[344 
, and 
and 
344] ON CERTAIN DEVELOPABLE SURFACES. 275 
15. But before discussing the Prohessian, I will further consider the developable 
itself. Regarding it as derived from the equation (a, 2b, Sc, 0, —27e][£, 1) 4 = 0, we have 
eTJ = (27) 3 {(ae — c 2 ) 3 + (— 3 ace + W-e — c 3 ) 2 } = 0, 
and observing that 
— 3 ace + 4 b 2 e — c 3 —c(ae — c 2 ) — 4e (ac — b 2 ), 
it appears that the equations of the cuspidal curve or edge of regression of the 
developable are ae — c 2 = 0, ac — b 2 = 0 (so that the cuspidal curve is a curve of the 
fourth order, the intersection of two quadric surfaces, or say a quadri-quadric curve). 
This is perhaps better seen by writing the equation of the developable in the form 
U —a (ae — c 2 ) 2 — 8c (ae — c 2 ) (ac — b 2 ) + 1 Qe (ac — b 2 ) 2 = 0, 
or what is the same thing 
U = (a, c, e\ae — c 2 , 4ac — 45 2 ) 2 = 0, 
where the discriminant of the quadric function is = ae — c 2 , which vanishes for the 
curve (ae - c 2 = 0, ac — b 2 = 0). 
16. Another form of the equation is 
U — a (ae + 3c 2 ) 2 — 8b 2 (3ace — 2b 2 e + c 3 ) = 0, 
which shows that the conic ae + 3c 2 =0, b — 0, is a nodal curve on the developable. 
And, again, another form is 
U = c 3 (9ac — 8b 2 ) + e (a 3 e + 6a 2 c 2 — 24a5 2 c + 166 4 ) = 0, 
which shows that the conic 9ac — 8b 2 = 0, e — 0, is a simple line on the developable. 
17. In my paper “On the Developable Surfaces which arise from Two Surfaces 
of the Second Order,” Camb. and Dub. Math. Jour., t. v. (1850), pp. 46—57, [84], I 
considered first the developable having for its edge of regression the intersection of two 
quadric surfaces; in the general case the developable is of the order 8; but if the 
two surfaces have an ordinary contact it is of the order 6 ; and if they have a 
singular contact (as there explained) it is of the order 5. And in the last mentioned 
case, if the equations of the two quadric surfaces are taken to be x 2 — 2wz — 0 : 
y 2 — 2zx — 0, then the equation of the developable was found to be 
4 z 3 w 2 + 12 z 2 x 2 w + 9 zx* — 24 zxy 2 w — 4 a?y 2 + 8 y 4 w = 0, 
which putting therein z = a, y = 2b, x — 2c, w = 2e, becomes 
a 3 e 2 + 6a 2 c 2 e — 24 ab 2 ce + 9ac 4 + 165 4 e — 8 b 2 c 3 = 0, 
which is the before mentioned developable U=0; the two equations x 2 —2wz=0, y 2 —2zx=0 
become by the same substitution ae — c 2 = 0, ac — b 2 = 0. We have in fact already seen 
that the developable U = 0 has this curve for its edge of regression. 
35—2