the extreme error of phase to be compensated falls to 1/4[lambda]. But,
as we have seen, such an error of phase causes no sensible
deterioration in the definition; so that from this point onwards the
lens is useless, as only improving an image already sensibly as
perfect as the aperture admits of. Throughout the operation of
increasing the focal length, the resolving power of the instrument,
which depends only upon the aperture, remains unchanged; and we thus
arrive at the rather startling conclusion that a telescope of any
degree of resolving power might be constructed without an
object-glass, if only there were no limit to the admissible focal
length. This last proviso, however, as we shall see, takes away almost
all practical importance from the proposition.

To get an idea of the magnitudes of the quantities involved, let us
take the case of an aperture of 1/5 in., about that of the pupil of
the eye. The distance f1, which the actual focal length must exceed,
is given by

/
\/ (f1^2 + R^2) - f1 = 1/4[lambda];

so that

f1 = 2R^2/[lambda] (1).

Thus, if [lambda] = 1/4000, R = 1/10, we find

f1 = 800 inches.

The image of the sun thrown upon a screen at a distance exceeding 66
ft., through a hole 1/5 in. in diameter, is therefore at least as well
defined as that seen direct.

As the minimum focal length increases with the square of the aperture,
a quite impracticable distance would be required to rival the
resolving power of a modern telescope. Even for an aperture of 4 in.,
f1 would have to be 5 miles.

A similar argument may be applied to find at what point an achromatic
lens becomes sensibly superior to a single one. The question is
whether, when the adjustment of focus is correct for the central rays
of the spectrum, the error of phase for the most extreme rays (which
it is necessary to consider) amounts to a quarter of a wave-length. If
not, the substitution of an achromatic lens will be of no advantage.
Calculation shows that, if the aperture be 1/5 in., an achromatic lens
has no sensible advantage if the focal length be greater than about 11
in. If we suppose the focal length to be 66 ft., a single lens is
practically perfect up to an aperture of 1.7 in.

Another obvious inference from the necessary imperfection of optical
images is the uselessness of attempting anything like an abs