500 
[86 
86. 
ON THE DEVELOPABLE DERIVED FROM AN EQUATION OF 
THE FIFTH ORDER. 
[From the Cambridge and Dublin Mathematical Journal, vol. v. (1850), pp. 152—159.] 
MÖBIUS, in his “ Barycentrische Calcul,” [Leipzig, 1827], has considered, or rather 
suggested for consideration, the family of curves of double curvature given by equations 
such as x : y : z : w = A : B : C : D, where A, B, C, D are rational and integral 
functions of an indeterminate quantity t. Observing that the plane Ax + By + Cz -f Diu — 0 
may be considered as the polar of the point determined by the system of equations last 
preceding, the reciprocal of the curve of Möbius is the developable, which is the 
envelope of a plane the coefficients in the equation of which are rational and integral 
functions of an indeterminate quantity t, or what is equivalent, homogeneous functions 
of two variables £, rj. Such an equation may be represented by U = a£ n + n£ w_1 g + ...=(), 
(where a, b, &c. are linear functions of the coordinates); and we are thus led to the 
developables noticed, I believe for the first time, in my “ Note sur les Hyper-deter 
minants,” Crelle, t. xxxiv. p. 148, [54]. I there remarked, that not only the equation 
of the developable was to be obtained by eliminating £, 77 from the first derived 
equations of JJ — 0 ; but that the second derived equations conducted in like manner 
to the edge of regression, and the third derived equations to the cusps or stationary 
points of the edge of regression. It followed that the order of the surface was 2(n— 1), 
that of the edge of regression 3 (n — 2), and the number of stationary points 4 (n — 3). 
These values lead at once, as Mr Salmon pointed out to me, to the table, 
m = 3 (n — 2), 
n — n, 
r =2(71-1), 
a =0, 
/3=4 (n- 3), 
9 =%(n-l)(n- 2), 
h = I (9w 2 — 53ti + 80), 
a: =2 (n - 2) (n - 3), 
V -2(«-l)(«-3).