464 MATHEMATICAL APPENDICES 56* 



But we can satisfy the first proposition, viz. that the trans- 

 ference of heat is not carried out by a transport of heated masses, 

 only by the assumption of some relation between N, B, and k. 

 After carrying out some easy integrations, we obtain the equation 



Ndx ' dxi 



For the performance of this integration there are two methods 

 of approximation, which have been already employed in the theory 

 of viscosity, of which the one consists in our putting the constant 

 mean value of B, the collision-frequency, in the place of B, while 

 in the second we substitute the mean value of the free path 

 I = w/B instead of its actual value. The former method gives for 

 the integral a value too large ; the second, one too small. By both 

 we arrive at a relation between the differential coefficients which 

 is of the form 



Q _ !_ dN _ h dk, 

 N dx " k dx ' 



here h = 1 according to the former method, and according to the 

 latter h ^. 



The true value of h must lie between these. It would then be 

 practically sufficient if, without seeking to exactly evaluate the 

 integral, we assumed the mean value h = f , and eliminated the 

 differential coefficient of N from by means of the equation 



1 . dN = 3 dk 

 N dx 4& dx' 

 We should obtain 



(f k~ l - mw 2 ) ^ rcoss 

 dx 



But it is also possible to calculate the value of h with exact- 

 ness, if we are not afraid of the tedious work of calculating by 

 a mechanical quadrature the values of both terms of the integral 

 which is above put equal to zero, just as, indeed, the similar in- 

 tegral in 48* occurring in the theory of viscosity was treated. 

 This calculation, too, has been made by W. Conrau, who has 

 communicated to me his result, viz. : 



h = 0-71066, 

 which gives for the value 



j 1 q: (2-21066 k~ 1 - mw 2 ) rcos s j . 



