260 



The Tide-generating Forces 



m 



_3MiR 



Then, if we take the horizontal component positive in the 



if 



g 2 E \a , 



direction towards the point of the earth's surface lying perpendicular beneath 



the moon (against the point Z) then 



K h = 



R8& 



= m sin2# , K v = 



dQ m, _ a . 



^ = -2(cos2^+|) 



(VIII. 5) 



The quantity m is characteristic of the tide-generating force 



m 



g 



2 E \a 



(VIII. 7) 



(VIII. 6) 



and one obtains: 



for the moon m/g = 8-57 x 10~ 8 , | 



for the sun m/g = 3-78 x 10~ 8 . ( 



If a more correct value for the tide-producing potential Q is required, 

 we have to keep more terms in the development in series of 1/q, and then 

 the best way is to develop 1/q according to spherical functions of the zonal 

 type P n {&). These P's are functions of & alone and are called zonal harmonics 

 or Legendre coefficients. We obtain 



We then have 



q a £j \a 



PJP) = 1 " 



p x (&) = cos^ , 



p 2 (&) = i(3cos 2 ^- 



p 3 (#) = i(5cos 3 ^-3cos^) , 



p 4 (&) = i(35cos 4 ^-30cos 2 ^ + 3) 



Pn{V) 



1) 



etc. 



(VIII. 8) 



Q 



MP? 



P£>) + *P&) + *P*W) + 



= Q 2 +Q, + Q i + 



(VIII. 9) 



As both P 2 and i> 4 are symmetrical to the great circle ■& = 90°, the same 

 applies for the parts of Q, which originate from these terms. An asymmetry 

 between the two hemispheres with the moon above or below the horizon can 

 only come from the terms Q^,Q b , etc. The term D 3 reinforces the i2 2 -part 

 on the hemisphere where the moon is in the zenith and weakens it on 

 the hemisphere where the moon is in the nadir. The most important term 

 by far is 42 2 , which corresponds to (VIII. 4). 



Equation (VIII. 3) describes the distribution of the tide-producing forces 

 by a body on the surface of the earth. The distribution of the total force 

 is shown in Fig. 110. The circle represents a cut through the earth and the 

 moon is at a great distance in the direction M. The arrows indicate the 

 direction and the strength of the total force which result from the conflicting 



