JuxE 6, 1 901] 



NATURE 



139 



particle a suitable force (which of course must be in the direc- 

 tion of the displacement and proportional to the difference of 

 the densities of the particle and of the ether) we could restore the 

 amplitude to the value it would have were the particle absent ; 

 under these conditions everything would go on as though there 

 were no particle in the ether, and consequently there would be 

 no scattered light, i.e., we should have neutralised the effect of 

 the particle by the application of this force. Hence, on the 

 other hand, we would have the same scattered light if the 

 particle were absent, and we should apply to this portion of the 

 ether this force reversed in direction, that is to say, each 

 particle acts as a centre of a certain harmonic force acting upon 

 the surrounding ether. Such a force will send out a plane 

 polarised wave, whose intensity is symmetrical about the direc- 

 tion of the force as axis ; it is zero in the direction of the 

 force, and a maximum in the plane perpendicular to this 

 direction. 



The exact effect of such a force has been investigated ana- 

 lytically by Stokes and also by Lord Kayleigh. The displace- 

 ment in the wave sent out by it is 



if the force is F cos 



ZTtht 



vhere ;■ is the distance from the 



centre of force to the point where the displacement is meas- 

 ured ; a is the angle between the direction of the force and the 

 line joining the point considered to the centre of force or the 

 mean position of the disturbing particle ; b is the velocity of 

 light ; D the density of the ether ; \ the wave-length of the 

 light sent out by the force ; and jr is the ratio 3.1416. 



If the displacement in the incident wave is A cos - — , the 



force we must apply to the particle to restore the displacement 

 to its natural value is 



Z-nbt 



T (D' - D) A 



;?)■■ 



where D' is the optical density of the particle and T is its 

 volume ; therefore, 



, , D'-D TrT . 2T ,,, , 



|=A — — — . — smacos — (bt-r), 

 ^ D rK- \ 



and the intensity of the scattered light is for each particle 

 V D y r-\' 



sm- 1 



Since the particles are in motion the light scattered from 

 different particles will have no definite phase relation ; hence, 

 to get the effect of a cloud of such particles we must add the 

 intensities of the light sent out by each separate particle. 



If the incident light is plane polarised, o will be a constant 

 for any given direction from the incident beam, and the total 

 intensity of the light scattered in this direction will be 



If the incident light is unpolarised, the intensity of the light 

 scattered at an angle j3 with the direction of the incident 

 beam will be 



'D'-DV- ir=(n-cos-fl)^T- 



: T> } 



A- I 



2-1,, 



T2 T- 



where 2-s- denotes the sum of ~ir for all the scattering particles 



in the line of vision. In none of this have we taken account 

 of the light that has undergone more than a single scattering. 



T T]- 



If we denote the me-an of the square of — by -r, and let N 



>■ r-c 



denote the number of particles in the line of vision, we can 

 write the expression for the intensity of scattered light in the 

 form 



,., /D'-DV- ^-(H-cos-/3) NTi= 

 ^'rD-J T* TV-'- 



What are the assumptions we have made in this treatment ? 

 They are : 



(I) Every scattering particle is so small that when a wave of 

 length A passes through the medium containing it the force is 

 the same at every point of the particle, i.e., each particle is 



NO. 1649, VOL. 64] 



small as compared with the culje of the shortest wave-length 

 of the incident light. 



(2) The particles are so far apart that their effect upon the 

 velocity of light through the medium is negligible ; ?'.<.'. the 

 particles are far apart as compared with the longest wave-length 

 with which we are dealing. 



In his discussion of Lord Rayleigh's equations, Crova claims 

 there is a third assumption, viz., that the number of particles 

 in unit of volume must be sensibly the same for all sizes of 



particles. He says : " La formule —^ est basce sur I'hypothese 



que le nombre N de corpuscules contenus dans I'unite de volume 

 d'air est sensiblement le meme pour toutes les dimensions de 

 ceux-ci." Mascart is of the same opinion. This is evidently 

 wrong. The expression 



D'-D \-7r'-'T-sin-a 



A'-^l 



applies to particles of all sizes, provided they are small in 

 comparison with the cube of the shortest wave-length. The 

 light from a cloud of such particles is merely the sum of the 

 light from the individual particles ; the relative number of 

 particles of various sizes does not enter into the consideration at 

 all ; indeed, the composition of the Hght is entirely independent 

 of all consideration of the number and size of the particles other 

 than as specified in the two assumptions we have named. 

 Particles of a size intermediate between these small ones and 

 those larger ones that reflect light regularly produce effects as 

 yet unknown, and are not amenable to this analysis. 



From Lord Rayleigh's expression for the intensity of the 

 scattered light we may conclude, if the manifold or multiply 

 scattered light may be neglected : 



(i) The scattered light is polarised in the plane of scattering 



and the amount of its polarisation is — , being a max- 



'^ H-cos^;3' 



imum (completely polarised) when the direction of scattering 

 is perpendicular to the direction of propagation of the incident 

 light. 



(2) The intensity of the scattered light varies —, times the 



intensity of the incident light. Its colour or wave-length is 

 independent of the direction of scattering. 



(3) The maximum intensity of the scattered light is in a 

 direction almost coincident with that of the incident light and 

 in the opposite direction, and the minimum is in the plane per- 

 pendicular to this. 



(4) The larger the particles (provided the assumptions above 

 are fulfilled), the more intense is the scattered light. 



As stated above, we know little, if anything, about the 

 action of particles that are just too large for this treatment to 

 apply, but in another of his papers Lord Kayleigh has solved 

 to the next approximation (on the electro-magnetic theory) the 

 special case of spherical particles, and finds that the light 

 scattered should vary as the inverse eighth power of the wave- 

 length. In the air there are surely some particles approxi- 

 mately fulfilling these conditions, and hence the sky should 

 appear bluer than indicated by the simple theory we have just 

 considered. But we have not yet bridged the gap between 

 "very small" particles and those large enough to give regular 

 reflection. 



We have thus far neglected the multiply scattered light, but 

 this increases in intensity as the square and higher powers of 

 the number of particles per unit volume, while the once- 

 scattered light increases as the first power only. Hence, for a 

 cloud of particles the multiply scattered light may easily 

 become appreciable. This again increases the proportion of 

 the blue. 



For all these reasons the colour of the light from the sky 

 should be expressed by the sum of a series of terms of powers 

 of the reciprocal of the wave-length; not by a single term, as 

 is ordinarily attempted. Crova, endeavouring to express the 



intensity by a single term of the form —'found values of « 

 varying from 2 to 6 under different conditions, the average 

 being about 4, as Lord Kayleigh and Captain Abney had found. 

 But in no case could n be determined so as to give more than 

 a fair agreement. As we have seen, values of n higher than 4 

 are to be expected ; the lower ones are to be accounted for by 

 the lateral scattering caused by the particles between the 



