339-342] Collisions with Law of Inverse Fifth Power 285 



and equation (666) becomes 



/ I \ 



(Wj + m. 2 ) ^ . 

 Thus the equations of motion can be written in the form 



* = -g^ etc (667), 



where 



<' = ^~p 3> = - (raj + ra 2 ) (668). 



341. With the centre of the first molecule as origin, let us take new 

 coordinates r, 9 in the plane of motion of the second. The motion of the 

 second particle in this plane is, by equations (667), an orbit described by 

 a particle of unit mass under the influence of a central force of potential <E>'. 



Hence we have the usual two first integrals 



r^ = h (669), 



I(r 2 + r 2 2 ) =<' + <? (670), 



where C and h are constants. Eliminating the time we obtain as the equation 



of the orbit, 



.. (Y9r\ 2 ) h 2 ,, ~ 



1 J I ] -1_ r 2 I __ = <f>' _L (7 



t Had/ ^ f (^4^^^' 



2 



If we take the direction of one asymptote for initial line this has the 

 integral 



r (j r 



ar ............... (672). 



2(7 2 



342. If V is the velocity at infinity (i.e. the relative velocity before 

 collision), and p the perpendicular from the centre on to the asymptote 

 described with velocity V, we have from equations (669) and (670), 



h=pV, G' = iF 2 (673), 



so that equation (672) can be written in the form 



-(' * 



If we now write t] for , this becomes 



Vr . d " .... -(674), 



