220 
NOTE ON THE DEGENERATE FORMS OF CURVES. 
[747 
“ there exists a quartic curve the penultimate of x-y' 2 = 0, with nine free summits, 
three of them on one of the lines (say the line y = 0), and which are three of the 
intersections of the quartic by this line (the fourth intersection being indefinitely near 
to the point x — 0, y = 0), six situate at pleasure on the other line x = 0; and three 
fixed summits at the intersection of the two lines.” Other forms have been con 
sidered by Dr Zeuthen, Comptes Rendus, t. lxxv. pp. 703 and 950 (September and 
October, 1872), and some other forms by Zeuthen; the whole question of the degenerate 
forms of curves is one well deserving further investigation. 
The question of the number of cubic curves satisfying given elementary conditions 
(depending as it does on the consideration of the degenerate forms of these curves) 
has been solved by Maillard and Zeuthen; that of the number of quartic curves has 
been solved by Dr Zeuthen.