123, 124, 125] CHARACTERISTICS OF POTENTIAL. 363 



2 nd . Its first partial derivatives are everywhere holomorphic. 



3 rd . Its second derivatives are finite. 



4 th . V vanishes at infinity to the first order, 



dV dV 

 ' ~d~' ~dz vamsn * secon d order, 



lim(E 2 ^} = - 



R = K\ GX) 



5 th . F satisfies everywhere Poisson's differential equation 



and outside of attracting matter, Laplace's equation 



Any function having all these properties is a Newtonian potential 

 function. 



The field of force X, Y, Z is a solenoidal vector at all points 

 outside of the attracting bodies, and hence if we construct tubes of 

 force, the flux of force is constant through any cross -section of a 

 given tube. A tube for which the flux is unity will be called a unit 

 tube. The conception of lines of force and of the solenoidal property 

 is due to Faraday. 



Since F is a harmonic function outside of the attracting bodies, 

 it has neither maximum nor minimum in free space, but its maximum 

 and minimum must lie within the attracting bodies or at infinity. 



In the attracting bodies the equation z/F=47tp says that 

 the concentration of the potential at any point, or the divergence of 

 the force from it is proportional to the density at that point, except 

 where Q is discontinuous. 



125. Examples. Potential of a homogeneous Sphere. 



Let the radius of the sphere be B, h the distance of P from its 

 center, 



Let us put s instead of r, using the latter symbol for the polar 

 coordinate, 



