214 FRESNEL. 
destroy each other, that darkness may result from the 
superposition of two portions of light. But when this 
Since, if we take the partial differentials in respect to ¢ and to a, 
lu du 
a = 7 008 (ni — ker) dg = #008 (nt — ker) 
ay dn 
Whence, du m2 dy 
a —— Baa 
And since that wave-function goes through all its changes while @, 
increases to e and the velocity v= + the time of the undulation 
2 
T= — and v _4 — An 
n T 27 
Whence, — and eal 
A A 
Or the formula becomes (adopting an arbitrary coefficient, a, for 
the amplitude of vibration which is wholly independent of the other 
quantities) 
2 
u=asin = (rt —2), 
Here it is to be observed, all depends on the coefficient being 
constant. To obtain a similar equation with a variable velocity or 
refraction is the object of the researches of M. Cauchy. 
The more extended views of M. Cauchy have led to the deduction 
of analogous, but more complex, equations, exhibiting resulting ex- 
pressions for the displacement, in three rectangular directions; besides 
including in the analysis a coefficient which expresses the variable 
relation of the velocity which gives the theoretical explanation of un- 
equal refrangibility. These forms thus include the deductjon of 
transverse vibrations, as a direct consequence of the first assump- 
tions, as to the constitution of an sethereal medium. But, with refer- 
ence to light, considered as homogeneous, the conditions admit of 
great simplification; which is best shown in that form of the investi- 
gation which was pursued by Sir J. Lubbock (Philos. Mag. Nov. 
1837), where, if the fourth powers of the disturbed distances of the 
molecules are neglected, the equations are at once reduced to the 
form above. 
The object of M. Cauchy’s researches here alluded to was to ex- 
