16 



Since the ratio uja is extremely small, its squares and higher powers 

 may be dropped, giving: 



dc[=2ira^u sin 6dd 



Substituting the expression for u in equation (16) and integrating: 



c[=fT(MayER^)(S cos^ 0-1) sin 9dd 

 = T(MayER^)(f3 cos^ d sin ddd-f sin Odd) 

 = Tr {May ER^)(-cos^ 0+cos e) + C (17> 



Taking 5=0 when 6—0, the constant of integration becomes zero, 

 and the expression for the tidal volume in the zone measured by the 

 angle is : 



g_=w{MayER'') (cos ^-cos^ 6) (18) 



This volume reaches a maximum when 3 cos^ 0—1=0 and is then 



iriMayER') (Vl/3- l/sVl/S) =2/9 T(MayER')^Js 



which is the volume of the positive tide over the zone P5P1P2 in figure 

 6. The volume of the negative tide is the same, as 2 reduces again to 

 zero when 0=90°. The condition of continuity is therefore fulfilled 

 by the expression for the equilibrium tide given in equation (16). 



30. Magnitude of the lunar equilibrium tide. — Assigning to the con- 

 stants in equation (16) their numerical values, the ratio, MjE, of the 

 mass of the earth to the mass of the moon is 1/81.45; a, the mean 

 radius of the earth, 3,959 statute miles; R, the mean distance to the 

 moon, 238,857 statute miles. The coefficient y2{MayER^)a then is 

 0.584 feet. The corresponding height of the lunar equilibrium tide 

 in feet is therefore: 



16=0.584(3 COS2 0-1) 



The factor 3 cos^ 0—1 has a maximum value of 2 when 0=0, and 

 a minimum value of —1 when 0=90°. The maximum range of the 

 lunar equilibrium tide is then 3X0.584 = 1.752 feet. This distortion 

 of the water surface of the earth is very small in comparison with the- 

 distortion due to the earth's rotation, since the latter, as measured by 

 the difference between the equatorial and polar radii, is 13.35 miles. 

 The tidal distortion is however superimposed upon and not measur- 

 ably affected by the distortion due to the earth's rotation. 



31. Solar equilibrium tide.— The solar equilibrium tide is, by trans- 

 posing in equation (16) ; 



Ui=y2iSayERi^)a(3 cos^ 0i-l) 



