THE DYNAMICAL THEORY OF GASES. 75 



sions in 77, , whose values will depend on the distribution of velocity among 

 the molecules. If the distribution of velocity is that which we have proved to 

 exist when the system has no external force acting on it and has arrived at 

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



and the equation of conduction may be written 



(144), 

 (145), 

 (146); 



[Addition made December 17, 1866.] 



[Final Equilibrium of Temperature.~\ 



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

 tained a term 2(/3 1)--^-, the result of which was to indicate that a column 



' p dx 



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, shews 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 temperature throughout. 



When I first attempted this investigation I overlooked the fact that 4 is 

 not the same as f . f ", 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, 



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



102 



