Polarization and Magnetization Electrons. 407 



It will now be our business to find the first variation 8Nw a 

 and the second variation 5 2 Nw a . 



§3. The first variation of the stream. 



The following conception of the stream-components will 

 greatly facilitate the evaluation of the variation. We keep 

 our eyes on the content of a space-eleme'nt dV, situated at 

 the point ,i ,(1) , x {2 \ oP\ at the time <2 ,(4) . Though physically 

 infinitesimal the element is supposed to contain a great many 

 particles so that NrfV is a great number. In an interval dt 

 these particles will in a four-dimensional space-time exten- 

 sion describe their so-called world-lines, that will fill up an 

 infinitesimal extension dVdt. Now sum up the components 

 of these lines in the direction of X", say. We find obviously 

 'NdV dx a . Dividing by the four-dimensional extension dV dt 

 we can say that the stream-component in the direction ofX a is 

 the sum of the components described by the individual particles 

 -per unit of volume per unit of time : 



OT da" ^ ■ 



-dTdT^ wa ' 



We hardly need say that the fourth component represents 

 the number of particles per unit of volume. It is obvious 

 that these components will satisfy the condition of continuity: 



2 ^-^r=° < ■ W 



By the displacements the component of each individual 

 world-line will chancre to 



if we neglect 6 2 . 



The sum of the components will thus grow to 



NciV f da* + 2(6)0 1~ <&» } . 



On the other hand we must be aware that the extension 

 occupied by the world-lines has changed also. We can find 

 the increase with the aid of the functional determinant of the 



