61 



114. The remaining part of the equihbrium argument is made up 

 of simple multiples of the angles v and ^, the longitude and right 

 ascension of the intersection, represented by the arcs UI and UJ, 

 figure 28. The arc UO is the longitude of the moon's node, conven- 

 tionally represented as A^^; the angle MOS=IOU is the constant angle 

 i between the moon's orbit and the ecliptic, and the angle lUO is the 

 constant angle co between the equator and the ecliptic. The value of 

 V in terms of N and these known angles may therefore be determined 

 by the solution of the spherical triangle 10 U and the value of ^ then 

 found from the right spherical triangle lUUi. As the moon's node, 

 0, makes the circuit of the ecliptic in its period of 19 years, the ascend- 

 ing intersection, /, moves to and fro over a comparatively small arc 

 on either side of the vernal equinox, U, the angle v increasing slowly 

 from to 13°. 02, then decreasing to —13°. 02 and increasing again to 

 zero. The angle ^ similarly fluctuates between the limits of 11°. 98 

 and —11°. 98. The maximum change in these angles during a year is 

 about 5°. The slowly fluctuating part of the equilibrium argument 

 formed by these two angles is conventionally designated by the 

 symbol u. The total equilibrium argument is then represented by 

 V+u. 



The value of A^ at the beginning of each calendar year at Greenwich 

 is tabulated with those of h, s, p, and pi. Its rate of change is 

 — 19°.326,19 per calendar year, or — 0°.002,206,41 per mean solar 

 hour. Its value at any instant is therefore readily found. The 

 values of v and ^ at that instant can then be found from a table giving 

 these angles for each degree of N. 



115. Components of the equilibrium tide. — If Vo is the value of V at 

 any given instant, taken as the origin of time, then at any time t there- 

 after, 



V=Vo+at, 



in which a is a constant whose value is: 



a=Wi0+n2^/+^3O'+^4co + n5wi (72) 



Each term of equations (68), (69), and (70) then has the form: 



y=A cos (V+u)=A cos [at+{Vo-\-u)] (73) 



The form of this expression shows at once that each term represents 

 a component of the equilibrium tide. For lunar components the 

 values of A and u change slowly with the longitudes of the moon's 

 node, but may be considered as constant during a limited period of 

 time such as a month or even a year. For solar components, A is 

 constant and u is zero. 



116. The numerical value of the speed of each component of the 

 equilibrium tide may be readily computed from the speeds of the 



