[From the Transaction! of the Royal Society of Edinburgh, Vol. xxvi.] 

 XLVIII. On the Geometrical Mean Distance of Two Figures on a Pl,>: 



[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 sy 

 of 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 

 logarithm 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 con- 

 centrated 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 



