324 Transactions of the Royal Microscopical Society. 
The last item obtained with a stop diameter, 
indicates an aperture of 50^^ by the usual methods, whereas the true 
angle is only 3J°, an error being shown in excess of more than 
fourteen times. The annexed diagraili demonstrates the cause of 
this error. The central angle shown is the true aperture assigned 
by the small stop at the base. The 
oblique angles represent the pencils, in- 
cluding the field of view, and showing 
light or an image at the eye-piece, up 
to an angular range of 50°. It is these 
outside rays which are superadded to 
angle of aperture taken by the usual 
method, and which are the cause of 
erroneous indications greatly in excess 
of the true angle. 
The foregoing is a mere illustration 
of excessive angles, indicated from ob- 
lique pencils, or angle of view alone, 
irrespective of true angle of aperture, 
for, of course, limiting stops of known 
diameter placed on the front lens serve no 
purpose for measuring full central angles. 
I The difficulty of estimating the degrees of 
these angles accurately, by such minute 
measurements as the focal distance, and 
working diameter of the front lens, may be avoided by halving 
this as an unknown quantity, and obtaining the value of the central 
angle by the differential results, shown between a half-obscured 
front surface and an entire one. That which is understood (and, in 
fact, alw^ays has been) as a definition of angle of aperture, in a 
micro object-glass, is a triangle having a base equal to the available 
diameter of the front lens, and a height equal to the focal length 
measured therefrom. Now it is clear that if half the front lens is 
stopped off diametrically close to the surface, only half the base of 
the triangle must remain, and consequently but half the existing 
angle or cone of rays in the axial direction. By the sector measure- 
ment such is not shown to be the case. The apparent degrees 
from a half diameter of incident front surface are much in excess of 
a half quantity, because the rays that form the oblique pencils 
extend behind and under the half stop, nearly to the margin of 
the field of view beneath. These are beyond the true axial angle 
of aperture, and are the cause of the false quantity always measured 
in excess of the proper angle, the rays including the angle of 
field must therefore be deducted up to the centre. This the half 
stop enables us to accomplish. Using the sector measurement the 
rule is this. Subtract the degrees shown by half the lens from 
