HARMOlSriC ANALYSIS AND PREDICTIOlSr OF TIDES. 



29 



If we ascribe the mean values to the distance d and d^ and substi- 

 tute the numerical values of the constants from Table 2, we obtain 

 from (74) the following, the unit of height being the foot : 



y = 0.584 (3cos^5— 1) (approximate lunar tide) . 



+ 0.010 (5 cos^ 6 — 3 cos 6) . . . (depending upon 4th power 



of moon's parallax) . 



+ 0.270 (3 cos^ ^1 — 1) (approximate solar tide) . 



+ 0.000,01 (5 cos^ d^ — 3 cos 6j) . (depending upon 4th power 



of sun's parallax). (75) 



From (75) it appears that the extreme ranges of the equilibrium 

 lunar and solar tides are approximately 1.75 and 0.81 feet, respec- 

 tively, and that the tides depending upon the fourth power of the par- 

 allax of the moon and of the sun are of little or no importance. 



9. DEVELOPMENT OF EQUILIBRIUM TIDE. 



In equation (74) an expression was obtained for the equilibrium 

 height of the tide in terms of the linear distances and the zenith dis- 

 tances of the tide-producing bodies. It is now proposed to develop 

 this equation in terms of variables which change uniformly with 

 time. Let us consider, first, the term representing the principal lunar 

 tide, which may be written 



y = 3/2 ^ I (cos^ 0-1/3) (76) 



In Figure 9 let represent the center of the earth and let projec- 

 tions on the celestial sphere be as follows: 



Fto. 9. 



lAB, the earth's equator; 



IM, the moon's orbit; 



CPA, the meridan of place of observation ; 



CMM' , the hour circle of the moon ; 



/, the intersection of moon's orbit with the Equator; 



P, the place of observation ; 



M, the position of the moon. 



