336 SUMMARY OF CURRENT RESEARCHES RELATING TO 



any other means to catch the pencil from air without loss of aperture, 

 a lens or any combination of lenses must exist which is capable of 

 collecting rays from a focus Q in air to a focus q in air, so that the 

 angle it* = 33°, and the amplification N at g = 2 • 33, as in the Stokes 

 front. The above formula would then be 



^^^^"" =2-33, 

 1 X sin 83° (m*) 



which would require an angle u for which 



sin ?< = 2-33 X 0-545= 1-26, 



that is, a sine greater than 1 ! 



The widest pencil which can be got out of a dry front under an 

 amplification of 2 • 33 is defined by the condition 



1 X sin 90° 



1 X sin M* 



= 2-33, or M* = 25° 30', i. e. 51° instead of 66°. 



Thus it is proved that the Stokes immersion-objective has a 

 larger aperture than any dry objective with the same back combination 

 can have. The same pencil (66°) which is readily got out of the 

 immersion front into the back combination cannot be got into it from 

 air, except with loss of amplification, i. e. of aperture. 



This disposes of the argument on the basis of Professor Stokes' 

 diagram. 



It does not prove, however, that no dry objective can have so large 

 an aperture as can be got with an immersion lens ; it only proves that 

 this is not possible with a dry objective under the assumption of the 

 same back combination. On page 321 will be found the demonstration 

 which proves that, as a general proposition applicable to all cases and 

 apart from all questions of particular constructions, a dry (Shadbolt) 

 objective can never equal in aperture an immersion (Stokes) objective 

 of wide angle. 



(b) The Simple Method. 



But Mr. Shadbolt may say that his mistake must have been a very 

 excusable one if the proof requires formulae which are not to be found 

 in English books, and we will therefore show how by the application 

 of the most elementary principle to be found in English optical books, 

 and without any calculation, the fallacy may be demonstrated. 



The loss of amplification with the Shadbolt front is obvious at a 

 glance, for the refraction at the spherical surface has been diminished. 

 A spherical surface (of refractive index n) amplifies an object which 

 is within the medium (for instance, the virtual object obtained from 

 the real object below a plane front surface) according as the distance of 

 such object from the vertex is increased. If the radiant coincides with 

 the centre the linear amplification will have a given value ; "f but if the 

 radiant is withdrawn from the centre to a point more distant from the 

 vertex the radiant of the emergent pencil is withdrawn still more. The 

 emergent pencil is reduced in divergence and, all other circumstances 

 remaining the same, this necessarily indicates increased amplification. 



t It will in fact be n, see ante, p. 327. 



