Fky — Impact in Three Dimensions. 81 



Hence, when S = Q, the representative point may move along a line 

 inclined at tan"' ft to the axis of R, whose projection on R = Q makes an 

 angle 6 with the axis of P, which satisfies F{Q) = 0, provided that this 

 value of Q makes <^[Q) negative. We shall now show that there is such a 

 unique root of F{d) = 0, if ^ is less than fi^ ; so that the possibility of 

 moving along this line and that of moving along the line of no sliding are 

 mutually exclusive. 



A root 6 of F[9) = 0, gives 



U\_ _ Uj_ _ Cr, cos + U., sin 6 

 cos ^ " sin 6* ~ cos' 6 + sin' ~ ^' 



[cifi - (f)) cosd + g = 0, [b/j, - (p) s'm d + f = Q, 



9- f 

 + = 1 



(a/x - <py {hfi - (py 



Tracing the curve 



y = — + — 1 



{afi - x)- (hfi - xy 



dx (ttfi - xf {hfx - xy' 

 y then increases from - 1 at .»■ = - oo to + co when x = fib, assuming a 

 greater than I ; it then diminishes from + co until 



at which point 



afji - X X -hfi af^ + hgi 

 -^--^ — = ^- or X = ^^j ^ . it, 



It then increases to + oo for x = fia, and then diminishes to - 1 for 

 2: = + CO . Thus there are always two values of x for which y = 0, one less 

 than fib, and the other greater than /x«, and there will be two other values 

 between jjA and jia if y is negative at the point where it is a minimum, or if 



/i' is greater than — ^-^ = U3-. 



(a - by 



This then is the value of fx referred to in section 7, such that if fx is 

 greater than fxi the circle x- + if = fi^ cuts the other branch of the 

 rectangular hyperbola H = 0. Of the two or four leal roots one only 

 can be negative, and the condition that there should be one negative root is 

 that y should be positive when « = 0, or that ix- should be less than 



