360 MATHEMATICAL APPENDICES 5* 



the collisions do not come into account in the state of equilibrium, 

 we have to deal only with the force which the particles that pass 

 backwards and forwards over the surface exert in virtue of their 

 own motion, and transmit from one half of the medium to the 

 other. What becomes of this motion after the passage from the 

 one half to the other whether it is destroyed or transformed is 

 of no moment, since, in the condition of equilibrium, for every 

 particle that loses its velocity another appears which acquires the 

 same motion by collision. 



We therefore obtain the momentum carried across the surface 

 dy dz in the unit of time in the positive direction of x from the 

 first half of the medium into the second by multiplying the above- 

 found number of particles which in unit of time cross the surface 

 with the velocities u, v, w, viz. 



NF(u, v, w)u du dv dw dy dz, 



into the components of the momentum of each of these particles, 

 which are 



mu, mv, mw, 



and then integrating the expressions so obtained. The summations 

 are to be carried out with respect to the possible values of the 

 velocity-components u, v, w, and their limits are determined by 

 noting that the passage from the first half of the medium to the 

 second in the positive direction of x can only occur with such 

 velocities as have a positive component u. Consequently, the 

 momenta carried over dy dz in unit of time in the positive 

 direction are 



|>00 pOO pOO 



dy dz Nm \ du\ dv\ dw u 2 F(u, v, ID) , 



J o J o J 



pOO pOO pCO 



dydzNm] dtij dv] dwuvF(u,v,w), 



poo poo poo 



dy dz Nm du dv\ dw uwFfu, v, w) . 



JQ J co J _ oo 



For the particles which pass from the first half to the second, 

 and for the momentum carried with them, the first half is compen- 

 sated by the passage that occurs in the opposite direction into it 

 from the second half. In this direction the passage can occur only 

 with negative values of u ; and the number of particles which 

 pass in this direction with velocity-components u, v, w being 



NF(u, v, w)u dii dv dw dy dz, 



