Calling the refractive index μ, we have as the critical value of ε,
ε = ½λ0 /μ sin α, (1),
λ0 being the wave-length in vacuo. The denominator μ sin α is the quantity well known (after Abbe) as the “numerical aperture.”
The extreme value possible for α is a right angle, so that for the microscopic limit we have
ε = ½λ0/μ (2).
The limit can be depressed only by a diminution in λ0, such as photography makes possible, or by an increase in μ, the refractive index of the medium in which the object is situated.
The statement of the law of resolving power has been made in a form appropriate to the microscope, but it admits also of immediate application to the telescope. If 2R be the diameter of the object-glass and D the distance of the object, the angle subtended by AP is ε/D, and the angular resolving power is given by
λ/2D sin α = λ/2R (3).
This method of derivation (substantially due to Helmholtz) makes it obvious that there is no essential difference of principle between the two cases, although the results are conveniently stated in different forms. In the case of the telescope we have to deal with a linear measure of aperture and an angular limit of resolution, whereas in the case of the microscope the limit of resolution is linear, and it is expressed in terms of angular aperture.
It must be understood that the above argument distinctly assumes that the different parts of the object are self-luminous, or at least that the light proceeding from the various points is without phase relations. As has been emphasized by G. J. Stoney, the restriction is often, perhaps usually, violated in the microscope. A different treatment is then necessary, and for some of the problems which arise under this head the method of Abbe is convenient.