456 Induction of Currents in Linear Circuits [CH. xiv 



Expression (454) can be transformed into 



so that when LN M* is neglected the energy becomes 



(Lii + Mi^f, 

 and this vanishes for the special case in which the currents are in the ratio 



-^ = =- . This enables us to find the geometrical meaning of the relation 

 ^2 L 



LN M 2 = 0. For since the energy of the currents, as in 501, is 



we see that this energy can only vanish if the magnetic force vanishes at 

 every point. This requires that the equivalent magnetic shells must coincide 

 and be of strengths which are equal and opposite. Thus the two circuits 

 must coincide geometrically. The number of turns of wire in the circuits 

 may of course be different : if we have r turns in the primary and s in the 

 secondary, we must have 



JL_M_r 



M~~ N~ s y 



and when the currents are such as to give a field of zero energy, each fraction 



is equal to ~ . 

 *i 



526. Let us next examine the modifications introduced into the analysis 

 by the neglect of LN M* in problems in which the value of this quantity is 

 small. We have the general equations ( 518), 



Ei-j^ + Mi^Rii ..................... (455), 



E 2 -j t (Mi l + Ni 2 ) = Si 2 ..................... (456). 



If we multiply equation (455) by M and equation (456) by L and sub- 

 tract, we obtain 



ME.-LE^EMi.-SLi, ..................... (457), 



an equation which contains no differentials. 



527. "To illustrate, let us consider the sudden making of one circuit, 

 discussed in the general case in 519. The general equations there obtained, 

 namely 



2 = 0, 



