ON CERTAIN DEVELOPABLE SURFACES. 
277 
544] 
where 9 is an arbitrary parameter; this is a surface having for a nodal curve the 
cuspidal curve of the developable; but if the discriminant of the quadric function 
vanishes, that is if 
3a 2 ce (ac + 2b 2 9) — 9b 4 c 2 9 2 = 0, 
for ae — c 2 = 0, ac — b 2 = 0, then the curve in question will be a cuspidal curve on the 
surface. But the last mentioned equation is 
3c [(a 2 e + 3b 2 c9 2 ) {ac — b 2 ) + ab 2 {(1 + 29) ae — 39 2 c 2 }] = 0, 
which for ae — c 2 = 0, ac — b 2 — 0, becomes 1 + 2(9 —3d 2 = 0, that is 9=1, which gives the 
Prohessian, or 9 = — i. 
22. For the latter value the surface is 
(9a 2 c, 3b 2 c, 3ace — 2b 2 e\ae — c 2 , lac — 4b 2 ) 2 = 0, 
or, expanding, the surface {9 = — £) is 
dee 2 
+ 
9 
dc 2 e 
+ 
30 
d 2 b 2 c 2 e 
— 
104 
d<f 
+ 
9 
ab 4 ce 
+ 
88 
ab 2 c 4 
— 
24 
b 6 e 
— 
36 
b 4 c s 
- 
24 
= 0, 
the before mentioned discriminant being 
= c {(3a?e — b 2 c) {ac — b 2 ) + ab 2 {ae — c 2 )}; 
but I have not further examined the geometrical signification of this surface, or 
inquired into its relation to the Prohessian. 
23. The equation of the Prohessian may be written 
PU = {ac— b 2 ) {16e {ac — b 2 ) 2 + 3a {ae — c 2 ) 2 } + b 2 U = 0, 
or what is the same thing 
P U = {ac — b 2 ) [a {ae + 3c 2 ) (3ae + c 2 ) — 32ab 2 ce + 166 4 e} + b 2 U = 0, 
the latter of which shows that the conic {ae + 3c 2 = 0, b = 0), which is the nodal line 
of the developable, is a simple line on the Prohessian. 
24. Consider the curve of intersection of the developable and the Prohessian; 
this is of the order 5x7, =35. We have ac—b 2 = 0, £7=0, or else 
16e {ac — b 2 ) 2 + 3a {ae — c 2 ) 2 = 0, £7=0. 
Consider for a moment the second system, this is 
3a (ae - c 2 ) 2 + 16e (ac - b 2 ) 2 = 0, 
3a (ae — c 2 ) 2 — 24c (ae — c 2 ) (ac — b 2 ) + 4Se (ac — b 2 ) 2 = 0,