The Abbe Diffraction Theory. Bij J. W. Gordon. 363 



that a cannot in any case be found in the same wave-front as 

 a' and a". If, then, they are in equal phase, they must lie in suc- 

 cessive wave-fronts, and a circle drawn from the centre B through 

 a and a" would cut the axis A B at a point lying exactly one wave- 

 length, or an integral number of wave-iengths, behind a. But why 

 should it ? The radius a' B is quite indeterminate, and depends only 

 on the particular focal adjustment of the instrument. It may be 

 varied by indefinitely small degrees so as to give any desired flatness 

 to the arc a a". But this readjustment of the upper focal plane 

 would make no difference to the distribution of a, a' and a" in the 

 principal focal plane, which, as our authors have themselves shown, is 

 quite independent of the distance of the diffraction grating from the 

 objective. Obviously the phase relation of a, a and a" must be quite 

 indeterminate, and the corner-stone of the Abbe theory crumbles into 

 dust. The rest of the demonstration is mere mathematics, quite 

 correctly worked out, but of course proving nothing when unsup- 

 ported by physical facts. 



(Summarising the results of this critical examination of the Abbe 

 theory, we arrive at this conclusion : — The calculation upon which the 

 theory rests is itself based upon two fundamental errors, one as to the 

 position of the real diffraction images in the principal focal plane 

 of the Microscope, the other as to the phase relation between the 

 direct and the diffracted rays. The theory itself, when extended 

 to determine the number of bright lines seen in the real image of a 

 diffraction grating, their position in the picture, and their individual 

 appearance, breaks down entirely, and yields only conclusions which 

 are wholly at variance with the facts. Upon theoretical ground this 

 may suffice. 



It must, however, be admitted that most of the authors who have 

 expounded the Abbe theory have based it not upon physics, but upon 

 observation, and certain phenomena now very familiar are pointed out 

 as the sufficient proof of the accuracy of Professor Abbe's conclusions. 

 It will be said : fi If you reject the Abbe theory, how are you to explain 

 these phenomena ? " But here again it will not suffice to place 

 implicit confidence in the writers of books. The eye sees what it 

 brings the faculty of seeing ; and no more striking illustration of this 

 principle is to be found than the limitation which prepossession in 

 favour of a theory has placed upon the vision of many able men who 

 have observed the Abbe phenomena. That no controversy may arise 

 as to the facts, I will not venture to put forward my own description 

 of these phenomena ; and, as among the descriptions which I have 

 seen, the best and best illustrated by far occurs in Dr. Zimmermann's 

 work, Das Mikroshojp* \ avail myself for this purpose of his text. 



Perhaps, however, I may be allowed to preface an account of the 

 phenomena with a description of the apparatus which is supplied by 



* 'Das Mikroskop,' von Dr. A. Zimniermann, Leipzig uml Wien (Franz Den- 

 ticke), 1895. 



