204 DR. J. BOTTOMLEY ON THE INTENSITY OF LIGHT AFTER 



Let %{£) = '{$ (f)dt. Then, integrating equation (2), we 



obtain 



log I =-^ (*)+<?, 



which may also be written in the form 



I=Ce -Mx(o (3 ) 



To determine the constant we must know the initial 

 values of I and t. Suppose that when t = o, I = I Q ; then 



I = Ce-^ (0) , 



and substituting for C in (3) we get 



I = I o€ ^ (o) e- MxW . 



This will be the general expression for the intensity of 

 light which has penetrated a length t of a medium of 

 variable density. 



I. Suppose the density to vary as the distance from the 

 plane of incidence ; then we have 



cj)(t) =nt 3 

 where n is some constant ; 



and 



x(0 = T 



I=Ce^. 



If I be the intensity when /=o ; we get 



Io=C. 



This determines the constant. If for nt we substitute d } 

 we obtain the following relationship between the intensity 

 and the density at any point :— «- 



