744 BELL SYSTEM TECHNICAL JOURNAL 



by a, as before, and the rate at which heat is generated per unit 

 volume by r; then: 



- r = (ma/e)J - dW/dx. (138) 



The current-density of heat is given by the formula (115), which I 

 repeat : 



W = hm Cv'cos^ dgdr. (139) 



Here g stands as always for the non-isotropic perturbation-term in 

 the distribution-function. This and the acceleration a are to be 

 determined from the two equations, 



a{dUldk) + kidUldx) = (v^/l)g = (v' cos d/l)g, (140) 



j/e = f ^gdr = fv^ cos2 dgdr, (141) 



which are the same as (117) and (118) except that the electric current 

 is no longer set equal to zero. 



Multiply both sides of (140) by ^mv- cos d, and integrate over the 

 entire velocity-space. The integral of the right-hand member is 

 W/l; developing the integral of the left-hand member, we find: 



W = ^ml{- 4anv + d{nv'')/dx). (142) 



Multiply both sides of (140) by cos d, and integrate over the entire 

 velocity-space; the integral of the right-hand member is J/le; de- 

 veloping the integral of the left-hand member, we get the equation 

 of which (127) was a special case, to wit: 



- 2amr' + d(nv)/dx = ZJiel. (143) 



Evidently these equations suffice to translate (138) into an expression 

 for r in terms of the mean free path, the universal constants, and 

 the averages of various powers of v. 



Postulating the Maxwell-Boltzmann distribution-function for /o; 

 working out the expressions for a and for dWIdx, and importing the 

 formulae for a (equation 114a) and k (equation 119), one finds: 



1 k dT J2 



(144) 



d I dT\ kdT 



dW dx =-r-K-7- ] -\- zJ --j— , 

 dx \ dx / e dx 



