190 



FRESNEL. 



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

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



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



Then, on the prin- 



each other; and in times which will be 



v v&amp;gt; 



ciple of &quot;least time,&quot; the condition is, 



-- h =minimum 



or, differentiating and multiplying by v v , 



vf dl + vdl = .... (1). 



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 



and substituting in (1) it becomes 



vl sini v sin r = 0, 



v 

 or sin i = sin r. 



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

 greater than v&amp;gt;: or the law of the sines fulfils the condition of &quot;least 

 time &quot; on the wave theory. 



On the other hand, the principle of &quot; least action&quot; requires, instead 

 of equation (1), that we have 



I v+l r v r = minimum, 

 or vdl+vldl =0: 



whence, by precisely the same process, there results 



sin i = sin r; 



v 



