252 Prof. F. Y. Edgeworth on the Application of 



the ?/'s take an #and any y at random. These considerations 

 may assist in interpreting the proposition that " the law of 

 distribution of velocities .... remains the same right up to 

 the extreme limiting case in which the spheres are packed 

 so tightly in the containing vessel that they cannot move " 

 (Jeans, ' Dynamical Theory of Gases,' §57, 3rd ed.). The 

 law of distribution does not hold true of contiguous molecules 

 in the same sense as when the independence of the velocities 

 is postulated. The proposition may be equally true, but it 

 is not equally informing as that which is based on the 

 postulate of independence. 



III. On the Lines of Maxwell. 



The third argument is based on the incidents of collisions 

 and encounters without the aid of " H" It is now attempted 

 to prove by this argument that the normal distribution is 

 necessary (as well as sufficient; I. c. p. 256). The attempt 

 is discouraged by high authorities (Boltzmann, ' Gastheorie,' 

 i. § 5 ; Watson, 'Kinetic Theory of Gases/ § 14) ; but it is 

 countenanced by Maxwell when he argues that the normal 

 distribution of velocities is not only "a possible form/' but 

 " the only form" (" Dynamical Theory of Gases," l Scientific 

 Papers/ vol. ii. p. 45). The proof which he gives in that 

 context is indeed very different from that offered here; 

 which is, rather, akin to the reasoning in Maxwell's paper 

 "On the final state of a system of molecules'" (loc. cit. 

 p. 351). 



1. Proof that the normal distribution is necessary to 

 stability. — Let us begin with the simple case of disks moving 

 in a plane (as above, I. 1) : two sets of disks of mass M 

 and m respectively, and radius at first supposed the same for 

 both sets, say R. Let N be the number of disks of each 

 type in a unit of area ; the unit being taken so that N is 

 large. The total area occupied by disks is 2N7rR 2 . Call 

 the ratio of this to the unit of area p, supposed a small 

 fraction, say 1/1000 or less. On a view of the medley at 

 any instant the number of disks which may be expected in 

 an assigned area A is pAir/4:. If a small area, say a multiple 

 much less than a thousand of U 2 , includes a disk of one set, 

 the frequencV with which (at the moment of inspection 

 within a unit area) it will include a disk of the other set is 

 of the order N/o. Likewise the probability that a disk taken 

 at random should have in' close proximity two disks (of 

 assigned sets) is of the order ~Np' 2 . These propositions are 



