TowNSEND — Geometrical Properties of the Atriphthaloid . 63 



points A and B. The two tangents OE and OF to which from the 



origin are given in position by the equation sec^ w = -- yj, and in 



2 

 magnitude by the equation OE = OF = - h, and the equal distances 



o 



OD of their chords of contact FF from the origin by the equation 



7,3 



/i 



4 h^ 

 "When -;: 7T = 1) then for the two tangents sec^ oj = 1, and the ovals 



27 B 



consequently contract into points whose equal distances from the ori- 



2 



gin = - A : the pairs of vertices A and B of the two ovals (see fig.) then 

 o 



coincide, and the equal distances OCof the two conchoidal vertices from 



the origin - -h. 

 " 3 



4 W' 

 When -z 7:; is < 1, then for the two tangents sec'^ w < 1, and there- 

 27 B 



fore w is imaginary. The two ovals then disappear altogether, and the 

 curve consists entirely of the two conchoidal branches, which never dis- 

 appear. 



The sum of the three roots r^, n, r^ of equation (1) being indepen- 

 dent of the value of w, and = h for all directions of r ; hence, for the 



