176 



distance from the eye of the observer. Then it is manifest, 

 as 6 is very small, that — 



6 varies as P, if 91 is constant. 

 6 .} » -—> if I* >, 



91 



And thence compounding — 



^ 91 ' 



Where ^ is a constant now to be determined — 



let P = 1, sin 1' =: sin 60" = nearly. 



Let the beads be 36,000 to the inch. 



Under 12,000 diameters, a bead 36,000th of an inch in 

 diameter subtends an angle (as P = 12,000). 



= 20-028 X i?^ 



00 



20628 



3 



= 6876" 



= 1° . W . 36". 



Assuming 6" as the smallest defining angle for a spherical 

 bead or string of beads of l-36000th in diameter, the power 

 can be found which represents a bead at this angle. 



For 6" = 20-628 x ^; 

 . • . P = 10-6 nearly. 



From this it appears that, with a perfectly aplanatic mag- 

 nifying power of 10'6, such beads will appear under an angle 

 of 6". But no microscope has hitherto been able to distin- 

 guish them with this low power at such an angle. 



These results can easily be tested by the well-known 

 value of 



Sin 1" = arc 1" = -OOOOOiSlS. 



