136 BOOK OF MATHEMATICAL PROBLEMS. 



at the points in which this line again meets them : prove that the 

 locus of the point of intersection of these tangents is the curve 



cr = 2ab {I + cos (0 + a-/?)} ; 



the second common point of the circles being pole, the common 

 chord (c) the initial line, a, b the radii, and a, ft the angles sub- 

 tended by c in the segments of the two circles which lie each 

 without the other circle. 



675. Two ellipses have a common focus, and axes inclined at 

 an angle a, and triangles can be inscribed in one whose sides 

 touch the other; prove that 



i * 2c S* = e i V + e c * ~ 2e , W* cs a, 



Cj, c a being the latera recta, and e t , e a the eccentricities. Also, if 

 0, <f>, \l/ be the angles subtended at the focus by the sides of any 

 one of the triangles, 



^ il/ c, 



4 cos cos cos = -s . 



VI. General Equation of the Second Degree. 



The general equation of a conic being 



ax' + by 9 + c + 2a'y + 2Vx + 2c'xy = 0, 

 the equations giving its centre are 



ax + cy + b' = 0, c'x + by + a' = 0. 



The equation determining its eccentricity may be found from 

 the consideration that 



a+ b 2c cos G> db c a 

 sin* o> sin" <o 



are unchanged by transformation of co-ordinates; and therefore 



