340 Transactions of the Society. 



no considerable magnifying power (it becomes in fact a plane mirror 

 •when n = n').* 



Now a Microscope has very little " depth of focus," that is to say 

 two points P and P' are not in focus at the same time ; but it does 

 give, by means of wide-angled pencils, a clear image of the central 

 portion of a small plane object when placed at a certain position and 

 normal to the axis. 



The law A must therefore be nearly true for a good Micro- 

 scope. 



Law B will then hold approximately for small values of u and 

 V and yields the condition 



M=N^- C. 



n 



This being premised, the aperture of the instrument is effec- 

 tively defined by Prof. Abbe, as the number of rays that it admits 

 in a plane passing through the axis of the instrument from a plane 

 standard object supposed in air. Further, the number of rays 

 emanating from an object or converging to the image are counted 

 by the angular breadth of the pencils of light emanating from, or 

 converging to, each small element of the object or image multiplied 

 by the linear dimensions of the object or image — it being supposed 

 that the pencils have small angular breadth and are of the same 

 intensity throughout. 



In other words, the aperture is proportioned to the square root 

 of the quantity of light admitted hy the objective under the given 

 conditions. 



Apply this definition of aperture and the convention of counting 

 rays to the telescope. If the method is a rational one it must yield 

 the same result whether the rays are counted as they enter the 

 objective or as they converge to the image. 



In the case of the astronomical telescope we are deahng with 

 objects subtending always a small angle at the instrument, of un- 

 known absolute dimensions in many cases, and always so distant 

 that the light proceeding from them is of equal intensity over an 

 area indefinitely larger than that covered by any instrument. 



The dimension of the object in this case must therefore be 

 measured by the angle that it subtends at the observer's position. 



Eeferring to law A, it can readily be shown that, if P Q is 



* Note by Prof. Abbe: — Mr. Hockin's new demonstration of the law of 

 the sines is remarkable for its simplicity and generality ; at the same time it 

 includes a very simple and clever demonstration of a proposition, which I 

 signalized in the article " Die Bedingungen des Aplanatismus," that a system 

 of lenses which collects wide-angled pencils, cannot be aplanatic for a continuous 

 row of foci along the axis, but can have only isolated pairs of conjugate aplanatic 

 foci. 



