KADIATION OF LIGHT IN DIFFEEENT MEDIA 



53 



as 1 : cos w). Consequently, if the assumption were true, b must 

 appear to be brighter than a, and the sphere would show increasing 

 brightness from the centre to the circumference. Close to the 

 margin the increase ought to be very rapid, and the brightness a 

 large multiple of that at the centre. 



This, as is well known, is not the case, the projection of the 

 sphere showing equal brightness. The quantity of light, therefore, 

 emitted from b within a given small 

 solid cone u f in an oblique direction 

 must be less than that which is emit- 

 ted from a within an equal solid cone 

 u in a perpendicular direction, and the 

 intensity of the rays must decrease in 

 the proportion of 1 : cos w when the 

 obliquity w increases. 



As then in one and the same 

 medium the number of rays conveyed 

 by a pencil and the photometrical 

 quantity of light are proportional, this 

 theorem of Lambert, established for 

 more than a century, is sufficient of 

 itself to overthrow the very basis of 

 the angular expression of aperture, 

 and to prove that, when we are dealing 

 with one and the same medium only, 

 the angle is not the sufficient expres- 

 sion, but that it is the sine of the semi- 

 angle which must be taken. 



We may pass now to the case of the 

 media being different, as air and oil, 

 and comparing the aperture of a dry 

 objective of 180 with that of an oil-im- 

 mersion objective of 100, the values of 

 n sin u (or the ' numerical ' aperture) 

 give 1'Oforthe former and 1*17 for the 

 latter, which is therefore represented to 

 have a larger aperture than a dry ob- 

 jective of the greatest possible angle. 



In this case also considerable per- 

 plexity has arisen. It has been assumed 

 that the total amount of light emitted 

 from a radiant point under a given 

 fixed illumination must be the same, 

 whether the point is in air, water, or oil, and that that being so, 

 the 180 admitted by the dry objective must represent a maximum 

 quantity of light, a * whole ' which cannot be exceeded, but only 

 equalled, by a water- or oil-immersion objective. The numerical 

 aperture notation giving figures in excess of I'O (which represents 

 180 in air) is consequently supposed to be clearly erroneous and 

 misleading. Here the whole difficulty lies in the absolutely false 

 assumption that there is identity of radiation in different media. 



FIG. 36. Diagram of a bright 

 spherical object emitting 

 light. 



