426 MATHEMATICAL APPENDICES 35* 



in unit time by the radius vector p, and we determine its value by 

 consideration of a point on the orbit so distant from the origin 

 that the attraction F(p) may be taken as vanishingly small ; the 

 velocity is then constant, being that of the relative velocity r with 

 which the particles began to approach each other in straight 

 paths, and the surface h is then equal to the product rb, where b 

 is the length of the perpendicular from the fixed particle on the 

 rectilineal part of the path of the moving particle. 

 The angular velocity is therefore 



f = brp~ 2 , 



and the first differential equation, on introduction of this value, 

 becomes 



p = &VV 3 - F( f >), 



which on integration gives 



The constant C may be determined by application of the equation 

 at an infinitely great distance p where the total velocity r is given 



by the formula 



r 2 = p 2 + ,o 2 f 2 , 

 while 



pf = h/p = 



from a former equation. We thus finally obtain 



dpF( P ). 



The shortest distance to which the particles approach each other 

 is determined by the vanishing of p, and thus by the equation 



= r 2 - 6VV 2 + 



A collision ensues if this distance is less than the radius s of the 

 sphere of action, and this occurs if the perpendicular distance b 

 which satisfies the equation 



is less than a limiting value, which we may put as 



d P F(p)} ; 



since for every value of p which falls within the sphere of action, 

 and is therefore less than s, we may assume that the function F(p) 

 is equal to 0, as this small distance is never reached. 



