PROCEEDINGS OF THE SOCIETY. 485 



to use no plane wave-fronts at all, but spherical wave-fronts for illumina- 

 ting his object, and when he desires to get very fine resolution he uses 

 not only spherical wave-fronts but spherical wave-fronts with an ex- 

 tremely short radius of curvature. Now this change in the state of 

 illumination brings about a complete change in the condition of the 

 spectra, and a change which is extremely material from the present point 

 of view. For it can easily be shown experimentally that the conclusions 

 which Prof. Everett has worked out this evening and demonstrated 

 mathematically by means of plane wave-fronts have no application when 

 the diffraction arises from the incidence of spherical wave-fronts upon a 

 grating. I have already shown in this room an experiment, described 

 on pages 366 to 373 of the Journal for 1901, which demonstrates this 

 proposition to the eye. Shortly stated it comes to this : When spherical 

 wave-fronts are used to illuminate the grating (or other object) the state 

 of resolution in which it is seen is wholly independent of the number of 

 spectra transmitted by the objective. It is possible by using a wave 

 front of short radius to collect as many as four or five orders of spectra 

 in a lens of low angle, and, indeed, of an angle so low that it is incapable 

 of yielding a resolved image of the grating. In that case the resolution 

 is not in the least improved by including a large number of spectra in 

 the image-forming beam. On the other hand, if the incident light is a 

 spherical wave-front of comparatively long radius of curvature, then it is 

 quite possible to get perfect resolution when only the central light is 

 passed and even the first diffraction spectra are shut out. Now these 

 are in fact the conditions under which the Microscope is used in practice, 

 and this experiment shows, as I venture to think, conclusively that under, 

 these conditions the presence or absence of particular spectra makes no 

 difference whatever to the state of resolution under which the object is 

 seen. 



There is a difficulty even more fundamental than this which appears 

 to me to be necessarily fatal to any theory by which the phase in one 

 focal plane is deduced from the phase in another not conjugate to the 

 first. It is this : the corrections necessary to render the instrument 

 aplanatic in the image plane involve its being non-aplanatic in the 

 principal focal plane, and indeed in any other plane situate on the same 

 side as the image plane of the principal plane of the objective. It is 

 therefore impossible to calculate the phase which would in fact result 

 from the formation of an imperfect linage of the source of light by 

 means of diffracted rays in the neighbourhood of the principal focus. 

 This circumstance does not render Prof. Everett's results uninstructive, 

 but it does show that they belong to the region of pure, and in no sense 

 to the region of applied, mathematics. It is therefore impossible to look 

 upon his paper as affording any proof of the Abbe theory in concrete 

 form and as applied to any actual objective ; but, within the limits to 

 which Prof. Everett himself has restricted them, his results are, if I may 

 say so with all humility, of very great interest and highly suggestive. 



Mr. Beck said, as he had been alluded to by Prof. Everett as having 

 lent the Abbe experiment, he would explain what it was. He then drew 

 a diagram on the board to represent the back focal plane of an object- 

 glass, and said that if they looked at the Microscope upon the table they 



