1207 



finite), and not to infinite. Also the absurdness of the infinitely great 

 value for Pr at the end of the collision is quite obviated now. Of 

 course the distribution remains finite, and during the collision there 

 comes no change in this at all. (On the assumption of e-^^'' the 

 density would decrease during the interval of collision, thought 

 infinitely small, from n X ^^'^ to n X 0, which is nonsense, because 

 when a molecule has once come to collision, the number of them, 

 as has been said, does not change again during the impact). 



Thus we are naturally led to a new method of calculation, which 



— in opposition to the usual one, the static one — might be called 

 the dynamic one. What happens to the molecules that pass each 

 other and that collide, will then have to be considered separately 



— though much can be simplified also with this mode of viewing these 

 things, and much can be brought under a comprehensive point of 

 view. 



And thus we have again i-eturned from Boltzmann to Maxwell 

 with the consideration of the separate paths or groups of paths. 



XV. The fundamental Path Equations. 



As may be seen from the subjoined 

 figure, the curved path now gets nearer 

 to (in the point B, with the minimum 

 distance r„,) than the straight path 

 [ji = GO, i. e. 7'= qd) in the point 0' 

 with the minimum distance 00' = r/=: 

 = a sin 6. The angle, at which the 

 path of the centre of the moving 

 molecule under consideration ((> is the 

 stationary centre) comes within the 

 sphei-e of attraction at A with respect 

 to the joining line AO, is namely indicated by 6. All these angles 

 lie on the circumference of a cone with A as vertex. As the path 

 is still undisturbed at A, the frequency of the angle 6 is as usual 

 =1 sin 6 do. Later on we shall have to integrate for all possible 

 angles 6. When the centres of the molecules lie on the circumference 

 of the sphere ?■ := .s\ they collide. The limiting angle O^ is that for 

 which the path just touches the circle r = .v (in D). All the paths 

 that enter under a smaller angle with respect to AO, give rise to 

 collision. 



It is self-evident that this limiting angle 6*,, in the limiting case 

 7^ = cx) (rectilinear paths) is given b}^ sin & = s: a, as AD then 



