Fry — Impact in Three Dimensions. 87 



<I>{B) negative, and when fi is greater than ju„ all the roots make ^{6) 

 positive. The tangent to ^ = at the origin is perpendicular to Z = 0, 

 and the line i = is further from the origin than any point of the ellipse, 

 for ^ + Z is positive for all values of x and y, so that no point inside 

 U ^ 0, for which U is negative, could be further from the origin than 

 L = 0, for which Z would be negative. Further, we may observe that the 

 line joining the origin to the centre of If = is inclined to the axis of x 

 at the same angle as the tangent to H = at the origin is on the other 

 side. The rectangular hyperbola H' = has its axes parallel to the axes 

 of X and y, passes through the four points 0, ; 0, - //{a - b) ; g/(a - b), ; 

 g/(a - b), - //{a - b), and occupies two positions according as / is greater or 

 less than g. In either case it follows that denoting the roots in order by 

 a /3 y S, F'(a) is always negative, and also F'(y) when y and S exist. 



Eef erring now to the four possible arrangements of the roots of F(d) = 

 given- in section 10 : — a lies in the first quadrant, between the perpendicular 

 on Z = and the axis oi y or Q; it makes f/<(0) negative and F'{6) negative ; 

 a lies in the same region ; either f3 or /3' lies in the third quadrant ; and 

 if they exist, y and S or 7' and B' lie in the fourth. When ju is very great, 

 a /3 7 S coincide with the direction of the axes. As II is positive at its centre, 

 therefore on any circle x" + y- = /x° it is positive from 8 to a or S' to a, 

 negative from « to /3 or a' to j3', positive from j3 to 7 or j3' to 7', and 

 negative from 7 to S or 7' to B' ; or if 7 S 7' B' do not exist, positive from j3 

 to a or /3' to a' and negative from a to /3 or a' to j3'. Thus moves 

 towards a or a if 7, B or 7', B' do not exist, and towards « or «' if 0^ lies 

 between B and /8 or B' and /3', but towards 7 or 7' if do lies between /3 and B 

 or /3' and S'. Now the two roots ^ and Z' say of /, = first appear when 

 fi is Inrge enough between and a, and remain in this position in the 

 arrangements (1) and (2) of the roots. Thus if d^ lies between jS and the 

 least of them, say Z. in (1), or between S and the least of them in (2), L is 

 first positive between ^„ and Z> then negative between Z and Z', and again 

 positive between Z' and a, so that K first diminishes, then increases, and 

 again diminishes. Accordingly it is possible so to arrange ZT^ Sg and dg 

 that ^will vanish three times. If 60 is between Z and Z', iT first increases and 

 then diminishes, and if 9^ is in any other position in (1) or (2) K diminishes 

 continually. The same holds for the arrangements (3) (4) until Z' crosses a. 

 Then if in (3) 6^ lies between Z' and /j' or Z and j3', or if in (4) d^ lies 

 between Z' and /3' or Z and S', K first diminishes and then increases, hence 

 Ko S„ and 6^ may be arranged so that in such cases ZT vanishes twice. As 

 n increases further Z' continues between a and j3' and Z comes between 

 7' and S'. If then 6q lies between Z' and /3', Zf first diminishes and then 



