as they fulfil the conditioDs with the upper or lower sign : in 

 this way we reproduce the foregoing results, which it is needless 

 here to repeat. 



Experimental Evidence. 



53. That the whole series of experimental results relative to 

 polarization are accurately represented by the foregoing formulas, 

 has been fully substantiated by the researches of Fresnel, Arago, 

 Brewster, and others; but these changes aiFord no distinctive 

 test between the several theories here adverted to. 



54. It may be desirable more particularly here to notice the 

 interpretation of the change of sign in these formulas. 



In general it is obvious, that in the elementary formula 



M=A sin -r— [vt^x); 



if we suppose x increased to ^+ -, we have 



At 



M' = Asin— fv/— a?— ^)=Asin —{vt—x) — ir \=—h%\n — {vt—x\ 

 or that a difference of sign indicates a change of 180° in phase, 

 and that this is equivalent to a change of jr in route. 



55. This conclusion includes the solution of the difficulty as 

 to the loss or gain of half a wave-length between reflexion at the 

 first surface of a dense medium represented by sin («— r), where 

 we have i > r ; and at the second, where we have i < r, which 

 gives —sin (i— r), — a difficulty which so long embarrassed the 

 early history of the theory, and which it is perhaps desirable to 

 state thus clearly, though of so elementary a character, as it 

 would sometimes appear to be still felt. 



There are, however, other cases when this change of sign is 

 material. 



Intensities of Light, 



56. The intensities of the reflected and refracted rays at dif- 

 ferent incidences, as compared with that of the incident light, 

 are measured by the vires vivce, that is, by the vibrating masses 

 multiplied by the squares of the amplitudes or velocities. 



Thus, considering the intensity of the incident ray as unity, 

 for (H) let 



: I=mA^ \^mfif, 



And for (K) let 



