390 Transactions of the Society. 



with the principal. It is thus possible, by adding to the apertures 

 in an Abbe diaphragm without change of spacing, to emphasise 

 the Abbe phenomena without otherwise modifying them, or by 

 changing the spacing to modify the phenomena as desired, an ex- 

 pedient to which recourse has already been had in the experiments 

 described above, in which ruled glass diaphragms replaced those 

 supplied by Zeiss with the Abbe apparatus. 



Coming now to the practical application of these laws, no further 

 explanation will be necessary to make plain the origin of the various 

 phenomena described by Dr. Zimmermann and illustrated in figs. 

 71 to 80. They result in every case from constructing the picture of 

 the Diffractions Platte from antipoints having the forms here shown 

 to be given to them by the various apertures which Professor Abbe 

 uses. This branch of the inquiry has been already exhausted by 

 the discussion based on fig. 86. 



Of greater interest is the solution of intercostals and such like 

 optical illusions produced in the ordinary use of the Microscope 

 when brought to bear on such objects as diatoms which are studded 

 with points of light. There is, for instance, the case of Pleurosigma 

 angulatum, to which my attention has been recently called by Mr. E. M. 

 Nelson. Much discussion has taken place concerning the appearance 

 of that diatom in the Microscope when exhibited by its own diffracted 

 light alone ; and it is interesting to see how any given theory of 

 resolving power illustrates the problems which such a case presents. 

 The Abbe theory has been brought to bear upon this case, and has 

 yielded a result which was brought before your Society in a paper 

 read by the late Mr. Stephenson {Jour. R.M.S., 1878, p. 186). How far 

 the explanation so given corresponds to the facts I do not know. 

 I am told that the bright points of the diatom are in that explana- 

 tion misplaced by an angle of 30°, but of this I have no personal 

 knowledge. But what is more surprising than either resemblance or 

 dissimilarity between the theoretical drawing and the actual image — 

 for these may on any hypothesis be accidental— is the fact that any 

 figure at all should have been put forward as the theoretical figure. 

 For, in fact, the figure may be made to vary indefinitely. Everything, 

 or almost everything, depends upon the adjustments. The angular 

 position of the bright dots is, of course, a definite fact which no 

 adjustment of the Microscope could alter. But the size and shape of 

 the antipoint can be varied indefinitely between very wide limits 

 by altering the power of the objective and the length of the draw- 

 tube. Like all calculations of the dimensions of the antipoint, the 

 algebraical operations involved in the solution of this problem are 

 very tedious, but a comparatively easy graphic solution can be 

 reached by treating the six beams of light as six equal coincident 

 forces inclined at angles in one plane proportionate to their relative 

 phases, and combining them by means of a polygon of forces as above 

 described. 



