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The effect of varying the exposure (e) is the same as of an inverse alteration in the transparency; it is the product et with which we really have to do. This refers to the first operation ; in the second, t" is dependent in like manner upon e't'. For simplicity and without loss of generality we may suppose that e = 1 ; also that e'/e = m, where m is a numerical quantity greater or less than unity. The equations which replace (1) and (2) are now
........................ (8)
and we assume that / is such that it decreases continually as its argument increases.    This excludes what is called in photography solarization.
We observe that if t, lying between 0 and 1, anywhere makes t' - t, then m must be taken to be unity.    For in the case supposed
and this in accordance with the assumed character of /cannot be true, unless m = 1. Indeed without analytical formulation it is evident that since the transparency is not altered in the negative, it will require the same exposure to obtain it in the second operation as that by which it was produced in the first. Hence, if anywhere t' = t, the exposures must be 'the same.
It remains to show that there is no escape from a local equality of t and t'. When t  0, t' = 1, or (if there be fog) some smaller positive quantity. As t increases from 0 to 1, t' continually decreases, and must therefore pass t at some point of the range. We conclude that complete reproduction requires m = 1, i.e. that the two exposures be equal ; but we must not forgot that we have assumed the photographic procedure to be exactly the same, except as regards exposure.
Another reservation requires a moment's consideration. We have interpreted complete reproduction to demand equality of t" and t. This Heeins to be in accord with usage ; but it might be argued that proportionality of t" and t' is all that is really required. For although the pictures considered in themselves differ, the effect upon the eye, or upon a photographic plate, may be made identical, all that is needed being a suitable variation in the intensity of the luminous background. But at this rate we should have to regard a white and a grey paper as equivalent.
If we abandon the restriction that the photographic process is to be the same in the two operations, simple conclusions of generality can hardly be looked for. But the problem is easily formulated. We may write
t'=Met),       *=V=/2(eY),   ..................... (9)
where e, e' are the exposures, not generally equal, and /lf /2 represent two functions, whose forms may vary further with details of development and intensification. But for some printing processes f% might be treated as a fixed function. It would seem that this is the end at which discussiontude in the relationship between t and t'; but a closer scrutiny seems to show that this is not the case.