Origin of the Radiation from Hot Bodies. 221 



from the large value /3i to zero, let us suppose that it 

 increases from zero quickly up to a very large value and 

 then quickly diminishes to zero again. Such a case could be 

 represented by supposing 



where a is small compared with \ 2 . In this case 



Ai 



fa — A I e «- cos q\ . d\. 



If X x is large compared with a, we may substitute -co and 

 + co for the limits, instead of — -£ and - l , without making 

 any appreciable error ; so that 



e « 2 cos q\ d\ 



= 2AV7T ae~ * . 



If this acceleration reduces a particle moving with a 

 velocity u to rest, 



A 1 €~* 2 d\ 



Hence 



= 2A\/7T«. 



q-cfi 



<j> 1 = ue~~; (5) 



and the energy radiated is 



8 e 2 _£fL 2 . qt 2 „. 



ln\T ire 2 sm ^^ W 



Since the acceleration is only appreciable for a time com- 

 parable with a, we may regard a as the time occupied by a 

 collision. 



Another form for /, which also has the property of being 

 very large at a particular time and diminishing very rapidly 

 on either side of that time, is 



where a is very small. This is very large when X is com- 

 parable with a, and small when \ is large compared with a. 



