y 
87 
398. 
ON A CERTAIN SEXTIC DEVELOPABLE, AND SEXTIC SURFACE 
CONNECTED THEREWITH. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), 
pp. ¿29—142 and 373—376.] 
I propose to consider [first] the sextic developable derived from a quartic equation, 
viz. taking this to be (a, b, c, d, e\t, l) 4 = 0, where {a, b, c, d, e) are any linear functions 
of the coordinates (x, y, z, w), the equation of the developable in question is 
(ae — 4 bd + 3c 2 ) 3 — 27 (ace — ad 2 — b 2 e 4- 2 bed — c 3 ) 2 = 0. 
I have already, in the paper “On a Special Sextic Developable,” Quarterly Journal 
of Mathematics, vol. vn. (1866), pp. 105—113, [373], considered a particular case of this 
surface, viz. that in which c was = 0, the geometrical peculiarity of which is that the 
cuspidal edge is there an excubo-quartic curve (of a special form, having two stationary 
tangents), whereas in the general case here considered it is a sextic curve. There 
was analytically the convenience that the linear functions being only the four functions 
a, b, d, e, these could be themselves taken as coordinates, whereas in the present case 
we have the five linear functions a, b, c, d, e. 
The developable 
(ae — 4 bd + 3c 2 ) 3 - 27 (ace — ad 2 — b 2 e 4- 2 bed — c 3 ) 2 = 0 
is a sextic developable having for its cuspidal curve the sextic curve 
ae — 4 bd 4 3c 2 = 0, 
ace - ad 2 - b 2 e + 2 bed - c 3 = 0, 
(say I = 0, J = 0, as usual), and having besides a nodal curve the equations of which 
may be written 
6 (ac-b 2 ) : S(ad-bc) : ae + 2bd-3c 2 : 3 (be-cd) : §(ce-d?) : 9 J 
= a : b : c : d : e : I,