OF TWO FIGURES ON A PLANE. 



283 



(7) The geometric mean distance of a circular line of radius a, from a 

 point in its plane at a distance r from the centre, is r if the point be without 

 the circle, and a if the point be within the circle. 



(8) The geometric mean distance of any figure from a circle which com- 

 pletely encloses it is equal to the radius of the circle. The geometric mean 

 distance of any figure from the annular space between two concentric circles, 

 both of which completely enclose it, is R, where 



(a, 2 - a, 1 ) (log R + 1) = a," log a, - a/ log a 2 , 



a, being the radius of the outer circle, and 2 that of the inner. The geometric 

 mean distance of any figure from a circle or an annular space between two 

 concentric circles, the figure being completely external to the outer circle, is 

 the geometric mean distance of the figure from the centre of the circle. 



(9) The geometric mean distance of all the points of the annular space 

 between two concentric circles from each other is R, where 



(a,' - a,')' (log R - log a,) = } (3a, 8 - a,') (a,' - a/) - < log J . 



a, 



When a,, the radius of the inner circle, vanishes, we find 



When a n the radius of the inner circle, becomes nearly equal to a 1} that of the 



outer circle, 



R = a r 



As an example of the application of this method, let us take the case of a 

 coil of wire, in which the wires are arranged so that the transverse section of 

 the coil exhibits the sections of the wires arranged in square order, the distance 

 between two consecutive wires being D, and the diameter of each wire d. 



Let the whole section of the coil be of dimensions which are small com- 

 pared with the radius of curvature of the wires, and let 

 the geometrical mean distance of the section from itself 

 be R. 



Let it be required to find the coefficient of induction 

 of this coil on itself, the number of windings being n. 



1st. If we begin by supposing that the wires fill up 

 the whole section of the coil, without any interval of 

 insulating matter, then if M is the coefficient of induc- 



362 



