10 Transactions of the Socidij. 



(Fig. 2), that the focus or radiant for an uncovered object is 

 exactly at its centre. The refraction is now aboKshed (in regard 

 to the pencil from the radiant) just as if there were homogeneous 

 ■p ^ immersion. If the above-mentioned view were 



correct, the consequence ought to be, that a dry 

 lens with such a fi-ont would utilize a wider 

 opening than an equal-power dry lens of the 

 ordinary plan, and the same opening as an 

 equal-power immersion glass of a balsam-angle 

 equal to the air-angle in question. But, of 

 course, the contrary is the actual fact. An ordinary dioptrical 

 computation shows that whenever such a dry objective with concave 

 fi-ont has the same power or focal length as a plane-front system 

 of equal aperture-angle, its opening must be also the same, exactly 

 — as the general piinciple of formula (7 j indicates. 



A misapprehension on this point has arisen thus : — If a homo- 

 geneous-immersion objective is taken, and its fi'ont-surface is ground 

 to a concave of the above description, xrhilst all other elements are 

 left unaltered, the angle admitted from air by the objective will be 

 no wider than that which was previously admitted from balsam, but 

 nevertheless the full opening will still be utilized. This seems 

 to prove, and indeed has been asserted to prove, that after alx)lishing 

 the front-refraction, a given aii'-angle will yield the same opening 

 as an equal balsam-angle. This, however, is a transparent fallacy. 

 According to well-known elementaiy ^propositions, a concave surface 

 diminishes an object at its centre in the proportion of the refractive 

 index 7i of the lens-substance. Consequently, the objective in ques- 

 tion has been changed into an w-times loicer powei ; and utilizing 

 BtiU only the same (and not a larger) back lens, it has necessarily 

 a smaller apertui'e. To restore the original focal length it would 

 be necessary to increase the depth of the posterior lens-surfaces in 

 such a way that the pencil should be by them contracted to the 

 same small diameter which otherwise it would have had with a plane 

 front. 



Whatever particular composition of objective is considered, the 

 result must always be the same. The relation between the aper- 

 ture-angles for different media and the corresponding openings of 

 the systems, as defined by proposition (7) cannot depend in any 

 way on the manner in which the pencils are refracted in the 

 system. A pencil from a radiant in air must always yield a smaller 

 aperture than an equal pencil from a radiant in balsam, whether 

 there is refraction or no refraction at the front surface of the 

 system. Consequently the difference of apeiiure mth equal angular 

 pejicils in different media must originate from a difference in the 

 pencils themselves, that is, must he founded on the different physical 

 nature of pencils in different media. 



