AND A SIMPLE APERTOMETER DERIVED THEREFROM. 3 



any wide-angle optical system, ichich satisfies the sine-condition for 

 a pair of conjugate foci, the equivalent refracting surface for 

 these foci is a part of a sphere. * 



In the case of the microscope objective, with which we are 

 principally concerned, the image is always formed in air, hence 

 pi = 1 in equation 1, and for a pair of conjugate rays meeting 

 in the vertex v, — 



b _ sin a _ p. 



a sin/2 M ' \ / 



Putting l for a + b, the distance p q, and u/p for b/a, we can 

 write equation 3 in the form — 



r- ^ LM . . (6) 



and equation 4 as — 



d = J~?- ■ • < 7 > 



An example will show the use and application of the last two 

 equations. A dry lens, of a focal length of 15*8 mm., gave in 

 a plane 205 mm. above the plane of the object, on the stage 

 of the microscope, a magnification of 11*5. Substituting these 

 values in equation 6, and remembering that /x = 1 in this case, 

 we have, for the value of the radius of the equivalent refracting 

 spherical surface — 



205 x 11-5 1Q 

 r = (11-5)* -1 = lbmm - 

 And obviously, so long as a<&, this surface must be convex on 

 its upper surface. By substituting in equation 7, we get for 

 the distance d of the centre of curvature o below the aplanatic 

 focus p, — 



d = (ii-5)»-i = 1 ' 6mm -; 



and again, so long as a<^b, o is below p. Thus, in a very 

 simple and practical way, it is possible to determine for any 

 aplanatic system, from the distance between the aplanatic foci 



* This proposition is well known for the particular case in which one of 

 the aplanatic foci is at infinity, as for a telescope object-glass ; but, so far 

 as I can discover, the general proof given above, simple though it is, and 

 important as it appears to be, does not occur in any English book on the 

 subject. Dr. von Rohr, of Jena, has, however, since the reading of the 

 paper, drawn my attention to an article by Mittenzwei in the Jahrbuch 

 fiir Photographie, 1888, pp. 317-20, which clearly anticipates my 

 proposition. 



