190 



FRESNEL. 



rived at the result, setting out from the ideas he had 

 adopted of the nature of light. And, lastly, Newton de- 



l P, described with the velocities vvt, which are in a constant ratio to 



each other; and in times which will be . Then, on the prin- 



v v' 



ciple of "least time," the condition is, 



I li 



1 =mimmum; 



v vi 



or, differentiating and multiplying by v vt, 



V dl + v dl' = Q . . . . (i). 



Then if x be the surface of the medium, taking equal increments 

 d x on each side of the point of incidence, and dropping perpendiculars 



to give corresponding increments d I d I', i and r being the angles of 

 incidence and refraction, we have geometrically 



sin i sin r 



dl ; dl> = : . . 



dx ax 



and substituting in (1) it becomes 



vl sin i v sin r = 0, 



(2); 



or 



sin i = sin r. 

 v' 



But, as i is necessarily greater than r, it follows that the v must be 

 greater than v': or the law of the sines fulfils the condition of "least 

 time " on the wave theory. 



On the other hand, the principle of " least action" requires, instead 

 of equation (1), that we have 



I v-t-V v' = minimum, 

 or vdl+vldl'=Q: 



whence, by precisely the same process, there results 



sin i = sin r ; 



v 



