Page:MillerTheory.djvu/8

The equality is exact for the central ray. For rays inclined as much as five minutes of arc on either side of the central ray,

${\frac {\cos \delta '}{\lambda '}}-{\frac {\cos \delta }{\lambda }}\ \leqq \ 0.2{\frac {v^{3}}{\mathrm {V} ^{3}}}.$

At azimuths other than those specified, the quantity ${\frac {\cos \delta '}{\lambda '}}-{\frac {\cos \delta }{\lambda }}$ is not greater than $0.3{\frac {v^{4}}{\mathrm {V} ^{4}}}$ for central rays, and not greater than $0.5{\frac {v^{3}}{\mathrm {V} ^{3}}}$ for extreme rays. We may set side by side the magnitudes of this disturbing effect for central rays at several azimuths according to rigorous computation and according to Dr. Hicks's approximate formula.

Azimuths.	Disturbing Effects Compared.
	Dr. Hicks's formula	Rigorous formula.
90°  90° ± 11°  90° ± 22°  90° ± 45°  90° ± 90° -90° ± 45° -90°	1.2 v3⁄V3 0.9⁠„ 0.65 „ 0.0⁠„ 0.0⁠„ 0.0⁠„ 1.2⁠„	0.0 v4⁄V4 0.3⁠„ 0.3⁠„ 0.0⁠„ 0.0⁠„ 0.0⁠„ 0.0⁠„

It will be seen that the effect detected by Dr. Hicks proves, by rigorous computation, to be entirely negligible for the central rays. Its extreme value for marginal rays is not greater than $0.5{\frac {v^{3}}{\mathrm {V} ^{3}}}$, which is entirely too small to influence the observations. This result is very satisfactory. It is proved for the specified adjustment of the angles, but it is easy to see that the rotation of mirror I about a perpendicular line in its surface, by a quantity like ten seconds of arc, will not change all relations of residual aberrations by important amounts. It is therefore established, at least for the adjustment specified, that the wave-fronts a and a' of fig. 4. intersect in the line B II, if the wave-fronts h and h" do, rigorously for eight principal positions, very approximately for all other positions. If, then, we can measure the linear distance between a and a' at some convenient position T, we may determine the angle between the wave-fronts a and a', which is the same as the angle between h and h", the