190 UNSULATORY THEORY OF LIGHT. 



is the same for the incident and reflected waves, their intensities are as 1^ and 

 v'^. When Ave compare the transmitted with the incident or reflected waves, we 



must consider the masses. By mnltiplyin,'^" y ^ by — —, and v? by • — ~-, their 



snit sin« 



sum will be found equal to — —, Avhich is raXl^ or the living- foi'ce of the 



sm; 



incident wave. 



For iK'ri)endicular incidence m=z — ^ and m':= — — become infinite: but their 



sni: • sni,o 



ratio remains finite ; and as they are not expressions for the absolute values of 



the masses, but only of their relative values, their ratio only is needed. By 



replacing sin: by its equivalent ?zsin/>, we have — 



cost cos/3 cos: 

 m : m : : — : — : - . : : - : cos/> : : cos: : 7icoBp ; 

 nsiup sin,o n 



which, when cos:=cosO^=:cos/?=l, gives m : m' : : 1 : n. And the sum of the 



intensities at perpendicular incidence is — 



lxv- + nxu-= ~~{+ ^ =-^—- = 1 = 1x1-, 



which is the intensity of the incident wave. 



If now we consider the general value of v^, given above, we shall see that v 

 increases with the increase of the angle of incidence, and becomes equal to the 

 total molecular velocity of the incident wave, when :=:90'^. For, at this inci- 

 dence, sin^::=^l, and cos:=:.0. Hence — 



( 



8=1; and w ^= 0. 



For intermediate incidence, we may transform the expression thus 



V [n^—\.) +cos^: — cos: 



V [n^— 1 ) + cos^: + cos:- 



The value of the radical diminishes Avith the increase of the incidence ; but 

 it diminishes less rapidly than cos:. For, the form of a binomial square being — 



if we put 2.T_y-|-?/= consta7it,\t is evident that as x diminishes, ?/must increase. 

 Flitting, therefore, cos: in place of x, we shall have — 



cos^: + 2v/cos: + 1/^=- cos^: + [11^ — 1), 

 or, 2v/cos:4-j/" = «~ — Inconstant ; 



and as cos; diminishes, the other part of the root of the radical increases, so 

 that the value of the entire radical diminishes less rapidly than cos:. The 

 numerator of the expression accordingly increases Avith increase of incidence, 

 and the denominator diminishes : and both these changes increase the value of 

 V. Hence, the amount of light reflected increases from incidence =0° to inci- 

 dence — 90^. It is worth observing that the expressions Ave have obtained 

 above for the molecular velocities of the reflected and refracted Avave, are also 

 deducible directly from the ordinary formulas for the impact of elastic bodies. 

 These formula;, (^employing m and m' for the masses, as aboA-e,) are — 



and u- 



VI + m' ' m + m' 



If, in place of m and m', Ave substitute the values found for the masses, v\z 



^__cos:^ and m'=^^^, the foregoing formulae will be reproduced, 

 sin: 8in/> 



