﻿Theory of Photographic Exposure. 263 



or, dividing by N and subtracting from unity, 



'-i-»(t)(-»)*-*o(-ir- 



N 

 i. e. ultimately 



-»-(»-ir m 



This is rigorously equivalent to or identical with (6) for any 

 m and X Now, for large JN", and any m, equation (6') in 

 which (1 — l/N)*=:l/e, asymptotically, gives at once 



k -™ 



-r T =l — e * = l — e^ va . 



This is the connexion between the rigorous arithmetical 

 formula and the exponential one. 



It is needless to insist that under the conditions prevailing 

 in all practicable experimental cases there is more than suffi- 

 cient mathematical accuracy in formula (4) 



k = N{l~e- na ) J 



which, apart from minor modifications, will henceforth be 

 used in what follows. 



This formula is of the familiar type proposed (1893) by 

 Elder, with the notable difference, however, that while his 

 exponent contained a free " parameter " or coefficient to be 

 evaluated empirically and principally depending upon "grain- 

 sensitivity " and wave-length, both coefficients in (4) are 

 completely determined, and the exponent moreover shows 

 an explicit and most essential dependence upon the size (a) 

 of the grain, and in the right sense too, i. e., giving an 

 increase of the " speed " with grain size. The comparison 

 with experimental facts of Elder's and of a number of other 

 formulae, constructed empirically, is too well-known to be 

 discussed here * Suffice it to say that, although it repre- 

 sents to a certain extent the photographic behaviour (the 

 " characteristic " curve) of some emulsions, and particularly 

 those with what is termed an " extended toe," it cer- 

 tainly shows considerable deviations from the observed 

 characteristic curves. Yet it will not be forgotten that all 

 these comparisons bore upon the resultant total photographic 

 densities, containing or integrating the effects of grains 

 belonging to a broad range of sizes (a), instead of equal 



* Cf. for instance, a paper bv Dr. F. E. Ross, Jonrn. Opt. Soc. Amer. 

 vol. iv. p. 255 (1920). 



