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XV. Contributions to the Theory of the Resolving Power of 

 Objectives. By Professor J. I). Everett, F.R.S* 



IN high class objectives, both of telescopes and microscopes, 

 the practical limit to the power of separating close points 

 (called resolving 'power or separating power} depends upon the 

 blurring due to diffraction. Owing to diffraction, the image 

 formed of a bright point is not a point, but a spot, brightest 

 in the central part, and falling off without any discontinuity 

 from the centre to the margin. In favourable circumstances 

 this spot is surrounded by a succession of bright rings. The 

 phenomenon is seen in its greatest perfection when small 

 aperture is combined with good definition. Blocking out the 

 central portion of the objective makes the spot smaller and 

 the surrounding rings relatively brighter. 



Dawes (Mem. R. Ast. Soc. xxxv. p. 158) made very 

 elaborate observations on double stars for the purpose of 

 investigating the separating power of telescopes ; and arrived 

 at the conclusion that the angular distance between the two 

 components, when they are nearly equal in magnitude, and 

 are just separated, is given by the formula — 



4*56 seconds, divided by diameter of objective in incites. 



The first calculation of the relative brightness at different 

 points of the spot and rings, which constitute the diffraction 

 image of a point formed by a lens symmetrical round an axis, 

 was published by Airy in 1836 (Camb. Trans, v. p. 283), in 

 a very clear and readable paper. His basis of procedure is 

 the very direct and intelligible one of considering the concave 

 wave-front which advances from the objective to the focus, 

 and computing, for its initial position, the " disturbance" which 

 it produces (according to Huy gens' principle) at any given 

 small distance measured laterally from the geometrical focus. 



Another principle of calculation, less obviously correct 

 but leading to precisely the Same result, is employed in 

 Mascart's Optigue and in Preston's Light. 



Both methods of procedure lead to one and the same 

 infinite series for the " disturbance" at given lateral distance 

 from the geometrical focus ; and this series is a Bessel's 



function of the first order. It is in fact — — — -, m denoting 



2ttR . . m 



— -rb, where R, is radius of aperture, /'focal length. 6 lateral 



^ •' 

 distance, and X wave-length. The calculation assumes 



identity of disturbance both in degree and in kind at all 



points of the wave-front. 



* Communicated bv the Physical Society ; read Feb. 28. 1902. 



