A SYSTEM OF PARTICLES. 271 



= ^ sin e cos e {Fy F"'y - F'y F"y) + cb cos 6 . F"y, 



and dividing by cos^, and equating coefficients of sin and of unity 

 to zero, we have 



FyF"'y- F'yF"y = 0, 

 The latter gives 



F-'y^-^Fy. 



Fy=e-\ 

 which also satisfies the former, and we obtain 



u = b sui —-{ct—x)e^^, 



A. 



7 27r, izr„ 



i' = 6 cos -- {ct — a-) e '\ * . 



A 



37. It is clear from these equations that if there be a motion 

 vertically, there must be a corresponding horizontal motion, and that 



when the vertical motion is zero, as when ct — x — — ov — the hori- 



4 4 



zontal is at its maximum, and vice versa. 



If we take the origin at a given depth h, and suppose the maximum 

 values of u and v at that point = m, we have m = b, and the greatest 

 velocity at the surface is 



u — me>^ ^ = V. 



38. Suppose now we have two sets of fluid, and that at the depth 

 h in one, and /?, in the other, the maximum velocity is m. 



Let u, Ui be the maximum velocities at the surface. 



