84, 85] NEWTONIAN POTENTIAL FUNCTION. 167 



space r (within S), and the surface integral that due to the 

 matter without. If the surface is equipotential, the surface 

 integral is 



19F 



The first integral is now equal to 4?r, so that 



V s being constant contributes nothing to the derivatives of V, so 

 that the outside bodies may be replaced by a surface layer of 

 density 



04) o- = ~~-- 



. 

 ~ 4?r 



The mass of the surface distribution 



n e being the outward normal, is the inward flux of force through 

 S, which is equal to minus the mass of the interior matter, and is 

 not, as in the former case, equal to the mass which it replaces. 



85. Potential completely determined by its charac- 

 teristic properties. We have proved that the potential function 

 due to any volume distribution has the following properties : 



1. It is, together with its first differential parameter, uniform, 

 finite, and continuous. 



2. It vanishes to the first order at oo , and its parameter to 

 the second order. 



3. It is harmonic outside the attracting bodies, and in them 



satisfies 



The preceding investigation shows that a function having 

 these properties is a potential function, and is completely de- 

 termined. 



