734 BELL SYSTEM TECHNICAL JOURNAL 



Since for electrons crossing in opposite senses ^ is of opposite signs, 

 the integral of this expression over all values of ^, multiplied by e, 

 gives the net current through da when the proper distribution-function 

 is inserted. Evidently the integral will vanish when / is isotropic; 

 there are enormous flows of charge both ways through da, but they 

 are balanced. Unbalance is brought about by the non-isotropic 

 perturbation-term in the distribution-function. Making the postulate 

 of Lorentz, we obtain for the net current through unit area perpen- 

 dicular to the field, the current-density /, the expression: 



J = e ( tmd^ = e r r r d^dr^d^-egiv). cno) 



t/ -J — tx «/ — 00 t/— 00 



Set for g{v) the expression in (217), and to effect the integration 

 transform into polar coordinates in the velocity-space, orienting the 

 polar axis along the axis of ^; integrating over the angles, one obtains: 



/ = Y' \ "i^' ~1~ dv. (Ill) 



m 6 1^ dv ^ ^ 



The final integration is easy if one chooses for/o the Maxwell function, 

 not quite so easy if one chooses that of Fermi. A further step will be 

 of some advantage. Integrating by parts, and noticing that v vanishes 

 at the lower and/o at the upper limit, we find: 



/ = - (87r/e2£/3m) f^vdv (112) 



and the integral remaining, divided by n the number of electrons per 

 unit volume, is seen to be l/47r times the mean value of ^'~^ — the 

 average of the reciprocal of the speed of the electrons, in the absence 

 of the field. Denoting this by v~^, we may write as the general 

 formula for conductivity: 



,=^ = \^fll-K (113) 



E 3 m 



The analogy with (91) is obvious, but we must not be misled into 

 identifying the average of the reciprocal speed with the reciprocal of 

 the average speed; they are not quite equal. 



The actual final formulae obtained by the old and the new statistics 

 — substituting, that is to say, the appropriate values of v~'^ from (72b) 

 and (72a), and putting G = 2 — are as follows: 



^ ^'^"^ (old), (114a) 



3 {IwrnkT}"-' 

 3 h^ 



{^^J" (new). (114b) 



