129, 130, 131] 



GAUSS'S MEAN THEOREM. 



375 



since the function is harmonic in the sphere considered. Accordingly 

 the formula reduces to the first term 



16) 



VdS. 



The surface integral represents the mean value of F on the surface 

 of the sphere multiplied by the area of the surface, 4^r 2 . Thus we 

 have the theorem due to Gauss. The value of the potential at any 

 point not situated in attracting matter is equal to the mean value of 

 the potential at points on any sphere with center at the given point 

 and not containing attracting matter. It at once follows from this 

 theorem that a harmonic function cannot have a point of maximum 

 or minimum, for making the sphere about such a point small enough 

 the theorem would be violated. 



131. 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 investiga- 

 tion shows that a function 

 having these properties is a 

 potential function, and is 

 completely determined. 



For we may apply the 

 above formula 5) to all space, 

 and then the only surface 

 integral being that due to 

 the infinite sphere, which 

 vanishes, we have 



Fig. 135. 



If however, the above conditions are fulfilled by a function F, 

 except that a certain surfaces S its first parameter is discontinuous, 



