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When a telescope is moving directly towards a star, Veltmann’s theorem^[1] shows that the motion will not affect the relative position of the focus to the first order. But a second order effect will remain outstanding, although it will be too small to be ascertained by focal settings. According to our present ideas, however, no effect of any order is to be expected, and our example shows how the compensation operates in a particularly simple case.

4. In what precedes we have contemplated only a change in one dimension of the moving system, or, as it may be expressed, a transformation of one coordinate in space. We have now to consider a related transformation to apparent or, as it is called, "local" time. Let axes be taken attached to the moving system, the measured coordinates being ξ, η, ζ. Let their position at the time t = 0 be the axes fixed in space, the corresponding coordinates being x, y, z. The motion of the system is supposed parallel to the axis of x or ξ.

We now imagine a new system of time τ, which depends not only on the absolute time t but also on the position in space. It is sufficient for our purpose to suppose that

$\tau =at-bx,\ \xi =cx-dt,\ \eta =y,\ \zeta =z.$

All optical phenomena which would be described by an observer at rest in space in terms of x, y, z and t will be described by an observer in motion in terms of ξ, η, ζ and τ. Now one result of the principle of relativity and of the constant velocity of light is that the spherical wave-front

$x^{2}+y^{2}+z^{2}=U^{2}t^{2}$

must appear to the moving observer as

$\xi ^{2}+\eta ^{2}+\zeta ^{2}=U^{2}\tau ^{2},$

for a spherical wave which actually converges to a point in reality must appear to converge to a point and to move with the velocity U. This requires

$(cx-dt)^{2}-U^{2}(at-bx)^{2}\equiv x^{2}-U^{2}t^{2},$

or

$c^{2}-b^{2}U^{2}=1,\ a^{2}-d^{2}-d^{2}U^{-2}=1,\ cd=abU^{2},$

which can be satisfied by

$a=c=\sec \beta ,\ bU=d/U=\tan \beta .$

Here β is arbitrary, but we must also have ξ= 0 identical with x = vt, or

$v=d/c=U\ \sin \beta ,$

1. Mon. Not., lxix. p. 504. Professor Whittaker considers that Veltmann’s paper added nothing to Fresnel’s original paper in Annales de Chemie, ix. p. 57 (1818). In attaching importance to Veltmann’s theorem I have followed Jamin (Cours de Physique).