﻿Deduction of Wiens Displacement Law. 923 



and i/r from to 7r/2, during a long time t ; and to do this we 

 must find the number of times that each of these effects is 

 produced on every wave, i. e. every element of the radiation 

 existing within the shell. 



4. We base our reasoning on the consideration that in a 

 very long time — though not in a short one — every element 

 of the energy within the shell must undergo reflexion from 

 M at the angles (^ i/r) just as many times as every other 

 element. The number of times that any particular effect of 

 reflexion at M is produced on each element of the energy is 

 therefore the ratio of the totalamount of energy thus affected 

 in the time t to the total amount present within the shell. 

 The changes of period in the ratios r a and r^ caused by 

 arrival and departure, occur alternately, and in finding the 

 total effect of a number of successive arrivals and departures 

 we have evidently to evaluate a product of the form 



r a r d . r a r rj . rd 1 rd 



But since multiplication is commutative, we shall get the 

 same result if we pursue the more convenient method of 

 treating all the arrivals by themselves, then all the departures 

 by themselves, and finally multiplying the two resulting ratios 

 together to get the combined effect of both arrivals and 

 departures. 



We start, then, with the arrivals. In any time t the 

 amount of energy which strikes M at angles between cp and 

 <j) + d(p is 



t . ~R^d\s cos (f> . 2-7T sin cf> dcp. 



The total amount present within the shell to be affected is, by 

 equation (1), 



4ttR^\ 



v . . 



Hence the number of times n that each element of the 

 energy arrives at M at angles between cf) and cf>-\-dcf> within 

 the long time t is 



_ t . K\d\s cos cp * 2-7T sin cf) d<fi 

 71 * "^ttE^aT 



cts 



= <p cos (f> sin cp dcj) * (4") 



