A]periures of Microseopie Objectives. By Prof. G. G. Stokes. 141 
a pencil of aperture 81^58'. A pencil of aperture in air no greater 
than 81° 58' is one which all parties allow can practically be brought 
to a focus ; it could be brought exactly to a focus by the use of 
surfaces other than spherical. 
Fig. 1. 
The spherical surface of no aberration accords with the form of 
the first lens to which the makers of immersion objectives have been 
led. By reducing somewhat the excessively large segment of a 
sphere represented in the figure, say reducing it to a hemisphere, 
the space gained in front (of thickness Q 0 if the reduction be to 
a hemisphere) is available for the cover or interposed balsam, which 
have both nearly the same index as the crown glass of the first 
lens ; and the aperture in glass, though reduced from the extreme 
of 180°, still remains very large. 
P.S. — The property of a circle employed in Fig. 2 admits of 
being proved in a few lines, and it might be convenient to the 
reader to have the demonstration. 
Let 0 (Fig. 2) be the centre of a circle of which A B is a diameter. 
In 0 B take a point Q, and in 0 B produced take a point q so that 
0 ^ is a third proportional to 0 Q and the radius. Let E be any 
point in the circumference, and join Q E, E, 0 E. 
Since the radius is a mean proportional between 0 Q and O q, 
we have in the two triangles Q 0 E, g' 0 E, which have a common 
angle at 0, Q 0 : 0 E : : 0 E : 0 g. Therefore the triangles are 
similar, and QE:g'E::OQ:OE; and also the angles 0 Q E, 
0 E are equal to g' E 0, Q E 0, respectively. Hence 
Sin. qB,0 : sin. Q R O : : sin. R Q O : sin. R 7 O :: ^R : QR OR : OQ. 
If then Q had been taken so that 0 Q : 0 B : : 1 : ya, where ya is 
