[332 
333] 
135 
uadric surface 
■ together the 
5 : w. We in 
satisfied. 
>, 
of contact is 
333. 
NOTE ON THE NODAL CURVE OF THE DEVELOPABLE DERIVED 
FROM THE QUARTIC EQUATION (a, b, c, d, e\t, 1) 4 = 0. 
[From the Philosophical Magazine, vol. xxvn. (1864), pp. 437—440.] 
Considering the coefficients (a, b, c, d, e) as linear functions of the coordinates 
x, y, z, w, then the equation 
Disct. (a, b, c, d, e\t, l) 4 = 0, 
or, as it may be written, 
(ae — 4bd 4- 3c 2 ) 3 — 27 (ace + 2bed — ad 2 — b 2 e — e 3 ) 2 = 0 
represents, as is known, a developable surface or “ torse,” having for its edge of 
regression (or cuspidal curve) the sextic curve the equations whereof are 
ae — 4 bd + 3c 2 = 0, 
ace + 2 bed — ad 2 — b-e — c 3 — 0 ; 
and for its nodal curve, a curve the equations whereof (equivalent to two independent 
relations between the coordinates) are 
ac — b~ ad — be ae + 2 bd — 3c 2 _ be — cd __ ce — d? 
a 26 6c 2d 
or, as these may also be written, 
a 2 d — Sabc + 26 3 = 0, 
a 2 e + 2abd — 9ac 2 + 66 2 c = 0, 
abe — 3 acd + 2b 2 d = 0, 
ad 2 — b 2 e = 0, 
ade — Sbce + 2bd 2 = 0, 
ae 2 + 2bde — 9c 2 e + 6cd 2 = 0, 
be 2 — Scde + 2d 3 = 0 ; 
e