108 Prof. R. Clausius on the Theorem of the Mean Ergal, 



which satisfies the requisite condition for the applicability of the 

 theorem of the mean ergal. 



To facilitate contemplation, we will not immediately treat of 

 .the molecules in confused motion and striking against one an- 

 other, but start from a simpler case. Given an infinitely ex- 

 tended space, furnished, in the manner spoken of in the last 

 section, with stationary molecules, between which very numerous 

 material points are circulating without affecting each other in 

 their motions, but rebounding from the spheres of action of the 

 molecules as often as they touch them. 



The force suffered by a point from the action of the rebound 

 we will conceive as proceeding from the surfaces of the spheres 

 of action, and assume that only in the immediate vicinity of 

 those surfaces does it become sensible, but then with still greater 

 nearness increases very rapidly. In this case, as long as the 

 point is at a sensible distance from all spheres of action, we may 

 regard its vis viva as constant, and put its share in the ergal 

 (which we will briefly call its ergal, since the motions of the 

 individual points are independent of one another) equal to nil. 

 But when the point comes close to the surface of a sphere of 

 action, then the part of its vis viva referred to the direction 

 normal to the surface very quickly diminishes to zero, and then, 

 after the reversal of the normal component of its motion, just as 

 quickly increases up to its original value. At the same time its 

 ergal increases from zero by just the same amount && the vis viva 

 diminishes, and then again sinks to zero. From this it follows 

 that the mean ergal of each point must be very small in propor- 

 tion to its mean vis viva, because only during a very short time 

 has the ergal specifiable values — and, further, that with a given 

 vis viva the mean ergal of a point must be proportional to the 

 number of collisions it suffers during unit time. 



For the number of collisions of one point, we have equation 

 (70), namely 



p/ __ N47r/3 2 t; 

 r - 4(V-N|7r/) 3 )- 



Let N' denote the number of moving points which are found 

 simultaneously in a volume of magnitude V ; and let us prelimi- 

 narily assume that they all move with equal velocity v ; then the 

 total number of collisions which take place within the volume V 

 during a unit of time will be represented by the product 



N'P'-N' N47r ^ - 

 4(V-Nf7rp 3 ) 



The same number holds also for another case. Given a vessel 

 in which a moving point only rebounds from the sides. In this 



