﻿relative to a Rotating Earth. 59 



and the pressure p is given by 



? = - V + i(a> 2 + 2a>n sin (/>) {.i< 2 + <j/-/3c) 2 } + & . (26) 



» r 



The approximate value of V' is +g[ 9T / + - ) , and p is 



the pressure at the reference point 0. In the motion indi- 

 cated above each particle of fluid moves in a horizontal 

 circle whose centre lies in the line (y=/3z, a? = 0). This line 

 lies in the meridian plane through point and is inclined to 

 the vertical at at an angle 6 towards the North, where 

 tan<9 = /3. When a) is very small in relation to H this in- 

 clined axis is almost parallel to the axis of the Earth ; and 

 when co is large this axis comes almost to coincidence with 

 the apparent vertical at 0. With co very large the motion 

 described above corresponds very closely with a uniform 

 rotation of the fluid about a vertical axis, as in the case of a 

 simple forced vortex. 



Another case of motion of incompressible fluid of interest 

 in the same connexion is that represented by 



.,, u=—coy; v = cox; w = 2X2 cos 6 . a?, . (27) 



with T 



J - = -V' + i (co 2 + 2vn sin (f>){% 2 +y 2 ) +2W cos 2 cb . x 2 + & , 



. . . (28) 



as the equation showing the distribution of pressure. This 

 motion differs from that first discussed in not being exactly 

 horizontal. The plane of motion of each particle of fluid 

 passes through the line OX, and is inclined to the horizontal 

 plane XOY at an angle 6 given by tan = 2Q cos <p/co. 

 When the angular velocity of rotation co is very large com- 

 pared with flcos</), the plane of motion of each particle is 

 practically horizontal, and the motion then corresponds very 

 closely with that of simple rotation of all the fluid as a solid 

 about, a vertical axis. When co becomes small, on the other 

 hand, the inclination of the plane of motion of each particle 

 of fluid to the horizontal increases. The two motions, repre- 

 sented by (24) and (27) respectively, are almost identical 

 when ft) is very large, and they differ entirely when co is 

 very small. It would be interesting to investigate the 

 manner in which a fluid, such as water, subsides to rest 

 from an initial condition of steady rotation about a vertical 

 axis. The solution represented by (24), (26), would appear 

 to be the exact solution for steady rotation of water in small 

 scale experiments. 



