Polarization and Magnetization Electrons. 409 



§ 4. The second variation. 



The second variation may be found without calculation by 

 a formal process. Indeed, we only have to substitute 8Nw a 

 for Nio a in formula (3) which gives 8Nw a to get : 



SB^iv^=X(b)0~ 1 \ r a S^sw^— r?>bl$w*~], 



8 2 mc« --= X (be) 6 2 3t [r* ~ \ r *Nw - r^ich X 



_ r bjL [ r a T$wc — r^w a 11.. (4) 



It is, however, very important to state that this formula 

 for the second variation implies the definition of the dis- 

 placements with an accuracy up to terms with 0' 2 as given in 

 formula (1). This can be verified by a direct calculation 

 following throughout the same line of argument as in the 

 preceding section, taking account of the terms of second 

 order everywhere. We shall not give the calculation at full 

 length, we may restrict ourselves to the indication that at 

 the last step, viz., in choosing the right starting-point from 

 where the displacement will carry us to the point-instant x a 

 under consideration, we have to be careful to take 



7 QX C 



and not x a —Ax a , as might be thought erroneously at first 

 sight. 



§ 5. The simultaneous displacements. 



As yet the displacements considered have been accompanied 

 by a shift in time. In view of the physical interpretation 

 of the formulae obtained, it will however be necessary to 

 realise the simultaneous positions of the electrons relative to 

 their nuclei. 



^ There is no objection to simplifying our formulae by drop- 

 ping 6 henceforth. Now, in a first approximation, we find 

 the electron belonging to the nucleus, which at the instant 

 x^ is in the point sP-\ x {2 \ x {z \ shifted to the point 



x^ + r il \ «»>+»#>, «<»>.+ 1*»; 



at the instant 



