90 Lord Rayleigh on Pin-hole Photography. 



invoked. If, on the other hand, the aperture be large, the 

 lens becomes indispensable. The size of the aperture should 

 accordingly be increased up to the point at which the lens is 

 sensibly missed; and this, as we have seen, will occur in the 

 neighbourhood of the value determined by (1). A more pre- 

 cise calculation can be made only upon the basis of a detailed 

 knowledge of the distribution of light in the image. 



The question of the best size of aperture for a pin-hole 

 camera was first considered by Petzval*. His theory, though 

 it can hardly be regarded as sound, brings out the failure of 

 definition when the aperture is either too large or too small, 

 and, as is very remarkable, gives (1) as the best relation 

 between r,f) and X. The argument is as follows : — If the 

 hole be very small, the diameter of the patch of light repre- 

 sentative of a luminous point is given by 



D=A/r, 



the measurement being made up to the first blackness in the 

 diffraction-pattern. " This formula is only an approximate 

 one, applicable when r is very small ; in the case of a larger 

 aperture, its diameter must be added to the value above given, 

 that is to say, 



~D = 2r+f\/r. 



From the last formula we can at once deduce the best value 

 for r ; in other words, the size of the aperture which corre- 

 sponds to the least possible value of D, and therefore to the 

 sharpest possible image. In fact, differentiating the last 

 expression, and setting in the ordinary manner, dD/tffr = 0, we 

 find at once 



w r hich corresponds to 



D = 2 N /(2/\)." 



The assumption that intermediate cases can be represented by 

 mere addition of the terms appropriate in the extreme cases 

 of very large and very small apertures appears to be inad- 

 missible. 



The complete determination of the image of a radiant point 

 as given by a small aperture is a problem in diffraction, solved 

 only within the last years by Lommelf. In view of the 



* Wien. Sitz. Ber. xxvi. p. 33 (1857) ; Phil. Mag. xvii. (1859) p. 1. 



f " Die Beugimgserscheinungen einer kreisruiiden (Effnung und ernes 

 kreisrunden Schirmchens," Aus den Abhandlnngen der k. buyer, Akademie 

 der Wiss. ii. CI. xv. Bd. ii. Abth. (Miinchen, 1884.) 



