the Inertia of Electric Convection. 229 



The well-known ambiguity, as to the expression under the 

 integral sign, disappears in our case, because we are only com- 

 paring the magnetic energy on two suppositions, and must 

 therefore consistently use in both cases the same expression 

 for the mutual energy of two current elements. As 13 is 



dr 

 laro-e, the value of i— for small values of s l will be sensibly 

 to 7 ds± 



equal to =P 1 according as s 2 is large on the positive or negative 



side, and consequently we find the above expression to be 



equal to 



,/ , C D 2ds, \ 



or 



2d Sl Qog(D+ Vr 2 + D 2 j-logr -i.). . . (2) 



We may substitute pd for D, and the above expression then 

 becomes, neglecting r Q compared with D, 



&1 ( 2 .o g ^-l). 



The bracket being independent of s x we conclude that the 

 central portions of the column have a magnetic energy which 

 for a current C and unit length is 



C 2 (2 log p + 2 log ~ - 1 ) = C 2 (2 log p + 1-996) . 



The excess of (1) over this will be the required difference in 

 the magnetic energy. Introducing the numerical value for B 

 this difference becomes 



Ki- 842 ) (3 > 



This expression being proved for any portion of the circuit 

 which can be considered straight for a length which is large 

 compared with the distance between the electrons, may be taken 

 to hold for the complete circuit, as, excluding sharp angles, 

 every circuit may be divided into portions satisfying this 

 condition. 



4. Some additional explanation is necessary for ordinary 

 conductors whose cross-section is many times larger than the 

 distance d between the electrons. The cross-s?ction of such 

 conductors may be divided into square elements, each square 

 having sides equal to d. If we imagine the electrons to 

 be placed at the centre of each square we may calculate the 

 mutual energy of any two parallel columns and thus obtain 

 further correcting terms. It is easily seen that these terms 



