DEVELOPERS AND THEORY OF DEVELOPMENT 349 



7. Thus there are two equations, 



D = D„{1 - e-A'O (3) 



and 



D = DJX - e--^'^'-'o)] (4) 



corresponding to the two similar equations for 7. 



Where a tie point exists but is depressed below the D = axis, the equations may 

 be written in the form 



Z> = £>„ - (Z)„ + D,)e-^t , (5) 



and 



D = D^ - {D„+ Do)e-K^*-'o) (6) 



where Do is the magnitude of the depression of the tie point, and, as usual, the con- 

 sideration is limited to points on the straight-line portion of the sensitometric curve. 



These equations are empirical in nature and difficulty will often be experienced 

 in trying to apply them to too wide a range of developing conditions, particularly if 

 emphasis is placed upon the very early stages of development. 



In many cases the existence of a tie point is doubtful or definitely disproved. 

 Under each conditions the similarity here apparent between 7 and D equations will no 

 longer remain. 



The practical value of any of these equations lies in the ability to use them to 

 interpolate or extrapolate from existing tests to other conditions. Thus, if we wish to 

 develop a negative to 7 = 0.9 and have tests showing the times required for, say, 

 7 = 0.7 and 7 = 1.0, interpolation is necessary to determine the correct time and may 

 be done by the evaluation of the first 7 equation. Of course, a worker who frequently 

 meets such problems as this relatively simple case will very quickly learn to estimate 

 correct times much more quickly than they can be calculated through the use of the 

 equations. The importance of the mathematical methods increases as processing 

 conditions are controlled more and more accurately, but for many amateur and com- 

 mercial procedures, high precision in interpolations of the type indicated is nullified 

 by poor technique and lack of the extreme care necessary to obtain reproducible 

 results. 



Considerable effort has been spent in the attempt to learn the true nature of the 

 development process and to identify the various stages with corresponding constants 

 in the equations. Thus the time of penetration of the developer into the emulsion, the 

 invasion phase or induction period, is considered the counterpart of the ^0 of 

 the equations. 



Diffusion of the developer in, and of products of development out of, the emulsion 

 must play an important part, and some efforts have been made to trace the course of 

 development through these processes. 



Adsorption theories of development have been advanced also, but none of these 

 attempts to study the development process has yet supplanted the much simpler 

 empirical relationships given above for practical interpretation of rate of development 

 data. 



A brief mathematical study of sensitometric curves and development data has 

 recently been published, ^ based upon approximations designed to represent statistically 

 emulsion conditions and development processes. The results obtained show unusually 

 good agreement between calculated and observed values. The mathematical forms 

 used depend upon many simplifying assumptions of a type which seem reasonable but 

 for which little direct experimental evidence exists. Hence the final fit may be viewed 



I Albersheim, W. J., J. Soc. Motion Picture Engrs., 29, 417-455 (1937). 



