344] 
267 
344. 
ON CERTAIN DEVELOPABLE SURFACES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vi. (1864), 
pp. 108—126.0] 
If U = 0 be the equation of a developable surface, or say a developable, then the 
Hessian HU vanishes, not identically, but only by virtue of the equation U = 0 of the 
surface; that is, HU contains U as a factor, or we may write HU=U.PU; the 
function PU, which for the developable replaces as it were the Hessian HU, is termed 
the Proliessian; and (since if r be the order of U the order of HU is 4r —8) we 
have 3?’ — 8 for the order of the Prohessian. If r = 4, the order of the Prohessian is 
also 4, and in fact, as is known, the Prohessian is in this case = U. The Prohessian 
is considered, but not in much detail, in Dr Salmon’s Geometry of Three Dimensions, 
(1862), pp. 338 and 426 [Ed. 4 (1882), p. 408] : the theorem given in the latter place 
is almost all that is known on the subject. I call to mind that the tangent plane 
along a generating line of the developable meets the developable in this line taken 
2 times, and in a curve of the order i— 2 ; the line touches the curve at the point 
of contact, or say the ineunt, on the edge of regression, and besides meets it in 
r — 4 points. The ineunt taken 3 times, and the r — 4 points form a linear system 
of the order r — 1, and the Hessian of this system (considered as a curve of one 
dimension, or binary quantic) is a linear system of 2r — 6 points; viz. it is composed 
of the ineunt taken 4 times, and of 2r—10 other points. This being so, the theorem 
is that the generating line meets the Prohessian in the ineunt taken 6 times, in the 
r — 4 points, and in the 2r — 10 points (6 + r — 4 + 2r — 10 = 3r — 8) ; it is assumed that 
r = 5 at least. 
The developables which first present themselves are those which are the envelopes 
of a plane 
(a, h, l) n = 0, 
Presented to the Royal Society and read 27 Nov., 1862, but withdrawn by permission of the Council. 
34—2