446 Lord Rayleigh on the Light emitted from a 



cos^ = by a similar factor in the numerator.. In the 

 integration with respect to % 



sin x d%= ~4 cos \x • d ( cos i%)- 



If we w T rite ty for 2&U cos J^, the mean sought may be 

 written 



yfr — yjr cosier) 2 



9w 2 r (sin 

 2PWJ ' 



iir' 



d+, . . . (37) 



the range for yjr being from to 2&R. The integration can 

 be effected by " parts." We have 



J (sin yjr — yjr cos ty) 2 sin 2 yjr — 2yfr sin yjr cos ty + yjr 2 



. . (38) 

 When yjr is small, the expression on the right becomes 



4" + "l8' 



so that the integral between and yfr is ^jlS simply. In 

 general, the mean intensity is 



9ft 2 2-v/r sillo/r cOSi/r — sin 2 -v/r — i/r 2 + ^/r 4 



— , . (<5y) 



8FK 2 t' 



in which -^ stands for 2kR. 



That the intensity, whether in one direction or in the 

 mean of all directions, should be proportional to n 2 is, of 

 course, what was to be expected. And, since the effect is 

 here a surface effect, it may be identified with the ordinary 

 surface reflexion which occurs at a sudden transition between 

 two media of slightly differing refrangibilities, and is pro- 

 portional to the square of that difference. If, as in a former 

 problem, we suppose the discontinuity of the transition to 

 be eased off, this reflexion may be attenuated to any extent 

 until finally there is no dispersed wave at all *, 



When we pass from the continuous uniform distribution 

 to the random distribution of n discrete and very small 

 obstacles, the term in n 2 representing reflexion from the 

 surface remains, and is now supplemented by the term in w, 

 due to irregular distribution in the interior. It is the latter 

 part only with which we are concerned in a question such as 

 that of the blue of the sky. 



It must never be forgotten that it is the " expectation >} of 



* Conf. Proc. Lond. Math. Soc. vol. xi. p. 51 (1880) ; Scientific Papers, 

 vol. i. p. 460. 



