JoLY — Veeto7' Expressions for Curves. 395 



Now 



^Oo a,ci> (0 - aj{t) 



^ % m 



^^05 rto<^(gi)-ao/(gi) 



by the method of partial fractions ; or, if 



- - p = — + i 



is the equation of the curve. Here €1, e^ . . . 6„ are the vectors parallel 

 to the asymptotes, and the construction is : — Take a system of n lines 

 thi'ough a point, and divide them homographically ; the locus of the 

 mean centre of corresponding points on the homographically divided 

 lines is a unicui-sal curve of the most general kind. If the lines are 

 real, and the homographic divisions also real, the curve has n real 

 asymptotes to which these lines are parallel. 



The line p = is homographically divided when e^ is given and 



t variable. The corresponding point on the line parallel to €2 is 



, and adding all these and dividing by w, the validity of the con- 



e^- t 



struction is evident. 



Suppose, however, that /(^, 1) = has a pair of conjugate roots, 



61 ± -y - 1 e-l. The terms arising from these are : 





Thus, when two of the roots of f{t, 1) = are imaginary, the cor- 

 responding homographically divided lines are imaginary also ; but they 

 combine into a real ellipse. In a similar manner, if two roots are equal, 

 a parabola replaces two of the lines. 



