66 OJST THE GENERAL PROBLEM OF PHOTOGRAPHIC REPRODUCTION [359 of light. We must suppose either that one kind of light only is employed, or else that t is the same for all the kinds that need to be regarded. The actual values of t may be supposed to range from 0, representing complete opacity, to 1, representing complete transparency. As the first step is the production of a negative, the question naturally suggests itself whether we can define the ideal character of such a negative. Attempts have not been wanting ; but when we reflect that the negative is only a means to an end, we recognize that no answer can be given without reference to the process in which the negative is to be employed to produce the positive. In practice this process (of printing) is usually different from that by which the negative was itself made; but for simplicity we shall suppose that the same process is employed in both operations. This requirement of identity of procedure in the two cases is to be construed strictly, extending, for example, to duration of development and degree of intensification, if any. Also we shall suppose for the present that the exposure is the same. In strictness this should be understood to require that both the intensity of the incident light and the time of its operation be maintained ; but since between wide limits the effect is known to depend only upon the product of these quantities, we may be content to regard exposure as defined by a single quantity, viz. intensity of light x time. Under these restrictions the transparency tf at any point of the negative is a definite function of the transparency t at the corresponding point of the original, so that we may write *'=/(*), .................................... (1) / depending upon the photographic procedure and being usually such that as t increases from 0 to 1, if decreases continually. When the operation is repeated upon the negative, the transparency t" at the corresponding part of the positive is given by - .............. (2) Complete reproduction may be considered to demand that at every point t" = t. Equation (2) then expresses that t must be the same function of if that if is of 1 Or, if the relation between t and t' be written in the form ') = 0, ................................. (3) F must be a symmetrical function of the two variables. If we regard t, t' as the rectangular coordinates of a point, (3) expresses the relationship by a •curve which is to be symmetrical with respect to the bisecting line t' — t. So far no particular form of /, or F, is demanded ; no particular kind of negative is indicated as ideal. But certain simple cases call for notice. Among these is t + t=l, ..................... ' ............ (4)ion.