500 Mr. C. Gr. Darwin on Collision oj 



From these we derive : — 



u = 2M 



M 



cos 0, 



1 



M + w 



, m sin 20 



tan G>= ^ — — -^? 



r M — mcos 2<9 



!■ 



(2) 



and 



v = 



{M cos + v/(»i 2 - M 2 sin 2 </>) } , 



1 



tan#: 



M + m 



m cot (f>±\/ {m 2 cosec 2 cf> — M 2 ) 

 M + m 



h-(3) 



J 



For the present we shall leave the ambiguities undetermined ; 

 but the two upper and the two lower signs go together. 



To find the frequency for a collision of this type we must 

 use the orbit. Now, in the case where the nucleus is heavy, 

 Itutherford * showed that when the « particle approaches 

 aloDg a line at distance p from the nucleus, the deflexion 

 is 2 /jl where ^ | 



tan/i= ^T 2 I" 



The apsidal distance is 



ptan(|_|). 



By a well-known process in the theory of orbits, this can be 

 adapted to the more general case by replacing 1/M by 

 1/M + l/m. The result then applies to the relative orbit of 

 the a particle. 



Thus if *E / 1 1\ 



the relative deflexion is 2//,, and so 



rt v sin <f> + u sin 6 

 tan 2ll= ~ n , 



r vcoscp — ucosU 



which, by use of (1) and (2), can be reduced to 



Hence = ^ — fi. 



Thus 



P 



tan 



£E/1 JL\ 

 V 2 \M + W 



and the apsidal distance is 



tan 26. 



■ W 



■ (5) 



Rutherford , loc. cit. 



