an Atmosphere containing Small Particles in Suspension. 377 



which we see the sun a little before sunset. The same 

 reasoning proved, of course, that the spectrum of even a 

 vertical sun is modified by the atmosphere in the direction of 

 favouring the waves of greater length. 



For such a purpose as the present it makes little difference 

 whether we speak in terms of the electromagnetic theory or 

 of the elastic solid theory of light ; but to facilitate compari- 

 son with former papers on the light from the sky, it will be 

 convenient to follow the latter course. The small particle of 

 volume T is supposed to be small in all its dimensions in 

 comparison with the wave-length (X), and to be of optical 

 density D' differing from that (D) of the surrounding 

 medium. Then, if the incident vibration be taken as unity, 

 the expression for the vibration scattered from the particle in 

 a direction making an angle 9 with that of primary vibra- 

 tion is 



— ^— _ S in cos — (bt-r)*, . . . (1) 



r being the distance from T of any point along the secondary ray. 

 In order to find the whole emission of energy from T we 

 have to integrate the square of (1) over the surface of a 

 sphere of radius r. The element of area being 2wr 2 sin 6d0, 

 we have 



^ S ^2nr*sm6d0 = ±7rP\m*ddd= ~; 



so that the energy emitted from T is represented by 



8tt s (D'-D) 2 T* f9 . 



"3" L>* \^ {Z) 



on such a scale that the energy of the primary wave is unity 

 per unit of wave-front area. 



The above relates to a single particle. If there be n 

 similar particles per unit volume, the energy emitted from a 

 stratum of thickness dee and of unit area is found from (2) by 

 introduction of the factor ndx. Since there is no waste of 

 energy on the whole, this represents the loss of energy in the 

 primary wave. Accordingly, if E be the energy of the pri- 

 mary wave, 



lc/E_ 87r 3 n (D'-D) g T a 



e dx ~ a w \ 4 ; • • • ( °) 



whence E = E 6- fe , (I) 



where ; _ 8tt^ (D'-D) 2 T^ 



8 D 9 - ~\ 4 ' ' ' * ^ 



* The factor n was inadvertently omitted in the original memoir. 



