344] ON CERTAIN DEVELOPABLE SURFACES. 271 
then if 6 = 3(\X + /jlY + vZ + pW), we have 
(3f, 3), 3, %8) = (0X + aU, BY + /3U, 0Z + ryU, BW + 8U). 
The function (X, Y, Z, WTft, l) 3 is the cubicovariant of {a, b, c, d\t, l) 3 and if for 
a moment these functions are represented by v, u respectively, and if we also write 
U = a?d? — &c. = U (a, b, c, d), then 
(a + 6X, b +BY, c + 9Z, d + 6WQt, l) 3 = u + 6v, 
and thence 
U (a + 6X, b + BY, c + BZ, d + BW) = Disct. (u + Bv), 
= (1 - B 2 Uf U, 
by a formula given in my “ Fifth Memoir on Quantics,” Phil. Trans., t. cxlviii. (1858), 
see p. 442 [156]; the function on the left-hand side thus contains U as a factor, 
and it at once follows that the function 
?7 (a + 3f, 5+3), c+3, d + 3[B); 
viz., the function obtained from U by writing therein (a + 3£, b + 3), c + 3, d + 2B) in 
the place of (a, b, c, d) respectively, contains U as a factor, and therefore vanishes if 
U = 0; that is a + T, 5 + 3), c + 3, d + are the coordinates of a point on the 
surface TJ = 0; they are in fact the coordinates of a point on the generating line 
through (a, b, c, d); this is a theorem which applies to any developable whatever, as 
appears by the following considerations. 
Remarks on the General Theory of Developables, Nos. 7 to 9. 
7. In general for any surface whatever, taking a point on the surface, the successive 
polars of this point (the last of them being the tangent plane) all touch at this point; 
and not only so, but the tangents to the two branches of the curve in which the 
surface itself (or any of its polars down to the quadric polar) is intersected by the 
ultimate polar or tangent plane, are respectively coincident. Suppose that for any point 
on the surface, the quadric polar becomes a cone: the vertex of this cone is not the 
point itself; hence the tangent plane at the point touches the cone along a generating 
line; that is the tangents to the curve of intersection with the surface, or with any 
of its polars, coincide with the generating line of the cone—and the curve of inter 
section of the tangent plane with the surface, or any of its polars, at the point of 
contact (instead of, as in general, a node) has a cusp. In particular the curve of 
intersection with the surface has at the point of contact a cusp. The condition that 
the quadric polar may be a cone is HU = 0, and when this differential equation is 
satisfied in virtue of the equation U= 0 (that is, when we have identically HU = U. PU), 
the surface is a developable. Now all that is proved in the first instance by the 
equation HU = 0 is that every point of the surface has the above mentioned property; 
viz., that the tangent plane at the point cuts the surface in a curve having a cusp 
at the point in question.