(411 ) 



^ 10. Let the plate be lioinogeneoiis, with its faces parallel to the 

 tirst and the socoiul principal direction. We shall take these for the 

 a.xes of .V and //, placing the origin (> in the front surface of the 

 plate, i. e. in the surface exposed to the rays, and drawing the axis of 

 z toward the outside. As has already been said, the absorption will 

 be calculated by means of the formula (30) ; it will therefore be deter- 

 mined by the components of ^ and by those of (?, on which they 

 depend. Now, our problem is greatly simplified, if we suppose the 

 thickness A of the plate to be infinitely small and if, in calculating 

 the absorption, we confine ourselyes to quantities of the first order 

 of magnitude with respect to A. The quantity w relating to unit 

 volume, we may then neglect all infinitely small terms in S and^Ü; 

 consequently, we need not attend to the changes of these vectors in 

 the plate along a line perpendicular to its faces. Moreoyer, in virtue 

 of the w^ell known conditions of continuity, the values of (?a and <£y 

 Avithin the plate will be equal to those existing in the aether imme- 

 diately before it; also, ^% will be 0, because it is so in the aether. 

 For Q'x and ^^ we may even take the values, existing in the inci- 

 dent beam, the reason for this being that the values belonging to the 

 reflected rays, (the vibrations reflected at the two sides being taken 

 together) are proportional to the thickness, if the plate is infinitely thin. 



It is seen by these considerations that in the case of a given 

 incident motion, ^a, (Ey, ^z are the only unknown quantities in the 

 three equations connecting the components of ^ and (1. We need not, 

 however, work out the solution of these equations. 



Finally, it must be kept in mind that, in the case of harmonic 

 vibrations, the mean value of w for a lapse of time comprising many 

 periods is given by 



"' = ^i«i(^.r + «.(V + «,(^^rh . . • (31) 



if (^a)» (^y)> (^-) ^I'G tlie amplitudes of the components of the current. 



§ 11. We shall in the first place assume that in the incident rays 



the electric force is parallel to the axis of x. Let its amplitude be a. 



Then, an element lo of the front surface will receive an amount 



of energy 



1 



-ca^o) ......... (32) 



per unit of time. 



Within the plate, there will be electric currents in the directions 

 of X and y. These will have amplitudes proportional to a, and for 

 which Ave may therefore write : 



