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



representing the total current in the positive direction, is zero. 

 If the flow is uniform in all the conductors, for any two conductors 

 there will be a term 



where 



S& log r M = jjljlogrdadbda'db', 



and r 12 is the geometric mean distance of all pairs of points in 

 the two cross-sections. For instance let us consider a single 

 circuit made up of conductors whose cross-section is denoted by 

 $j and conductors carrying the return current whose cross-section 

 is S a . We then have 



T= jjw'H'da'db' + ijj u/H'da'db', 



while if we divide H into two parts, H due to the conductor 1, 

 and H^ due to the conductor 2, 



HI C l 2 1 1 fj.w log rdadb, 

 fli 



H 2 = (7 8 2 1 1 /u<; log rdadb, 



T becomes the sum of the integrals 



<20) T = ^jjw'da'db' -jjjj^ww' log r dadbda'db' 



+ Qjjtfdafdb' -jjjLww' log r dadbda'db' 



S 2 S, S 2 



+ ^jjw'da'db' -jjjjfjLWw' log r dadbda'db' 



S, S z S,. 



+ Q jjw'da'db' -jjjLww' log r dadbda'db'. 



Si S z S 2 



The first and third integrals, being the constant Ci/2 multiplied by 

 the direct and return currents respectively, cancel each other, and 

 so do the fifth and seventh. The fourth and sixth are equal to 

 each other, and their sum is 



- 2/jLlJ 2 log r 12 = 2/*/ 2 log r u , 



