276 
ON CERTAIN DEVELOPABLE SURFACES. 
[344 
18. But in the paper just referred to, it is also shown that considering the 
developable which is the envelope of the common tangent planes 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 it 
is of the order 5. 
In the last mentioned case the surfaces may without loss of generality be reduced to 
conics, and their equations may be taken to be (y 2 — 2zx = 0, w = 0) and (x 2 —2ziv=0, y=0)( 1 ), 
and this being so the equation of the developable is 
32 z 3 w 2 — 32 z 2 x?w + 72 zxy % w + 8zx i — 27 y 4 ia — 4a? y 2 = 0. 
This is really a developable of the same kind with the first mentioned developable of 
the order 5; for writing x = 12c, у = 8b, z = 3a, w = — 8e, the equation becomes 
a 3 e 2 + 6a 2 c 2 e — 24ab 2 ce + 9ac 4 + 166 4 e — 8 b 2 c 3 = 0, 
which is the before mentioned developable U=0. The equations of the two conics 
become (9ac — 8& 2 = 0, e = 0) and (ae + 3c 2 = 0, b = 0), and the developable is thus the 
envelope of the common tangent planes of these two conics. It has been seen that 
the first conic is a simple line, but the second conic a nodal line, on the developable. 
19. Recapitulating, the developable of the fifth order U— 0, which is the envelope 
of the plane (a, 2b, 3c, 0, — 27e) (t, l) 4 = 0 is the locus of the tangents of the quadri- 
quadric curve (ae — c 2 = 0, ac — b 2 = 0), and it is also the envelope of the common tangent 
planes of the conics (9ac — 8b 2 = 0, e = 0) and (ae + 3c 2 = 0, b = 0). 
20. Returning now to the Prohessian, its equation may be written in the form 
PU = (3a 2 c, — 3b 2 c, ace + 2b 2 e\ae — c 2 , 4ac — 4b 2 ) 2 = 0, 
and the discriminant of the quadric function is 
3 a 2 ce (ac + 2b 2 ) — 9 b 4 c 2 , 
which is 
= 3c {(a 2 e + 3b 2 c) (ac — b 2 ) + 3 ah 2 (ae — c 2 )}, 
and recollecting that the equations of the cuspidal curve or edge of regression of the 
developable are ae — c 2 = 0, ac—b 2 — 0, it thus appears that the curve in question is also 
a cuspidal curve on the Prohessian. 
21. Consider for a moment the surface 
(3a 2 c, — 3b 2 cd, ace + 26b 2 e'§ae — c 2 , 4ac — 4b 2 ) 2 = 0, 
1 These are in fact the conics made use of, p. 57 in the paper above referred to [and p. 495 in the 
reprint], but the equations are by mistake given as {x 2 - 2yz = 0, to—0), (y 2 - 2zw = 0, x = 0), that is, x and y are 
interchanged.