NUMERICAL APERTURE. 



1G3 



that the rays emerge approximately parallel, so they give a clear 

 idea of an opening or aperture. 



A convenient i)lace to choose for the proof I am giving is the 

 moment when they have just emerged. 



Let G C and H D (Fig. 4) be the exterior and interior faces 



iP^i^ 



a ^ 



Fig. 4. 

 of an aplanatic system on the plane of the paper and the line B A 

 represent the princi[)al axis of the system, meeting each face in 

 C and D. Let A and B be two conjugate foci, and A G any ray 

 of light which passes through the system, and is refracted along 

 H B to B ; then by the Abbe-Helmholz law of aplanatic con- 

 vergence, 



Sin B 



=a constant=r, 



Sin A 

 as long as the same focus and system are considered. Let the 

 refractive index on the A side be in and on the B side unity, 

 that is, air ; then if a and b represent the ratio of the arc to the 

 radius of two very small conjugate angles lying close to the axis, 

 and E represent the magnification of the image at B by the 

 Lagrange Helmholz law, 



a VI 



^ ~ K 

 If the focal length of the objective be / and the image is pro- 

 jected to a distance, ^, 



E= — (nearly) 



a mf 

 ' ' ' l^=~d; 

 but as the sines of very small angles are equal to the ratio of the 

 arc to the radius, and as tlie first equation holds good for all 



angles, 



mf 



