212 EQUATIONS OF MONOCHROMATIC LIGHT IN MEDIA 



electrical phenomena,* we shall not introduce at first the hypothesis 

 of Maxwell that electrical fluxes are solenoidal.t Our results, however, 

 will be such as to require us to admit the substantial truth of this 

 hypothesis, if we regard the processes involved in the transmission of 

 light as electrical. 



With regard to the undetermined questions of electrodynamic in- 

 duction, we shall adopt provisionally that hypothesis which appears the 

 most simple, yet proceed in such a manner that it will be evident exactly 

 how our results must be altered, if we prefer any other hypothesis. 



Electrical quantities will be treated as measured in electromagnetic 

 units. 



2. We must distinguish, as before, between the actual electrical 

 displacements, which are too complicated to follow in detail with 

 analysis, and which in their minutiae elude experimental demonstration, 

 and the displacements as averaged for spaces which are large enough 

 to smooth out their minor irregularities, but not so large as to oblite- 

 rate to any sensible extent those more regular features of the electrical 

 motion, which form the subject of optical experiment. These spaces 

 must therefore be large as measured by the least distances between 

 molecules, but small as measured by a wave-length of light. We 

 shall also have occasion to consider similar averages for other quan- 

 tities, as electromotive force, the electrostatic potential, etc. It will 

 be convenient to suppose that the space for which the average is 

 taken is the same in all parts of the field, J say a sphere of uniform 

 radius having its center at the point considered. 



Whatever may be the quantities considered, such averages will be 

 represented by the notation 



L JAve' 



* It has, perhaps, retarded the acceptance of the electromagnetic theory of light that 

 it was presented in connection with a theory of electrical action, which is probably more 

 difficult to prove or disprove, and certainly presents more difficulties of comprehension, 

 than the connection of optical and electrical phenomena, and which, as resting largely 

 on a priori considerations, must naturally appear very differently to different minds. 

 Moreover, the mathematical method by which the subject was treated, while it will 

 remain a striking monument of its author's originality of thought, and profoundly 

 modify the development of mathematical physics, must nevertheless, by its wide depar- 

 ture from ordinary methods, have tended to repel such as might not make it a matter 

 of serious study. 



fA flux is said to be solenoidal when it satisfies the conditions which characterize 

 the motion of an incompressible fluid, in other words, if u, v, w are the rectangular 



components of the flux, when 



du dv dw_ n 



dx dy dz~ ' 



and the normal component of the flux is the same on both sides of any surfaces of 

 discontinuity which may exist. 



J This is rather to fix our ideas, than on account of any mathematical necessity. For 

 the space for which the average is taken may in general be considerably varied without 

 sensibly affecting the value of the average. 



