282 
ON CERTAIN DEVELOPABLE SURFACES. 
[344 
which is found to be divisible by (ae—4bd) 3 : and we thus obtain for PU the form 
PU = — 108 (a 1 2 # + 4 abde — 14<b 2 d 2 ) (acl 2 + b' 2 ef + (a 2 e 2 + 28 abde — 20 b 2 d 2 ) (ae — 4 bd) 3 , 
which puts in evidence that the cuspidal line (ae — Abd = 0, ad 2 + b 2 e — 0) of the 
developable is also a cuspidal line of the Prohessian. 
32. Writing the equations of the developable and the Prohessian under the forms 
A 3 -21 B 2 = G, 
LA 3 - 108 MB 3 = 0, 
and substituting in the second equation A 3 — 27 B 2 , it becomes B' 2 (L — 4>M) = 0, that 
is the intersection is made up of A 3 = 0, B 2 — 0, which is the cuspidal curve taken six 
times (order 36), and of the curve A 3 — 27 B 2 = 0, L— 4>M — 0 (order 24). But sub 
stituting for L, M their values, the equation L — 43/=0 becomes 
that is 
d 2 e 2 — 4 abde — 12 b 2 d 2 = 0, 
(ae + 2bd) (ae — Gbd) = 0, 
so that the last mentioned curve is composed of the intersections of the developable 
by the two quadric surfaces 
ae + 2 bd — 0, ae — Gbd = 0. 
33. Now combining with the equation of the developable the equation ae 4- 2bd = 0, 
and observing that in consequence of the last mentioned equation we have 
(ae — 4>bd) 3 = (— Gbd) 3 = — 216 b 3 d 3 = + 108 ab' 2 d‘ 2 e, 
the equation of the developable gives (ad 2 — b 2 e) 2 — 0, or we have (taken twice) the curve 
ae + 2bd — 0, ad 2 — b 2 e = 0, which is a curve of the sixth order made up of the lines 
(a = 0, b = 0), (d = 0, e = 0), and of a quartic curve (an excubo-quartic x ) the nodal line 
on the developable. If in like manner with the equation of the developable we combine 
the equation ae — Gbd= 0, then from this equation we have 
(ae — 4 bd) 3 = (2 bd) 3 = 8 b 3 d 3 = %ab 2 d 2 e, 
and the equation of the developable then gives 
(ad 2 + b 2 e 2 ) — g 4 T abhPe = 0 ; 
that is, 
ad 2 + 0b 2 e = 0, ad 2 + ^b 2 e = 0, if 6 + ^ = 2 — 
0 
1 A quartic curve which is the complete intersection of two quadric surfaces is termed a quadriquadric; 
a quartic curve of the kind which is not such complete intersection but can only be represented by means 
of a cubic surface is termed an excubo-quartic.