6 F. J. CHESHIRE ON ABBE'S TEST OF APLANATISM, 



consider the ray pcq intersecting the second (back) principal 



focal plane of the objective, at a distance h from the axis ; and 



let the distance of this plane from the point q = i^. Then, q 



being in air, we have from equation 5, — 



n usin a /ox 



sin /? = C (8) 



m v ' 



Since the angle /?, in a microscope system, never exceeds a few 

 degrees, its tangent may be taken as equal to its sine ; hence — 



■k/8-ri ( 9 ) 



L l 



and the magnification at Q is equal to the distance L l5 divided by 

 the back focal length of the objective system ; or — 



« - y (10) 



Combining equations 9 and 10 we obtain — 



sm/3 = — -; 

 m/ 



and substituting in equation 8 — 



~7- = fx sin a = N.A (11) 



a well-known result which tells us that for objectives of a 

 given focal length their N.A.'s vary directly as the effective 

 diameters in the upper focal planes. Imagine now the point p in 

 air (fx = 1), and the ray cp produced backwards until it 

 intersects, at a distance r from the axis, a plane normal to the 

 axis, and at a distance A from the aplanatic focus p ; and further, 

 let us suppose that rays can only enter the system through a 

 very small stop at p. Then to find the radius r of a circle 

 which, placed normal to the axis and at a distance A from the 

 aplanatic point p, shall project so as to completely fill the effective 

 opening in the upper focal plane of an objective with a given 

 N.A., we have only to remember that the N.A. = sin a, and that 

 r/A = tan a, to obtain the desired equation, — 



R= A 'tan (sin -1 n.a.) (12) 



A circle drawn with such a radius, and placed at the distance 

 A) will fill any objective with the given N.A., no matter what its 

 focal length may be. 



Agreeing, then, upon some convenient value for A? it is a very 

 simple matter to calculate the various values of r for a series 

 of circles which shall correspond, in the way described, to N.A.'s 



