56 



104. Mathematical derivation of lunar equilibrium tide. — To derive 

 the formulae for reducing the components of the actual tide to their 

 mean values, for converting these mean values to the amplitudes 

 applicable at any given period, and for determining the epochs of 

 the components, it becomes necessary to develop the mathematical 

 expressions for each component of the equilibrium tide in terms of 

 astronomical constants. The expressions for the lunar and solar 

 equilibrium tides in terms of the hour angle and declination of the 

 moon and sun, derived in equation (24), afforded the means for 

 developing the characteristics of the tide, and for inferring therefrom 

 the speeds of most of the components. The much more elaborate 

 expression necessary to develop the coefEcients and phase relations 

 of the components of the lunar and solar equilibrium tides will now 

 be developed in outline. 



105. In figure 28, A^is the north pole of the earth's axis on the celes- 

 tial sphere, UISiMiPi the 

 celestial Equator, UOS 

 the ecliptic (the path of 

 the sun), U the vernal 

 equinox and S the posi- 

 tion of the mean sun 

 at a given instant; lOM 

 the moon's path (orbit), 

 / its intersection with 

 the Equator, the 

 moon's node, M the 

 position of the moon, 

 P the zenith of a tidal 

 station, NPPi its celes- 



the hour circle of the moon, NSSi the hour 

 and L'l the foot of the great circle drawn 



Figure 28. 



tial meridian, NMMi 

 circle of the mean sun 

 through the vernal equinox perpendicular to the moon's orbit. 

 Then: 



d (theta), the arc PM, is the zenith distance of the moon. 



8 (delta), the arc MiM, the declination of the moon. 



X (lamda), the arc PiP, the latitude of the station. 



H, the angle PNM, the hour angle of the moon. 



N, the arc UO, the longitude of the moon's node. 



I, the angle MJM, the inclination of the moon's orbit. 

 The symbols conventionally assigned to other arcs and angles, 

 and to pertinent astronomical constants are: 



T, the arc PiSi, the hour angle of the mean sun. 



s-\-k, the arc UiM, the true longitude of the moon. 



s, the mean longitude of the moon; i. e., the longitude which it 

 would have it if travelled at the average rate along its orbit. 



