328 SUMMARY OF CURRENT RESEARCHES RELATING TO 



combination. Eeing therefore a higher power (^ inch) with the same 

 back combination and diameter of the emergent pencil as the lower 

 power (^ inch) it necessarily had a larger aperture than the latter ! 



We doubt whether there is on record any more extraordinary 

 scientific fallacy than this, propounded as it was with all possible 

 seriousness. That we may not be supposed to intend any personal 

 reflection on one side, we may point out that the strangeness is, if 

 anything, enhanced by the fact that the other side did not discover the 

 mistake, and considerations and arguments have been brought forward 

 on the action of such a hemisphere involving the greatest absurdities. 

 Had the true position been appreciated, it would at once have been 

 seen that the angular aperturists had in fact given a clear and simple 

 demonstration that their own vieiv was lorong, for they had shown that 

 equal angles in air and glass gave not the same but different aper- 

 tures, the latter being the larger. The dry objective when used with 

 the hemisphere was of course converted into a true immersion objective, 

 the balsam-angle of which was the same as the original air-angle 

 of the dry objective. 



This problem may also be used to demonstrate that an immersion 

 objective can have an aperture exceeding the maximum aperture 

 of a dry objective. For it follows from the formula (p. 322) that 

 no dry front can exist which can be substituted for the hemispherical 

 immersion front without loss of aperture or loss of amplification ; for 

 if there could, such a dry front must give an emergent pencil of 90° 

 under an amplification of I'b. The formula, however, shows that the 

 widest cone which can be got out of any lens receiving the rays from 

 air under an amplification of 1 • 5 is circa 82° only. 



(b) The Concave Hemisphere. 



This arises in the following way. 



When the angular apertm-ist is confronted with the fact that the 

 emergent beam of a wide-angled immersion objective is wider than that 

 of any dry objective, he contends that whilst admittedly when the 

 object is in air, a wide emergent beam from an angle of 120° contains 

 more rays than a narrower one from an angle of 60°, yet that when 

 the front medium is changed the increase in the emergent beam can 

 no longer be treated as representing an increase in the rays taken up 

 from the object, for that must necessarily be so in consequence of the 

 action of the plane surface of the front lens, which reduces an air- 

 pencil of 180° into one of 82° only, when it passes into the glass 

 (cf. Fig. 55). With the immersion fluid, however, the reducing 

 action of the plane surface is abolished — the pencils from the radiant 

 are no longer reduced to 82°, but can expand to the fullest extent 

 which the objective will allow, the expanded pencil, however, still 

 representing, it is supposed, no more than the reduced one. 



As proof of this view, he considers a dry objective (say ^ inch, of 

 140° air-angle) with a plane front. The pencil of 140°, after being 

 contracted within the front to one of 76 • 5° (» = 1 • 52), will require, 

 in order to be transmitted through the system, a given clear diameter 

 cf the back combination. Assuming the plane surface removed, and 

 a concave one with the centre of curvature at the radiant substituted, 



