FiJY — Impact in Three Dimensions. 83 



Thus the value of S depends on t^^"''', where 



and therefore is zero, finite, or infinite, according as yp(u) is positive, zero, or 

 negative. 



Similarly the magnitude of B depends on 



!v If (a) \ 



and so £ is finite, if \p (a) is positive, and infinite, if if, («) is zero or negative. 

 Now, when F{9a) is positive, 6 advances towards the next root of F(d) = 0, 

 and at that root F'{0) is negative, as F(d) is then decreasing ; on the other 

 hand, when F(6^) is negative, 9 moves backward, and again at the first root 

 which it meets F'(0) is negative. Also ^ (9) is positive for all the roots 

 of F(9) = except one, or for all the roots, according as fi is less or greater 

 than /ii. Thus, when the special root for which {9) is negative exists, for it 



'^^"^ I>i")-P(a) 

 is negative, p (a) being positive for all values of a ; also we note that 

 for it F'{a) = <j){a) - p [a] is negative ; hence, if 9o is adjacent to this root 

 on either side, 9 moves towards this root, but never reaches it, as at it H would 

 be infinite. For any other root a, (j> (a) is positive, and we saw that F'{a) is 

 negative for the root which is being approached, and so B is finite. 



For the special root, *S^ would be infinite, in all other cases when 6 becomes 

 equal to the proper adjacent root, ^ = 0. In the particular case of ^u = n,, 

 the special root gives <p (a) = 0, so that although S would be finite for it, 

 B would be infinite, and so 9 does not take up such a value. For the 

 double root a, when n = ni, S = 0, and B is finite, as F"(a) is positive. 



There are four possible arrangements of the roots F{9) = 0. (1) If /z is 

 less than ;ii and less than fi^, there are two roots a, /3, of which we take a to 

 be such that (p (a) is negative. (2) If fi is less than ;Ui and greater than fi^, 

 there are four roots a, /3, 7, S, of which we take a to be the root for which 

 <f>(a) is negative. (3) If fi is greater than ^1 and less than ^2, there are two 

 roots a, /8' of which we take u to be the root for which F'' (a) is negative. (4) If 

 fi is greater than /xi and greater than H2, there are four roots a, /3', j, S', of 

 which we take a, y' to be the pair which make F' {6) negative. It will 

 be proved in section 12 that referred to the particular axes described at 

 the end of section 8, a and a are always in the first quadrant, j3 and (3' in 

 the third, and the other two 7, S, or j', S', when they exist, always in the 

 fourth. 



R.I. A. PROC, VOL. XXXIII., SECT. A. [13] 



