346 VHI. NEWTONIAN POTENTIAL FUNCTION. 



If we consider at a point x, y, z all directions r and take the 

 mean of --T for these directions, inasmuch as the mean of any 



product of cosines is equal to zero, because for every cosine that 

 appears the negative also appears, the terms with products disappear. 

 Also from symmetry, denoting the mean value by the bar, 



'<? = 0* = y\ 

 Since always a 2 + ft 2 + y 2 = 1? we have 



and 



O 



Accordingly 



V_ I >> 



dr* s W "*" a</ 2 """ 



therefore 4 V is equal to three times the mean of the second derivative 

 of V in a definite direction for all possible directions leading from 

 the point in question. This interpretation is due to Boussinesq. 1 ) 



By means of this result we may obtain a third interpretation 

 connecting the value of z/F at a point with the mean excess of 

 values on the surface of a small sphere, with center at the point, 

 over the value at the center. 



If F denote the value at the center, the value at a distance R 

 in any direction is given by Taylor's theorem, 



Integrating over the surface of a sphere of radius R, the deriva- 

 tives of V varying with the direction, since dS = R 2 d&, dividing by 

 the constant JR 2 



30) //(F- F ) da, = *ff 



Now since 



the terms in the first integral depend upon the directions simply 

 through the direction cosines of r, which on account of symmetry 

 cause the integral to vanish. If V is the mean of V on the surface 

 the equation then becomes 



1) Boussinesq, Application des Potentiels a I' etude de Vequilibre et du mouve- 

 ment des solides elastiques, p. 45. 



