in the Kinetic Theory of Gases. 91 



the whole, and is therefore independent of the time t, but not 

 that the ratio of this number to the total number of the 

 minority molecules is independent of x. I put here, side-by- 

 side with the dictum of De Morgan, this other, that it is just 

 in the calculation of probabilities that one must not be satisfied 

 with general statements, but must sharply prove each assump- 

 tion at the risk of becoming dull as the result of clear logical 

 developments. The bare expansibility of mathematical deve- 

 lopments can hardly be taken to be the test of their applicability 

 to Physics. 



But with Prof. Tait's proof of Maxwell's law of distribution 

 of velocities there falls also his proof of Avogadro's law, which 

 implies the firsu law ; or at least it becomes superfluous, since 

 in Maxwell's second and my proof of the first law, Avogadro's 

 law is, without this, proved at once. Another proof of 

 Maxwell's law of distribution of velocities, however, does not 

 yet exist. 



(If the gas were supposed to be in constant motion, then we 

 must understand by x the difference of the component velocities 

 of a molecule from the mean velocity of all the molecules ; 

 and the like must hold for y and z, in order that the distribu- 

 tion of velocity among the molecules of the minority may be 

 the same function of x, y } z as with a gas at rest.) 



§ 3. On the Mean Length of Path. 



The mean length of path of a gaseous molecule is most 

 naturally defined, according to Maxwell's proposal, as the 

 arithmetic mean of all the paths which all the molecules in the 

 unit volume describe between one impact and the next. Prof. 

 Tait, on the other hand, proposes (in the first paper of the 

 ' Edinburgh Transactions,' p. 74) to fix attention upon all the 

 molecules in the unit volume at a particular instant, and to 

 observe what path each of these molecules describes from the 

 given moment until it next comes into collision, and of all 

 these paths to take the arithmetic mean. In Maxwell's 

 method we take into the arithmetic mean so many paths of 

 each molecule as it makes impacts in the unit of time ; in 

 Tait's method, on the other hand, only a single path of each 

 molecule is counted. Since the swifter molecules come into 

 collision more frequently, and generally describe a longer path 

 from one impact to the next than the slower ones, in the first 

 method the longer paths are counted relatively oftener, and 

 therefore the mean must come oat greater than in the second. 

 In order to calculate the mean path according to the first 

 method, we observe all the impacts which each molecule 



