356 MATHEMATICAL APPENDICES 1 



where F denotes a kind of probability-function, with the deter- 

 mination of which we shall proceed in Appendix II. 10*-15*. 



Just as the values of the velocity for different particles are in 

 general different, so also will the times be different during which 

 the molecules move in the same direction with unchanged speed. 

 Let the length of this time that is, of the interval between two 

 successive collisions be t, and let the probability of a molecule's 

 moving for the time t in a straight line with unchanged velocity, 

 whose components are w, v, lu, and then colliding in the following 

 infinitesimal interval dt be 



f(t,u,v,w)dt. 



This function /, like the other F, I will not here more closely 

 examine, as an obvious property of it is sufficient for our present 

 purpose. Since every path must certainly come to an end, the 

 sum of all the probabilities must be certainty, or 



With this notation, therefore, the number of molecules in unit 

 volume, which move with the velocities u, v, 10, but only for the 

 time t without collision, is 



NF(u, v, w)f(t, u, v, iu}dt du dv dw. 



At the end of time t this group begins a new path in a new 

 direction with a new velocity. 



But if we assume that the state of the gas does not change 

 with the time, then for every molecule that at any moment loses 

 the velocity-components u, v, w another molecule must at the 

 same moment gain these velocities as the result of collision. After 

 the interval t, therefore, the number of molecules above given 

 move again out of the unit volume under the same circumstances 

 and with the same probability of collision. Thus the number of 

 paths which in the unit of time are newly begun by the N 

 molecules in unit volume with velocities u, v, w, and which are 

 ended after time t, is 



NF(u, v, w)f(t, u, v, w) d --du dv dw. 

 t 



2*. Summation 



We use this formula to determine the number of particles 

 which in a unit of time cross any surface-element within the gas 

 with a given speed in a given direction. 



