152 MICEOSCOPIC VISION. 



- = tan a' ; therefore ^^ ^ , = ^ =^ ; as the angles are so 

 p' tan a p J 



small the angles may be written for their tangents thus — 



We have seen that radiation in different media varies 

 as (n^), the square of the refractive index, (this was demon- 

 strated by Prof. R. Clausius in 1864,) and therefore the 

 radiation for a small plane angle a in any medium will 

 vary as n a. So when there are different media on either 



side of the lens formula (i.) becomes — — =^; but in the 



n'a J 



microscope w' is always I'O for air, therefore 



-=-^ ........ (11.) 



a nf 



Now the law of convergence for aplanatic systems, to 

 which images formed by wide-angled pencils conform, is 

 the equality of the ratio of the sines of the angles of the 

 inclination of conjugate pencils to the axis. Thus, let a be 



a very small angle, as before, and 6, c, u 



increasingly larger angles of inclination of incident pencils 



to the axis, and a' h' c' u' those of the 



emergent pencils, then 



sin a _ sin h _ sin u ,... . 

 sin a' sin 6' sin if' 



In the case of the very small angles we may suppress the 

 sine and write the angle itself, thus by (ii.). 

 sin u _CL _ p' 



sin li a' nf 



(iv.) 



s 



Above we had »' = -. ; now as u' in the microscope 



tan u 



with its present nosepiece cannot exceed about 3°, we may 

 legitimately write the sine for the tangent, thus— 



