TRANSMITTED THROUGH AN ABSORBING MEDIUM. 201 



In the above remarks I have supposed we are dealing 

 with homogeneous light, or with white light which has 

 penetrated a medium containing soluble black in solution. 

 To apply the formulae generally we must prefix to them 

 the sign of summation. 



In seeking a priori the law of transmitted light, we 

 might have reasoned as follows, which involves less 

 assumption than Herschers reasoning. 



Suppose we have a column of any length; conceive it 

 divided anywhere into two lengths oc and y by an 

 imaginary plane. Let I be the initial intensity of light; 

 after penetrating the column x we shall have 



I x =I Q <f>{w). 



But if light of intensity I x penetrate a column of length y, 

 the transmitted light will be I x <f>{y) , or by substitution 

 l <b(x)<f>{y). Since the length of the column is %-\-y, 

 the emergent light will also be expressed by \ Q (\>{oc-\-y). 

 Equating these two expressions for the same quantity, 

 there results 



<t>(%)<l>(y)=<t>(x+y)' 



It is well known that this functional equation is satisfied 

 by an exponential form. 



SER. III. VOL. VIII. 



