498 W. B. STOKES ON RESOLUTIONS OBTAINED WITH DARK-GROUND 



Now, the inclusion of the second order spectrum may require 

 considerable obliquity on the part of the incident light. Such 

 obliquity, and consequently the resolving power of the objective, 

 will be limited by the numerical aperture of the illuminator. 



The arrangement of spectra formed by a plane grating can 

 be determined by the equation 



N = (/A sin Uq /J. sin Un) ^/n, 

 where 



N = numbei' of intervals per unit o length in the object, 

 k = waves ,, of light in air, 



fx refractive index of the medium, 

 7/^ = angle with normal b}^ spectrum of zero order, 



'^^n = )? 55 J ^''th ,, 



Assuming, as is usual, that the normal coincides with the 

 optic axis, we may note two deductions of importance. 



(1) Maximum resolution with dark-ground illumination will 

 be obtained when spectra of first and second orders just enter 

 the objective on opposite sides, that is 



fjL sin ?/, = (N.A.) = ^ sin n.,, 

 objective 



whence 



fjL sin 21q = o fjL sin w,, 



That is to say, the N.A. of the illuminator must be three times 



that of the objective if maximum resolving power of the 



objective is to be obtained.* The resolving power thus obtained 



would be 



N = 2(N.A.)^, 

 objective 



(2) Failing the condition (1) for maximum resolution the 

 finest structures resolvable will be those which send a spectrum 

 of the second order to the edge of the objective, the first order 

 being easily included on the other side. In this case 



N = (^ sin Uq /ii j-in u.^) Ji/2, 

 = (N.A. + N.A.) 1-/2, 

 illuminator objective 



the angle U2 having a negative sign relatively to kq. 



Now, here we have a definite conclusion derived from the Abbe 

 theory. Putting practicable values for the numerical apertures 

 and the wave-frequency, it will be found that the fineness of 



* This has been pointed out by M. I. Cross, Knoivlcdge (p. 37), Jan. 

 1912. See pp. 477 and 480 supra. 



