Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/842

Fig. 6.

A telescope receiving light from $\mathrm {S}$ and moving from $\mathrm {OE}$ to $\mathrm {O} _{1}\mathrm {E} _{1}$ while light traverses $\mathrm {OE} _{1}$.

A stationary telescope receiving the same ray at the same instant would have had to occupy the position $\mathrm {OE} _{1}$, and would have looked straight at the object (with a slightly greater focal length). Hence the angle $\mathrm {O} _{1}\mathrm {E} _{1}\mathrm {O}$ or $\mathrm {OSO} _{0}$ is the angle of aberration, the amount by which the object appears to be displaced in the direction of motion. A telescope which had been revolving round the source, instead of being translated, would have gone from $\mathrm {AB}$ to $\mathrm {OE} _{1}$ in the time, and have rotated through this same angle. Call it $e$; it is such that

${\frac {\sin e}{\sin \phi }}={\frac {\mathrm {EE} _{1}}{\mathrm {OE} _{1}}}={\frac {u}{V}}=\beta$, say,

the medium, remember, being stationary.

The focal length of the moving telescope differs from that necessary for a fixed one; being $\mathrm {OE}$ instead of $\mathrm {OE} _{1}$, or

$f^{\prime }=f\left(\cos e-\beta \cos \phi \right)$;

but this is best regarded as part of the Doppler effect, since its principal term represents radial motion. With a non-achromatic lens the change of refrangibility due to motion tends to compensate^[1] this effect. But whereas the change of refrangibility is produced equally by motion of source or motion of receiver, this change of focal length seems to be caused only by motion of receiver. It is a shortening of focus as a telescope recedes from the light. I suppose it is too small to observe, else it would seem able to discriminate motion of earth from motion of star, and give absolute motion of telescope through the ether.

A terrestrial source (e.g., a sodium flame) might be used, and a perfectly achromatic lens; but surely no focussing could be delicate enough to discriminate such sort of difference as exists between the two sodium emissions?

The way in which motion of receiver to or from source causes an apparent change of frequency, i.e., a real change in the frequency with which waves are received, is too well known and simple to be more than mentioned. Its amount in any direction is

$\log {\frac {n}{n^{\prime }}}=\log \left(\cos e+\beta \cos \phi \right)\approx \beta \cos \phi$,

where $\beta =u/\mathrm {V}$, $\sin e=\beta \sin \phi$, and $u$ is the velocity of the telescope at angle $\phi$ with the ray.

1. This was originally written “exaggerate.”