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XXIII. — On the Geometrical Mean Distance of Two Figures on a Plane. 

 By Prof. J. Clerk Maxwell, F.R.S. 



(Received January 5th ; read January 15th, 1872.) 



There are several problems of great practical importance in electro-magnetic 

 measurements, in which the value of a quantity has to be calculated by taking 

 the sum of the logarithms of the distances of a system of parallel wires from a 

 given point. The calculation is in some respects analogous to that in which 

 we find the potential at a point due to a given system of equal particles, by 

 adding the reciprocals of the distances of the particles from the given point. 

 There is this difference, however, that whereas the reciprocal of a line is com- 

 pletely defined when we know the unit of length, the logarithm of a line has no 

 meaning till we know not only the unit of length, but the modulus of the system 

 logarithms. 



In both cases, however, an additional clearness may be given to the state- 

 ment of the result by dividing, by the number of wires in the first case, and by 

 the number of particles in the second. The result in the first case is the loga- 

 rithm of a distance, and in the second it is the reciprocal of a distance ; and 

 in both cases this distance is such that, if the whole system were concentrated 

 at this distance from the given point, it would produce the same potential as it 

 actually does. 



In the first case, since the logarithm of the resultant distance is the arith- 

 metical mean of the logarithms of the distances of the various components of the 

 system, we may call the resultant distance the geometrical mean distance of 

 the system from the given point. 



In the second case, since the reciprocal of the resultant distance is the 

 arithmetical mean of the reciprocals of the distances of the particles, we may 

 call the resultant distance the harmonic mean distance of the system from the 

 given point. 



The practical use of these mean distances may be compared with that of 

 several artificial lines and distances which are known in Dynamics as the radius 

 of gyration, the length of the equivalent simple pendulum, and so on. The 

 result of a process of integration is recorded, and presented to us in a form 

 which we cannot misunderstand, and which we may substitute in those ele- 

 mentary formulae which apply to the case of single particles. If we have any 

 doubts about the value of the numerical co-efficients, we may test the expression 



VOL. XXVI. PART IV. 9 D 



