182 HANDBOOK OF PHOTOGRAPHY 



infinity. The value of gamma infinit}' is of considerable significance in both the 

 practical and theoretical aspects of sensitometry. 



The time-gamma curves are often useful in the processing of photographic mate- 

 rials. It has already been indicated that a desirable relation between the density of 

 the silver deposit and the brightness of the image is attained for a value of gamma 

 equal to unity. Therefore, if a time-gamma curve for a particular type of photo- 

 sensitive material and developer is available, reference to the y-Td curve will indicate 

 immediately the development time required to give this value of gamma. The time- 

 gamma curv'^es for the same photographic material for two values of k are given in Fig. 

 19. The ultimate value of gamma attained, however, -/„, is shown as being the same 

 in both cases, although this is not always necessarily true. 



If development proceeds in accordance with a simple law of physical chemistry, 

 as it does for many materials, at least approximately, it can be shown that, theoretically, 

 the time-gamma curve is related to the maximum value of gamma through the 

 equation 



7 = T»(l - e-i^) (27) 



where k = the constant of development; 

 t = the time of development; 

 7„ = the maximum value of gamma to which the photographic material can be 

 developed. 

 This equation holds for many tj'^pes of materials, although the validity of these 

 theoretical relationships in practice depends upon the degree to which the actual H 

 and D curve conforms with the theoretical or ideal familj^ of H and D curves. 



From this last equation, the value of y^ can be determined if we know the value of 

 7 which is obtained for a development time t, when development has been carried 

 on with a developer whose development constant is k. Thus 



T« = :r^^ (28) 



Often, however, the value of the development constant is not known with sufficient 

 precision to be useful in the above equation for the determination of gamma infinity. 

 In such cases 7« may be determined from the measurements made on two density 

 strips, both of which have been processed together in the same solution, but for differ- 

 ent lengths of time. For these conditions, we have, for the first sensitometric strips 

 processed for time ti, 



7, = 7=c(l - e-**i) (29) 



and for the second strip developed for time t^, 



72 = 7cc(l - e-^-.) (30) 



If we process the second strip twice as long as the first strip, then t2 = 2ii, and the 

 equation for 72 becomes 



72 = 7»(1 - e-^'^'i) = 7«[1 - {e-''h)^] (31) 



Combining these two equations for 71 and 72, we obtain 



72 — 71 



-— n — kt , 



(32) 



71 



from which the development constant is found to be, 



A- = 1 log, ( 'L\ (33) 



t\ \72 — 71/ 



