310 SUMMARY OF CURRENT RESEARCHES RELATING TO 



thirds of the way up, and being identical with that corresponding 

 to 82° balsam-angle. 



The diagram, Fig. 54, will serve to indicate more plainly the pro- 

 gressive increase in the diameters of the emergent pencils of objectives 

 of any given power from an angular aperture of 60° in air to the 

 highest oil-angle of 180°, and it will be seen that the pencil which 

 emerges from a dry objective of 180° air-angle is less in diameter 

 than that emerging from a water objective of 180° water-angle, or 

 an oil objective of 180° balsam-angle, in the ratio of 1*0 to 1-33, 

 or 1 • 52, the intensity of the light being approximately * the same in 

 all. The dotted circles in the latter two cases are of the same size 

 as that corresponding to the 180° air-angle and are added for ready 

 comparison. 



The diameter of the pencil emergent from the dry objective is, 

 moreover, found to remain the same whether the object is mounted dry 

 (the radiant pencil being then of largo angle) or in balsam (with a 

 much reduced angle at the radiant), so that the fallacy of the notion 

 that the balsam cuts down not merely the angle but the apertm-e also 

 becomes apparent. 



When the fact of this regular increase is recognized, it is en- 

 deavoured to avoid the necessary consequence of the admission by 

 alleging that although after the 180° air-angle is reached the emer- 

 gent pencil still increases, yet that such increase does not mean the 

 same above the 180° as it did helow, for that when 180° air-angle is 

 passed, and the balsam-angle of 82° substituted, the plane surface of 

 the dry lens no longer exercises any reducing effect — the large air- 

 angles in front of the lens are no longer compressed within 82° in 

 the lens, wuth a necessarily reduced emergent pencil, but are allowed 

 to expand to their full natural extent, with a 

 Fig. 55. proportionatelyenlarged emergent pencil. Thus 



in Fig. 55 the larger (inner) air-angle in front 

 of the lens is refracted at the plane surface on 

 its entrance into the glass, and becomes less 

 than 82°. The smaller (outer) angle, assuming 

 oil to have been substituted for aii", is not 

 reduced by refraction at the first surface, but 

 passes into the glass with its original angular 

 extension. The larger emergent pencil is therefore, it is supposed, 

 fully accounted for without there being necessarily any larger 

 aperture in the proper sense of the term ! 



One of the plainest of optical considerations disposes of this idea 

 of the action of the plane surface ; for, on abolishing the refraction 

 at the jjlane surface of a dry lens by substituting a concave one, it is 

 seen that the relative " opening " of the lens remains precisely the 

 same, and is not greater.f 



As, therefore, in regard to the measure of the " openings " we 



* That is, less only the loss of 10 or 12 per cent, by reflectiun at the first 

 surface of the front lens. 



t See II. " Angular-Aperture FalluciLS, No. 1 — The Hemisphere Puzzles, 

 (6) the Concave Hemisphere." 



