118 BOOK OF MATHEMATICAL PKOBLEMS. 



Under this condition, therefore, the ellipses 



f , '<* 



will be such that an infinite number of triangles can be described, 

 whose sides touch the first, and whose angular points lie on the 

 second. If this condition be not satisfied, the two given conies 

 and the locus will be found to have four common tangents, real 

 or impossible. 



Again, for the reciprocal problem, " If a triangle be inscribed 



x z y* 

 in the ellipse -75 + ~ = 1, and two of its sides touch the ellipse 



x* y s 



-, + ^a = 1, to find the envelope of the third side." 



If a, /?, y be the angular points, and (a, /?), (a, y) the sides 

 which touch the second ellipse, we have 



=cos _, 



_ 



and a like equation in a, y. Hence, as before, 



,8 + y ,j8+y 



C' cos V C' 



and the third side is 



cos 



X 



a 7 



cos 



wherefore the envelope is 



A' 

 which -coincides with 



