272 
ON CERTAIN DEVELOPABLE SURFACES. 
[344 
8. What really happens in the case of a developable is more than this; viz. the 
curve of intersection is made up of the generating line taken twice, and of a curve of an 
order less by 2 than the order of the surface. Let (x, y, z, w) be the coordinates of the 
point on the developable, U= 0 the equation of the developable, (A, B, G, P, F, G, H, L, M, N) 
the second derived functions of U, (21, 23, (£, @, 2, 93?, 9?) the inverse system, 
K the determinant formed with the second derived functions, so that we have K=HU=0. 
K = 0. 
cone 
are 
given by 
= 21 
£ 
@ 
: 8, 
= & 
23 
g 
: 93?, 
= @ 
% 
(S 
: 9?, 
= 8 
93? 
9? 
: 
quivalent 
to 
each 
itrary multipliers, 
21, 
£ > 
(S 
& 
23, 
93? 
©, 
8 , 
e, 
9? 
93?, 
9?, 
$ 
cone 
will 
be as T 
3) : 3 : 933, and hence observing 
that the absolute magnitudes of these quantities are arbitrary, x + $ : y + 3) : z + 3 : w/4-933 
will represent the coordinates of any point on the line joining the point (x, y, z, w) 
with the vertex of the cone, that is, the generating line through the point (x, y, z, w); 
which is the theorem in question, the coordinates being in the present investigation 
denoted by {x, y, z, w) instead of the (a, b, c, d) of the example. 
9. Reverting to the developable U = a-d 2 — &c. = 0, the results previously obtained 
show that the coordinates of the vertex of the cone which is the quadric polar of the 
point (a, b, c, d) are as X : Y : Z : W (these quantities denoting as above the coefficients 
of the cubicovariant), and thence also that the coordinates of any point on the gene 
rating line will be as a + 6X : b + 6Y : c + 0Z : d + 6 W, where 6 is arbitrary. 
Special Quintic Developable, Nos. 10 to 25. 
10. We have, secondly, the developable of the fifth order 
U = a 3 e 2 + 6a 2 c 2 e — 24 ab 2 ce + 9 ac i + 166 4 e — 8 b~c 3 — 0, 
derived from the quartic function (a, 26, 3c, 0, —27e'$t, l) 4 , or, what is the same thing, 
at 2 + Sbt 3 + 18ci 2 — 27c = 0, where it will be observed that, as well in the quartic function 
as in the equation of the developable, the sum of the numerical coefficients is = zero; 
it was on this account that the foregoing form of the quartic function was selected 
in preference to the form (a, b, c, 0, e\t, l) 4 . The last mentioned form has for its 
discriminant 
(ae + 3c 2 ) 3 — 27 (ace — b-e — c 3 ) 2 , = e (a 3 e~ — 18a 2 c 2 e + 54a6 2 ce + 8lac 4 — 27b 2 e — 546 2 c 3 ),