504 



Mr. C. Gr. Darwin on the Collisions of 



plate, so that if! H strikes this plate, it bounces off elastically. 

 The plate is perpendicular to the direction of the initial 

 motion of H. The orbit is again given by {$'2), where X 

 has the same meaning as before. If there is no impact, the 

 angle between the asymptotes is therefore 2\. An impact 



occurs if r<b when <£=^, that is if y <1 — cot A, or if we 



substitute for \ its value in terms of p and solve, when 



^V'-lf^W4 



(8-1) 



In this case it is easy to see that the second half of the 

 orbit is exactly what it would be without the impact, if H 

 had approached along the same line, but from the other end. 

 So the angle between the asymptotes is it— 2\. The relation 

 between p and 6 is therefore 



p = fjucot6 or j9 = /-6tan#, .... (S'2) 



according as p does or does not satisfy (8*1). 



10° A 20° 30° 40 J 50° 60° 70° 80° 90 c 



Collision Delation for Elastic Plate of radius b. The numbers 

 on the (p, 6) curves refer to the yalue of fx/b, that is are pro- 

 portional to 1/V 2 . The points of discontinuity are marked by 

 a cross stroke, where they are not obvious. The dotted curve, with 

 its firm line continuations, gives the relation between p and 6 for 

 fjL/b=0~2. 



If the incident velocity is so low that /a > &/4, there are no 

 collisions at all. If fi<bj4i we encounter the complication 

 that p is not a single-valued function of 6. The case where 

 fir=0'2b is shown by the dotted curve in fig. 7. In these 



