396 Transactions of the Society. 



centre. The law therefore derived from this example is perfectly- 

 general, and if fi v be the diameter of an image formed in a vacuum 



and j3 q be the diameter of the corresponding image formed in 



glass, we shall have the numerical relation between them expressed 

 by the equation 



7p= n g . . . . (2) 



Pg 



Eecurring now to equation (1) above (p. 393) we may give to 

 it a more general form. As it stands it applies only to the case 

 in which both the conjugate images are formed in the same 

 medium. If we now consider the general case in which in front 

 of the aperture we have a medium in which the wave-length is \ e 

 and behind it one in which it is \ v we shall have for the magnifying 

 power. 



M = bill.* . . . (3) 



This is Helmholtz' first result, but he expresses it somewhat 

 differently. For the purpose that he has in view it is convenient 

 to express the magnitude r in terms of the divergence angle u. A 



glance at fig. 82 (p. 392) will show that r = A ' ' m A X 

 Therefore equation 3 may be written 



sin u 



M _ V _ \ sin u e _ n ( sin u 



/3 f X e sin u v n v sin u 

 . • . n e sin w € /3 e = n v sin u^ /3„. . . (4) 



which is the form that the equation takes in Helmholtz' paper. 

 In this expression n € sin u e is what is now known as the numerical 



aperture of the objective, and n^ sin u the numerical aperture of 

 the image formed in the instrument, or as the case may be, in the 

 observer's eye. 



It would probably simplify the understanding of the signifi- 

 cance of numerical aperture by microscopists who do not happen 

 to be also mathematicians if they were told that the sine relation 

 or numerical aperture law amounts only to the very familiar pro- 

 position that the magnifying power of a lens varies inversely as its 



* This expression is a little more symmetrical than the equivalent expression 



m = 55. 



".) r < 



