Tue Microscope. DAT 
angular aperture. The working distance of an objective is the 
cosine of one half its angular aperture, the focal length being 
the radius of said angle; that is, supposing the lens to be with- 
out thickness at the edge at least. According to this the work- 
ing distance of the hypothetical 10 inch lens of 170° will be less 
than *% of an inch. <A 7s objective of 170°, presumably of no 
thickness, would have less than tv of an inch working distance. 
In this way the—practically impossible—maximum working 
distance of any objective can be found by calculation from the 
following formula: 
F' cos A= W. 
F being the equivalent focal length; A, one-half the angle 
of aperture; W, the working distance. But, since objectives, 
to be of sufficient angular aperture, have to be of considerable 
thickness, the real working distance can only be a comparatively 
small part of the theoretical maximum. Therefore, the deter- 
mination of the theoretical maximum werking distance of an 
objective of certain focal length and angular aperture can only 
serve to find what part of said maximum the real working 
distance is. Indeed, the estimation of the quality of an object- 
ive, in regard to working distance, should always in the first 
place relate to the proportion of its real working distance to the 
theoretical maximum. (And the angular aperture should 
always be determined for the medium through which the 
objective works.) To determine this proportion by a number, 
the unit of which, to be the theoretical maximum working 
distance, the following formula will serve: 
B 
——— 
F cos A. 
V, being the proportion of the real working distance to the 
unit; B, the real working distance; /, the equivalent focal 
length; A, one-half the angle of aperture. 
If the different kinds of objectives are compared according 
to the above principle it will be found that the dry working, 
and, to some extent, the water immersion objectives of wide 
angle are most generally of extremely short comparative work- 
ing distance ; far shorter than the thickness of the system can 
account for. The reason of this is, that the front lens is much 
