28 U. S. COAST AND GEODETIC SUEVEY. 



For the tangent of the angle between the radius vector and the 

 resultant of the forces under consideration we have from (24), (25), 

 (30), and (61) 



3/2 ^,4- (5 cos=^ e-l) smd • , , 



(.2 



3/2 ^^-^ (5 cos^ ^-3 cos 6) 



since — and — -n^ — are each very small compared with unity. 



Comparing (72) with (71), it is found that the angle made by the 

 resultant force with the radius vector is the same as the angle made 

 by the normal at the same point. 



Taking d as the mean distance of the moon and substituting nu- 

 merical values from Table 2 for the coefficient in (71) we obtain 



tan 1^' = 0.000,000,001, 4 (5 cos^ d-\) sin (73) 



in which ^' has a maximum value of about 0.0004'' when ^ = 31.1°. 

 The maximum deflection of the normal due to the fourth power of 

 the sun's parallax is only about 0.000,000,5". 



Expressions similar to (44) and (64) may be formed for the equilib- 

 rium neight of the tide due to the sun, letting 



8 = mass of sun, 



d^ = distance from center of earth to center of sun, 

 6^ = angle at center of earth between line to sun and to point 

 of observation on earth. 



Then for the height {y) of equilibrium tide due to combined action of 

 moon and sun we have 



Ma? . . • 



y = ^ -p js (3 cos^ 6—1) a (approximate lunar tide). , 



4- i -E' ;t4 (5 cos^ d — 3 cos 6) a . . (tide depending upon 4th 



power of moon's paral- 

 lax). 



S a^ 

 + ^ E^j^ (3cos^0i — 1) a . . . . (approximate solar tide). 



S a* 

 "^ ^ F /T* (^ ^^^^ ^1 — 3 cos ^i) a . (tide depending upon 4th 

 ^ power of sun s parallax). 



(74) 



In (74) it will be noted that M, E, S, and a are constants. The dis- 

 tance d and d^ vary within certain limits because of the eccentricity of 

 the orbits of the moon and earth. The angles 6 and ff^, which are 

 practically the same as the zenith distances of the moon and sun, re- 

 spectively, vary with the declinations and hour angles of these bodies 

 and also depend upon the latitude of the place of observations. 



