TRANSMITTED THROUGH AN ABSORBING MEDIUM. 199 



absorbing columns for the transmitted light in one case, 



we shall have 



I x =lo0(ff), (i) 



and in the other case 



Wo<£(2/)> ( 2 ) 



(j> being the unknown function which it is required to 

 determine. Since these two intensities are equal, 



I *(*)=I'o*(y) (3) 



If both columns receive an equal increment, the inten- 

 sities will again be equal. Let this common increment be 

 denoted by k, then 



I o <£(a? + *)=I' o 0(y + *) (4) 



These equations will still hold if y=o. 



In this case the duller surface is observed directly, and 

 the length of the absorbing column over the brighter 

 surface is increased until they are brought to the same 

 intensity. Since when y = o, V 9 =V OJ therefore 0(o) = i. 



Equations (3) and (4) will become 



Io#*)=r<„ (5) 



I o <f>(ff + «)=I' o 0(«) (6) 



Expand (6) in terms of k, 



Io 



=r (i+f(o)/ C +^l/c 4 +&c). 



Since the first terms of the expansion on each side are 

 equal, the equation may be written 



1 



