342 Transactions of the Society. 



Now apply the same method to the aperture of the Microscope. 

 The instrument is suited for viewing at one time a small plane 

 object, and a small plane disk must be taken as standard object. 

 Again, this object may be in air or in some other medium, and a 

 method of counting the rays must be adopted that will yield the 

 same result whether they are counted in air or in any other 

 medium through which they may pass. Let P Q be a section of a 

 small body in a medium of refractive index n, ^ q its image in a 

 medium of index 7*', A B a small portion of the surface bounding 

 the media, and C the centre of curvature of the arc A B (fig. 3). 



The rays coming from P Q are measured by a number pro- 

 portional to 



P Q X Z^ A P B, 



and equal, suppose, to 



AxPQx Z'APB. 



The same rays reach p q, and in the medium n' they are measured by 

 a X jiq X Z'' ApB,and these two expressions are to be the same. 



A.PQ. Z°A PB ^ J J 



A.^g'.Z^A^B 



but 



PQ = APxZ^PAQ 2 



p q = Ap X Z.^p Aq 3 



Suppose P A to make an angle 6 with the normal A C, and p A 

 to make an angle 6' with the normal A C ; then, as usual, 



n sin = w' sin 6' 4 



Again, P A Q is a small increment of the angle 6 and pAq is the 

 corresponding increment of the angle 6\ 



.•. n cos 6 Z.^F AQ — n' GOsO' Z.^p Aq ... 5 



also 



A P . Z" A P B = B n = A B cos ultimately . . 6 



and 



Aj) . Z^AjpB = A w ' = A BcosO' ultimately . . 7 



A PQ Z^APB _ A AP.Z'PAQ ABcosS Ap 



' ' B ' YI ' /."ApB ~ B * Ap Z^pAq ' AP ' A B cos 6' 



_ A cos e Z'^ P A Q 

 ~ B cos Q' ZJ p Aq 



_ A «/ _ 

 - B ' n ~ ' 



or we may put 



A = M 



B = n\ 



