RELATION OF APERTURE AND POWER IN THE MICROSCOPE, 211 
Though this proposition allows a comparison of those objectives 
only which have equal apertures and are of similar construction, 
it leads to important inferences. Firstly, it shows that the charac- 
teristic constant quantity #, which is the ‘measure of perfection ” 
of the objectives, does not require a separate determination for 
every single system. If the value of ~ be known for one system, 
it is known for all systems of the same kind, 2.e., for all which have 
the same aperture, are constructed on a similar formula and with 
an equal degree of technical skill. Secondly, the proposition in- 
dicates the method by which a direct comparison of objectives of 
different focal lengths may be obtained in regard to the guality of 
images of equal amplification. 
Suppose the angular dissipation of the light—the constant angle 
‘u—to be given for a particular kind of objectives of definite aper- 
ture. If any one of these objectives has a focal length =/, its normal 
re l ; 
amplification is [N] = Fr: (7= 250 mm. or Io inches). If now the 
total amplification of the microscope is required = N, the necessary 
super-amplification to which the said system must be subjected 
will be 
N WN 
’ 
[Ny 2 
and this super-amplification will introduce an angular dissipation of 
the rays at the ultimate images, which is shown by the expression 
U=vua=—/f/u. 
Z 
Consequently: For objectives of the same kind and the same 
aperture, but with different focal lengths, the manifestation of the 
inherent defects and aberrations under a given power of the micro- 
scope is always in direct proportion to the focal length (or in the 
inverse proportion of the objective-power) by which such amplifica- 
tion is obtained. 
2. If we could suppose objectives of ideal perfection—absolutely 
free from all technical defects and all unavoidable aberrations—the 
in the demonstration above, will not be in full accordance with the actual cir- 
cumstances. In practice the ve/ative accomplishment of smaller lenses will be 
inferior to that of larger ones. According to the experience of the author, the 
difference is, however, not very considerable except when the dimensions are 
very minute. Though some difficulties of exact workmanship are increased 
with smaller dimensions, there are others which are diminished; and the 
relative amount of those defects which cannot be overcome by careful and skil- 
ful work, may therefore be considered as nearly equal for all focal lengths, 
within rather wide limits. 
