182 Lord Rayleigh : Hydrodynamiccd Notes. 



time *. Again, it follows from (2) that k^v—a^ in (3) may 

 be replaced by ^dx, as is exemplified in (4) and (6). 



When the waves move under the influence of a capillary 

 tension T in addition to gravity, 



^=gk+Tk 3 /p, (7) 



p being the density, and for the wave-velocity (Y) 



Y 2 = a 2 /k 2 = 9 /k + Tk/ P , (8) 



as first found by Kelvin. Under these circumstances Y has 

 a minimum value when 



* 2 =<V/T (9) 



The group-velocity U is equal to da/dk, or to d(kV)/dk; 

 so that when Y has a minimum value, U and Y coincide. 

 Referring to this,, Kelvin towards the close of his paper 

 remarks " The working out of our present problem for this 

 case, or any case in which there are either minimums or 

 maximums, or both maximums and minimums, of wave- 

 velocity, is particularly interesting, but time does not permit 

 of its being included in the present, communication." 



A glance at the simplified form (3) shows, however, that 

 the special case arises, not when Y is a minimum (or 

 maximum), but when U is so, since then d 2 a/dk 2 vanishes. 

 As given by (3), rj would become infinite — an indication 

 that the approximation must be pursued. If k=k 1 + f;, we 

 have in general in the neighbourhood of k u 



Kx — Gb = k v T — cr,t+ \x-t-~-\p 

 \ dkj * 



1.2 dk, 2 * 1.2.3 d*! 8 *' ' " " { } 



In the present case where the term in f 2 disappears, as well 

 as that in f, we get in place of (3) when t is great 



cos (&!# — atf) f +0 ° o , , 1t v 



V= ri , — ~ 1 \ COS a 3 , da, . . . (11) 



varying as t~^ instead of as t~K 



The definite integral is included in the general form 



( + cosa»du=-r(-)cos^, . . . . . (12) 

 * Cf. Green, Proc. Koy. Soc. Ed. xxix. p. 445 (1909). 



