1893 - 94 .] Mr G. A. Berry on Focus of Concavo-Convex Lenses. 195 
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From the formula (in which /x = 1 *5) we see that by mak- 
ing t = 3R, F = 2R. 
These measurements have been given to this lens (shown), which 
is therefore focussed when in contact with the object looked at if 
the convex surface be next the eye, but has to be withdrawn to 4/3 
its thickness if looked through in the opposite direction. This is 
shown as illustrating, in the most exaggerated manner, the difference 
in the focus as measured from each surface. 
The great spherical aberration of a lens whose surfaces are of 
equal curvature must render it of little or no practical use. Pos- 
sibly, under certain conditions, such a lens might be used to 
counteract the aberration of a much stronger concave lens. 
If two piano-cylindrical lenses (of the kind in general use for the 
correction of simple astigmatism) of equal strength and opposite 
signs be combined so that their axes coincide while their plane 
surfaces are in contact, the result is, of course, a lens having the 
effect of a much weaker convex cylinder. Owing to greater 
spherical aberration, a cylindrical lens of this kind produces a 
greater astigmatic effect than an ordinary piano-cylindrical lens of 
the same strength. For instance, by the combination of convex 
and concave piano-cylinders of 20 cm. focus and thickness 5 mm., 
the focus according to the above formula is 12 metres. Yet such a 
lens causes a greater astigmatic effect than a piano-cylinder of 
4 metres focus. 
The two concavo-convex cylindrical lenses shown have respec- 
tively for each surface a radius of 5 cm. with thickness 7T5 mm., 
and therefore focus 2‘3 metres, and a radius of 10 cm., thickness 
4*45 mm., and focus 13 '5 metres. 
From recent observations with Javal’s ophthalmometer, it has 
been shown that in many cases of corneal astigmatism the degree 
of astigmatism at different parts of the cornea is far from being the 
same. It seems just possible that in some cases of this kind a 
better correction might be got by means of a concavo-convex 
cylinder, so that in this direction there might be a practical 
application. 
