10 G\NYTiiER, Range of Progressive Waves in Deep Water. 



The Expansion of the Coefficients in the 

 Series. 



Although the formal expressions for x,y, and ti (or <p) 

 have now been obtained in terms of /.t and therefore of /, 

 it is still necessary to express these coefficients in ascend- 

 ing powers of H^ and /^, , before the series add anything 

 to our knowledge of the motion of the fluid in Stokes' 

 problem. 



It is unlikely that this step can be performed by any 

 method without a considerable amount of labour. In 

 order to retain, in all cases, and as conveniently as may 

 be, expansions which proceed in sines and cosines of 

 multiples of 0, I adopt the method of expanding 



cos /-(£", sin ^ + ^o sin 20 + ...), 



and 



smr{E^s\n f + E.,sm 2(p + ...), 



by means of series in which the coefficients are Bessel's 

 Functions. 



The relations which I continually employ are 



cos (/\ sin ■^) = Jo{X) + 2E.7„,„(\) cos 2m\p 

 and 



sin (X sin \p) = 22J^,,„_j(X) sin {2m - i)\p. 



I shall not give the details of the work, but write 

 down the full expansions to the 6th degree of approxima- 

 tion. The only simplification which I make is to replace 

 Jo(A) by unity in cases where X' occurs to a sufficiently 

 high power to justify my doing so. 



The expansions are 



cos r[EiS\n f + E.2sln2<j) + ... +^Gsin 6^} 

 = JXrEy.,{rE.;)J,{rR) - 2J.lrEyirE.). 



