190 



FRESNEL. 



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

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



l V, described with the velocities vvf, which are in a constant ratio to 

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

 ciple of "least time," the condition is, 



III . . 

 1 =mmimum; 



V VI ' 



or, differentiating and multiplying by v v', 



V dl + v (III = . . . . (1). 

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

 dxon each side of the point of incidence, and dropping perpendiculars 



to give corresponding increments dldV, i and r being the angles of 

 incidence and refraction, we have geometrically -, 



sm J 



al^- dll=:: 



ax 



dx 



■ (2); 



and substituting in (1) it becomes 



d' sin i — i; sin r = 0, 



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

 greater than «': 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 



lv-\-l' vi = minimum, 

 or vdl+vldli=0: 



whence, by precisely the same process, there results 

 . . vi . 



