442 Dr. W. F. G. Svvanii on the Expression for 



facts, even in the matter of" the relation between the thermal 

 and electrical conductivity, as has formerly been supposed. 

 The method adopted by Lorentz, Richardson, and others, in 

 calculating the conductivity are based on quite different 

 principles to that adopted in calculating (1) and (2), the 

 persistence of the velocities playing, at any rate implicitly, a 

 primary part in the development of the theories. The value 



obtained by Lorentz for a is \ / -^— —*—, where u is here 



■the square root of the mean square velocity. The point of 



the present paper is not necessarily to suggest that (1) is far 



•wrong in itself, but that the assumptions on which it is based 



■do not lead to it but to (2). 



It is perhaps worth while here uttering a word of warning 



against any feeling which one may have towards assuming 



that since all methods of calculating the conductivity lead 



ne "X/u 

 to — — multiplied by some numerical factor, therefore the 



assumptions employed in these methods are all more or less 

 equivalent. If we set out on any hypothesis to calculate a 

 an terms of n, e, X, u, a#, we are practically bound to find 



/ ne~ l ~\,u 



—Q- multiplied by a numerical factor, since this is prac- 

 tically the only way in which these quantities can be combined 

 so as to result in the dimensions of a conductivity, unless we 

 make some most improbable combinations. When we set out 



nc A.?/ 

 to calculate a in terms of n, e, X, u, ad the ■•-— ^— part is little 



au 



more than the result of pure algebra, so that it would be 



.obtained if all the physics and all the dynamical principles 



in our arguments were wrong. The correctness of the 



numerical factor is in fact the only criterion for the truth of 



the theory. 



It will be convenient first to give the formal proof of (2), 

 and then to put the proof into such a form as to indicate the 

 exact points in which the discrepancy between (1) and (2) 

 has arisen. One or two mathematical details will be re- 

 legated to an appendix. 



In Part II. of the paper expressions are deduced for the 

 .electrical and thermal conductivities on the same assumptions 

 as those on which (2) is deduced, with the exception that the 

 -electrons are supposed to move in the absence of the field 

 with the velocities given by Maxwell's law. A method of 

 improving the theoretical value of k\cr by making the free 

 ;path a suitable function of the velocity is also discussed. 



