TRANSMISSION THROUGH AN ABSORBING MEDIUM. 203 



on a surface for which the coefficient of transmission is 

 e~ m . Let us take a plate of the medium of thickness At, 

 and let 6 _< > l + Am ) De the coefficient of transmission at the 

 upper surface. If V be the transmitted light, 



i<lt wA ', 



>±e . 



Expanding, and putting AI for I'— I, this becomes 



AI Af 

 ^r< —mAt + m z &c. 



1 2 



At 2 - 

 > — (m + Am) At+ (m + Am) 1 &c. 



Ultimately we get 



c?logI=— mdt (i) 



Now suppose m to be some function of t ; if this be 

 some integral form, then we may determine I. 



If light has passed through a homogeneous medium 

 t units long, then I = I e -m *. If the length be unity, 

 I = I e -n \ If the unit length contain q units of colouring- 

 matter, we also have I = I e - ^. Equating these two values 

 of I, there results the equation 



m=fiq. 



Let d be the density of the colouring-matter, then d will 

 be proportional to q. If the unit of density be that due 

 to the distribution of the unit mass through unit volume, 

 then we may replace q by d. Now suppose the density 

 variable and some function of t, so that we may write 



d=<f>(t). 



Substituting in equation (i) we get 



<n<>gi=— ^0(*)<# (2) 



p2 



