86 
ME. CLEEK MAXWELL ON THE DYNAMICAL THEOEY OE GASES. 
and neglecting terms of the forms and g 3 and g?? 2 when not multiplied by the large 
coefficient Jc u we find by equations (75), (77), and (54), 
(143) 
f |(3(P+^+|,«)+/3 ± • f(l 4 +SV+?T)-(3(e+^+? 2 )| - w|l 
= -3VA#{»+^+5? 2 )- J 
The first term of this equation may be neglected, as the rate of conduction will rapidly 
establish itself. The second term contains quantities of four dimensions in g, jj, £, 
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), 
?=Sp.F=3*b (144) 
!¥=?■?= (145) 
i ( 146 ) 
and the equation of conduction may be written 
(147) 
[Addition made December 17, 1866.] 
[ Final Equilibrium of Temperature^] 
[The left-hand side of equation (147), as sent to the Royal Society, contained a term 
2(/3— 1) - jr, 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 temperature throughout. 
When I first attempted this investigation I overlooked the fact that f 4 is not the same 
as g 2 . g 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 thermo- 
dynamics. 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 conclu- 
sion that the temperature would increase with the height. This does not lead to mecha- 
* The last term on the left-hand side was not multiplied by j8. 
