ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 387 



and if v l be the actual velocity, then the number will be v^a. ; therefore 



where v 1 is the actual velocity of the striking particle, and v its velocity 

 relatively to those it strikes. If v t be the actual velocity of the other particles, 

 then v = -Jv? + v?. If v l = v t , then v = 72 v l , and 



Note*. M. Clausius makes a = %irs 1 N. 



PROP. XL In a mixture of particles of two different kinds, to find the 

 mean path of each particle. / 



Let there be JV, of the first, and N t of the second in unit of volume. 

 Let s, be the distance of centres for a collision between two particles of the 

 first set, 5, for the second set, and s for collision between one of each kind. 

 Let v 1 and v t be the coefficients of velocity, M w M t the mass of each particle. 



The probability of a particle J/ t not being struck till after reaching a 

 distance x l by another particle of the same kind is 



* [In the Philosophical Magazine of 1860, Vol. I. pp. 434 6 Clausius explains the method by 

 which he found hia value of the mean relative velocity. It is briefly as follows : If u, v be the 

 velocities of two particles their relative velocity is /w* + if 2uv cos 6 and the mean of this as 

 regards direction only, all directions of v being equally probable, is shewn to be 



lu' IS 



v + 3 when u <= v. and u + = when u > v. 

 3 v 3 u 



If v-u these expressions coincide. Clausius in applying this result and putting u, v for the 

 mean velocities assumes that the mean relative velocity is given by expressions of the same form, 

 BO that when the mean velocities are each equal to u the mean relative velocity would be ^u. 

 This step is, however, open to objection, and in fact if we take the expressions given above for the 

 mean velocity, treating u and v as the velocities of two particles which may have any values between 

 and oo , to calculate the mean relative velocity we should proceed as follows : Since the number of 



4 _2? 

 particles with velocities between w and u + du is N 3 / u'e ' du, the mean relative velocity is 



16 



rr (*+*)/ iv*\, 16 r / . ,(-%.*)/ IA , . 



/ wVe \a- >' I u + g I dudv + -^r- I I u'v'e V a V/ I + 5 I du dv. 

 Jo Jo \ 3 u ) a'/JV /o JO \ / 



This expression, when reduced, leads to -j= Jo. 1 + /3*, which is the result in the text. Ed.] 



492 



