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10                                           HYDRODYNAMICAL  NOTES
In the numerator of (3) c^ and &, are functions of x and t.    If we in< what change (A) in so with t constant alters the angle by 2?r, we find
so that by (2) A = 27T/&,, i.e. the effective wave-length A coincides with of the predominant component in the original integral (1), and a like r holds for the periodic time*. Again, it follows from (2) that k^oo- crji
may be replaced by    k^dao, as is exemplified in (4) and (6).
When the waves move under the influence of a capillary tension addition to gravity,
p being the density, and for the wave-velocity ( F)
as first found by Kelvin.    Under these .circumstances V has a minii value when
The group- velocity [/"is equal to da-jdk, or to d(kV)jdk; so that win has a minimum value, U and V coincide. Referring to this, Kelvin tov the close of his paper remarks " The working out of our present probler this case, or any case in which there are either minirnums or maximuti both maximums and minimums, of wave-velocity, is particularly interes but time does not permit of its being included in the present comtnunicat
A glance at the simplified form (3) shows, however, that the special arises, not when F is a minimum (or maximum), but when U is so, since d^a/dkj2 vanishes. As given by (3), rj would become infinite  an indici that the approximation must be pursued. If k = k^ + , we have in ge: in the neighbourhood of k1}
In the present case where the term in f2 disappears, as well as that in. get in place of (3) when t is great
varying as.i"'^ instead of as t~^>
The definite integral is included in the general form
r+                       9    / 1 \        T-
coBa.m.da=-r(~}GOB~ ,
.' _oo                       m    \m/       2m
* Of. Green, Proc. Eoy. Soc. Ed, Vol. xxix. p. 445 (1909).ing term in 77 has an expression equivalent to