450 
A MEMOIR ON CUBIC SURFACES. 
[412 
Reciprocal Surface. 
201. This is 1*lxzw — y 3 = 0, viz. it is a cubic surface of the form XXI = 12 — SB 3 . 
There is no nodal curve, b' = 0, and no cuspidal curve, c = 0. Moreover B' = 3. 
Article No. 202. Synopsis for the foregoing sections. 
202. I annex the following synopsis, for the several cases, of the facultative lines 
(or node-couple curve) and of the spinode curve of the cubic surface; also of the 
nodal curve and the cuspidal curve of the reciprocal surface. It is to be observed 
that in designating a curve, for instance, as 18 = 4x5 — 2, this means that it is a 
curve of the order 18, the partial intersection of a quartic surface and a quintic 
surface, but without any explanation of the nature of the common curve 2 which 
causes the reduction, viz. without explaining whether this is a conic or a pair of lines, 
and so in other cases; this may be seen by reference to the proper section of the 
Memoir. 
Facultative lines. 
Nodal curve. 
Spinode curve. 
Cuspidal curve. 
1=12 
27 
27 
12 = 3x4 
24 = 6x4 
II = 12 - C 2 
15 
15 
12 = 3x4 
18 = 4x5-2 
III = 12-£ 3 
9 
9 
12 = 3x4 
16 = 4x5-4 
IV = 12 - 2<7 2 
7 
7 
10 = 3x4-2 
12=4x4-2-2 
V=12 - B i 
7 = 5 +edge twice 
7 = 5 + rec. of edge twice, 
rec. of edge tacnodal 
10 = 3x4-2 
12 = 4x4-4 
VI= 12-£ 3 -<7 2 
3 
3 
9=3x4-3 
10=4x4-4-2 
VII = 12-£ 5 
3 = 2 + edge 
3=2 +rec. of edge, 
rec. of edge is cuspnodal 
9 = edge + unicursal 
8-thic 
10=rec. of edge + 
unicursal 9-thic, 
rec. of edge is cuspnodal 
VIII=12 - 3C 2 
3 
3 
6 = 2x3 
6 = 2x3 
IX = 12 - 2B 3 
none 
none 
8 = 4 conics 
8 = 4 conics 
X = 12 - B 4 - C 2 
3 = 1 + edge twice 
3 = 1 + rec. of edge twice, 
rec. of edge is tacnodal 
6 = 2 x 3 
6 = 2 x 3 
XI = 12-B 6 
3 = edge 3 times 
3 = rec. of edge 3 times, 
rec. of edge is oscnodal 
6 = 3 conics 
6 = 3 conics 
XII = 12- U 6 
3 
3 
6 = 2x3 
6=2x3 
XIII = 12 - B 3 - 2C 2 
1 
1 
4 = 2x2, nodal qua- 
driquadric 
4 = 2x2 quadriquadric 
XIV = 12 - B 5 - C 2 
1 = edge 
l = rec. of edge, 
rec. of edge is cuspnodal 
4 = 3 + edge 
4=3 +rec. of edge, 
rec. of edge is cuspnodal 
XV=12- U 7 
1 
1 
4 = 2x2, nodal qua- 
driquadric 
4 = 2x2 cuspidal qua 
driquadric 
XVI = 12-4 C 2 
3 
3 
none 
none 
XVII = 12-2 B 3 -C. 2 
none 
none 
2 = conic 
2 = conic 
XVIII = 12- J5 4 - 2C 2 
3 = 1 + edge twice 
1 + rec. of edge twice, 
rec. of edge tacnodal 
none 
none 
XIX = 12 - B 6 - C. 2 
3 = axis 3 times 
3 = ree. of axis 3 times, 
rec. of axis oscnodal 
none 
none 
XX = 12- U 8 
none 
none 
2 = conic 
2 = conic 
XXI = 12-3 B 3 
none 
none 
none 
none 
I pass now to the two cases of cubic scrolls.