ON FINDING THE EQUATION OF THE CHARACTER- 
ISTIC BLACKENING CURVE FOR A PHO- 
TOGRAPHIC PLATE 
P. S. HELMICK 
In 1890 Hurter and Driffield ^ proposed the following equation 
for the density ^ “D” of a photographic plate exposed for a time 
“t” to light of constant intensity : 
D^l/a Loge(b--[b— l]e-«^) 
where “a”, “b”, and ‘‘c’' are parameters. 
This equation has since been quite generally employed by other 
investigators,^ but its one great disadvantage lies in the difficulty 
of finding numerical values of the parameters for an experi- 
mentally determined plate-curve. Sheppard and Mees * give meth- 
ods of approximating values of “b” and “c” only when “a” 
equals 1. 
This contribution points out two ways of qbtaining values of 
the constants from an empirical curve which has the form of the 
equation given above by (1) an algebraic and (2) a graphical 
process. 
Assuming that the experimental curve is of the form above, 
and selecting four points (L, DQ, — — , (L, D4), so that 
L— ti+At, t3=:ti+2At, and t4=ti+3At, by elimination it can 
readily be shown that 
(eal>2 _ _ ('gaDi _ _ ^aD,) ^ 
1 1 
c = Loge ( e^®i — e^^2)At — Eoge ( e^^^ — , e^®^) ^ 
and 
b = 
gaDi g— cti 
1 — 
so the parameters are thus determined in terms of D^, , D4i 
and At. 
The graphical method, superior to the algebraic method in ac- 
tual practice, can be very conveniently established. 
ijourn. Soc. Chem. Ind.. 9, 455; 1890. 
2 Density of a plate equals logarithm to the base 10 of the ratio of the incident 
light to the light transmitted by the plate. 
3 See, for example, “Theory of the Photo. Process.” Sheppard and Mees, p. 288; 
1907. 
4 Loc. cit. 
