86 Proceedings of the Royal Irish Academy. 



It is instructive also to exhibit the sign of <^ (0) for each of the roots 

 by noting that 



fi<^ [%) = E = a:r^ + hi/- + gx +/y, if x = n cos 0, ij == fi sin 0, 

 so that the sign of <l> (6) will be known by seeing that it is negative when 

 inside the ellipse -£' = and positive when outside. Similarly the sign 

 of F'(ti) may be shown by the position of the point jj. cos 6, /isind with 

 respect to the rectangular hyperbola 



H' = (a~b){x--,f) +gx+f!/ = 0. 



In order that all cases may be represented by one figure, we make 

 Use of the axes obtained at the end of Section 8, for which a is greater 

 than h, h = 0, and / and g are each negative. For such axes the rotation 

 from P to § may turn out to be clockwise or counter-clockwise; but if 

 one figure is drawn, the other possible one is the same as the first when 

 viewed from the opposite side of the paper, and the description of the 

 figure by quadrants applies equally well to both. 



Substituting - /, - g for / and g in H, E and L, 



E = a^--g..hf-fy = «(- - K^" ^ " £ " S = '' 



L = c - gx - fy. 



3=0 passes through the origin, has its asymptotes parallel to the axes, 

 and they meet in its centre !j/(a - h), - //(a - b). The branch through the 

 origin is confined to the first and third quadrants, the other branch to the 

 fourth quadrant, so that as n varies, one root of E{6) = is always in the 

 first quadrant, one in the third, and the other two in the fourth, but they 

 exist only when ju is greater than f^i. The ellipse passes through the origin 

 cutting H = there at right angles, cuts off from the_ axes lengths equal 

 to gla and f/b, and as its axes are parallel to the axes of P and Q it 

 passes through the point g/a, f/b, which point also lies on 11=0, and the 

 origin and this point are the only points common to iT = and E = 0, as 

 may be seen from the forms B = 17,7/ - JJix = 0, E = TJ^x + f/ji/ = 0, so 

 that any point which satisfies H = 0, E = 0, besides the origin, makes 

 (x' + 2/') Ui = 0, and (ar + ?/') Ut = 0, and therefore makes U^ = 0, Ui = 0. 

 As the ellipse and the rectangular hyperbola intei-sect in the origin and 

 at g/a, f/b, and have no other point of intersection, so that the other 

 branch of the hyperbola lies completely outside the ellipse, it is quite clear 

 from the figure that when /j, is less than ^i, one root and one only makes 



