154 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



79. Characteristics of Potential Function. We have 

 now found the following properties of the potential function. 



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

 continuous. 



2nd. Its first partial derivatives are everywhere holomorphic. 



3rd. Its second derivatives are finite. 



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



lim(JKF)Jf; 



.8=00 



-*-,... vanish to second order, 

 das 



1 I T"fcf> " " \ 



hm LR 8 3--) = - 



R=oo V OX 



5th. V satisfies everywhere Poisson's differential equation 



8 2 F 8 2 F 



++ = ~ p) 



and outside of attracting matter, Laplace's equation 



| 8 2 F | = 

 dec 2 dy 2 dz 



Any function having all these properties is a Newtonian 

 potential function. 



The force X, F, Z is a solenoidal vector at all points outside 

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

 the flax 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 V 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 AF=477y> says that 

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

 force from any point is proportional to the density at that 

 point. 



