190 FRESNEL. 
rived at the result, setting out from the ideas he had 
adopted of the nature of light. And, lastly, Newton de- 
lV, described with the velocities vv!, which are in a constant ratio to 
l 
each other; and in times which will be og eS Then, on the prin- 
ciple of “least time,” the condition is, 
l 
v 
or, differentiating and multiplying by vv, 
vdi+vdU=0.... (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 
toleas 
+ 5) = minimum ; 
al 
= dx dz 
dl, 
KG? 
a 
_- 
incidence and refraction, we have geometrically 
sini —sin r 
= dl! —————"—" «| 6 18% 2 $ 
ar dic (2)5 
and substituting in (1) it becomes 
v sint—v sin r=0, 
os 
int =—sin 7. 
or sin 4 7 
But, as 7 is necessarily greater than 7, 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. og 
On the other hand, the principle of “ least action”’ requires, instead 
of equation (1), that we have 
lv+/ o = minimum, 
or vdi+vidl=0: 
whence, by precisely the same process, there results 
! 
sin? = —sin rT; 
~ 
