470 Transactions of the Society. 



of the microscopical image depends solely on the diameter of the 

 pencils at their emergence from the ocular, and is in the direct 

 proportion to the square of that diameter. This diameter (d) is 

 strictly expressed by the simple formula 



if N denotes the amplification of the ultimate image for a distance 

 of vision = I, and a the numerical aperture of the system. If 

 we have a narrower illuminating pencil, which does not fill the 

 whole opening of the system, the numerical aperture corresponding 

 to the angle of the illuminating pencil must be substituted for a, 

 instead of the full numerical aperture of the system. The diameter 

 of the emergent pencils, and consequently the illuminating power, 

 is entirely independent of the particular composition of the 

 Microscope (objective, ocular, and length of the tube), and is 

 solely determined by the aperture and the total amplification (the 

 acsidental losses of light by reflection and absorption being dis- 

 regarded). By giving values to I and ^. we obtain from the above 



formula the diameter d in millimetres. Under the assumptions 



made in the computation of the figures of the second column of 



Table II. (i. e. \ = • 55 /x and v = 2') we have a constant ratio 



of a : N, viz. :— 



a _ l t 



N "^2 (264-5) ; 



and taking I = 250 mm. and substituting those values in the above 

 formula we have 



250 

 d = 26T5 S °' 95mm ' 



consequently the same diameter d for all powers, and always the 

 same brightness of the image therefore, provided the different 

 . apertures are fully utilized by the incident illuminating pencils. 

 By increasing the values of a assigned for every power N by 

 Table II. we enlarge the diameter of the emergent pencils in the 

 same proportion ; we should, for example, have d = 1 • 9 mm. 

 throughout, if the apertures — so far as this is possible — were in- 

 creased in the ratio of 1 : 2, which would correspond to the assump- 

 tion of a visual angle v = 1' for the least accessible detail. It is 

 obvious, however, that larger apertures can be of advantage only 

 so long as the value of d does not exceed the diameter of the 

 pupil of the eye under the actual conditions of microscopical 

 observation; for if this should happen, the iris of the observer 

 must stop-off the marginal part of the lens-opening exactly in 

 the same way as if a diaphragm were placed on the objective. 



