258 The Tide-generating Forces 



It is interesting to compare the tide-producing force with a force familiar 

 to us, e.g. the force of gravity. If E is the mass of the earth, the gravity 

 for a mass element /u on the earth's surface is in sufficient approximation 



xju(E/R 2 ) 

 and the ratio 



maximum tide-producing force of the moon _ -MR 3 _ 

 force of gravity E a 3 



81-5 60 3 



i.e. the tide-producing force of the moon is approximately a nine-millionth 

 part of the force of gravity; it reduces gravity by that amount at the points N 

 and Z and, therefore, this reduction of gravity is extraordinarily small. 



One may get an idea of its minuteness considering that, under its action, 

 the end point of a spiral spring, which is extended one meter by a 1 kg weight 

 suspended to it, is displaced by the tide-producing force as little as 



1 l in-3 l 



m — n 10" 3 mm = ^.ju . 



9 million 9 9 



In the considerations put forth thus far on the tide-generating forces, the 

 rotation of the earth around its axis in 24 h has been disregarded, since it 

 does in no way influence the derivation of the tide-producing forces. It causes 

 only relatively small changes in the shape of the earth's surface (revolution 

 ellipsoid instead of a sphere) and a small change in the apparent weight of 

 the individual unit of mass. Gravity is now the resultant of the attractive 

 force and the centrifugal force of the rotation, but there is no modification 

 in the magnitude and the direction of the tide-generating forces. The derivation 

 of the tide-producing forces, as done here, was made by considering the 

 "revolution" of the bodies around their common centre of gravity. This 

 derivation seems to be the most simple and at the same time the most logical. 

 Incorrect derivations, which are mainly due to a wrong interpretation of 

 the influence exerted by the revolution or rotation of the two interacting 

 bodies and which lead to forces of a different order of magnitude, have ap- 

 peared in literature but soon disappeared (see Muller, 1916). 



On the untenable flood theory by Galileo, developed before Newton, which 

 regarded the tides as an effect of the relative motion between the solid earth 

 and the oceans, see Mach (1904). 



2. The system of Tide-producing Forces as a Function of the Zenith Distance 



The distribution of the tide-generating forces can be derived for any 

 arbitrary point on the earth, by extending the considerations given above 

 for the points Z and N. Let B be such a point with the unit mass (/u = 1) 

 (see Fig. 109), E the centre of the earth and M the centre of the moon. Then 



