358 MATHEMATICAL APPENDICES 2* 



the total number which after a shorter or longer time reach the 

 mark, we have to integrate with respect to dt from t = to = oo. 

 Hence 



dy dz du dv dwFj(*d*NF(u % v, iu)f(t, u, v, w) 



is the required number of particles which pass through the 

 element dy dz in unit time with the velocity whose components 

 are u, v, w. 



3*. Keaction 



This formula, however, only gives the number of particles 

 which cross the surface-element from one-half of the medium to 

 the other in the positive direction of x, that is, with a velocity 

 such that its component u is positive. 



If in like manner we obtain the number of the particles which 

 pass in the reverse direction from the second half of the medium 

 to the first, the value of the relative coordinate r, which must be 

 positive since for this motion the component u is negative, must 

 satisfy the condition 



4 x 4 ut. 



Therefore the number of particles crossing in this opposite direc- 

 tion is 



dy dz du dv dw f ^ [ dfNF(u, v, w)f(t, u, v, w) 



JO t Jut 



= dy dz du dv dw [ [**dfNF(u t v, w)f(t, u, v, w). 



Jo i Jo 



In form this formula is distinguished from the other only by its 

 sign, but if the functions N,F,f depend not only on the given 

 variables t, u, v, w, but on the position as well, it may also differ 

 by reason of a difference in the signification of these functions. 



4*. Summation Carried Out 



The integrations can be immediately effected in the case 

 wherein these functions do not depend on position, but when the 

 state of the motion is the same at every point of the gaseous 

 medium. With this assumption we obtain, by aid of a theorem 

 given in 1*, 



f dt f ut 



+ dy dz du dv dw NF(u t v,w)\ - f(t, u, v, w) \ df 



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



