Fky — Impact in Three Dimensions. 77 



Similarly we get 



5; sin = ;S„ sin Q„ - hP - bQ - fK, (2) 



K =^ Ko-gP-fQ- cK, 



a 



1 



1 



A 



+ 



Hi- 



p 





+ 





^^ 



I'l 

 6' 



f 



A2A3 



^ P 



+ 





3 



- + 



A2 Xi 

 A' 



+ 



P' 



1 

 + - 



C 



' ^ 



where 



with corresponding values of h, c, g, h, so that it is easily seen that a, h, c, 

 he - /", ca - g^, ctb - h-, and A = abc + 2fgh - af- - hg^ - ch- are all 

 positive. 



3. During the course of the impact an instant ought to arrive when -ST, 

 the velocity of compression, vanishes. If Pi be the value of R at that 

 instant, it is an experimental law that the impact is over when P becomes 

 equal to (1 + e)Pi, where e is the coefficient of restitution. In two 

 dimensions K vanishes only once dui'ing the impact, but in three 

 dimensions I find that K may vanish once or thrice. This result no 

 doubt conflicts with our preconceived ideas about impact, but so do other 

 results which undoubtedly hold in two dimensions also, for instance : — S may 

 begin by increasing, and K also may begin by increasing. 



When K vanishes three times, and P does not attain the value (1 + e)Pi 

 while K is negative, that is between the first and second vanishing of K, then 

 we must take Pi to be the value of P when K vanishes for the third time, so 

 that /r shall be negative when P attains the value (1 + e)Pi. Between the 

 second and third vanishing of P, K is positive, and the impact could not 

 be over when the bodies are still compressing each other. 



4. When the bodies are perfectly smooth, P = 0, § = during the 

 impact, so that Pi = K^lc, which is positive, as K^^ and c are both positive. 

 The final values of the velocities of rotation and translation are then 

 obtained by putting P = 0, § = 0, P = (1 + e)Ri in the equations (1). 



5. If the coefficient of friction n is very large, it is commonly supposed 

 that when ^ = 0, (S = also, so that 



AA'i = {ab - ]v) K^ + (hf - bg) S^ cos 0^ + (gh - af) S^ sin 6^. 



If, however, we treat the problem in this way, as K^, /S',,, d^ may have any 

 values provided Ko and So are positive, we can arrange an impact such 

 that Pi is negative, which is absolutely impossible. For instance, taking 

 0„ and Oo + tt, the factor multiplying S^ changes sign, so that if (ab - h'') K^ is 

 taken less than S^ multiplied by the absolute value of that factor, K^, S„, and 

 either ^o or 0„ + ir give an impact for which Pi is negative. This proves 



[12-] 



