y 



CHAPTER II. 



GRAVITY AND GRAVITY POTENTIAL. 



6. Gravity. The exterior force upon which the conditions of equilibrium and 

 motion in the atmosphere and in the sea depend is gravity. By gravity without 

 further specification we mean the force the intensity of which is found by the pen- 

 dulum experiments. It is the resultant of two different actions the attraction of 

 the earth and the centrifugal torce due to the earth's rotation. But in practical 

 application we shall never make use of this decomposition of the force into the two 

 components of different origin. 



A first condition for the solution of concrete problems relating to the equilib- 

 rium and the motion of the air and the sea is therefore a knowledge of the intensity 

 of gravity at ever)' point of the space filled by these two media. This knowledge 

 is founded on the actual measurements of the intensity of gravity at the earth's 

 surface. But it is not necessary for us to take into consideration all the small 

 irregularities in the variation of this force as they present themselves in geodetic 

 investigations. Where no measured values of the intensity of gravity are at hand it 

 will suffice to work with the " normal " values, as they can be calculated by the 

 general formulae of geodesy. They will give an approximation far closer than that 

 by which we can find the values of any other force upon which the atmospheric or 

 oceanic equilibrium or motion depends. 



We shall therefore write down the formulae necessary for the calculation of this 

 normal intensity of gravity, and give a complete tabulation of these formulae. 

 According to the common terminology, we shall call the tabulated quantity the 

 acceleration of gravity. But it should be remembered that it represents, as already 

 mentioned (section 2), at the same time the intensity of gravity measured statically 

 by the weight per unit-mass of the heavy body. 



7. Normal Decrease of Gravity in the Atmosphere. Let the numerical value 

 g x of the acceleration of gravity be known at a point of the earth's surface. Its value 

 g can then be calculated at any height z above this point from the decrease of 

 the attraction with the increase of the distance from the attracting masses, and from 

 the increasing influence of the centrifugal force with the increasing distance from 

 the earth's axis. Setting aside quantities of the order of magnitude of the square of 

 the ratio z/r, z being the height and r the radius of the earth, we find, according to 

 Helmert,* as the best expression for the decrease of the gravity with the height, 



() g (g\ 0.000003086^) 



* Helmert : Ueber die Reduction der auf der physischen Erdoberflache beobachteten Schweerebeschleuni- 

 gungen auf ein gemeinsames Niveau, zweite Mitteilung. Sitzungsberichte der Akademie der Wissenschaften, 

 Berlin, 1903, p. 650. 



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