Fry — Impact in Three Dimensions. 85 



positive, as K„ and c are each positive. During the course of the impact R 

 continually increases, so that the representative point must reach this plane, 

 but so far as this reasoning goes, it might be when R became infinite, and if 

 so the whole theory would collapse. We shall first examine the variations 

 of K during the initial stage. During this stage K is given by the formula 



K = K„ -gP -fQ-cR--^ K, - k 



{gn cos Q + fn sin -i- c) ^ (&) dQ. 



The sign of this integral depends entirely on the sign of 

 giJL cos Q +f/t sin 6 + c, 

 because during the initial stage of sliding we saw that kdO and x(^) ^^^ e^-ch 

 positive. When (i is approaching the root a, the impact terminates before a 

 is reached, so that in this case we must prove that ir may become zero 

 before 6 equals o. Now (j)(6) + g/n cos + fn sin + c is positive for all 

 values of Q, and near = a, f (6) is negative, so that near a 



gn cos 9 +fii sin 6 + c 

 is positive, and if we examine the value of the above integral, we shall see 

 that it therefore becomes equal to positive infinity, when becomes equal 

 to a, in just the same way in which we examined the value of R. Thus the 

 integral can become equal to any positive quantity as 6 approaches a, and so 

 for some value of 6 between 6^ and o it becomes equal to lu and K vanishes. 



During any other initial stage of sliding K is given by the same formula 

 and remains finite until S = 0. H fi (/' + g') ' is less than c, 



c + giii cos + ffi sin 6 



is always positive, so that K diminishes during the initial stage of sliding ; 

 but if fxl/'+g')^ is greater than c, then during the initial stage, by 

 varying (^„, we can arrange that K may first decrease and then increase 

 and in special cases again increase, so that by varying the values of S„, K„ we 

 can arrange that K shall vanish once or twice or thrice during the initial 

 stage. 



12. To elucidate this point — a point which caused more trouble than 

 the whole of the rest of this investigation — it is necessary to show how 

 the roots of F{d) =0 and 



g/j. cos 6 + ffj, sin + c = 

 are arranged in order as fi increases. This is done by recognising as in 

 section 7 that such roots are the angular co-ordinates of the points of 

 intersection of the circle x- + y- = /c with the rectangular hyperbola 



11= {a - i) xy + gij - fx = 0, 

 and with the line 



L=gx^fy\c = ^. 



[13*] 



