of Refraction, Reflexion, a»d Extinction. 275 



and the extreme variety of innumerable secondary dis- 

 turbances. The nature of! this ultimate state of equilibrium 

 must now be elucidated, 



We commence by briefly recapitulating the most im- 

 portant assumptions which we shall adopt. We will suppose, 

 in the first place, that the greatest linear dimensions of the 

 vibrators are exceedingly small compared with the mean 

 distance between neighbouring vibrators or N — 3, N being 

 the number of vibrators per unit volume It follows from 

 this that the vibrators, being outside each other's sphere of 

 action, are free to vibrate vvith practically perfect inde- 

 pendence. On the other hand, in order to arrive at definite 

 results we have to admit (inasmuch as our mode of argument 

 essentially involves spacial averaging) that the mean distance 

 between neighbouring vibrators is negligible in comparison 

 with the shortest wave-length with which in the applications 

 of theory we shall be concerned. In speaking in future of 

 material media we shall suppose that these conditions are 

 amply fulfilled. 



§ 4. In the interior of the medium imagine a stratum 

 bounded by the planes z = z and z = Zo + dz , so that dz 

 represents the thickness of the stratum. By what has gone 

 before, we have to consider the stratum as permeated by 

 vibrators each of which determines a secondary field around it; 

 hence, at a p int M(.r, //, z), a resultant electric field is origi- 

 nated under the operation of the stratum. We now intend 

 to evaluate its components [E? } ] etc. For this purpose we 

 have to effect a summation over the vibrators, say V, V ]? Y 2 , . . . 

 contained in the stratum. In lieu of this we shall find it con- 

 venient to follow a different course. From the point V draw 

 lines VMu YiV. . . para llel respectively to VjM, V 2 M, 

 and such thatVM 1 = V 1 M ; VM^VjM, etc. The points M, 

 M lf Mo, .... fill a second stratum parallel to, and situated at 

 a distance z — z from the first. Instead of adding the com- 

 ponent fields arising at M under the joint operation of 

 V, Vi, V 2 . . . ., we obtain the desired result if we combine the 

 fields which V determines at M, M 1? M 2 , ... The assumption 

 on which we proceed is here evidently that every wave under 

 consideration must be exactly plane. When this condition is 

 fulfilled, the advantage of introducing the auxiliary points 

 Mj, M 2 , . . . . is obvious. 



To find [E^ 2) ] we now refer to cylindrical coordinates 

 whose axis coincides in direction with the former axis of z 

 Let p be the radius vector to the point M drawn perpendicular 



