Change of Form of Long Waves. 427 



where 



'."2 



2 7 3^ 23^3* 2\W 



N i_*22_ i/KA + * 5/ ay _ i ay 



24 y ^ 4 8\d#v 6 a^*a^ 3 " 24 3^3*' 



*~±f&+L'££ ay_A/ayv j_y ay 1 ay 



720 y 3# 6 48 Bar 2 ^d 4 72 VdW 1203a?' 3# 5 720 3^3*' 



By differentiation with respect to x equation (5) may be 

 written 



+ ^g + 6P^^ + ... + ?g 1 =0. . (6) 

 Moreover, a second equation must hold good at the surface, 



viz. 



_ % |£i +Vl _<^_i= (7 ) 



2 a^ a* 



In order to satisfy equations (6) and (7) by the method of 

 successive approximations, we put ?/ 1 =Z + 97, f=q -\-/3, where 

 Z and <7 are supposed to be constants, and rj. and /3 small 

 functions depending upon x and t. Dealing, then, with the 

 fact that for long waves, whose wave-length is great in com- 

 parison with the depth of the canal, every new differentiation 

 with respect to x gives rise to continually smaller quantities, 

 these equations become as a first approximation: — 



3/3, 3/3 , 3*7 n 



3. + 3^3£ 

 * ox 3^ ox 



and are satisfied by taking 



and £ = V^, • (8) 



where a is an arbitrary constant which we will suppose to be 

 small. 



