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IV. — On the Question of a Theoretical Limit to the Apertures 
of Microscopic Objectives. 
By Professor G. G. Stokes, M.A., D.C.L., LL.D., Sec. E.S., 
Lucasian Professor of Mathematics in the University of Cambridge. 
{Read before the Royal Microscopical Society, Ju7ie 5, 1878.) 
I HAVE just received from Mr. Mayall, jun., a photograph of Professor 
E. Keith’s computations relative to an immersion i microscopic ob- 
jective by Mr. ToUes. I have not at present leisure to go through 
this long piece of calculation, which I am the less disposed to do as 
the calculation is perfectly straightforward, and has evidently been 
made with great care, and I can see no reason to question the result. 
The only reason for scepticism as to the results of such calculations 
seems to be a notion derived from a priori considerations, that it is 
impossible to collect into a focus a pencil of rays emanating from a 
radiant immersed in water or balsam of wider aperture than that 
which in such a medium corresponds to 180° in air, or, in other 
words, than 2 7, where 7 is the critical angle. 
I do not wish to enter into controversy on the subject, or to 
criticise the arguments by which this statement has been sustained ; 
I prefer to show directly that it has no foundation. 
To disprove an alleged proposition, the shortest and least 
invidious plan is often to show by one or more particular instances 
that it is untrue. 
Suppose a pencil of parallel rays is incident upon a refracting 
medium of index p, and let it be required that it be brought 
without aberration to a focus q within the medium. By a well- 
known proposition, the form of the surface must be that of a prolate 
ellipsoid of revolution generated by the revolution of an ellipse of 
which q is the further focus, and the eccentricity, about its major 
axis, which is parallel to the incident rays. Conversely, if be a 
radiant within the medium, the emergent rays are parallel to the axis. 
The limit of the incident parallel rays in any section through 
the axis is the pair that touch at the extremities of the minor axis. 
Consequently in the reversed pencil the limiting rays are those that 
proceed from q to the extremities of the minor axis. If we suppose 
the index to be 1 • 525, for which 7 = 40° 59', the extreme rays 
will be inclined to the axis at the complementary angle 49° 1'. 
Hence a radiant within glass may send a pencil of aperture 98° 2', 
which by a single refraction shall be brought accurately to a second 
focus at infinity. The double of the critical angle is only 81° 58', 
so that the aperture exceeds that supposed limit by 16° 4'. 
If it were required that the pencil after the single refraction 
should converge to a real focus, the surface would have to be gene- 
rated by the revolution of a cartesian oval instead of an ellipse. If 
