ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 
363 
any surface diminishes in proportion to the cosine of the inclination of 
the rays to the normal. This means that the amount of light radiating 
from a point in a homogeneous medium varies as (sin w) 2 , where u is the 
semi-angle of the solid cone. The next point is that the radiation of 
energy, such as light and heat, in different media, varies as ( n 2 ), where 
n is the refractive index of the medium. Therefore the total effect of 
radiation in any medium is proportional to (n sin w) 2 , i.e. the square of 
the numerical aperture. 
The Lagrange-Helmholtz-Abbe theorem may be called the Magna 
Charta of Microscopy. If u and u ! be the angles of convergence of any 
ray on either side of a given system, n and n' the refractive indices of 
the media on either side, and M the magnifying power, then 
u 
Lagrange (1803) showed that - = M, the media on both sides of the 
u 
system being the same, and the aperture small. 
Helmholtz (1866) that = M, the media different, but the aper- 
ture small. 
Abbe (1873) that 
' »i sin u 
M for any aperture or media. 
Mr. Nelson outlines the proof of the last result, and shows that the 
following useful formulae may be obtained : 
(■•) 
\ back lens _ 
= / i 
N.A. 
(ii.) 
M x J back lens 
distance 
N.A. 
This gives a simple way of measuring the equivalent focus of any 
objective without an apertometer. The image of a stage micrometer is 
projected without an eye-piece on a screen, say 2 or 3 ft. from the back 
lens, and M the magnifying power is measured ; this, when multiplied 
by half the diameter of the back lens, and the product divided by the 
projection distance, gives the N.A. 
In discussing the Abbe theory that coarse structures are imaged 
according to ordinary dioptric laws, and fine ones according to diffrac- 
tion phenomena, Mr. Nelson points out that the line of demarcation 
between coarse and fine was placed at 1/2500 of an in., but that this 
position is untenable, because the diffraction pencils from gratings such as 
wire sieves, linen threads, or ruled scales, where the intervals are at least 
1/50 in., can be easily seen without any special apparatus. With suitable 
apparatus the spectra arising from much larger gratings have been made 
visible. 
The following simple but important experiments prove that the 
Fraunhofer diffraction law applies even to large objects which can be 
seen without instrumental aid. 
(1) When a scale on a carpenter’s rule is examined through a dia- 
phragm held close to the eye, the hole being 0*011 in. in diameter, some 
divisions on the rule can just be perceived at in. What is the fineness 
of the divisions ? 
Let a be the diameter of the hole, and \ the wave-length, say 
