THE 

 LONDON, tii'INBUIUH, am. DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[SIXTH SBRIEiS] ['\ : 



MA Y 1917. 



AP/7 •< 



XXXYlTI. On Periodic Irrotational Waves at the Surface of 

 Deep Water. By Lord Rayleigh, O.M., F.R.S * 



^j^HE treatment of this question by Stokes, using series 

 A proceeding by ascending powers of the height of the 

 waves, is well known. In a paper with the above title f it 

 has been criticised rather severely by Bu inside, who con- 

 cludes that "these successive approximations can not be 

 used for purposes of numerical calculation...". Further, 

 Barn side considers that a numerical discrepancy which he 

 encountered maybe regarded as suggesting the non-existence 

 of permanent irrotational waves. It so happens that on this 

 point I myself expressed scepticism in an early paper J, but 

 afterwards I accepted the existence of such waves on the 

 later argument- ot Stokes, M c (Jowan§, and of Korteweir and 

 D- Vries|| In 19111" I showed that the method of the 

 early paper could be extended so as to obtain all the later 

 results of Stokes. 



The discrepancy that weighed with Burnside lies in the 

 fact that the \alue of/3 (see equation (1) below) found best 

 to satisfy the conditions in the case of a = ^ differs by about 



* Communicated by the Author. 



f Proc. Loud. Math. Soc. vol. xv. p. 26 (1915). 



t P il. Mag. vol. i. p. 257 (1876) ; ' Scientific Papers/ vol. i. p, 261. 



§ Phil. Mag. vol. xxxii. pp. 45, 553 (1891 ). 



I] Phil. Mag. vol. xxxix. p. 442 (1895). 



«j[ Phil; Mag. vol. xxi. p. 183 (1911). 



Phil. Mag S. rt. Vol. 33. No. 197, May 1917. 2 D 



