24 
On the Limits of the Optical 
From this theorem it follows — 
Firstly, that when a ray, B, proceeding from a luminous point 
has an absolute smaller divergence angle than the ray A, the 
divergence angle of B will, after subsequent refraction, remain 
always less than that of A, because the product obtained by our 
theorem for B is from the beginning less than that obtained for A, 
and for the same reason must continue to be smaller after each 
refraction. 
Secondly, when two rays, starting from the same point on the 
axis, with equal angles of divergence, but following planes which 
extend in opposite directions through the axis, their divergence 
angles continue to be equal after each refraction, a result which 
appears indeed at once evident from the symmetrical disposition of 
a lens system round its axis. 
If now we imagine the illuminating rays, on their way to the 
object, to be circumscribed by interposing a diaphragm pierced with 
a circular opening whose centre coincides with the axial line, the 
plane of the diaphragm being at right angles with the optical axis, 
then those rays which pass through the opening close to its margin 
have all alike the largest divergence angle, and retain the same 
relation after each fresh refraction. These rays obviously occupy 
the exterior outline of cones having a circular base, and whose axis 
is the optical axis of the lens system, and they constitute the 
boundary of the cone of light proceeding from the luminous point. 
The divergence angle of these border rays is, in this case, through- 
out their entire course, the angle which the semi-aperture of the 
conical surface bounding the illuminating cone measures. 
^^^^^^^^^^^^ From this there follow, firstly, certain 
^^^^^^^^^^^H important results in regard to the photo- 
^^^^^I^^^^^H metric conditions of the microscope image. 
^K^^KB^^^t^^ According to known laws of photo- 
^HHHHHHB^I metry, we may equate L the quantity of 
^HH^^^BH^H light sent forth from the luminous point 
^^H|^I^H^^H c^S upon another point ds, whose distance 
^^^HHB|^^^H is r, as follows, where (r, N) and (r, n) 
^^^HflBH^^^H represent the angles formed between the 
^^^^HlH^^^H line T and the normals N and n. 
L = J '- — - . COS. (r, N) . COS. (r, n). [6] 
If now we understand hy ds the cir- 
cular aperture of the cone of rays at one 
of the refracting surfaces, and by <^ 8 a 
luminous point intersected by the axis so that r falls in the axial 
line, 
