The Tide-generating Forces 



259 



the force of attraction A = x(M/q 2 ) acts in the direction BM, the centrifugal 

 force Z = (4ji 2 a/Tl) in the direction parallel to EM. According to the third 

 law of planetary motion by Kepler, which states that the squares of the 



• M 



Fig. 109. Determination of the potential of the tide generating forces. 



periods of revolution of the planets around the sun are in the same ratio 

 as the cubes of their mean distances from the sun {a z \T\ = const. = xM/4n 2 ) 

 we can also write for Z : Z = x(M/a 2 ) and we obtain in the point B with 

 the zenith distance of the moon & for the horizontal and vertical components 

 of the forces of attraction: 



. M . Q , . M , 



Ah = x—zsmv- A v = h — cos# ; 



and of the centrifugal force: 



_ M . Q M 



Zh = x — sin*? , Z v = x — cos u . 

 a 2 fl- 



it can easily be found from the triangle EBM that 



cos#' 



acos&—R 



sin# = 



a sin & 



, Q = fll+-i 



R 2 2R 



\l/2 



COS?? 



a J 



If, in developing \/g z into a series, we neglect the term of a higher order, 

 then we have 



— = -i H cos# 



q° a°\ a 

 and we obtain for the two components 



„ A 3 Mi? . 



A A = A h — Z h = ~x^- sin 2^ 

 2 a 3 



A.^ — A id £<$ - — JX 



These forces have a potential 



MR I 



cr \ 



cos 2 # — 



1 



31' 



O = ~« — — (x — cos 2 # 



:(^-cos 2 # 



(VIII.3) 



(VIII.4) 



17* 



