590 Mr. J. H. Jeans on the Conditions 



energy-function E of § 4 the analysis of § 4 must hold, and 

 the system moves only over a single stream-line, not over a 

 complete energy surface. Now suppose the field o£ force to 

 continuously change so that ultimately the potential is in- 

 finite over the lines a, b, c . . ., and is zero over the rest of 

 the plane. This ultimate state is an exact mathematical re- 

 presentation of the case in which the motion is disturbed by 

 rigid boundaries placed over the lines a, b, c . . . However 

 near the field of force may be to this ultimate state the 

 argument of § 4 must be admitted to be valid. Hence unless 

 we assume the ichole argument in some way to become invalid, 

 when we finally pass to the limit, it would seem that the 

 theorem cannot possibly be true for the case in question. I 

 cannot, for myself, see any reason for treating this limit as 

 an exceptional case, and Lord Kelvin's recent experiments* 

 seem to bear out this view. 



§ 7. The same argument will, I think, apply to any case 

 in which the motion is determined for all time by the state 

 of the system at a given instant. For example, it applies if 

 we try to replace our typical system by amass of gas whether 

 inclosed within rigid boundaries or not. When, however, 

 the subsequent career of the system is in some way fortuitous 

 the objection does not hold, and this class of exceptions in- 

 cludes the important case in which the systems are molecules 

 of a gas, in which the disturbance of the path arises from 

 fortuitous collisions with other molecules. 



Application to Molecules of a Gas. 



§ 8. Let us now suppose the exactly similar dynamical 

 systems of § 2 to be the molecules of a gas. 



Suppose that each molecule is surrounded by an imaginary 

 sphere, and let it be supposed that these spheres are of such 

 a radius that two molecules exert no action upon one another 

 except when their spheres intersect. When two such spheres 

 intersect an " encounter " is said to take place, lasting until 

 the spheres again become clear of one another. 



Binary Encounters. 



§ 9. We shall begin by considering binary encounters 

 only ; that is to say, we assume that the event of a sphere 

 being simultaneously intersected by two other spheres is so 

 rare that it may be neglected. 



We treat this case as follows : — As soon as an encounter 



* Kelvin, Phil. Mag. [6] ii. p. 1. 



