176 Mr Havelock, On the continuous spectrum. 



The method is based upon the expression of a pulse of radiation 

 as a Fourier integral, this corresponding to its physical analysis 

 into periodic constituents ; thus we may put an impulse into the 

 form 



/•OO 



(f>(t)= I R cos utdu (5), 



io 



where u is proportional to frequency, 



and J? is a function of u. 



And as a consequence the energy of the pulse can be expressed 



by 



E= R 2 du (6). 



Jo 



Consider, for instance, the type of pulse given by 



4>{t) = A<r** (7). 



When c is very large this pulse is intense and narrow, while for 

 smaller values it becomes less abrupt. To express it as a Fourier 

 integral, we notice that for large values of c all the effective part 

 is contained within a small region, so that without appreciable 

 error we may take ± oo to be the limits for the time t. Then we 

 have 



/•GO /-GO 



7T(f> (t) — I I <p(co) cos u(o) — t) dudco 



J J -oo 



/•GO /•OO 



= 2 I cos utdu I A e~ c "' M ' 2 cos ucodco 

 jo Jo 



= — I e 4c 2 du (8), 



c Jo 



and the energy of the pulse is given by 



A 2 f°° _^! 

 E=——\ e w du (9). 



C 2 7T 2 Jo 



Now the heat radiation of a substance is supposed to consist of 

 an irregular sequence of pulses, thus giving rise to a continuous 

 spectrum; and it seems probable that this also holds for a gas, 

 the individual impulse being given out during the short time of 

 an encounter between two molecules. Any such impulse, when 

 it reaches the observer, may be considered to be a function of 

 the relative velocity of the two molecules and of the temperature 

 of the gas as a whole; thus taking the type of pulse given in 

 (7) and introducing r and 6 in such a way as to conform to the 

 general character of a pulse due to an impact, we have 



(j>(t) = Ar m e-» w (10), 



