92 Prof. L. Boltzmann on some Questions 



contained in the unit volume suffers during an entire second, 

 and note all the paths which each molecule describes between 

 two consecutive impacts, and take the arithmetic mean of all 

 these paths. If, according to Prof. Tait's notation, there are 

 n molecules in the unit volume, of which n . n v have a velo- 

 city lying between v and v + dv, so that 2??„ = 1 ; and if these 

 latter molecules come into collision with other molecules 

 N„ times in the second, and describe a mean path p v , so 

 that ~$$ v .p„ = v, then, since N„ gives at once how many 

 paths each of the n . n D molecules describes in the second, 

 the mean path, according to Maxwell's definition, is 



_ Xnn v N„ v _ 2n v v 

 1 _ %n?i v N„ ~ *Zn v : p v ' 



This is exactly the formula which Maxwell*, 0. E. Meyer f, 

 &c. use, and which gives X, 1 = 0"0707X,. Here \ is the mean 

 path which a molecule would have if all the others, without 

 alteration of magnitude and number, included iu the unit 

 volume were at rest. According to the method proposed by 

 Prof. Tait, we have to take of each of the n . n„ molecules, 

 not N„ members, but only one member, p v , in the arithmetical 

 mean ; and so we obtain 



Znn v r 



In accordance with what has been said, it will be seen that 

 Prof. Tait is decidedly in the wrong if he designates Maxwell's 

 definition of mean path as an erroneous one ; on the contrary, 

 it seems to me more natural than either of Prof. Tait's new 

 definitions. For those who care for numerical results, I may 

 remark that twenty years ago, in the course of similar inves- 

 tigations, which remained unpublished, I found 



= 0-650511, 



Jo xe-* + (2.v 2 + l)^e~'\lx 



i 



o xe- x,2 + (2x 2 + l)fe- x2 dx 



4x 5 e~* 2 dx 

 o xe +(2x 2 + l )fe~* 2 dx 



= 0-677464, 

 =0-838264, 



* Phil. Ma<?. ser. 4, vol. xix. (18G0). 



+ Theorie der Gase, Breslau, 1877, p. 294. 



