560, 561] Rapidly alternating currents 489 



respectively, and arranged so as to have very little magnetic leakage, so 

 that LNM 2 is negligible (cf. 525). We then have approximately 



L_M_N 

 r' 2 rs s' 2 ' 

 and equations (524) become 



r s r 



so that the currents will flow in opposite directions, and either may be greater 

 than the current in the main circuit. By making s nearly equal to r and 

 keeping the magnetic leakage as small as possible, we can make both 

 currents large compared with the original current. But when s = r exactly, 

 we notice from equations (524) that the original current simply divides itself 

 equally between the two branches. 



Rapidly alternating currents. 



561. This last problem illustrates an important point in the general 

 theory of rapidly alternating currents. In the general equations (519), 



d T\ dT 



let us suppose that the whole system is oscillating with frequency p, which 

 is so great that it may be treated as infinite. We may assume that every 



variable is proportional to e ipt , and may accordingly replace ^- by the multi- 

 plier ip. The equations now become 



(520) ' 



and all the terms on the left hand may be neglected in comparison with the 

 first, which contains the factor ip. The terms on the right cannot legitimately 

 be neglected because X, //,, ... are entirely undetermined, and may be of the 

 same large order of magnitude as the terms retained. If we replace X, /JL, ... 

 by ip\' t ip/j,', ..., the equations become 



7\T 



-.- + X'a s + fib 8 + . . . = 0, etc. 



in which X', //, ... are now undetermined multipliers. These, however, are 

 exactly the equations which express that T is a maximum or a minimum 

 for values of x 1} x 2 , ... which are consistent with the relations (cf. 559) 

 necessary to satisfy Kirchhoff's first law. Since T can be made as large as 

 we please, the solution must clearly make T a minimum. 



Thus we have seen that 



As the frequency of a system of alternating currents becomes very 



