214 
A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. [340 
Quartic Scrolls, Article Nos. 36 to 50. 
36. We may consider, first, the quartic scrolls $(1, 1, 4). The section is a quartic 
curve having an a-tuple point and a /3-tuple point, where a + /3 = 4; that is, we have 
a = 2, /9 = 2, a quartic with two nodes (double points), or else a = 3, /3 = 1, a quartic 
with a triple point. But the case a = 2, /3 = 2 gives rise to two species: viz., in general 
the quartic has only the tw T o double points, and we have then a scroll with two nodal 
(2-tuple) directrix lines, and without any nodal generator; the section may however 
have a third double point, and the scroll has then a nodal (double) generator. For 
the case a = 3, /9 = 1, the section admits of no further singularity, and we have a 
quartic scroll with a triple directrix line and a single directrix line. 
37. Next for the quartic scroll $(1, 1,4). The section is here a quartic curve with 
an a. (+ /3) tuple point, where a +/3 = 4; that is, cl —2, ¡3 = 2, or else a=3, /3 = 1. In 
the former case the section has a 2 (+ 2) tuple point, that is, a double point where 
the two branches have a common tangent—otherwise, two coincident double points: say 
the curve has a tacnode; the line of approach is the tangent at the tacnode. We 
have here a scroll with a twofold double line; there are however two cases: viz., in 
general the section has, besides the tacnode, no other double point ; that is, the scroll 
has no nodal generator: the section may however have a third double point, and the 
scroll has then a nodal (double) generator. In the case a — 3, /9 = 1 the section has 
a triple point, and the line of approach is the tangent at one of the branches at the 
triple point; the scroll has a twofold, say a 3 (+ 1) tuple directrix line: as the section 
admits of no further singularity, this is the only case. The foregoing enumeration 
gives three species of quartic scrolls /9(1, 1, 4), and three species of quartic scrolls 
S (1, 1, 4), together six species, viz. these are as follows : 
Quartic Scroll, First Species, S (1 2 , 1 2 , 4), with two double directrix lines, 
and without a nodal generator. 
38. Taking (x — 0, y = 0) and (z = 0, w — 0) for the equations of the two directrix 
lines respectively, the equation of the scroll is 
(*$#, y?{z, w) 2 = 0. 
Quartic Scroll, Second Species, S' (1 2 , 1 2 , 4), with two double directrix lines 
and with a double generator. 
39. This is in fact a specialized form of the first species, the difference being that 
there is a nodal (double) generator. Supposing as before that the equations of the 
directrix lines are {x = 0, y — 0) and (z = 0, w = 0) respectively ; let the equations of the 
nodal generator be (x + y = 0, z + w — 0) ; then, observing that for the first species the 
equation may be written (*\x, y) 2 (z, z + w) 2 = 0, it is clear that if the terms in z 2 and 
z(z + w) are divisible by (x + y) 2 and (x + y) respectively, the surface will have as a new