ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 
241 
lutely essential to the production of a correct image, the only way to 
ensure such an image is to collect all the spectra produced. But as the 
structures examined are generally unknown, and since we know a priori 
neither the number nor the arrangement of the spectra produced, we can 
never know if we collect them all, and consequently we can never know 
if the image which we observe represents exactly the structure, or even 
if we do not see the image of a structure which does not actually 
exist. 
From this follows the theoretical conclusion that we can draw from 
the examination of a microscopical image no mathematically correct 
deduction as to the real structure of the object which has produced it, if 
the dimensions of the details of the structure are less than 0*10 /x . As 
an example we have in the case of Pleurosigma angulatum six spectra, 
and only six, whatever may be the obliquity of the illumination. But it 
is known that a grating composed of two series of lines cutting at 60° 
gives in the Microscope, beside the central pencil, two concentric series 
of spectra, the first of six, the second of twelve. By varying the number 
and arrangement of the spectra admitted, different images can be obtained, 
but by eliminating the second series altogether an arrangement of spectra 
similar to that of Pleurosigma angulatum is obtained, and an image is 
produced similar to the structure of that diatom, which, however, does 
not correspond to the real stru dure of the grating. Since, then, in the 
case of Pleurosigma angulatum we do not know whether there may not 
exist a second series of spectra sufficiently separated to escape our 
present objectives, we cannot affirm that the structure of this diatom is 
really that of which we observe the image. From these facts two con- 
clusions can be drawn : 
(6) It is important to collect the greatest number of diffraction 
pencils, and as these pencils diverge from the point where they are 
produced, it is necessary to employ objectives of the greatest possible 
aperture. 
(7) The power of an objective is a direct function of its aperture. 
The brightness of the image depends upon two factors, — the aperture 
of the illuminating pencil and the magnification. Calling w the aperture 
of the objective and / the focal length, the brightness of the image will 
be proportional to 
/ 2 w 2 
so that in order to obtain the same degree of brightness, as / diminishes 
t o must be increased. 
The maximum aperture which can be practically attained corresponds 
to an angle in air of 135°-140°. 
This angular aperture can be easily applied to an immersion homo- 
geneous objective of 1/8 in. focus, and gives a numerical aperture of 
about 1*40; but the maximum will be reached if the magnification be 
pushed much farther, and, instead of augmenting it will be necessary to 
reduce the aperture. Thus in the catalogues of opticians, e. g. Powell 
& Lealand, we find the aperture 1*50 applied to objectives of 1/6, 1/8, 
1/12, and 1/20 focal length ; but to 1/25 only 1*38 aperture and to 1/50 
only 1*33 can be given. Beyond a certain limit, then, there is no 
advantage in augmenting the magnification of the objective. The author 
