GEOMETRICAL MEAN DISTANCES. 



758. GEOMETRICAL MEAN DISTANCES IN A PLANE. The value 

 of R defined by equation (4) is the geometrical mean distance* of 

 the point P from the different points of the surface S. Let us in like 

 manner consider two surfaces S and S' situate in the same plane, 

 and let r be the distance of the two elements dS and dS' ; if we put 



(S) 



the integration extending to the surfaces S and S', the length Rj is 

 the geometrical mean distance of the two surfaces. 



Finally, if the two elements ^S and dS' belong to the same 

 surface, the value of R 2 given by the equation 



(6) S 2 /.R 2 



is the geometrical mean distance of the surface S that is to say, that 

 of all the points taken in pairs. 



The geometrical mean distances play an important part in 

 calculating the coefficients of self-induction. 



The geometrical mean distance from a point to a figure is 

 manifestly comprised within the greatest and least of the distances 

 from this point to the different elements of the figure. The same 

 is the case for the geometrical mean distance of the two figures 

 A and B. 



The following property is a direct consequence of the definition. 



If R a c and Rj c are the geometrical mean distances of two figures 

 A and B to a third C, and R a +j c the geometrical mean distance of 

 the sum of the two figures A and B from the third, we have 



(A + B) / . R (a+ b)c = A/ . R ac + B/ . R 6c . 



This equation enables us to determine the value of R for a complex 

 figure, when we know it separately for different parts of the figure. 



The calculation of R in different cases reduces to a question of 

 analysis ; we shall give a few examples. 



i st. For a straight line of length a, and at a point at a distance x 

 from the line, the geometrical mean distance is defined by the 

 equation 



al. R = 



= dsl. r = 



* MAXWELL. Trans. Roy. Soc. Edinburgh. 1871-72. Electricity and 

 Magnetism. Vol. II., 691-693. 



