288 Transactions of the Society. 



the difference between the annulus in A and that in G- amounts 

 to more than 25 per cent, of the illuminating power of the whole 

 objective. 



If any black dots should appear in the disk, as at K, the 

 learner must understand that either his substage condenser is 

 racked up within its focus, or if in focus there must be spherical 

 aberration somewhere in his illuminating beam. It may be that 

 his substage condenser is defective, or a bullseye has been placed 

 between the source of light and the substage condenser and not 

 properly focused so as to give parallel rays. 



It was mentioned above that Mr. Smith's annular illumination, 

 K, differed from mine. This kind of illumination is very likely 

 to double periodic structures. The following test should be applied 

 when this kind of illumination is used.* Illuminate the periodic 

 structure by means of a very narrow axial cone, an objective of 

 wide angle being used. Example. — Let the objective have an 

 aperture of N. A. 1 ■ 0, and let its back lens appear as in L, where 

 we see that the four diffraction beams are 0'25jST.A. apart; con- 

 sequently we know that the structure under this objective must 

 have a periodicity of about 24,000 per inch. Now, if by manipu- 

 lating the illumination, or by introducing spherical aberration into 

 the objective by means of an improperly adjusted tube-length, a 

 structure of 48,000 per inch is resolved, it should be recognised at 

 once that this resolution must be false ; for if the structure had 

 had a periodicity of 48,000 per inch the back lens of the objective 

 would have presented an appearance as in M, and certainly not as 

 in L. 



This test consists, then, in roughly measuring in terms of the 

 estimated N.A. the angular divergence of beams diffracted by 

 periodic structures ; then by means of the table, printed on the fly- 

 leaf of this Journal, the periodicity of the structure will be found. 

 It is obvious that the scheme of this test may be extended by 

 making the narrow illuminating beam so oblique that it touches 

 the edge of the back lens ; then if there are two spectra their 

 distance apart will be ■ 5 1ST. A., and the structure will be 48,000, 

 but if only one, then it will be 96,000 ; if the separation is equal 

 to about two-thirds of the back lens, then the structure would be 

 60,000, and so on. Of course, if the objective used for the test has 

 some N.A. other than 1*0 N.A., its N.A. must be multiplied by 

 the fraction, whatever it may be, as in the W.A. in the above table. 

 This test has obviously nothing whatever to do with the Abbe 

 theory, because the physical fact that light in passing through a 

 minute grating is bent off at a certain angle, which depends on 

 the fineness of that grating, was known before Professor Abbe was 

 born ; therefore any one could have employed the aperture of an 



* Bristol Naturalists' Society, viii. pt. 2 (1897) p. 163. 



