34] DEFINITE INTEGRALS. 65 



Now since the volume of a sphere is J the product of the 

 surface by the radius, we have, on making R approach zero, 



A T7 . T . (Mean V on surface F at center} 

 A F= 3 Lim l ->- . 



.8=0 -ft 



The negative scalar parameter A F was accordingly called by 

 Maxwell the concentration of F, being proportional to the excess of 

 the value of F at any point over the mean of the surrounding 

 values. It is evident from this interpretation of AF that if the 

 concentration of a function vanishes throughout a certain region, 

 then about any point in the region the values at neighbouring 

 points are partly greater and partly less than at the point itself, 

 so that the function cannot have at any point in the region either 

 a maximum or minimum with respect to surrounding points. A 

 function that in a certain region is uniform, continuous, and has no 

 concentration is said to be harmonic in that region. The study of 

 such functions constitutes one of the most important parts, not 

 only of the theory of functions, but also of mathematical physics. 



By means of the same theorem we may obtain another repre- 

 sentation of A F. Let us apply the theorem to the space included 

 between two small concentric spheres of radii R L and R 2 = jR x + h. 

 Then at the outer sphere 



IF 



and the surface integral being taken over the surface of both 

 spheres, with the normal pointing in each case into the space 

 between them, 



As we make h approach zero, the first term of the second integral 

 destroys the first, and 



[fdVja T . rr 9 2 F 7 , 



- ^-dS = Lim ^ hdS, 

 JJ dn fc =0 }JM> 9r 2 



so that fff&VdT = Lim ([ ~kd& 



JJJ h=v HE, dr* 



Now hdS is the element of volume dr, so that AF may be 



9 2 F 

 defined as the mean value of the second derivative ---- for all 



W. E. 5 



