146 Mr Richardson, The Theory of the 



motion of the centre of gravity of the two particles is uniform and 

 uninfluenced by their mutual action. 



Let us consider two particles of mass m 2 and m 2 respectively, 

 and let us measure all the coordinates from their centre of gravity. 

 The distance between the two particles at any time t is p = p 2 + p 2 

 where m 1 p 1 = m 2 p 2 and p 1; p 2 are the respective distances of the 

 particles from the origin. Then if f be the angular velocity of 

 each particle about a straight line through the origin moving with 

 the uniform velocity of the centre of gravity of the system, the 

 equations of motion of the two particles are 



nhp" - WjpiT 2 = - F (p! + p a ) = m 2 p 2 - m 2 p 2 r 2 , 



where F(p x + p 2 ) is the attraction between the two molecules 

 when their centres are p — p 1 -\- p 2 cms. apart ; together with 



! (*■*) = 5 (^>=o. 



Integration of the last two equations gives 



if l-\ — a\ where /\ is twice the area swept out by the vector 



7)l 2 J 



p x per second. Moreover if V be the relative velocity of the two 

 molecules when they are so far apart as to be uninfluenced by 

 their mutual attractions, and b be the least perpendicular distance 

 between the straight paths they originally pursued, then 



7 W 

 a 2 



Substituting these values for f the first equations of motion 

 reduce to one, viz. 



m{p" - m l b 2 V 2 p- 3 = - aF{p). 



The integral of this is 



/•GO 



±p 2 = C-ib 2 V 2 p- 2 + ^- F{p)dp. 

 Further, when p = oo , C=^V 2 , 



/.QO 



so that p 2 = V 2 - b 2 V 2 p~ 2 + 2—1 F(p) dp. 



The smallest distance within which the molecules approach is 

 given by p = 0, 



ie.,by V 2 -b 2 p- 2 V 2 + ^ F( P )dp = 0. 



