Fky — Impact in Three Dimensions. 79 



curve ; so that there are always two real points of intersection, and four if ju 

 is large enough to make the circle cut the other branch of the hyperbola ; 

 this value of ^ we shall call ^2. 



If at any instant during the impact % should become equal to one of 

 the roots of -f (^) = 0, we should expect the representative point to move as 

 described in this section ; but as a matter of fact the conditions set down here 

 never occur unless they do so initially, for we shall see that whenever ^ 

 becomes equal to a root of F[Q) = 0, at the same time jS = 0, so that we are 

 brought to the consideration of what will happen when (S = initially or at 

 any moment during the impact. 



This discussion of the roots will be further developed and also another 

 discussion given in section 9 ; both of which will show that when fx is less 

 than the value \x\ necessary to make rolling possible, one root exists for 



which (S) is negative, so that when sliding is along it *S increases as --^ 



do 



is positive, and one or three other roots exist for which cp (d) is positive ; 



and that when fi is greater than /it, <p (6) is positive for the two or four roots 



which then exist, so that when sliding takes places initially along any of 



these other directions, S decreases continually. 



8. To simplify further discussion we shall show (see Eouth's Rigid 

 Dynamics) that by turning the axes of P and Q about the axis of B through 

 an angle y, h may be made zero. 



For after the axes are turned through y, let the values of P, Q, S, 6 for 

 the new axes be denoted by the same letters with dots, and we get 



S'cos 9' = S cos cos -y + (S sin 9 sin y = [Su cos 0^ - aP - hQ - gB] cos y 



+ [So sin Oo- hP - hQ -fR) sin y 

 = aSj'cos 0/ - (a cos -y + 7i sin y) (P'cos y - Q'sin y) 



- (A cosy + 5siny) (P'siny + ^'cosy) - (5' cosy +/&m.y)R 

 = (S/cos 0/ - (a cos'y + '2h cos y sin y + 5 sin'y) P' 



- f — ^ sin 2y + A cos 2y\Q' - [g cos y + /sin y) P 

 similarly 



»S"sin 6' = /S/sin 9,' - ( -^ sin 2y + h cos 2y j P' 



- (a sin^y - 2h sin y cos y + 6 cos'y) Q' - (/cos y - g sin y) P. 

 Thus A = if y has any of the four perpendicular directions given by 



tan 2y = . 



a - 



