116] SECOND DIFFERENTIAL PARAMETER. 345 



The integrals in x, y, z, yz, zx, xy, being proportional to the co- 

 ordinates of the center of gravity and products of inertia of a homogeneous 

 sphere, all disappear from symmetry, while those in x 2 , y 2 , z* are all 

 equal and represent the moment of inertia of the sphere with respect 



to the diametral plane, which by 75 is ^^-K 5 - Accordingly if V 



denotes the mean value of V in the sphere of radius E, the above 

 integral equation becomes 



terms of higher order 

 Dividing by R 5 and taking the limit for E = 0, 



27) limff-^ = 



that is, the excess of the mean value of V throughout the volume 

 of a small sphere over the value at the center is proportional to the 

 value of z/F at the center and is of the second order of small 

 quantities. This interpretation is due to Stokes. 



From this point of view Maxwell calls z/F the concentration 

 of F, since it is proportional to the excess of the value of F at a 

 point over the values at neighboring points. It is evident from this 

 interpretation of z/F that if the concentration of a function vanishes 

 throughout a certain region, then about any point in the region the 

 values at neighboring 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. 



Another interpretation of z/F may be obtained as follows: we 

 have by the rule for the derivation of any function in any direction r, 

 with direction cosines, cos (rx) = a, cos (ry) = /5, cos (rz) ==-- 7, 



d d Q 3 d 



-d-r = a Wx + Pdy + r^' 



dV 

 Applying this to the function -^ we obtain 



a 2 F d dV / d d c\ dV.dV 3V 



28 ) 



- 



oxoy 



