278 Theory of the Tides 



by the condition, that the volume of the ocean is constant and the integration 

 must extend over the entire ocean surface S. If the ocean covers the whole 

 earth, then C = 0, by the general property of spherical surface harmonics. 

 However, inc ase of a limited ocean C depends on the distribution of land 

 and water on the globe. If the tide-producing potential according to (VIII. 17) 

 consists of terms of the form Afi cos (at + sX) where A is the numerical coeffi- 

 cient and G the Goedetic coefficient and S is an integer and if we abbreviate 



P = I Gcos(sfydv , Q = | Gs'm(sfydv 

 s s 



we obtain for the partial tide produced by this term 



rjs = — 



P 

 Gcos{sX) — -= 



COSoY- 



Gsin((M)-^ 



sin at) . (IX. 3b) 



The integrals P, Q and also S, the surface of the oceanic area, can be computed 

 by mechanical quadratures, and certain corrections are added to the previous 

 values, which also depend upon the relative motion of the disturbing body. 

 These corrections were primarily derived by Thomson and Tait (1883 para. 

 808) and, later on, they were examined more closely by Darwin and Turner 

 (1886), and it was then found that they are quite unimportant with regard to 

 the actual distribution of land and water. However, it became apparent that, 

 in this improved equilibrium theory "corrected for the continents", the time 

 of high water does no longer coincide with the maximum of the tide potential, 

 i.e. that there is an "establishment ,, which is different for each locality. 



A further improvement consists in considering the mutual attraction of 

 the water particles. To the tide potential of the disturbing body comes the 

 gravitation potential of the elevated water-masses in the tidal spheroid. If the 

 ocean covers the entire earth Q is increased by 



3 Q - 



~5V gv 



in which q is the density of the water, g m the mean density of the earth 

 and g/g m =018. Thereby the amplitude H increases in the proportion 



1 



(Q/Qm) 



which factor is 112. The consideration of the gravitation potential of the 

 elevated water increases the equilibrium tide by 12%. In addition to this, 

 Poincare (1910, p. 60) has taken into account the distribution of land and sea. 

 The calculations then became very intricate and the resulting improvement 

 seems to be so important that the above-mentioned formulae are no longer 

 accurate, even approximately. 



