86] 
ON THE DEVELOPABLE DERIVED FROM AN EQUATION &C. 
501 
where the letters in the first column have the same signification as in my memoir 
in Liouville, [30], translated in the last number of the Journal. The order of the nodal 
line is of course 2 (n — 2) (n — 3); Mr Salmon has ascertained that there are upon this 
line 6 (ft — 3) (n — 4) stationary points and |(ft — 3) (n — 4) (n — 5) real double points, 
(the stationary points lying on the edge of regression, and with the stationary points 
of the edge of regression forming the system of intersections of the nodal line and 
edge of regression, and the real double points being triple points upon the surface). 
Also, that the number of apparent double points of the nodal line is 
(n — 3) (2n 3 — 18ft 2 + 57ft — 65). 
The case of U a function of the second order gives rise to the cone ac — If- = 0. 
When U is a function of the third order, we have the developable 
4 (ac — b 2 ) (bd — c 2 ) — (ad — be) 2 = 0, 
which is the general developable of the fourth order having for its edge of regression 
the curve of the third order, 
ac—b 2 — 0, bd — c 2 — 0, ad — bc= 0, 
which is likewise the most general curve of this order: there are of course in this 
case no stationary points on the edge of regression. In the case where U is of the 
fourth order we have the developable of the sixth order, 
(ae — 4bd + 3c 2 ) 3 — 27 (ace + 2bcd — ad 2 — b 2 e — c 3 ) 2 = 0 ; 
having for its edge of regression the curve of the sixth order, 
ae — 46d + 3c 2 = 0, ace + 2 bed — ad 2 — b 2 e — c 3 = 0, 
with four stationary points determined by the equations 
abed 
b c d e' 
The form exhibiting the nodal line of the surface has been given in the Journal 
by Mr Salmon. I do not notice it here, but pass on to the principal subject of the 
present paper, which is to exhibit the edge of regression and the stationary points of 
this edge of regression for the developable obtained from the equation of the fifth order, 
U — af 5 + bb^rj + 10cp?7 2 + 10 d£ 2 r) a + oe^rj 4 + fr) 5 = 0; 
viz. that represented by the equation 
□ = 0 = a 4 / 4 + 160a 3 ce/ 2 + ... — 40006 2 c 3 e 3 , 
[I do not reproduce here this expression for the discriminant of the binary quintic] 
a result for which I am indebted to Mr Salmon.