Lord Rayloigli on Pin-hole Photography. 91 



practical application to pin-hole photography, I have thought 

 that it would be interesting to adapt Lommel's results to the 

 problem in hand, and to exhibit upon the same diagram 

 curves showing the distribution of illumination in various 

 cases. For the details of the investigation reference must be 

 made to Lommel's memoir, or to the account of it in the 

 Encyclopaedia Britannica, art. "Wave Theory/' p. 444. But 

 it may be well to state the results somewhat fully. 



In the following formulae a is the distance from the radiant 

 point to the aperture, b from the aperture to the screen upon 

 which the image is formed. The circumstances being sym- 

 metrical about a line through the radiant point and the centre 

 of the circular aperture (radius r), the illumination I 2 will be 

 the same at all points of the screen equally distant J from the 

 axis, and the problem to be solved is the determination of P 

 as a function of ? for given values of a, b, r, and X. Lommel 

 finds that 



F =^ C2 + S2 )> • • • • (3) 

 where 



C = J* J cos (-i- Kp 2 — Ip cos <f>) .pdp dcj>, ... (4) 



S= ffsin(J/cp 2 — lpcos(j>) .pdpd<j), ... (5) 



and the following abbreviations are introduced : — 



2-7T a + b 1 277-f , 



(6) 



The above corresponds to an incident wave whose intensity 

 at the aperture is measured by 1/a 2 . The integration is to be 

 taken over the area of the aperture, that is from </> = to 

 <$> — 2^7, and from p = to p = r. If we introduce the notation 

 of Bessel's functions, we have 



C = 2tt J (lp) cos (±/cp 2 ).p dp, .... (7) 



S = 2tt \ r j (lp) sin (ifcp 2 ).p dp (8) 



By integration by parts of these expressions Lommel develops 

 series suitable for calculation. Setting 



KJ 



V lr=g, (9) 



