96 Prof. L. Boltzmann on some Questions 



Strictly speaking, those molecules ought to be excluded 

 from the integration which during the time Bt come into 

 collision with each other. But since the number of these 

 molecules is infinitely small, like Bt, and the integral moreover 

 is multiplied by 8t, there result thus only terms of the order 

 of magnitude (Bt) 2 , which may be neglected. 



(3) If %$ suffers a change in consequence of the impacts 

 which occur during the time Bt, this change also, which 

 we will denote by &£<f>, must be taken into account. I 

 proceed exactly as Lorentz does in the paper referred to. 

 We choose from all the impacts which occur in the paralelle- 

 piped dx dy dz during the time Bt only those for which, before 

 impact, the velocity-components of one of the impinging 

 molecules lie between the limits (2), whilst the velocity of the 

 centre of gravity of the impinging molecules lies between the 

 limits 



u and u + du, v and v + dv, iv and w + dw; . (10) 



and, lastly, the direction of the line of centres of both molecules 

 for the impacts chosen at the moment of impact lies within 

 an infinitely small cone of given direction in space and of 

 aperture d\. If a denotes the diameter of the molecules, V 

 their relative velocity, and the acute angle of the directions 

 Y and C, then the number of selected impacts is 



dn= a 2 ff-^f cos, 6 do dw dp d\Bt ; . . . (11) 

 where 



f=A*', y, z , & V, £ 0, f x =f(x, y, z, u - ft v -v, w - £ 0, 

 dp = dudvdw (12) 



Since we have assumed the masses equal, the velocity- 

 components of the second of the impinging molecules before 

 impact differ infinitely little from u — £, v— 77, w — £. After 

 impact the velocity-components of the first of the impinging 

 molecules must lie between the limits 



randf + c/f, rf and 17^ + dfj , ) ? and (T +<*$*; • (13) 



those of the second molecule then differ infinitely little from 



W-f , V- 7)', w—g. 



The function <p for the two molecules before impact has 

 then the values 



4>(jCj y, z, £ V, K, and <$>(x, y, z, «-£, v-tj,w-^ t), 



for which we will write <£ and <j>i for the sake of brevity. But 



