74 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Let I = height above the earth's surface, 

 p = density of the gas, 

 p = pressure " " 

 g — gravity acceleration, 

 R'= gas constant for unit mass, 

 T = absolute temperature. 



dp p 



Then — ln = 9P = ^Wt> whence 



— = - Wj,^ and so log p = - ^,1 + log k, 



__£*_ 

 or p = ke R ' T , where k = pressure at the earth's surface. 



Then for any two points A and B, with pressures 2>i and p%. respec- 

 tively, and molecular concentrations n\ and n 2 respectively, we have 



^ — — = e R'T = e R'T 



Pi n\ 



In this equation W is the work of lifting unit mass from h to ^ against 

 the pull of gravity. If we wish to deal with a single molecule of mass 

 m, we can write 



Z* = — = e R'mT = e RT 



Pi fh 

 which is the Boltzman equation as used by Richardson. 



Thermoelectric Equilibrium in a Detached Wire 

 (with consideration of (B) electrons only). 



In dealing with the free electrons in a metal unequally heated 

 we have a case somewhat like the one just discussed but considerably 

 more complicated. In place of g we must now put// the resultant of 

 attractions and repulsions per gram of free electrons, this resultant 

 being called positive when it is directed along the path of diminishing 

 I, and we must consider /' as a variable. In place of R' for a gram of 

 gas we must put R' for a gram of free electrons, and take R' as a varia- 

 ble according to equation (1). Moreover T is now to be taken as a 

 variable along /. We shall Avrite 



/? = dT -r- dl, 



and shall treat /3 as a variable. 



