. Al =-^ 2 (l + S^), 



mk 2 C 2 \ 2 P J 



328 Mr. C. G. Darwin on the 



and also the reaction of their radiation on their motion. 

 We thus get a set of equations 



P 3 



where {3 U is a quantity depending on the arrangement and 

 is of the order of unity. The whole amplitude of the radia- 

 tion scattered to a distant point in the yz plane is 



r 



The character of the solution of the simultaneous equations 



,2 



e- 1 



depends on the magnitude of -v^ -3 . If it is small we get 

 A x = T*r&> so tliat tne scattered radiation is proportional 



e 



m 

 e 



mk 2 C 2 ! 



to v -—,. On the other hand, if 72n2 -o is large we can 

 mC 2 mkrkj p* 



neglect the terms on the left, and we have a set of equations 



v Q A 



of the form 1+2 ^V"* =0. Whatever the solution may 



2 p z 

 be, it will give Aj independent of k and proportional to p\ 

 so that the scattered radiation will be proportional to PpK 

 This is exactly the result found by Lord Rayleigh* in his 

 theory of the light of the sky. 2 1 



Thus the question turns on the value of J2n2 -g. Taking 



ink \j p 



light of wave-length 5xl0~ 5 cm. and p = 10~ 3 cm., we 

 represent the effect of a molecule of the atmosphere. This 



e 2 1 

 gives , 2 p 2 -3=18, which is probably large enough. For 



our inner ring of electrons we take /o = 5xl0~ 10 and for 



soft X rays 2tt/&=10- 8 . Then -4™ \ = 6 x 10~ 3 . This 

 J mk 2 \j- p 6 



is small enough, so that we may assume that an atom scatters 

 long waves to the same extent as short. 



13. The "Excess " Scattering. 



The waves scattered by the electrons in an atom combine 

 to a certain extent so as to give a scattered radiation greater 

 in intensity than is simply proportional to their number. 

 For example, the electrons at distances 5 x 10" 10 cm. apart 

 would exert an effect almost proportional to the square of 



* Cf. Rayleigh, ' Sound,' vol. ii. p. 149. 



