14 



spherical equilibrium surface produced by the action of gravity alone,, 

 affords a yardstick for measuring the effect of the tide-producing 

 force of the moon, and is called the lunar equilibrium tide. 



27. Equation of the tidal surface of equilibrium. — ^Let r be the dis- 

 tance CP (fig. 6) from the center of the earth to any point P on the 

 equilibrium surface, 6 the angle between CP and the axis of the sur- 

 face, and as before E and M the masses of the earth and moon, respec- 

 tively, R the distance between their centers, a the radius of the earth, 

 and M the coefficient of gravitational attraction. Let Vt and Vg be, 

 respectively, the lunar tide-producing potential and potential due to 

 gravity at P. 



The force of gravity becomes zero when r is infinite. The gravity 

 potential is then, from the definition in paragraph 22 : 



Vg=r{Efxy)dr=Ei^/r. (13) 



Since, as shown in paragraph 24, the total potential at all points on 

 the surface of equilibrium is constant : 



V,+ Vg=C. 



Substituting the expression for Vt found in equation (11), and for 

 Vg in equation (13), the equation of the surface of equilibrium be- 

 comes : 



}iM(ji(r'IR') (3 cos^ 0- 1) + Eyilr=C. (14) 



28. An indefinite number of surfaces are given by this equation as 

 various values are assigned to C. If the oceans were continuous, the 

 particular surface to be chosen would have a volume equal to that of 

 the sphere with radius a, since the volume cannot be altered by the 

 tidal disturbance. It will be shown that this condition is fulfilled by 

 the surface whose radius vector is equal to the earth's radius where the 

 tide-producing potential is zero, i. e., where cos^ 0= 1/3. Such a surface 

 will intersect the sphere in the small circles P1P2, and P3P4 in figure 6, 

 The resulting value of the constant is found by placing r=a and cos- 6— 

 1/3 in equation (14), giving: 



Eixla=C 



and the equation of the surface of equilibrium is therefore: 



%MiJ. (r'/R^) (3 cos^ 6-1) + E^i/r = Efija 



which reduces to : 



YiiMa^/ER^) (3 cos^ e-l)=a'(r-a)lr^ 



Kepresenting the height of the equilibrium tide by u, it follows from 

 the definition in paragraph 26 : 



r—a=u. 



