Problem of Photographic Reproduction. 737 



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. 



The effect o£ 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 = l; 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 



t'=f(t), t = t"=f(mt'); .... (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 /, lying between and. 1, anywhere 

 makes t' = t, then m must be taken to be unity. For in the 

 case supposed 



t=At)=f{mt) ; 



and this in accordance with the assumed character of / 

 cannot be true, unless ??i=l. 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 ex- 

 posures must be the same. 



It remains to show that there is no escape from a local 

 equality of t and t' . When £ — 0, t' = l, or (if there be fog) 

 some smaller positive quantity. As t increases from to 1, 

 t' continually decreases, and must therefore pass / at some 

 point of the range. We conclude that complete reproduction 

 requires ra = l, i. e. that the two exposures be equal ; but we 

 must not forget 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 seems 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 photo- 

 graphic 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. 



