to a wave-length ([lambda]). In virtue of the general law that
the reduced optical path is stationary in value, this retardation may
be calculated without allowance for the different paths pursued on the
farther side of L, L', so that the value is simply PL - PL'. Now since
AP is very small, AL' - PL' = AP sin [alpha], where [alpha] is the
angular semi-aperture L'AB. In like manner PL - AL has the same value,
so that

PL - PL' = 2AP sin [alpha].

According to the standard adopted, the condition of resolution is
therefore that AP, or [epsilon], should exceed 1/2[lambda]/sin [alpha].
If [epsilon] be less than this, the images overlap too much; while if
[epsilon] greatly exceed the above value the images become
unnecessarily separated.

In the above argument the whole space between the object and the lens
is supposed to be occupied by matter of one refractive index, and
[lambda] represents the wave-length _in this medium_ of the kind of
light employed. If the restriction as to uniformity be violated, what
we have ultimately to deal with is the wave-length in the medium
immediately surrounding the object.

Calling the refractive index [mu], we have as the critical value of
[epsilon],

[epsilon] = 1/2[lambda]0/[mu] sin[alpha], (1),

[lambda]0 being the wave-length _in vacuo_. The denominator [mu] sin
[alpha] is the quantity well known (after Abbe) as the "numerical
aperture."

The extreme value possible for [alpha] is a right angle, so that for
the microscopic limit we have

[epsilon] = 1/2[lambda]0/[mu] (2).

The limit can be depressed only by a diminution in [lambda]0, such as
photography makes possible, or by an increase in [mu], 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 [epsilon]/D, and the angular resolving power is given by

[lambda]/2D sin[alpha] = [lambda]/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 tel