The Helmholtz Theory of the Microscope. By J. W. Gordon. 397 



focal length — subject to two provisos : (1) That focal lengths shall 

 be measured in time, that is to say, taken to be proportional to 

 the optical path; and (2) That if the image be formed in a 

 medium where the wave-length is short its diameter is to be 

 shortened in the same proportion as the wave-length. 



Having obtained this equation (4), Helmholtz proceeds to apply 

 it in various ways to the examination of the Microscope. And first 

 he proposes by its aid a system of rating objectives by what he 

 calls Normal Magnifying Power. Normal magnifying power, for 

 which Helmholtz proposes the symbol N — may I presume to 

 suggest M — may be explained thus : — 



Fig. 85. 



Let u ( be the divergence angle of the wave-front which enters 

 the objective from the stage of the Microscope, and u be the 



divergence angle of that which focusses in the observer's eye. Then 



ft n sin u e 



__ r- ' = M, the magnifying power of the instrument. In 



P € Hy Sill 10^ 



place of the actual eye Helmholtz proposes a conventional eye, to 

 be taken as a standard of comparison. This conventional eye 

 would have a pupil 3 mm. in diameter and a focal length, in an, 

 and measured of course from the edge of the pupil, of 250 mm. ; 



1*5 



therefore its N.A. is w^k = 0*006. It will of course be recognised 



that this is not even an approximation to the numerical aperture of 

 the actual human eye, which has a focal length of less than one inch, 

 and focusses in a medium having about the refractive index of 

 water. But any standard, if adopted, would serve almost as well 

 as any other for the making of comparisons between different objec- 

 tives, and on Helmholtz' plan an objective of N.A. = 0*006 would 

 have a normal magnifying power M = 1. Accordingly a lens with 



N.A. = 1 will have a N.M.P. = n AAtJ = 166*7, and so on. 



0*00b ' 



In a simple Microscope, say, for example, a pair of spectacles, 

 the N.A. of the lens must be obtained by dividing 1 " 5 mm., the semi- 

 diameter of the pupil of the eye. by the focal length of the lens, 

 Aug. 19th, 1903 2 d 



