the earth as a whole, together with a similar differential with respect 

 to the attraction of the sun, are the tide-producing forces. 



15. The tide-produciiig force of the moon. — In figure 1 ,JJ is the center 

 of the earth, the center of the moon, and P any point at or within 

 the earth's surface, r the distance CP, a the radius of the earth, E 

 the distance between the center of the earth and the center of the 

 moon, D the distance from P to the center of the moon, Q (theta) the 

 angle between CO and CP, and P the angle between PO and CP 

 produced . 



Let M be the mass of the moon , 



IX (mu) the gravitational attraction betw een two units of mass 

 at one unit's distance. 



The attraction of the moon on a unit of mass at th,e point P is then 

 Mn/D'^ acting m the 

 direction of PO, and 

 its component in the 

 direction C P is 

 (Mfi/D') cos P. Sim- 

 ilarly the attraction 

 of the moon on a unit 

 of mass at the center 

 of the earth is M/jl/E^ 

 acting in the direction 

 CO, and its component in the direction CP is {M'lx/E^) cos 6. The com- 

 ponent of the difference of these forces, in the direction CP, is: 



fr= {MiJilD') cos P- {Mix/E') cos d (2) 



Let A be the foot of a perpendicular from the center of the moon, 

 0, to the line CP produced. Then: 



D cos P=PA, E cos d=CA=PA+r. 

 whence: 



E cos d=D cos P-\-r. cos P=(E cos d—r)ID. 

 Giving: 



fr=Mfi [{E cos d-r)/D'- cos O/E'] 



=Mfx [(cos e-r/E) (PV^')-cos d]/E-. (3) 



From the triangle POC: 



D^=E'i'r'-2Er cos 0. 

 Whence: 



DyE^ =1-2 (r/P) cos 6 + (r/Ey. 



Figure 1. 



