222 SCIENCE PROGRESS. 



but becomes zero in the limiting case when -£ becomes 



infinite. 



35. We can show secondly that the value of the expec- 

 tation, though positive, is less than — . For if w, xf, be the 



vector velocities of the two spheres before collision, their 

 vector velocities after collision are in the notation of quater- 



'° + ^ 1 i_ • 1 • 1 r 



nions ± p, where p is a vector whose tensor is that 01 



(1) --» \P, and for which all directions are equally probable. 



Consequently the average direction for either is that of 



, (0 + 1^ . . 



the vector ; and since hw^ is on the average negative, 



the resolved part in direction w is on average less than -• 



36. We might call the original velocity w the parent, 

 and the two after-collision velocities the children, and so in 

 relation to the velocities acquired in subsequent collisions, 

 10 is the ancestor and the others descendants. And we 

 might say that the quality of the ancestor, viz., positive 

 velocity in x, survives to the descendants with intensity 

 diminished at each generation, but this is true only on aver- 

 age, because at each generation is introduced a large ele- 

 ment of chance in the vector p. If, therefore, sphere A 

 have positive velocity w in x, the chance that another sphere 

 B shall have positive velocity in that direction is increased 

 if it be given that (in the above notation) the velocities of 

 A and B have a common ancestor. Just as the expectation 

 that a man now at Cambridge may become a distinguished 

 mathematician would be increased if we knew that he and 

 Professor Tait had a common ancestor. 



$J. To apply this doctrine to our elastic spheres. Sup- 

 pose a number of them to be contained within a small ima- 

 ginary spherical surface S. If they be very closely packed, 

 so to speak, they will for the most part not escape into the 

 surrounding medium without first undergoing many colli- 

 sions with one another, and so without their velocities, to 

 continue my metaphorical language, becoming related to one 

 another. And their velocities will be no longer independent. 



