ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 
259 
Before concluding it will bo as well to investigate for the sake of 
comparison the aberration of the commonest form of mirror, which is 
found in every Microscope, viz. a silvered shell of glass, in other words 
a lens mirror, where — r = — s. Formula (ii.) shows that when — r 
= — s, r = 2 F ; this is precisely the focus of a metallic concave 
mirror, therefore the glass part of the system has no influence at all. 
The equation for aberration, however, indicates a reduction in the 
aberration from 
silvered shell. 
125 that for a concave mirror, to 
it 
that for the 
F 7 ’ ” " F’ 
Practical experiments with thick shells yield somewhat 
F 
different results ; with a thickness of it will be found that the focus 
7*3’ 
F 
has been lengthened by — , and the aberration slightly reduced to 
o 
•085 t 
Thickness therefore is, in this case, an advantage. 
Aplanatic lens mirrors will be found useful for lieberkuhns, side 
reflectors, superstage and substage illuminators ; they are especially 
adapted for parallelizing wide-angled beams, and would therefore ad- 
vantageously take the place of bull’s-eyes or parallelizing condensers. 
They would of course be invaluable for search lights and lighthouses.* 
As the manufacture of these aplanatic lens mirrors can present no 
difficulties but what are met with in the most ordinary optical work, I 
can see no reason why they should not be largely used, and I trust that 
this effort of mine may be the means of not only improving, but also of 
cheapening a common and useful portion of microscopical apparatus. 
Note . — The radius of the silvered surface s is throughout this paper 
considered negative, because it is measured from the surface to the 
centre, that is from right to left, but in the table it is entered positive, 
so that it may agree with the practice of manufacturing opticians. 
Fig. 22 is drawn to scale from one of the formulas in the third block 
of the table, it therefore illustrates an aplanatic lens mirror of the 
thickest type. 
The following are some of the angles of the lens mirrors of the 
middle block having an aperture of 100°. The extreme ray parallel to 
the axis makes an angle of 46J° with its normal at the incident surface. 
The corresponding angle in glass is 28^°, viz. 13° less than the critical 
angle. This ray undergoes a reflection of 28^°, therefore this reflected 
ray in glass is parallel to its normal at the incident surface. In the 
thin lenses of the first block the angles of the extreme rays are slightly 
greater, but in the third block they are a trifle less than 28J° in glass. 
* The following are the radii, &c., of large mirrors : — 
D 
100° 
/ 
t 
— r 
+ s 
12-0 
6-62 
0*82 
8*11 
12*0 
12 0 
6-40 
1-20 
8*25 
12-36 
12-0 
6-15 
1-54 
8*35 
12-71 
Two and three foot mirrors will of course have twice and three times those values 
respectively. 
