MICROSCOPIC VISION. 



lol 



either side of a given system, n and n' the refractive in- 

 dices of the media on either side, and M the magnifying 

 power ; then — 



u 



Lagrange (1803) showed that ' — 31, the media on both 



It 



sides of the system being the same and the aperture 

 small. 



Helmholtz (1866) that — — / = il/, the media different but 



the aperture small. 



7i sin 2L 



Abbe 1 (1873) that — -, — -. , ^ 31, for any aperture or 



media. 

 Prof. Helmholtz^ also proved this last result, but was 

 preceded in publication by Prof. Abbe by six months. 



The following is a condensed epitome of the proof. Let 



p and p' be two conjugate points on the axis, and let u 



represent the angular divergence of an immergent, and u' 



the convergence of an emergent pencil ; and a, a', very 



small angles with the same significance. Let air be on 



both sides of the system. When p' is large, ^:> = /very 



s 

 nearly. Then if s is the semi-aperture, ~ = tan a ; and 



1 Prof. Abbe's theorem, J. R. M. S., ser. ii. vol. i. (1881), p. 392. 

 Vide also an interesting paper, /. R. M. S., vol. iii. (1880), p. 509. 



2 Prof. Helmholtz' theorem, Proceedintjs Bristol Nat. Soc, New 

 Ser., vol. i., part 8; and in Monthly Microscopical Journ.^ vol. xvi. 

 (1876), pp. 22, 25, 26. 



