Theory of the Resolving Power of Objectives. 167 



A simple calculation (given at p. 277 o£ my DeschaaeU 

 Part iv.) shows that the extreme difference of optical path, 

 for disturbances coming from different points o£ a concave 

 wave-front to a point at lateral distance b from the geo- 

 metrical focus (the centre of the sphere to which the wave- 

 front belongs), is 26 sin a, a denoting the angular radius of 

 the wave-front as seen from the focus, When the extreme 

 difference of path is X, we have therefore 



6 = 5-£- (1) 



2 sin a 

 Comparison with observation shows that this value of b 

 represents with fair accuracy the limit of separation. The 

 angle subtended by the distance b at the second nodal point 

 of the objective, which is identical with the angle subtended 

 by the corresponding distance in the object, as seen from the 

 first nodal point, is 



f 2/ sin* D ? W 



/ being the focal length, and D the diameter of the objective. 

 This formula X/D for the least distance between the com- 

 ponents of a double star, agrees with Dawes's value above 

 quoted, if we put X= "000022 inch= # 5G micron. The wave- 

 length for the brightest rays is usually taken as *55 micron, 

 which is as good an agreement as could be desired. 



Passing now to the case of the microscope, and supposing 

 the same formula for the minimum distance b in the image to be 

 still applicable, we may conveniently transform it by means 

 of the equation (which we shall discuss later) 



fj, 1 7/ 1 sin 6 l = fi. 2 y. 2 <m d 2 .... (3) 

 applicable to any optical system which gives sharp flat images. 



In this equation, 

 i/ l y 2 denote the distances of a point of the object, and the 

 corresponding point of the image, from the axis of 

 the system ; 

 Pi fi. 2 the indices of the first and last media ; 

 Oi 0-2 the angles made with the axis by any incident ray and 

 the corresponding emergent ray. 

 The ratio (/x : sin 0i)/(fi 2 sin Q 2 ) is equal to the magnification 

 y-i\y\' and is therefore the same for all values of 0. This 

 constancy is called by Abbe the sine condition. 



In the present case Q 2 is a., y 2 is b, /jl 2 is 1 ; and if a, denote 

 the obliquity of an extreme incident ray. the equation gives 

 /*! y x sin ot l = b sin a, 

 sin oe. j _ sin a. A, X \j f . 



J 1 ii x sin a x fa sin a x * 2 sin a 2/* i sin a, * ; ain ~ ' 



