PROFESSOR CAYLEY ON CUBIC SURFACES. 
321 
Reciprocal Surface. 
201. This is 27xzw— y 3 =0, viz. it is a cubic surface of the form XXI=12 — 3B 3 . 
There is no nodal curve, b'=0, and no cuspidal curve, c'=0. Moreover If =3. 
Synopsis for the foregoing sections. Article No. 202. 
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-I 2 -B 3 
9 
9 
12=3x4 
16=4x4-4 
IV— 12— 2C 2 
7 
7 
10=3x4-2 
12=4x4-2-2 
V=12-B 4 
7=5+ edge twice 
7=5+rec. of edge twice, 
rec. of edge tacnodal 
10=3x4-2 
12=4x4-4 
VT=12— B 3 — C 2 
3 
3 
9=3x4— 2 
10=4x4-4-2 
VII=12— Bj 
3=2+edge 
3=2+ree. 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 a 
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=l+edge twice 
3=l+rec. of edge twice, 
rec. of edge is tacnodal 
6=2x3 
6=2x3 
XI=12-B 0 
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 
XTV=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— 4C 2 
3 
3 
none 
none 
XVn= 12 - 2 C 3 -C 2 ... 
none 
none 
2 = conic 
2 = conic 
XVni=12-B 4 -C 2 ... 
3=1+ edge twice 
1 + rec. of edge twice, 
rec. of edge lacnodal 
none 
none 
XIX=12— B 6 — C a ... 
3= axis 3 times 
3= rec. of axis 3 times, 
rec. of axis oscnodal 
none 
none 
XX=12— U 8 
none 
none 
2 = conic 
2 = conic 
XXI=12— 3B 3 
none 
none 
none 
none 
I pass now to the two cases of cubic scrolls. 
mdccclxix. 2 x 
