Problems connected ivith the Flow of Electricity in a Plane. 373 



Now it has been shown by Boltzmann that if, in such a gas, 

 the number of variables in each molecule be n, the total energy 

 T is to the energy of translation T x as n to 3, or 



T 3T 



We have therefore 



, 2T civ dW ' /n . 



pclv= -Y—dv. (9) 



* n v dv 7 v ' 



and 



^=rflogT+?dlog«-^^:^+g f nogT. . (10) 



Or, observing that T x and not T is here the measure of tempe- 

 rature, we may write 



^ = |dlogT 1+ |dlog,-^-^- / ^+gdlogT 1 , (11) 



the condition necessary to make -^ or -^ an exact differen- 

 tial remaining the same as in the case first examined, as ex- 



dW 

 pressed in equation (5), except that — — - dv is not the varia- 

 tion of the total potential upon the hypothesis of uniform ex- 

 pansion at any instant, but only such variation of the potential 

 of the forces between the molecules. 



XLY. On some Problems connected with the Floiv of Electricity 

 in a Plane. By Oliver J. Lodge, B.Sc* 



THE following paper may be regarded as a sort of appendix 

 to a paper by Professor Gr. C. Foster and myself, entitled 

 | On the Flow of Electricity in a uniform plane Conducting 

 Surface," and published in the Philosophical Magazine for May, 

 June, and December 1875 ([IV.] vol. xlix. pp. 385-400 and 

 453-471, and vol. 1. pp. 475-489), in which the known laws 

 of the flow of electricity in an unlimited plane were deduced 

 from the fundamental idea that the effects of any number of 

 poles could be obtained by simple summation of the effects due 

 to each pole separately, without using any mathematics higher 

 than elementary algebra. Several questions arose in the course 

 of the paper which it was not thought advisable to consider 

 then, but some of which seemed to deserve a separate treat- 



* Kead before the Physical Society on the 26th of February and the 

 25th of March, 1876. Communicated by the Society. 



