On the Estimation of A^perture. By Frof. E. Abbe. 15 



the increase of the admitted rays with increased opening is very 

 simply accounted for. We see the additional portions of the solid 

 cone from the radiant, which correspond to the additional portions 

 of the enlarged opening. But if in any other case (for instance, 

 when the medium is different) we see that a certain solid cone A 

 from a radiant is transmitted through a certain opening a, and that 

 another solid cone of rays B cannot be transmitted through the 

 same opening a, but requires a wider one /9, whilst all other 

 circumstances, except those of the radiant, have remained the same, 

 we can of course only conclude that the pencil B must contain 

 rays which are not contained in A, even if the admitted cone is not 

 increased in size. For the additional portion (j5 — a) of the wider 

 opening (3 conveys rays to the image which are certainly not 

 conveyed by the smaller opening a. Whence can this surplus 

 come if not from the radiant ? Obviously the pencil B, which 

 requires the additional opening, must embrace more rays, even if 

 it should not be of greater angle. 



Now the fact is, that a given objective may collect the rays 

 from a radiant in air almost to the entire hemisphere (as, for 

 instance, in the case of an immersion lens when focussed on a dry- 

 mounted object close to the covering-glass) and it then utilizes a 

 definite opening, double its focal length. But when the radiant is 

 in balsam (without any other alteration), the same opening is seen 

 to be utilized by the rays which are within a smaller cone of not 

 more than 82^, and rays which are outside this cone require 

 a surplus of opening, which is never required for rays in air. 

 This holds good, as has been shown, whether there be refraction or 

 no refraction at the front surface of the system ; the difference is 

 based solely on the difference of the medium. Consequently we 

 arrive at the conclusion that the solid cone of 82° in balsam 

 embraces the same rays which in air are embraced by the whole 

 hemisphere ; and every wider cone in balsam, exceeding the 82^, 

 conveys more rays from the object than are admitted by the whole 

 hemisphere of radiation in air. 



The definitive inference from the foregoing consideration is 

 obvious. There is no way of reconciling the seeming contradiction 

 between these two facts, (a) that a cone of > 82° from a radiant in 

 balsam embraces 7nore rays than a cone of 180° from a radiant in 

 air, and (b) that the angular extension of the former cone is less 

 than that of the latter, except by admitting the physical fact 

 that the same rays which in air are spread over the whole 

 hemisphere, are closed together, or compressed, in balsam within a 

 narrower conical space of 41° around the perpendicular ; and all 

 rays which travel in balsam outside this cone constitute a surplus 

 of new rays, which are never met with in air, that is, are not 

 emitted when the object is in air. 



