OF EVERY DEGREE OF TRANSPARENCY. 217 



depend upon the period of vibration, that is, upon the color of the 

 light.* 



We may therefore write in vector notation 



[E] Ave = *[U] Ave -f^[U] Ave (7) 



where 3> and "& denote linear functions.! 



The optical properties of media are determined by the form of 

 these functions. But all forms of linear functions would not be 

 consistent with the principle of the conservation of energy. 



In media which are more or less opaque, and which therefore 

 absorb energy, must be of such a form that the function always 

 makes an acute angle (or none) with the independent variable. In 

 perfectly transparent media, must vanish, unless the function is 

 at right angles to the independent variable. So far as is known, the 

 last occurs only when the medium is subject to magnetic influence. 

 In perfectly transparent media, the principle of the conservation of 

 energy requires that <3> should be self-conjugate, i.e., that for three 

 directions at right angles to one another, the function and independent 

 variable should coincide in direction. 



In all isotropic media not subject to magnetic influence, it is prob- 

 able that <> and ^ reduce to numerical coefficients, as is certainly 

 the case with <3? for transparent isotropic media. 



9. Comparing the two values of [E] Ave , we have 



A.Z 



5 Pot[U] Ave - V[g] Ave = $[U] Ave +*[U] Ave . (8) 



Jr 



This equation, in connection with that by which we express the sole- 

 noidal character of the displacements, if we regard them as necessarily 

 solenoidal, or in connection with that which expresses the relation 

 between the electrostatic potential and the displacements, if we reject 

 the solenoidal hypothesis, may be regarded as the general equation of 

 the vibrations of monochromatic light, considered as oscillating elec- 

 trical fluxes. For the symbol Pot, however, we must substitute the 

 symbol representing the operation by which electromotive force is 

 calculated from acceleration of flux, with the negative sign, if we are 

 not satisfied with the law provisionally adopted. 



* The relations between the displacements in one of the small spaces considered and 

 the average electromotive force is mathematically analogous to the relation between the 

 displacements in a system of a high degree of complexity and certain forces exerted 

 from without, which are harmonic functions of the time and under the influence of 

 which the system vibrates. The ratio of the displacements to the forces will in general 

 vary with the period, and may vary very rapidly. 



An example in which these functions vary very rapidly with the period is afforded 

 by the phenomena of selective absorption and abnormal dispersion. 



t A vector is said to be a linear function of another, when the three components of the 

 first are homogeneous functions of the first degree of the three components of the second. 



