Mr. J. C. Maxwell on the Dynamical Theory of Gases. 215 



when the system has no external force acting on it, and has ar- 

 rived at its final state, we shall have, by equations (29), (31), (32), 



|5 = 3|1.|5_3^ ) (144) 



|V= P-?= jj> (145) 



g*?* = p .? = £; (146) 



r 



and the equation of conduction may be written 



[Addition made December 17, 1866.] 

 [Final Equilibrium of Temperature.] ' 

 [The left-hand side of equation (147), as sent to the Royal 



on dp 



Society, contained a term 2(/3 — l) L -j-, the result of which was 



to indicate that a column of air, when left to itself, would assume 

 a temperature varying with the height, and greater above than 

 below. The mistake arose from an error* in equation (143). 

 Equation (147), as now corrected, shows that the flow of heat 

 depends on the variation of temperature only, and not on the 

 direction of the variation of pressure. A vertical column would 

 therefore, when in thermal equilibrium, have the same tempera- 

 ture throughout. 



When I first attempted this investigation I overlooked the fact 

 that £ 4 is not the same as f 2 • £ 2 , and so obtained as a result 

 that the temperature diminishes as the height increases at a 

 greater rate than it does by expansion when air is carried up in 

 mass. This leads at once to a condition of instability, which is 

 inconsistent with the second law of thermodynamics. I wrote 

 to Professor Sir W. Thomson about this result, and the difficulty 

 I had met with, but presently discovered one of my mistakes, 

 and arrived at the conclusion that the temperature would increase 

 with the height. This does not lead to mechanical instability, 

 or to any self-acting currents of air, and I was in some degree 

 satisfied with it. But it is equally inconsistent with the second 

 law of thermodynamics. In fact, if the temperature of any sub- 

 stance, when in thermic equilibrium, is a function of the height, 

 that of any other substance must be the same function of the 

 height. For if not, let equal columns of the two substances be 

 * The last term on the left-hand side was not multiplied by /3. 



