Change of Form of Long Waves. 443 



e=-bhk; ce=bhk; a x = -6(/i— h) ; ft = (— W + JcXA— *). . (69) 

 Substituting as a tffo'rd approximation : — 

 h l -=h—bhk + e l ', k x = k + £M + o^, a = l— 5(/i — ^) + a 2 ? 



we obtain finally, 



ei =-L c M(-/ i + 2Q; ce^lchk^h-k) ; a 2 =(b 2 -%c){h 2 -hk+k 2 ). (70) 



Hence the equation of the surface of the waves is, including 

 all terms of the third order : — 



v =[l-b(h-k) + {b 2 -2 G )(h*-hk + k*)] Vl +[b + (--2b < > 



+ ±c)(h-k)-] Vl z + (b* + lc) Vl ' + . • (71) 

 where 



% = A 1 cnH^i/4^+^(M = y^ /7 A^). . . (60) 



J ll ^h-bhk + y7ik(-h+2k); k^k + bhk + lchk^h -*). (72) 

 Here, according to (59), 



■i-F( l -S^r + ->^a+-«- • (73) 



whereas the value of c and more correct expressions for a and 

 5 could only have been obtained by means of still more tedious 

 calculations, which we have not executed. 



If we confine ourselves to that degree of approximation for 

 which all the calculations have been effected, we may write 

 for the equation of the wave-surface : — 



M =( 1 -SX/^ ™ 



For the solitary wave, when k=-0, we have * 



i-[i-3]* + V ( 77 > 



% = / iS echH(l-g)^^/3A. . . (78) 

 January 1895. 



* Another close approximation of the surface-equation of this wave has 

 been' deduced by McCowan, Phil. Mag. [5] vol. xxxii. (1891), p. 48. 



