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



could not have passed from the section (U, V ; u, v)s, unless 

 daring r a new positive couple of that variety had been 

 created. But this could only happen through a very 

 improbable event ; not only must there be initially (within 

 the unit area) a disk belonging to the same class as one of 

 the specified variety — say of mass M with velocities U, Y — 

 with two disks in the neighbourhood, one of them of the m 

 type, but also the position and velocities of those second and 

 third disks must be such that they shall collide with each 

 other during the time t and one of them of the m type 

 should acquire the velocities u, v and the orientation with 

 respect to the first disk which the specified variety connotes. 

 By parity of reasoning the whole initial content of the 

 reciprocal section (u, v\ U, Y)s will have passed out of that 

 section during the time t; and the final content of the 

 reciprocal section will consist of the disks comprised initially 

 in the section (U, Y ; u,v)s. But on the hypothesis of 

 stability the fin;il content of the reciprocal is equal to its 

 initial content. Thus initially and constantly the content of 

 (TJ, Y ; u, v)s is equal to that of (u', v' ; U', Y')s. Let the 

 relative velocity of the partners in a positive section be w ; 

 and accordingly in the reciprocal negative, — w. Let the 

 breadth of a section be a = A.scos#, where 6 is the angle 

 made by the line of relative velocity with the line joining 

 the centres at the moment of impact. Then, if at first the 

 distributions F and / are supposed independent, we have, 

 equating the content of the section to that of its reciprocal, 



ivrct AU A Y Au Av F (U, Y) f(u, v) 

 =WT*ATJ'AY , Au r Av t F{W,Y ! )f(u',v'). 



The differential factors, the product of small finite dif- 

 ferences on the two sides, being equal, by elementary 

 dynamics (I. c. pp. 257-267), we obtain the equation 



F(U,V)./(«,c) = F(U',V) f(n',v'); 



subject to the condition 



M(U' 2 + Y' 2 ) + m(u' 2 + v' 2 ) = M(U 2 + Y 2 ) -4- m (it 2 + v 2 ). 



The well-known solution of this functional equation is 



F(U,V) = Const, exp -\M(U 2 + Y 2 )~ 

 f(u, v) — Const, exp — \m(u 2 + v 2 ) 



where X- is a constant to be determined from the given 

 mean energy of the system. 



But if the independence of F and /is not assumed in the 



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