﻿Quantum Theory of Photographic Exposure. 1367 



by the fundamental formula (7), first paper, 



k = 



| f(a)[l-e-" a ]da, 



where f (a) da is the number of targets of size a to a + da 

 originally present. Now, if lot is of the order of .1/2 or even 

 l^ 2 , we can take/(a)=C07is£. within the integration interval 

 with sufficient accuracy for all of* our experimental emulsions. 

 Thus, denoting by JV the original number of targets in the 

 whole class, so that 



we shall have g = 1 ^ ' 



or, writing simply a for the average a = -^-(a 1 -fa 2 ), and there- 

 fore, « 2 = a + a j a i — a ~~ u > 



.N 2noc 



Remembering that i(e na — e~ na ) = sinh (net), writing for 

 brevity 



r = lo £iy~p • " ( 13 ) 



and replacing a iu the chief term by 



a' = a[l- Vo>]' 2 *, 

 we have ultimately the required formula 



, , sinh(na) nA , 



v = na — log — - (14) 



& not ' 



Notice that the correction term depends only on not, that is 

 to say, for \ = const., on the product of the exposure and the 

 class breadth. If this product is a fraction, such as one-half 

 or even two-thirds |, we can write, up to {not)*, 



v = na' — ^(nu) 2 (14a) 



If, as explained, all the contemplated targets of the 

 emulsion are divided into classes of equal breadth 2«, the 



# This is accurate enough provided <r/a is small. In the correction 

 term the semi-breadth a requires practically no amendment. 



t If a = lfx' 2 and the exposure is as in the previous concrete cases, the 

 value of this product does not exceed 06. 



