of Gases and Molecular Force. 511 



force in increasing the number of collisions is fundamental; 

 and we will proceed to determine the number of collisions 

 that occur amongst the molecules of a gas when molecular 

 force is in operation, in comparison with the number when 

 there is no molecular force. Let b be the perpendicular 

 distance from the centre of molecule C to AB the asymptote 

 to the relative path of the molecule D, both molecules being 

 supposed to be spheres of radius a ; let V be the relative 

 velocity of D when it is so far from as to be moving almost 

 along the asymptote, then with the usual notation for orbits 

 under central forces h = bY, where h is twice the area described 

 in unit time by the vector CD denoted by r, and 1/r being 

 denoted by u. 



Let ?n 2 J?(u) be the molecular attraction, and m 2 f(u) be the 

 mutual potential energy of two molecules of mass m at dis- 

 tance r apart, then the usual differential equation of the 

 orbit is 



d 2 u m¥ (u) 



with its first integral the equation of energy, 



{(S/ +m2 } =i»*= m /M + i y2 > 



v being the velocity at any reciprocal-distance u. 



Now when this orbit is such that there is no collision, we 

 can determine the nearest distance to which the molecules 

 approach one another (an apsidal distance) by the condition 

 du/dO=0; denote the reciprocal of this distance by w, it is 

 then given by 



ihV = mf(w)+iY*, 

 or 



mflvj) - ±b 2 Y 2 w 2 + 1 V 2 = 0. 



Now there will be a collision if 1/w is less than 2a, that is, 

 if iv is greater than l/2a ; hence the greatest value of b for 

 which a collision is possible is given by 



mf(l/2a)-ib*Y 2 /(2ay + iV* = * 

 y = (2a) ^ 1+ 2m/(l/2a)j [, . . (1) 



and there is a collision for every value of b from up to that 

 given by the last equation. Testing this assertion by applying 

 it to the case when there is no molecular force, we put 

 mf(l/'2a)=0j and find that there is a collision for all value? of 

 b from up to 2a, which is correct. Hence, molecular force 



rA 2 



