Complete Radiation at a given Temperature. 467 



impulsively from rest divide themselves into two groups, con- 

 stituting progressive waves in the two directions, and that the 

 whole energy of each of these waves is the half of that com- 

 municated initially to the system in the kinetic form*. 

 The application of (21) to (11), where 



ffives 



r^e-''^'^'^da^=\re-''''''\hi, . . . (22) 



J -co Jo 



as ma}'" be easily shown independently. The intensity, cor- 

 responding to the limits u and u + du, is therefore 



c e ' an ; 



and this, since 7i and n are proportional, is of the form (5). 



If an infinite number of impulses, similar (but not neces- 

 sarily equal) to (8), and of arbitrary sign, be distributed at 

 random over the whole range from — oo to +<X) , the intensity 

 of the resultant for an absolutely definite value of n would be 

 indeterminate. Only the prohahilities of various resultants 

 could be assigned. And if the value of n were changed, by 

 however little, the resultant would again be indeterminate. 

 Within the smallest assignable range of n there is room for an 

 infinite number of independent combinations. We are thus 

 concerned only with an average, and the intensity of each 

 component may be taken to be proportional to the total 

 numbei- of impulses (if equal) without regard to their phase- 

 relations. In the aggregate vibration, the law according to 

 which the energy is distributed is still for all practical pur- 

 poses that expressed by (5). 



If we decompose each impulse (8) in the manner explained, 

 we may regard the whole disturbance as arising from an 

 infinite number of simultaneous elementary impulses. These 

 elementary impulses are distributed not entirely at random ; 

 for they may be arranged in groups such that the members of 

 each group are of the same sign, and are, as it were, aimed at 

 the same point under a law of error ; while the different 

 groups are without relation, except that the law of error is 

 the same for all. It is ob'saously not essential that the 

 different groups should deliver their blows simultaneously. 

 Further, it would have come to the same thing had we 

 supposed all the impulses to be delivered at the same point 



* ' Theory of Sound,' vol. ii. § 245. 



