12 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



In order to calculate the correction in this case, we shall consider the earth as 

 a sphere of radius r, and make use of the well-known theorem in the theory of 

 attraction that a spherical shell of constant density does not exert any influence on 

 a point inside it. In the depth z below sea-level we have therefore only to take 

 into account the attraction of the mass contained within a sphere of radius r z. 

 M being the whole mass of the earth and m that of the shell, we have for the 

 acceleration of gravity at sea-level 



and at the depth z below sea-level 



M ' in 

 g= \r-zf 



Neglecting squares or products of the small ratios m\ J\T and z/r, we conclude 



from these equations 



z in 



g = g'o+ 2 g --goJjf 



Denoting by p m the mean density of the earth, and by p that of the spherical shell, 

 we have 



M '= $irr a p m in = q-rrr 2 zp 



and thus 



w *-*+?H) 



For the factor 2g- /r we have to use, according to Helmert, the value 0.000003086. 

 For the density of the spherical shell we shall use as an average value p = 1.05, cor- 

 responding to the density of the sea-water at the depth of nearly 5000 meters 

 (compare table 14 h). Choosing finally p m = 5.5 as the probable value of the 

 average density of the earth, we get the formula 



(b) g = g a + 0.000002202^ 



by which we shall calculate the normal values of the acceleration of gravity in 

 the sea. 



The values of the correction term 0.0000022022 are given in table 2 H of the 

 Hydrographic Tables. 



Of course the normal values of the acceleration of gravity, which we are thus 

 able to calculate, will generally slightly differ from the real local values, as a conse- 

 quence of the local distribution of mass. It must also be remembered that the 

 spherical shell does not consist exclusively of water, but also contains the land- 

 masses below the continents. For this reason we might have chosen a still greater 

 value for the mean density of the shell. But this heterogeneity of the shell will 

 have different effects near the coasts and in the middle of the open sea, and we 

 therefore leave it out entirely, the more so as the "normal" value of gravity gives 

 a precision amply sufficient for the discussion of the dynamics of the sea in the 

 present state of development of this science. 



