th the method of a material pointer, attached to the
parts whose rotation is under observation, and viewed through a
microscope, is of interest. The limiting efficiency of the microscope
is attained when the angular aperture amounts to 180 deg.; and it is
evident that a lateral displacement of the point under observation
through 1/2[lambda] entails (at the old image) a phase-discrepancy of a
whole period, one extreme ray being accelerated and the other retarded
by half that amount. We may infer that the limits of efficiency in the
two methods are the same when the length of the pointer is equal to
the width of the mirror.

[Illustraton: FIG. 5.]

We have seen that in perpendicular reflection a surface error not
exceeding 1/8[lambda] may be admissible. In the case of oblique
reflection at an angle [phi], the error of retardation due to an
elevation BD (fig. 5) is

QQ' - QS = BD sec [phi](1 - cos SQQ') = BD sec [phi] (1 + cos 2[phi]) = 2BD cos [phi];

from which it follows that an error of given magnitude in the figure
of a surface is less important in oblique than in perpendicular
reflection. It must, however, be borne in mind that errors can
sometimes be compensated by altering adjustments. If a surface
intended to be flat is affected with a slight general curvature, a
remedy may be found in an alteration of focus, and the remedy is the
less complete as the reflection is more oblique.

The formula expressing the optical power of prismatic spectroscopes
may readily be investigated upon the principles of the wave theory.
Let A0B0 be a plane wave-surface of the light before it falls upon the
prisms, AB the corresponding wave-surface for a particular part of the
spectrum after the light has passed the prisms, or after it has passed
the eye-piece of the observing telescope. The path of a ray from the
wave-surface A0B0 to A or B is determined by the condition that the
optical distance, [int] [mu]ds, is a minimum; and, as AB is by
supposition a wave-surface, this optical distance is the same for both
points. Thus
_ _
/ /
| [mu]ds (for A) = | [mu]ds (for B) (4).
_/ _/

We have now to consider the behaviour of light belonging to a
neighbouring part of the spectrum. The path of a ray from the
wave-surface A0B0 to the point A is changed; but in virtue of the
minimum prope