Lord Rayleigh : Hydrodynamical Notes. 183 



giving 



j cos a 2 . dx= \/\%) ; \ cos a3 - du ~ ~7r^{k) 



(13) 



The former is employed in the derivation of (3). 



The occurrence of stationary values of U is determined 

 from (7) by means of a quadratic. There is but one such 

 value (Uo), easily seen to be a minimum, and it occurs 

 when 



P={ V ''*-l}4j? =-1547 f. . . . (W) 



On the other hand, the minimum of V occurs when k 2 =gp/T 

 simply. 



When t is great, there is no important effect so long as 

 x (positive) is less than Uo£. For this value of oc the Kelvin 

 formula requires the modification expressed by (11). When 

 x is decidedly greater than Uft, there arise two terms of the 

 Kelvin form, indicating that there are now two systems 

 of waves of different wave-lengths, effective at the same 

 place. 



It will be seen that the introduction of capillarity greatly 

 alters the character of the solution. The quiescent region 

 inside the annular waves is easily recognized a few seconds 

 after a very small stone is dropped into smooth water *, 

 but I have not observed the duplicity of the annular waves 

 themselves. Probably the capillary waves of short wave- 

 length are rapidly damped, especially when the water-surface 

 is not quite clean. It would be interesting to experiment 

 upon truly linear waves, such as might be generated by the 

 sudden electrical charge or discharge of a wire stretched just 

 above the surface. But the full development of the peculiar 

 features to be expected on the inside of the wave-system 

 seems to require a space larger than is conveniently available 

 in a laboratory. 



Periodic Waves in Deep Water advancing without 

 chamje of Type. 



The solution of this problem when the height of the waves 

 is infinitesimal has been familiar for more than a century, 

 and the pursuance of the approximation to cover the case of 

 moderate height is to be found in a well-known paper by 



* A checkered background, e. g. the sky seen through foliage, shows 

 the waves best. 



