140 
Transactions of the Royal Microscopical Society. 
the distance of the point of convergence were considerable compared 
with the dimensions of the glass, it is evident that the oval would 
not differ much from the ellipse considered in the first instance, nor 
would the extreme aperture in glass fall much short of the limit 
assigned above. Or again, the rays emerging from the ellipsoid 
might be brought to converge to a second focus q in air by 
receiving them on a prolate spheroid of which q is the further 
focus and fjb~^ the eccentricity, and allowing them to emerge from 
the glass by a spherical surface of which q is the centre. Or the 
parallel rays might be brought to a focus without sensible aberration 
as is done in telescopes. 
I do not, of course, propose this as a practical construction of a 
microscope. It is intended simply and solely to show the fallacy of 
the supposed limit of 2 7 assigned to the aperture, within a medium, 
of a pencil which can be brought without sensible aberration to a 
focus in air. As the sphericity rather than spheroidicity of the 
surfaces employed does not enter in any way into the arguments 
by which the limit in question is attempted to be established, the 
spheroidal or cartesian surfaces are quite admissible in argument. 
Nevertheless, as an example of what can be done without going 
beyond spherical surfaces, and as bearing in a very direct way on 
actual practice, I will take another instance. 
Let it he required to make a pencil diverging from a radiant 
point Q in glass diverge from a virtual focus q after a single 
refraction into air. 
If P be a point in the required surface, ya Q P — qV must he 
constant, which gives, according to the value we arbitrarily assign 
to the constant, an infinity of cartesian ovals, any one of which, by 
its revolution round Q q, would generate a surface which may be 
taken for the bounding surface of the glass. In one particular 
case the oval becomes a circle, namely, when the constant = 0, in 
which case we have a circle cutting the line internally and 
externally in the ratio of I to yu. 
This case is represented in Fig. 1, in which 0 is the centre 
of the circle HAL, which by revolution round the line ^ Q A 
generates the sphere. Pays diverging from Q within the glass 
proceed after refraction at the surface of the sphere as if they came 
from q. To find the limit of the pencil, we have only to draw the 
tangents ^ H K, L M, and H K, L M will be the extreme rays after 
refraction. The incident rays Q H, Q L corresponding to these are 
inclined to the normals 0 H, 0 L at the critical angle. It is easy to 
prove (as will appear from the postscript) that the lines Q H, Q L 
are prolongations of each other, so that the aperture in glass of the 
pencil which, after refraction into air, diverges without aberration 
from q is 180°. The aperture H g L of this pencil, after refraction 
into air, is 2 7, which with the above value of 7, for which the 
figure is drawn, comes to 81° 58'. Setting aside chromatic variations, 
the refracted rays proceed, of course, as if they came from q, forming 
