452 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. 



has supposed that the external medium is homogeneous, but since 

 there is no field there, the value of the inductivity is immaterial. 

 If z, p be rectangular coordinates parallel and perpendicular to 

 the axis in the plane of any orthogonal cross-section of the tore, 

 the whole flux of induction through the section is 



(3) 



p l and p 2 being the least and greatest values of p on the contour 

 of the section for a given value of z, and being given a^ functions 

 of z by the equation of the contour. 



If another circuit be wound in any manner about the tore, 

 embracing it n' times, the flux through it is n' times that just 

 found, and the mutual electrokinetic energy of a current / 2 in it 

 and a current / x in the former winding is, according to the last 

 sentence of 226, 



(4) T 



The mutual inductance of the circuits is accordingly 



(5) J 



If the second coil coincide with the first the flux through itself is 



so that the self-inductance of the toroidal coil is 

 (6) . L = 2 K 



The electrokinetic energy is 



'dpdz 

 P 



For a coil of square cross-section whose side is 2a and whose mean 

 radius is R, 



R 4- n. 



(8) L 



