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 t and to x, 



du du 



= n cos (nt kx) = k cos (nt KX) 



dt dx 



d-n 



TT 

 dx* 



Whence, d^-u n'* d?u 



' 



And since that wave-function goes through all its changes while t, 



increases to and the velocity v = -7- the time of the undulatioa 



k 



27T A 7,n 



T = and v = - = - 

 n T 27T 



__, 27ru 2/r 



VV hence, n = - and k = 



A A 



Or the formula becomes (adopting an arbitrary coefficient, a, for 

 the amplitude of vibration which is wholly independent of the other 

 quantities) 



27T 



u = a sin (i't x). 



A: 



n 

 Here it is to be observed, all depends on the coefficient^- being 



K 



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 deduction of 

 transverse vibrations, as a direct consequence of the first assump- 

 tions, as to the constitution of an rethereal 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- 



