■(a). 



A SYSTEM OF PARTICLES. 269 



individual particle is partly in the same direction and partly in a 

 vertical direction, which we will call that of y. There is no motion 

 in the direction of s. 



It is evident that we are here considering the motion of waves on 

 water or of the tide waves. 



The equations of motion are 



dx ^\ dif' 

 which give one as their result, viz. 



^ d{v) _ d^ d{u)^ 

 dx dt dy dt '" 



also we have the equation -,- h ^ = 



dx dy 



36 Now ^'^^'^^ _ du du du 

 dt dx dy dt 



d{v) _ dv dv dv 



~dr~~cnc^' ^d7/^Tt' 



and the additional hypothesis that the motion is a uave motion, in- 

 troduces the further conditions 



« = asm ^(c^ - x)fy, 



V = bcos~{c( - x)F{y), 

 fiy) and F{y) being any functions of y. 

 By substitution in the equation 



du dv 



dx dy ~ ' 

 we get 



- — rt cos — {ct - x)fy + bcos~{cf - x) F'y = ; 



M M 2 



