A comparison with 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°; and it is evident that a lateral displacement of the point under observation through ½λ 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.
We have seen that in perpendicular reflection a surface error not exceeding 1⁄8λ may be admissible. In the case of oblique reflection at an angle φ, the error of retardation due to an elevation BD (fig. 5) is
QQ′ − QS = BD sec φ(1 − cos SQQ′) = BD sec φ (1 + cos 2φ) = 2BD cos φ;
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, ∫ μ ds, is a minimum; and, as AB is by supposition a wave-surface, this optical distance is the same for both points. Thus
∫ μ ds (for A) = ∫ μ 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 property the change may be neglected in calculating the optical distance, as it influences the result by quantities of the second order only in the changes of refrangibility. Accordingly, the optical distance from A0B0 to A is represented by ∫(μ + δμ)ds, the integration being along the original path A0 ... A; and similarly the optical distance between A0B0 and B is represented by ∫ (μ + δμ)ds, the integration being along B0 ... B. In virtue of (4) the difference of the optical distances to A and B is
∫ δμ ds (along B0 ... B) − ∫ δμ ds (along A0 ... A) (5).
The new wave-surface is formed in such a position that the optical distance is constant; and therefore the dispersion, or the angle through which the wave-surface is turned by the change of refrangibility, is found simply by dividing (5) by the distance AB. If, as in common flint-glass spectroscopes, there is only one dispersing substance, ∫ δμ ds = δμ·s, where s is simply the thickness traversed by the ray. If t2 and t1 be the thicknesses traversed by the extreme rays, and a denote the width of the emergent beam, the dispersion θ is given by