732 



PROFESSOR CLERK MAXWELL ON THE GEOMETRICAL 



centric 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 be- 

 tween two concentric circles from each other is R, where 



« - a 2 2 ) 2 (log R - log flj) = j (3« 2 2 - a?) (a x 2 - a?) - a 2 * log ^ . 



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



R = ae~* . 



When a 2 , the radius of the inner circle, becomes nearly equal to a lt that of the 



outer circle, 



R = a x . 



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 

 beR. 



Let it be required to find the co-efficient 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 co-efficient of in- 

 duction of a linear circuit of the same shape as the coil 

 on a similar parallel circuit at a distance R, the co-efficient of induction of 

 the coil on itself will be 



rc 2 M . 







2d, The current, however, is not uniformly distributed over the section. 

 It is confined to the wires. Now the co-efficient self-induction of a unit of 

 length of a conductor is 



C - 2 log R , 



o 



o 



o 



o 



o 



o 



o 



o 



o 



where C is a constant depending on the form of the axis of the conductor, and 

 3 the mean geometric distant 

 Now for a square of side D, 



R is the mean geometric distance of the section from itself. 



logEj = log D +3 log 2 + |- - Y2 



