Fey — Impact in Three Dimensions. 89 



aud so is finite, and is above or below the plane i2 = 0, according as 



K„ + ^ Sa cos ^0 + V ^1 sin H„ 



is positive or negative, since the denominator is positive = ^/ab. When 6 

 becomes equal to a or y the representative point is on the line of no sliding, 

 and so if it has not crossed the plane of no compression or has crossed it twice, 

 it is still on the same side of the plane of no compression as the origin, and we 

 shall prove that by moving along the line of no sliding it will meet the plane. 

 If it has already crossed the plane of no compression or crossed it three times, 

 we shall prove that it will not meet it again. As before we can prove more 

 than is required, for if P'Q'B' is any point on the same side of ^= as the 

 origin, we shall prove that by drawing a line through it parallel to the line of 

 no sliding and moving up the line so that Ji increases, we shall cross the 

 jilane K = 0. We have then to show that 



F+-^ B, Q'+ 4-^, B'+B 

 a 



satisfies TT = vi'ith R positive. Substituting in K = 0, 



B = — ^ — - — -^ = positive quantity 



c- ^- + 4" 

 a 



as the numerator is given to be positive, and the denominator 



= A/a5 = positive quantity. 



The same formula for B also proves that if the point P'Q'B' is on the 

 opposite side of the plane K = to the origin, or if the numerator is negative, 

 then by moving up the line so that B increases we do not meet the plane K = 0. 

 Always the plane of no compression is crossed once or thrice during the impact. 

 If the representative point after first crossing the plane of no compression 

 with B = Bi attains a position for which B = {1 + e) Bi before it crosses it 

 again, the impact is over ; but if not the impact will be over when 



B= {1 + e]Bi 



where Bi is the value of B when the plane is crossed for the third time. 

 We are compelled to modify the experimental law in this way, otherwise we 

 would have the absurdity of taking the impact to be finished when compression 

 was still taking place. 



15. Impact in Three Dimensions when fx is very great. — Eef erring P and Q 

 to the axes for which h = 0, a is greater than b, and / and g are each negative, 

 we see that as fx is very great we have an extreme case of the arrangement (4) 



