80 Proceedings of the Royal Irish Academy. 



I further note that we may take a greater -than &, for if a is less than h, 

 interchange the axes of P and Q, also that we may take / and g to be each 

 negative, for if not by reversing the direction of the axis of P we change 

 the sign of g, and by reversing the direction of the axis of Q we change 

 the sign of / Thus there is one unique pair of axes for which li = 0, 

 a is greater than t, and / and g are each negative, but the rotation from 

 P to Q for this pair may turn out to be clockwise or counter-clockwise. 



9. If <S = initially or at any instant during the impact, as we take >S' to 

 be positive, its direction being given by %, it must either remain zero or 

 increase. If it remains zero, the representative point moves along the line 

 given by the intersection of the planes 



aP + gU = S^ cos 0g hQ + fit = S^ sin On, 

 h being taken = 0. Along this line 



adP + gdB^O, bdQ+fdB = 0, 

 and as n\dlif must be greater than or equal to {dPf + {dQf, motion along 

 this line, called the line of no sliding, is possible only if /i is greater than or 

 equal to /u,, where 



If S increases from zero, as 



*^r^«' '-! = -*«'■ 



initially F(d\ = 0, and by differentiating 



drb_ dS m_ _ dFjti) dd 

 dP'^ dP'dR ' ~~dM~ ' dB ' 

 Now, 



^'(^) = - SE^ = 3L(^isin0 -U,cos6) = ?7, cosfl + f/jsinO - assure -b/icos^e 

 da du 



= <p{6) - x>{&), where p(d) is positive for all values of 6, 



■■ ^dR'^r-dB'p^^yrdB^^' 



.-. as iS' = initially, and -5^ and »(^) are each positive, -7^ = initially; 



dB dB 



differentiating again 



d^Q 

 .: -p— = initially, and similarly it may be shown that all the derived 

 dB' 



functions of 6 with respect to B vanish initially, so that 6 remains constant. 



