Fi4Y — Impact in Three Dimensions. 93 



sliding or rolling ; but if K is positive for ~, the plane ^ = has not been 



crossed, and the solutioii is as in (1). 



(4) If the signs of ^r,/ are -, +, we have the case of motion in the fourth 

 quadrant, and putting g cos +/sin = (/' + g-)i sin [Q -Z), ^ increases 



TT 



until Q = Z,, if Bo is less than f, and then diminishes until Q = -^- Also if 



TT 



So is greater than ?, K diminishes until = -=. In either case K will vanish 



TT 



once or will not vanish, according as K for - is negative or positive. If 



negative, the end of the impact may be in the sliding or rolling stage ; if 

 positive, it must be in the rolling stage. Thus if 



^0 - So ( — COS0O + — sin 6, 

 \a a 



which is the value of ^for = — , is positive, the solution is obtained by taking 



S= when K= 0, except in the peculiar case occurring in (2), in which case 

 and in all other cases the solution depends on the solution for of the 

 equation K= 0. Also we see that, no matter how large ^u may be, sliding may 

 not cease in certain cases by the end of the impact, and in all such cases H may 

 be taken = when we proceed to find to v w oo^ Oy w;, etc. 



In this discussion /u is taken to be very large but still definite, so that 

 P and Q vanish if R vanishes. We would have to deal with a different 

 problem, if we could assume prominences on one surface to fit into depressions 

 on the other, so that the two bodies interlock, and P and Q can have values 

 when R =0. In this case if Ri is negative, or the point of intersection of the 

 line of no sliding and the plane of no compression is below the plane R = 0, 

 by taking 22 = 0, and finding P and Q from S = 0, we arrive at a point on the 

 line of no sliding for which K is negative, and conclude that the impact is 

 then over. 



