Fresnel's Reflexion of Light Theory. 315 



us x— ( */ n + ^K^/n— 1), which is < 1 whatever positive 

 integral value n may have, hence +v'<Z applies to the case 

 of //,>!. Similarly, — y/q applies to fx< 1. 



It has now to be shown that the negative sign of ^/p is 

 applicable only when the value of /n lies beyond the limits 

 2+^/3, and some statement of the reason for this will enable 

 the argument to be more easily followed. 



Fresnel's tangent formula shows that the reflexion of light 

 vibrating in the plane of incidence diminishes from normal 

 incidence to the incidence tan -1 /*, but his sine formula, for 

 the vibration perpendicular to that plane, gives a continual 

 increase of the light reflected as i increases. The combined 

 formula for common light gives ratios of reflected light 

 which generally increase with i throughout the whole range 

 of i in which the reflected light can vary. This statement 

 regarding common light, however, requires to be qualified to 

 the extent that it only holds good while ft lies within the 

 limits 2± i/3. When fju lies beyond these limits, the 

 quantity of reflected common light diminishes at first as i 

 increases from 0°. We have seen that the ratio of reflected 

 common light, in terms of x and //,, is 



and differentiating this expression we get 



At normal incidence x = fi~ l , and we find / 7 (yu, _1 ) = 0, showing 

 that the gradient of /(%) at normal incidence = 0. 



But, 



(x) =4[Vx 4 + Vr(^Y-2)(^-6^+]) + ^ % (^-f i)(2 M v-/xY+ 8) 



and / / V 1 )= -VO* 2 -^ 1 ). 



i£/ w 0*- 1 )=0, then ^=2+^3. 



i 



Therefore, f'(fjr x ) changes from positive to negative as fi 

 passes out through the limits 2+ ^3 ; wherefore, when fi lies 

 beyond these limits, /(#) diminishes at first as % increases 

 from jju- 1 , or, as i departs from the normal. 



This decrease in /(%) with increase of x can only be ex- 

 pected to continue through a limited range and the following 



