210 THE MICROSCOPICAL NEWS. 
the necessary super-amplification will be= 20, and all aberrations 
and other defects inherent to the system will appear in the image 
under twice the visual angle of that in the other case. 
(2) In order to compare the performance of different objectives 
under various powers, it will be necessary and sufficient, according 
to the foregoing theorems, to determine for any given system the 
constant quantity , by which the inherent dissipation of the rays 
is measured. 
One part of this problem may be settled by means of the 
following proposition :—With objectives of equal aperture, similar 
construction, and equal degrees of technical excellence, the con- 
stant visual angle of the dissipation-circles is always the same and 
independent of the focal length. . 
This may be proved by a very simple consideration. Suppose a 
system A of a certain aperture and given focal length f to be 
brought to the best possible correction of which the construction 
may admit for a certain distance of the image. Another system B 
of exactly similar composition may now be obtained by reducing 
the linear measures of a// the elements (all radii, diameters, dis- 
tances, &c.) and all technical defects of figure and positions of the 
lenses in the same proportion (say, ¢g., of 2:1), just as if the 
diagram of the system and the transmitted rays had been drawn on 
a reduced scale. The focal length will thereby be changed in the 
same proportion (f : 3 f) and also the distances of the conjugate foci 
of best correction ; but the aperture will not be changed, and the 
angles of all emerging rays—of regular or irregular transmission— 
will be the same as the angles of the corresponding rays in A, by 
virtue of the strict geometrical similarity of all the elements. If 
now the space over which the rays of ome pencil are dissipated at 
the image of A, subtends a certain angle x in regard to the posterior 
principal focus of A, the same angle ~ must obtain for the image 
of B in regard to the corresponding principal focus of B; and that 
angle ~ must persist, as has been shown, if B should afterwards 
project an image to any other distance (e.g., at a corresponding dis- 
tance to A), provided the best correction for the new position of 
the conjugate foci be obtained. Consequently the angular value 
of the dissipation-circles (~) is the same for all sémélar systems, 
however different the focal lengths may be.* 
* Strict similarity cannot of course obtain, except when the distances of the 
conjugate foci, for which the objective is corrected, are proportional to the focal 
length. We may, however, disregard all differences of construction which could 
be affected, or undone, without introducing new essential aberrations ; and those 
changes of a system which are necessary in order to compensate for different 
positions of the image, belong to that kind. The proposition will therefore 
hold good for all objectives of a similar formula and equal aperture. 
The assumption of a proportionate reduction of the geometrical defects 
(defects of figure, centering, &c.) with decreasing focal length, which is implied 
