550 Transactions of the Society. 



the triangular pattern is at a disadvantage when a still further 

 increase in the number of spectra admitted is aimed at ; it evidently 

 requires a very considerable increase of aperture to bring into 

 action any of the outer circle of spectra. Both these peculiarities 

 of a triangular design are well exemplified in the case of Pleuro- 

 sigma angulatum. Any objective which resolves the structure at 

 all — and a numerical aperture equal to 0*55 will do this — will 

 show the familiar dots, provided the objective be well corrected. 

 On the other hand, it is very difficult to attain a pronounced 

 advance on that image, even with oil-immersion objectives. 



It would be useless to attempt a very precise discussion of the 

 image of any dot-pattern by simple reasoning ; this must be 

 left for another occasion, when I propose to treat concrete cases 

 mathematically. 



We will instead try to draw some further conclusions from the 

 above general discussion. 



In the previous paper I showed that one important advantage 

 resulting from the use of an extended source of light, or of a wide 

 illuminating cone, was that the want of focus of an elementary 

 diffraction-image was overcome and replaced by a well-defined 

 focus, such as one expects with an optical instrument. The study 

 •of dot-patterns enables us to see another and even greater ad- 

 vantage. In order to obtain extreme resolution with a narrow 

 beam of light, we must let it enter obliquely, through the marginal 

 zone of the object-glass. But that gives us the high resolving 

 power in one direction only — along the diameter of the object- 

 glass having the direct light at one of its ends ; it leads to the 

 formation of a misleading image, inasmuch as fine detail is shown 

 in that one particular direction, whilst detail no finer, perhaps even 

 considerably coarser, in other directions is not even hinted at. A 

 well-centred illuminating cone overcomes this; it gives us equal 

 resolving power in all directions, and thus brings into view every- 

 thing that a given objective can resolve, no matter in what 

 directions the structural details may be arranged. Here, then, is a 

 full explanation of the necessity of a uniformly bright and well- 

 centred cone of illumination. Any want of centring, any dark or 

 coloured portions in the circle of light at the back of the objective, 

 imply a want of symmetry in the image, and a corresponding 

 danger of misleading images. It will, indeed, be found, when the 

 nature of " critical illumination " is impartially examined, that the 

 type of image looked for with such illumination is invariably 

 obtained when, on looking down the tube, a uniformly bright and 

 perfectly centred circle of illumination is seen — no matter how 

 obtained — and that the critical image is as invariably absent when 

 examination of the back-lens shows any want of uniformity or 

 symmetry in the said circle, no matter how brought about. 



