KELATIVE APERTUEES 



49 



from the radiant, and u*, U* the angles of the same rays on their 

 emergence ; then we shall have always 



sin U* : sin u* : : sin U : sin u ; 



sin U* sin u* 



or, _^^ = _ -- = const. =c; 



sin U sin u 



that is, the SMWS of the angles of the conjugate rays on both sides of 

 an aplanatic system always yield one and the same quotient c, what- 

 ever rays may be considered, so long as the same system and the 

 same foci are in question. 



This proposition holds good for 'every arrangement of media, and 

 refracting surfaces that may go to the composition of the system, and 

 for every position of object and image. It is the law upon which de- 

 pends the delineation of an image by means of wide-angled pencils. 



When, then, the values in any given cases of the expression 

 n sin u (which is known as the ' numerical aperture ' and expressed 

 by N.A.) has been ascertained, the objectives are instantly compared 

 as regards their aperture, and, moreover, as 180 in air is equal to 

 1-0 (since w=l-0 and the sine of half 180 or 90=1'0), we see with 

 equal readiness whether the aperture of the objective is smaller or 

 larger than that corresponding to 180 in air. 



Thus, suppose we desire to compare the relative aperture of three 



tains in our Universities, and is thoroughly understood amongst University men. 

 But to those unaccustomed to mathematical formulae confusion might easily arise 

 from the juxtaposition of different symbols meaning precisely the same thing. To 

 meet the possible necessity of these this footnote is inserted with an accompanying 

 diagram to illustrate the identity of ' n sin u ' with ' fj. sin <J>.' 



The student who has mastered Snell's Law of Sines, given and illustrated on p. 8 

 (fig. 1), will by a glance at the figure Al on p. 48 understand the meaning and import- 

 ance of the expression ' N.A.' (numerical aperture) and at the same time will grasp 

 wherein it differs from ' angular aperture ' (q.v.). He will also perceive how it comes 

 to pass that an angular aperture of 70 in glass is equivalent to an angular aperture of 

 120 in air. 



In the figure the upper hemispherical lens represents the front of a homogeneous 

 immersion objective. It is supposed to be focussed on an object in contact with the 

 lower side of a cover-glass. Between the plane front of the lens and the upper surface 

 of the cover-glass is a drop of oil of cedar-wood, whose refractive index is 1*5, being 

 thus identical with the cover-glass and the front lens. 



It is understood that no slip is used, and that there is nothing between the object 

 and the front lens of the condenser. 



In this case the axis A B is the normal (p. 5, fig. 2) ; on the left-hand side there 

 is a ray which makes an angle of 35 with the normal in glass issuing into air on the 

 right-hand side of the normal. By Snell's formula (p. 3) 

 fj. sin <f> = /a' sin </>' ; 

 sin*'= ^gH^ = l-5x'573 = . 86; 



fj. I'O 



<' = 60 (from Table I.) 



Therefore the ray on emerging from the under surface of the cover-glass will make 

 an angle of 60 from the normal. 



The dotted lines show the path of the ray where the German symbols are used. 

 n sin un* sin u* 



sin u*= _ 



n* 1-0 



?,* = 60 (from Table I.) 



Numerical aperture, therefore, is the sine of half the angular aperture multiplied 

 by the refractive index of the medium. 



It will be observed that the rays passing through the oil of cedar enter the front 

 lens without refraction ; this is due to the fact that the media in which the rays are 

 travelling are of the same refractive index, i.e. they are homogeneous. 



E 



