Prof. Lorenz on the Theory of Light, 411 



of his differential equations. This theorem, taken as one of 

 general application, can be easily disproved by a simple example. 

 The following differential equation may serve as such : — 



where we will assume a to be greater than b, and a very small. 



The mean value of the periodic coefficient a + b cos — is a, and 



the mean value of the integral would therefore, according to the 

 above theorem, be approximately 



c 



<j)=e 

 But this result is incorrect ; for integration gives exactly 



<f) = e°~J a+bcos h 

 and afterwards approximately, if a is very small, 



a value which differs essentially from the foregoing, inasmuch 

 as the constant of periodicity b enters into it. 



The theorem, therefore, cannot be maintained in all its gene- 

 rality; neither is it any more specially applicable to Cauchy's dif- 

 ferential equations. In the first place, not only circular polari- 

 zation, but also double refraction, if the equations are otherwise 

 correct, must result from the periodicity of the coefficients. In 

 order to prove this, we have no need to lose ourselves in endless 

 calculations, for nature herself has carried out the calcula- 

 tion. Brewster and, more recently, M. Schultze (Verhandl. d, 

 rheinl. Gesellsch. 1861) have in fact shown that transparent sub- 

 stances in thin layers are doubly refracting. This important fact 

 therefore demonstrates that a periodicity in the interior of bodies 

 must involve double refraction. 



In the second place, it is at once obvious that the thickness of 

 the layers of a periodically heterogeneous body must exert an 

 influence upon the course of the ray which will depend upon its 

 w ave- length, and therefore that, whatever differential equations 

 we take as our basis, chromatic dispersion at least is deducible 

 from the periodicity of the coefficients. This explanation of 

 chromatic dispersion shows moreover very simply why it is that 

 chromatic dispersion is peculiar to material bodies and is not 

 possessed by a vacuum. 



If, however, we now find that theory requires us, on the one 

 hand, to assume a periodicity in the internal structure of bodies, 



2E2 



