THE GEOMETRICAL MEAN DISTANCE OF TWO FIGURES ON A PLANE. 281 



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 

 for the mean distance by taking the point at a great distance from the system, 

 in which case the mean distance must approximate to the distance of the centre 

 of gravity. 



Thus it is well known that the harmonic mean distance of two spheres, each 

 of which is external to the other, is the distance between their centres, and that 

 the harmonic mean distance of any figure from a thin shell which completely 

 encloses it is equal to the radius of the shell. 



I shall not discuss the harmonic mean distance, because the calculations 

 which lead to it are well known, and because we can do very well without it. 

 I shall, however, give a few examples of the geometric mean distance, in order 

 to shew its use in electro-magnetic calculations, some of which seem to me to 

 be rendered both easier to follow and more secure against error by a free use 

 of this imaginary line. 



If the co-ordinates of a point in the first of two plane figures be x and y, 

 and those of a point in the second and TJ, and if r denote the distance between 

 these points, then R, the geometrical mean distance of the two figures, is 

 defined by the equation 



log R . \\l\dxdy dgdrj = //// log r dxdydgdrj. 



The following are some examples of the results of this calculation : 

 (1) Let AB be a uniform line, and a point 

 not in the line, and let OP be the perpendicular 

 from on the line AB, produced if necessary, then 

 if R is the geometric mean distance of from the 

 line AB, 



AB . (log 72+ l) = PB . log OB-PA log OA + OP. AOB. 

 VOL. ii. 36 



