8 Transactions of the Society. 



The quotient appearing in the expression of p is thus shown to he 

 nothing else but the equivalent focal length of the system ; and we 

 have now 



p = f(n sin u), or ^ = n sin m = a.f (7) 



The ratio of the lineal' semi-opening of any system to the 

 focal length of the system is expressed by the value of a or by the 

 "numerical aperture." The value of n sin u is the aperture- 

 equivalent of every objective whatever may be the medium in which 

 the radiant is placed. 



III. — Inferences from the Aperture-equivalent. 



The simple result of the foregoing demonstration may he 

 summarized as follows : — 



(1) There exists a definite ratio between the linear opening and 

 the focal length of a system, which must be entirely independent of 

 the composition and arrangement of the system, and solely deter- 

 mined by the above-mentioned aperture-equivalent of the admitted 

 cone of rays. When this equivalent is the same, we have always 

 the same proportion of opening to focal length, whatever may be 

 the particular arrangement of refracting media in the system. 



(2) A purely angidar determination of aperture is shown to 

 he irreconcilable with any rational notion of a term which must be 

 defined essentially in relation to opening. Aperture it is seen 

 cannot be expressed by an angle, nor by any mathematical function 

 of an angle alone, but must be determined by a composite function 

 of the angle and the refractive index of the medium to which the 

 angle belongs. 



(3) Even with one and the same medium at the radiant, aper- 

 ture does not increase or decrease in proportion to the angle, but 

 with the sine of the semi -angle (or the chord of the whole angle). 

 If the angle is changed from 60° to 180°, the aperture is not 

 changed in the proportion of 1 : 3, but of 1 : 2 only. 



t The above formulae hold good in perfect strictness, if the distance / of 

 the image is tttken not from the accidental plane of the back-surface, but rather 

 from the posterior principal focus of the system (i. e. the place where are depicted 

 distant objects in front of the system). The equation (7) will therefore afford a 

 strict expression for the semi-diameter of the emergent pencil at the plane of the 

 posterior principal focus of the system. In microscope-objectives of the ordinary 

 type of construction that focus is always very near to the back lens of the system, 

 and the difference may be disregarded practically. 



At first sight it migiit appear to be more convenient to define the aperture- 

 equivalent by 2 n sin u = 2 a, instead of a, in order to express the ratio of the 

 diameter of the opening (instead of the sewM-diameter) to the focal length. In 

 matliematical dioptrics, however, the angles of the rays with the axis, and, 

 correspondingly, the distances of points from the axis are always given as the 

 effective elements. To introduce the double of tliese angles and distances 

 is not only unnecessary, but would give rise to a somewhat inconvenient com- 

 plication of all mathematical expressione. 



