On the Classes of Progressive Long Waves, 30 •' 



to the fourth and also to the sixth order the general solution 

 for/' will thus be in terms of elliptic functions ; but this will 

 cease to be the case to the eighth order. Having drawn these 

 conclusions I shall now omit from (13) the terms (in square 

 brackets) of the eighth order and proceed to the integration 

 to the sixth order. 

 The solution gives 



2('c 2 -^)/ i -y" 2 = A(c 2 -(//0 3 + B(c 2 -^) 4 



_ / 2 (5c*-gh)(c*-gh) \ 

 V 3 : Q ghV ) CJ 



+ 4/' 4 (14) 



The presence of two constants in this equation is rather 



2 7 



apparent than real ; for if we take /j, — for the value of/ 



which makes/" vanish, we can find both A and B in terms 

 of fi. Thus 



and the equation (14) may be written 

 2(V-^W'' 2 



4/*V-l) (15c 2 - Ugli) { h)t f fl K*-gh) \ 



/. 2 (5o 2 -^)( C 2 -, 9 /Q\ f , 3 . /iV-jyA)' ! 



c 2 J 



