of Refraction, Reflexion, and Extinction. 285 



the purposes of the present problem, the scheme of relations 

 laid down in our preceding calculations requires modifica- 

 tion. It may be readily seen that all the results established 

 in § 1 above will here still hold true ; but in order that they 

 may thus preserve their generality, we must replace (1), § 2, 

 by the following equation of definition : 



¥(r, = - % (Pe- inb *o + QJnKynit-ar)^ > # ^ 



where P and Q are new constants, to be determined here- 

 after. It follows that the quantities A, B, C (on which the 

 calculation of secondary effects ultimately depends) are 

 expressible in the form 



ahi 2 



+ B = - — (Pe- inbz « + Q € "^ yn(t-ar) , (1 __ R2 ) • + R}? ^ 

 -C=—(¥e- inb ** + QJ nbz °)M t - ar Hi + K}, . . . (6) 



where R = l/anr as before. 



We now pass on to the calculation of the secondary effects 

 to be expected, in the present case, upon the principles 

 established in §§ 4 and 5 of this communication. Imagine 

 a thin stratum in the plate, at right angles to the axis of z ; 

 let dz Q be tiie thickness of the stratum. We have to 

 evaluate the resultant electric field, at a point M(a*, ?/, z) 

 of the field, due to the presence of vibrators occupying the 

 stratum. Let [E^ 2) ] represent the component along x of 

 this resultant field ; proceeding as in § 4 we easily find : 



[Ef ] = -2iran ^ N(Pe-»' wfe o + Q 6 ^-o) 6 *4^-a I *-*o I ]. (7) 



By a like process, for the component along y of the resultant 

 magnetic field excited bv the stratum, we obtain 



[Hf ] = + 27rrtn dz^(Ye~ mg o + Q e ™bz o yn[t-a | z-z Q | ] . ^ 



the upper sign is to be taken when z — z is positive and the 

 lower sign when this difference is negative. 



Let the point M be chosen within the substance of the 

 plate, in a plane z = z at right angles to the axis of z. We 

 proceed to consider the electric field ^measured along x) 

 which arises at M under the simultaneous operation of 

 secondary waves travelling across the plane in the positive 



Phil Mag. Ser. 6. Vol. 38. No. 225. Sept. 1919. X 



