EXPOSURE AND EXPOSURE DEVICES 213 



in the plane of the entrance pupil, and directly proportional to its luminous intensit.y 

 /. Let the distance between P and R be X. Then the intensity of light falling upon 

 the entrance pupil will be proportional to I/X^. The distance X may be considered 

 as being made up of two components. One of these is the distance from P to the 

 interior principal focal lengthL, which distance is given by —L/M where M is the linear 

 magnification produced by the lens system. The negative sign is required because 

 of the inversion of the image. The second component of the distance X is the 

 distance LR from the plane of the principal focus to the plane of the entrance pupil. 

 Since the principal-focus and the entrance-pupil planes are never very far removed 

 from one another, the distance LR may be expressed by L{1 + q) where 9 is a small 

 positive or negative decimal. Neglecting the negative sign required because of the 

 image inversion, the distance from P to R may be expressed as 



X=L[{l+q)+^] 



(2) 



The intensity /' of the light at the point P' is proportional to the cone whose half 

 angle is 6. The maximum diameter of this cone at the exit pupil is determined by 

 the area of the aperture, which is given by 



where d = the diameter of the aperture. 



The cone of light emerging from the exit pupil comes to a focus at P' and produces 

 an image of P whose size is proportional to the linear magnification of the system M. 

 The intensity of the image at P' is inversely proportional to the area of the image. 

 But the area of this image is 



« = -4- (4) 



so that /' is proportional to 4/TrM'. 



Finally, the intensity of the image at P' is reduced by absorption and reflection 

 by the separate elements of the lens system. Of the light incident upon the lens, 

 some is absorbed, but a greater part is reflected from the lens surfaces, especially if 

 these are uncemented. The quantity of the emerging light is always less than that 

 incident upon the system and is proportional to the incident light and the trans- 

 mission of the lens system T. Consequently I' is proportional to T. 



Having discussed briefly the separate factors which influence the intensity of 

 the image, we may now combine the separate effects. Thus, for an object P on, or 

 very near to, the optical axis of the lens system, the intensity of the image is 



X^a ^^^ 



where fc is a numerical constant depending upon the units of measurement. By sub- 

 stituting for A, X, and a, the values already determined, and by simplifying, the 

 expression becomes 



IL'[M{1 +q) + ipj ^''^ 



Since g is a small fraction, little error is introduced if it is neglected, and for practical 

 purposes the above equation may be simplified to 



