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Full text of "Scientific Papers - Vi"

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),
and
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
ancl               ^F^"!^-"^5
where Z7 is the group- velocity. Accordingly,
(1
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