258 
SUMMARY OF CURRENT RESEARCHES RELATINO TO 
allowance was made for thickness in his calculations. Let r be the 
radius of the incident, and s that of the silvered surface, t the thickness, 
and d , D, the diameter when the mirror has an aperture of 80° and 100° 
respectively, F being the true focus, and the refractive index /jl = 1*516, 
then the following are the formulas for aplanatism : 
t 
80° 
100° 
r 
s 
d 
D 
- 1*086 F 
+ 1*652 F 
F 
5 
1*26 F 
1*56 F 
~ 1*086 F 
+ 1*627 F 
F 
1*3 F 
1*58 F 
6*3 
- 1*089 F 
+ 1*611 F 
F 
9 
1*32 F 
1*61 F 
It must be remembered that the focus F, given in the above formulae, 
is the true focus, measured from the silvered surface, but, to save 
those intending to construct a lens-mirror trouble, I have drawn out the 
appended table (see p. 260) of the actual radii, thicknesses, and diameters 
of such lens mirrors as are likely to be useful to the microscopist. In 
this table /, the working focus, viz. the focus measured from the incident 
surface, is given, instead of F, the theoretical or true focus, which is 
measured from the silvered surface. The diameters given in the table 
are the lengths of the chord of the incident surface, and as the edge of 
the lens should be a radius of the silvered surface (see fig. 22) the 
diameter of the silvered surface will be somewhat larger. If the lens is 
edged parallel to its axis the aperture of the mirror will be reduced. It 
is clear that the mirror must have a diameter greater than that portion 
which subtends the angle of aperture. For example, the extreme ray in 
the fig. makes an angle of 50° with the axis, but if the mirror had only 
just sufficient aperture to include that angle the ray would never pass 
out of the lens. 
In the aplanatic lens mirror the chromatic aberration is less than in a 
bull’s eye, because the greater part of the angular bending of the ray is 
performed by reflection instead of by refraction. In the following apla- 
natic lens mirrors of 100° of aperture, - of the total bending is accom- 
plished by reflection without any dispersion. 
If an aplanatic lens mirror be compared with a bull’s-eye of two 
pianos of best form its superiority is at once manifest. 
First, the bull’s-eye cannot approach the lens mirror in aperture ; 
secondly, in the bull’s-eye all the bending is accompanied with dis- 
persion, whereas in the lens mirror only - of the total bending suffers dis- 
persion ; thirdly, the spherical aberration, even with two pianos of the 
w 2 
best form, amounts to no less than *36 fourthly, the diameter of the 
bull’R-eye soon reaches a limit, because of the thickness of the lenses. 
