olute
destruction of spherical aberration. An admissible error of phase of
1/4[lambda] will correspond to an error of 1/8[lambda] in a reflecting
and 1/2[lambda] in a (glass) refracting surface, the incidence in both
cases being perpendicular. If we inquire what is the greatest
admissible longitudinal aberration ([delta]f) in an object-glass
according to the above rule, we find

[delta]f = [lambda][alpha]^-2 (2),

[alpha] being the angular semi-aperture.

In the case of a single lens of glass with the most favourable
curvatures, [delta]f is about equal to [alpha]^2f, so that [alpha]^4
must not exceed [lambda]/f. For a lens of 3 ft. focus this condition
is satisfied if the aperture does not exceed 2 in.

When parallel rays fall directly upon a spherical mirror the
longitudinal aberration is only about one-eighth as great as for the
most favourably shaped single lens of equal focal length and aperture.
Hence a spherical mirror of 3 ft. focus might have an aperture of 21/2
in., and the image would not suffer materially from aberration.

On the same principle we may estimate the least visible displacement
of the eye-piece of a telescope focused upon a distant object, a
question of interest in connexion with range-finders. It appears
(_Phil. Mag._, 1885, 20, p. 354) that a displacement [delta]f from the
true focus will not sensibly impair definition, provided

[delta]f < f^2[lambda]/R^2 (3),

2R being the diameter of aperture. The linear accuracy required is
thus a function of the _ratio_ of aperture to focal length. The
formula agrees well with experiment.

The principle gives an instantaneous solution of the question of the
ultimate optical efficiency in the method of "mirror-reading," as
commonly practised in various physical observations. A rotation by
which one edge of the mirror advances 1/4[lambda] (while the other edge
retreats to a like amount) introduces a phase-discrepancy of a whole
period where before the rotation there was complete agreement. A
rotation of this amount should therefore be easily visible, but the
limits of resolving power are being approached; and the conclusion is
independent of the focal length of the mirror, and of the employment
of a telescope, provided of course that the reflected image is seen in
focus, and that the full width of the mirror is utilized.

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