On the Estimation of Aperture. By Prof. E. Alhe. 391 



opening {at the plane of emergence) atid the focal length of the 

 system. 



There is no other rational way of defining the admission of 

 rays to an objective, and consequently no other definition of aper- 

 ture which agrees with this fundamental idea. I need hardly say 

 that this suggestion is nothing new. It is a matter of general 

 consciousness ; for every one will agree that the aperture of a given 

 objective is altered when the utilized diameter of the back lens is 

 changed by the application of different stops ; and that a clear 

 opening of say 3 mm. in a |-, is less aperture than the same clear 

 opening in an |. 



On the other hand, it is true that the apertures of objectives 

 may be compared as regards equality or inequality merely by the 

 angles of the admitted pencils, if the medium at the radiant is 

 the same, because under this condition equal angles indicate an 

 equal admission of rays, and different angles diff'erent admission. 

 The assumption, however, that apertures can be defined or com- 

 pared by the angle alone, is an entirely arbitrary one unless it were 

 proved that the admission of rays is always in proportion to the 

 angle, and does not depend on any other element. As no 

 attempt at a proof has been brought forward in support of this 

 hypothesis (it being in reality, as will be seen hereafter, opposed 

 to the fact), the proper way of obtaining a correct expression of 

 aperture hy means of the angle will be to investigate ivhat 

 expression must be taken, in order to define the same thing as is 

 denoted by the ratio of opening and focal length. 



Until a comparatively recent period the above assumption has 

 persisted as a dogma — without any investigation of the subject. 

 The author may claim to have been the first to put this dogma to 

 the test of scientific principles and to point out its fallacy by the 

 indication of the unequal aperture-equivalent of objectives. 



The demonstration of the general validity of this fact is 

 given here in detail for the benefit of those who may care for 

 such a treatment of the question. 



II. — Determination of the relative Openings of Systems hy the 

 Aperture-angle and the Befractive Index of the Medium. 



In 1873 the author and — quite independently — Professor 

 Helmholtz established a general relation between the pencil of rays 

 admitted by an optical system and the pencil emerging from it ; 

 a relation which pertains to the angles of convergence in both 

 pencils, and must always obtain whenever a system is aplanatic, 

 or is capable of depicting an object by means of wide-angled pencils. 

 The proposition is : — 



Let and 0* (Fig. Ill) be the conjugate aplanatic foci of a 



