362 VIII. NEWTONIAN POTENTIAL FUNCTION. 



is the potential of the earth's attraction. But by Poisson's equation, 



so that we have, 



74) 

 Now by the divergence theorem, 



=-//!>. 



so that 



Now if the volume of the earth be v, its mean density g m , the volume 

 integrals are respectively equal to g m v and v, so that, multiplying by 



this becomes 



7A x & 2 



76 ) ^m = 



Thus if we know the value of g at every point on an equipotential 

 surface, we obtain the value of the product yq m in terms of the 

 angular velocity, and the surface integral of g. Using a formula 

 given by Helmert representing the results of geodetic determinations 

 of g, Woodward 1 ) finds for the value of j>p m 



= 3.6797 x 10 

 Richarz and Krigar-Menzel 2 ) obtain, in a similar manner, 

 y$ m = 3.680 x 10~ 7 see" 2 . 



Combining this result with Boys's value of 7, p. 30 (see erratum), we 

 obtain for the mean density of the earth the value 



124. Characteristics of Potential Function. We have 

 now found the following properties of the potential function. 



1 st . It is everywhere holomorphic, that is, uniform, finite, con- 

 tinuous. 



1) Woodward, The Gravitational Constant and the Mean Density of the 

 Earth. Astronomical Journal, Jan. 1898. 



2) F. Richarz und 0. Krigar-Menzel,, Gravitationsconstante und mittlere 

 Dichtigkeit der Erde, bestimmt durch Wagungen. Ann. der Phys. u. Chem. 36, 

 p. 177, 1898. 



