373] 
ON A SPECIAL SEXTIC DEVELOPABLE. 
517 
and we have Plucker’s six equations, which may be considered as included in the 
three equations 
to = r(r — l) — 2y — 3 (w + b), 
/3 = Sr {r — 2) — 6y — 8 (n + S-), 
r = to (to — 1) — 2h — 3/3. 
These two systems constitute together a system of six equations between the ten 
quantities to, r, n, a, /3, b, g, h, x, y. Considering to, r, x, b as arbitrary, the six 
equations determine the remaining quantities n, a, /3, h, x, y. 
The curve 
ae — 4 bd + 3c' I 2 = 0, ace + 2 bed — ad? — b?e — c 3 = 0, 
is a sextic curve, the edge of regression of the sextic torse 
(ae — 4bd + 3c 2 ) 3 — 27 (ace + 2bed — ad? — №e — c 3 ) 2 = 0, 
and we have in this case, as is well known, 
m, r, n, a, /3, Sr, g, h, x, y 
= 6, 6, 4, 0, 4, 0, 3, 6, 4, 6. 
But putting as above c = 0, then instead of the sextic curve we have the excubo- 
quartic curve ae — 4bd = 0, ad? + l?e = 0, which is a curve having two stationary tangents, 
viz. these are the lines (a=0, 6 = 0) and (d = 0, e = 0), which are in fact given along 
with the curve, by the foregoing equations ae — 4bd = 0, ad 2 + b 2 e = 0. We have in this 
case Sr = 2, and the system is thus found to be 
w, r, n, a, /8, S-, g, h, x, y 
= 4, 6, 4, 0, 0, 2, 3, 3, 4, 4, 
it was in fact the consideration of this case which led me to take account of the new 
singularity of the stationary tangent lines. 
I take the opportunity of referring to a most valuable and interesting paper by 
Schwarz, “De superficiebus in planum explicabilibus primorum septem ordinum,” Crelle, 
t. lxiv. (1864), pp. 1—16. The author, after referring to my paper “ On the deve 
lopable derived from an equation of the fifth order,” Cambridge and Dublin Mathe 
matical Journal, t. v. (1850), pp. 152—159, [86], enters into the enquiry there suggested 
as to the means of ascertaining the degree of the c planarity’ of a developable surface. 
He starts from certain theorems derived from Riemann’s theory of transcendental 
functions, viz.: If an algebraical (plane) curve of the order r has |(r-1) (r -1) - p 
double points (nodes or cusps), then the coordinates of a point of the curve may be 
expressed rationally 
If p = 0, that is, if the curve has the maximum number of double points, by a 
single parameter.