44 ABERRATION IN A DISPERSIVE MEDIUM
When m is zero, the value of /* is n/7; and this is true approxin when m is small. Thus, from (5),
with sufficient approximation.
Now in (8) the difference Vz-Vl corresponds to a change in the coeffi of t from n + mv to n - mv. Hence, denoting the general coefficient of t of which 7 is a function, we have
and (8) may be written
Z-_l _ 2m ~7| 7
Again, 7=<r/&, ff-dir/dk,
, _,, o- <i7 . cZ7 o- cZ/c
and thus -fr- -7— = /? -r~ = J- — 7- -*- ,
V dcr da- /c dcr
, a- dV _<r dk __ F
where Z7 is the group- velocity. Accordingly,
expresses the aberration angle, as was to be expected. In the present prol the peculiarity impressed is not uniform over the wave-front, as imv supposed in discussing the effect of the toothed wheel; but it exists nc theless, and it involves for its expression the introduction of more than frequency, from which circumstance the group- velocity takes its origin.
A development of the present method would probably permit the soli.; of the problem of a series of equidistant moving apertures, or a single mo aperture. Doubtless in all cases the aberration angle would assume value v/U.of t IH it+mv or u—mv. Thus