M8 Trof. Clausius on the Conduction of Heat by Gases. 



If we now suppose that, instead of being at rest, the other 

 molecules are all moving with a common velocity in a given 

 direction, the likelihood of the molecule in question striking 

 another during the time dt will plainly be represented by the 

 same formula, if we substitute, for the absolute velocity, v } of this 

 molecule, its velocity relatively to the other molecules. Let V 

 be the common velocity of the other molecules, $ the angle 

 which the direction of their motion makes with the direction 

 of the molecule under consideration, and R the relative velo- 

 city; then 



R = •V a +t>*-2Vi>cos& . . . . (29) 

 and with this value we can put 



= 7rp 2 NR (30) 



Einally, let us suppose that the other molecules move, not all 

 in the same direction, but in various directions, and with velo- 

 cities which are not necessarily equal to each other; in this case 

 the velocities of our molecule, relatively to the several other 

 molecules, will be various, and we must use in the equation the 

 mean value of the relative velocities. Denoting this mean value 



by R*, the equation for a becomes 



0=7rp 2 Nl, (31) 



and thence we obtain as a result of (27), the following equation 

 for a : — 



«=^ 2 Nj. ...... (32) 



1 § 18i We have now 'to determine the mean relative velocity of 

 a given molecule moving in our stratum, as compared with all 

 the other molecules simultaneously existing therein. 



The velocity of the given molecule relatively to another given 

 molecule, whose direction forms the angle <£ with its own direc- 

 tion, and whose velocity is V, is determined by equation (29). 

 If we now consider all the molecules which are moving in the 

 same direction, their velocities, as we have already seen in § 8, 

 are not, exactly equal to each other; and hence the velocities of 

 the given molecule relatively to them are also somewhat unequal. 

 We will accordingly, in the first place, introduce a mean relative 

 velocity, denoted by R, for each separate direction. 



In order to be able to present in a tangible form the conside- 

 rations which further regard the various directions in which the 

 movements take place, we will, as before, imagine a spherical 

 surface described with the radius 1, and regard the various direc- 



* The reason for putting here two horizontal strokes over the letter R, 

 instead of only one, as in former cases, will become evident immediately. 



