1911] ON THE GENERAL PROBLEM OF PHOTOGRAPHIC REPRODUCTION 67 which obviously satisfies the condition of symmetry. The representative curve is a straight line, equally inclined to the axes. According to (4), when t = 0, t' — 1. This requirement is usually satisfied in photography, being known as freedom. from fog—no photographic action where no light has fallen. But the complementary relation t' = 0 when t = 1 is only satisfied approximately. The relation between negative and positive expressed in (4) admits of simple illustration. If both be projected upon a screen from independent lanterns of equal luminous intensity, so that the images fit, the pictures obliterate one another, and there results a field of uniform intensity. Another simple form, giving the same limiting values as (4), is *2 + ^'=l; .................................(5) and of course any number of others may be suggested. According to Fechner's law, which represents the facts fairly well, the visibility of the difference between t and t + dt is proportional to dt/t. The gradation in the negative, constituted in agreement with (4), is thus quite different from that of the positive. When t is small, large differences in the positive may be invisible in the negative, and vice versd when t approaches unity. And the want of correspondence in gradation is aggravated if we substitute (5) for (4). All this is of course consistent with complete final reproduction, the differences which are magnified in the first operation being correspondingly attenuated in the second. If we impose the condition that the gradation in the negative shall agree with that in the -positive, we have dtjt^-dt'lt',.................................(6) whence t.t' = C, .......................................(7) where 0 is a constant. This relation does not fully meet the other requirements of the case. Since t' cannot exceed unity, t cannot be less than G. However, by taking C small enough, a sufficient approximation may be attained. It will be remarked that according to (7) the negative and positive obliterate one another when superposed in such a manner that light passes through them in succession— a combination of course entirely different from that considered in connexion with (4). This equality of gradation (within certain limits) may perhaps be considered a claim for (7) to represent the ideal negative; on the other hand, the word accords better with definition (4). It will be remembered that hitherto we have assumed the exposure to be the same in the two operations, viz. in producing the negative and in copying from it. The restriction is somewhat arbitrary, and it is natural to inquire whether it can be removed. One might suppose that the removal would allow a greater latitude in the relationship between t and t'; but a closer scrutiny seems to show that this is not the case. 5—2