438 Dr. G. J. Stoney on Microscopic Vision. 



cone. And, finally, all these V's, since they travel in pre- 

 cisely the same direction, will, as is proved in the last section, 

 coalesce into a single resultant beam W travelling along the 

 axis of the cone, which single resultant may accordingly be 

 substituted for the elementary sheaf of beams. 



The general conclusion is : — The whole of the light emitted 

 from the objective field may, by Theorem 1, be resolved into 

 beams of uniform plane waves ; these beams may be divided into 

 small groups, each an elementary sheaf of beams; and each 

 elementary sheaf of beams may have a single beam substituted 

 -f or { tm — In every subsequent step of our investigation we need 

 only deal with these resultants — these secondary beams as they 

 may be called — which, though many, are limited in number. 



30. Another proof of Theorem 2. — Theorem 2 may be 

 proved in many ways, and a proof which carries the analysis 

 of an image down to its simplest elements will be found 

 instructive. Describe a hemisphere in front of the objective 

 field and round its centre. Gall the point where the optic 



Fiff. 2. 



axis of the microscope pierces this hemisphere, its pole. 

 Planes passing through the optic axis may be called the 

 meridional planes ; and the objective plane, being perpendicular 

 to the axis of the microscope, will be its equatorial plane. 

 Divide the equator of our hemisphere into seconds of arc, 

 i. e., into 1,296,000 parts, which will afford sufficiently 

 minute divisions upon which the bases of elementary cones 

 may abut. Draw parallels of latitude also at intervals of a 

 second ; and draw meridians as in the figure, marking out in 

 conjunction with the parallels of latitude the bases of the 

 elementary cones, or rather pyramids. These become 

 narrower the higher the latitude, and as soon as they have 

 shrunk to half a second horizontally every alternate meridian 

 may be omitted, until they have shrunk again till other 

 meridians may be omitted without any of the little sectors 



