92 Proceedings of the Royal Irish Academy. 



During the initial stage, as R is negligible in comparison with P and Q, 



+'^(cote) , 



a ) 



or 



T^ n- Q (9 a f ■ q\ o /'tan0„\'^ sin0o (f/ « / • o) 

 K = Ko- SJy- cos d„ +■'1 sin 00 + SJ - — jr -^—jr - cos + - sin0 . 

 V* & / \tanf^y sin ff [a a \ 



Also K = K^-ki ig cos e+ /sin 6) (cot 6)'"' 'cosec' ddO. 



J Bo 



(1) If the signs of g and / are each negative, we have the case of motion 

 in the first quadrant of the special axes, and we see that K increases during 

 sliding, so that to get J?,- for K = 0, rolling must have commenced, and 



Hi 



Ko - -S'o I - cos 6/„ + -^sin 0„ 

 a 



(2) If the signs of g, f are +, -, we have the ease of motion in the second 

 quadrant, and if we put /sin + ^ cos = - (/' + g'')^ sin {Q - Z,'), we see 

 that if 00 is greater than ?', K increases all the time, as K is initially positive ; 



TT 



.-. in this case it is positive for = ^, or 



K, - S, (^ cos 0„ + -^ sin 0„ 



\ « ft 



is positive, and the solution is as in (1) ; but if 0„ is less than Z\ I^ first 

 diminishes and then increases, so that K may become zero once or twice 



between 0,, and r ; if it has become zero once, K for = - is negative, and 



remains so as we proceed along the line of no sliding, and so we find 

 6 for K = 0, then get i?,-, and find when E becomes equal to (1 + e) Pi ; if 



TT 



IC has not become zero or has become zero twice, K for = - is +, and by 



proceeding along the line of no sliding K = a. third time, and the solution 

 is as in (1), unless, in the case of K vanishing twice, the value of E when K 

 vanishes for the second time is equal to or greater than (1 + e) times its value 

 when K vanishes first. 



(3j If the signs of g, f are + +, we have the case of motion in the third 



quadrant ; K diminishes until 0=9, and if K for ^ is negative the plane of 



no compression has been crossed, and the end of the impact may be during 



