378 Lord Rayleigh on the Transmission of Light through 



If we had a sufficiently complete expression for the scattered 

 light, we might investigate (5) somewhat more directly by 

 considering the resultant of the primary vibration and of 

 the secondary vibrations which travel in the same direction. 

 If, however, we apply this process to (1), we find that it fails 

 to lead us to (5), though it furnishes another result of interest. 

 The combination of the secondary waves which travel in the 

 direction in question have this peculiarity, that the phases 

 are no more distributed at random. The intensity of the 

 secondary light is no longer to be arrived at by addition of 

 individual intensities, but must be calculated with considera- 

 tion of the particular phases involved. If we consider a 

 number of particles which all lie upon a primary ray, we see 

 that the phases of the secondary vibrations which issue along 

 this line are all the same. 



The actual calculation follows a similar course to that by 

 which Huygens' conception of the resolution of a wave into 

 components corresponding to the various 

 parts of the wave-front is usually veri- 

 fied. Consider the particles which oc- 

 cupy a thin stratum dx perpendicular 

 to the primary ray x. Let AP (fig. 1) 

 be this stratum and the point where 

 the vibration is to be estimated. If 

 AF = p, the element of volume is 

 dx . "27rpdp, and the number of particles 

 to be found in it is deduced by intro- 

 duction of the factor n. Moreover, if 



Fisr 1. 



OP = 



AO = 



S—„2 



= x 2 -\-p 2 , and 



pdp = rdr. The resultant at of all the 

 secondary vibrations which issue from 

 the stratum dx is by (1), with sin equal to unity 



.dx 



p» D'-D ttT 



277- 



cos — - (bt — r) 2irrdr } 



or 



7 D'-DttT . 2tt,, , 



n ax . — y\ — sm — [bt — x) 



U A A, 



(6) 



To this is to be added the expression for the primary wave 



27T 



itself, supposed to advance undisturbed, viz., cos — (bt — x), 



and the resultant will then represent the whole actual dis- 

 turbance at as modified by the particles in the stratum dx. 

 It appears, therefore, that to the order of approximation 

 afforded by (1) the effect of the particles in dx is to modify 

 the phase, but not the intensity, of the light which passes 



