234] ELECTROMAGNETISM. 461 



(5) equal to 



C - 2/*wS log r = C - 2fjLl log r, 



where r is defined by the equation 



(11) S logr = 1 1 logrdadb. 



But from the interpretation of a definite integral as a mean, 23, 

 we see that logr is the arithmetical mean of the logarithms of 

 the distances of all the points of the cross-section from the fixed 

 point x, y. Now defining the geometric mean of n quantities as 

 the nth root of their product, we see that r is the geometric 

 mean of the distances of the points of the area from the point 

 #, y, for its logarithm is the arithmetical mean of their logarithms. 

 If TI and r. 2 be the geometric mean distances of a point from two 

 areas Si and S Z) r 3 the geometric mean distance of the point from 

 both areas taken together, we have by the definition, (n), 



( 1 2) (S 1 + &) log r, = S! log n + S 9 log r 2 . 



By means of this principle we may find the geometric mean 

 distance from a complex figure if we know it for the various 

 parts of the figure. This method is due to Maxwell*. We shall 

 first find the geometrical mean distance from a circular ring of 

 infinitesimal width. Let p be the radius, e the width of the ring, 

 and h the distance of the given point from its center. Inserting 

 polar coordinates in the equation ( 1 1 ), 



ffcr 



(13) 27r/>e log r = \ log (h 2 + p 2 2hp cos <) ped<j>. 



Jo 



This integral assumes different forms according as h is greater or 

 less than p. Taking out from the parenthesis the square of the 

 greater of these, and integrating, we get 



(14) \ogf = logh+-^j, h>p, 

 or 



(15) lo * = lo S ? + 4^ ^ Q ' P >h > 



where J is the definite integral 



T27T 



J (a) = I log (1 + a 2 - 2a cos <) d<f>, 



* Trans. Roy. Soc. Edinburgh, 18712. 



