449] SKETCH OF RECENT RESEARCHES UPON QUARTIC AND QUINTIC SURFACES. 245 
The references, by the name of the author, and number (if any) of his paper, are 
to the subjoined list of Memoirs. 
As to the scrolls, we have Cayley (3) and (4), and Cremona ; the division into 
12 species is, I believe, complete : see post, the remarks upon Schwarz’s paper on 
quintic scrolls. 
As regards the non-scrolar surfaces : 
1. Without a singular curve. The surface may be without a cnicnode (conical 
point), or it may have any number of cnicnodes up to 16, Cayley (7) : the cases of 
singularity higher than a cnicnode are probably very numerous, but they have been 
scarcely at all examined. The memoir just referred to relates chiefly to the several 
cases of not more than 10 nodes; the cases of 11, 12, 13, 14, 15, 16 nodes are con 
sidered incidentally, Rummer (2), but it was not the object of his paper to make an 
enumeration, and there may be cases which are not considered ; the discussion of the 
cases considered is very full and interesting. The case of 16 nodes is also considered, 
Rummer (1). As to the surface with 16 nodes, it is to be remarked that the wave- 
surface, or generally the surface obtained by the homographic deformation of the wave- 
surface—called, Cayley (1), the “ tetrahedroid ”—is a special form of surface with 16 
nodes : its relation to the general surface is explained, Cayley (2). 
2. Quartic surface with nodal line : considered incidentally, Clebsch (2) and (3). 
There are through the nodal line 8 planes, each meeting the surface in a line-pair : 
considering any 7 of these, and taking out of each of them a line, the 7 lines are 
met by a conic which also meets a determinate line out of the remaining line-pair ; 
there are thus on the surface 2 7 , = 128, conics; viz., these form 64 pairs, each pair 
lying in a plane, and being the complete intersection of the surface by such plane ; 
the number of these planes is of course = 64. 
Although not properly included in the present case, I mention the quartic surface 
which is the reciprocal of the cubic surface XIX = 12 — B 6 — C 2 , Cayley (5): the nodal 
curve is here an oscnodal line counting as three nodal lines. 
3. Quartic surfaces with nodal conic. Such a surface may be without cnicnodes, 
or it may have 1, 2, 3, or 4 cnicnodes ; the cases, other than that of 3 cnicnodes, are 
mentioned, Rummer (3) ; but the question is examined, and the remaining case of 3 
cnicnodes established, Cayley (6). 
The general case of the nodal conic without cnicnodes is elaborately considered, 
Clebsch (1): it is shown that there are on the surface 16 lines, each meeting the 
conic, and which in their arrangement are strikingly analogous to the 27 lines on a 
cubic surface ; viz., if on a cubic surface we select at pleasure any one of the 27 
lines, and through this line draw a plane which besides meets the cubic surface in a 
conic ; then, disregarding the line in question and the ] 0 lines which meet it, the 
remaining 16 lines each meet the conic, and are related to it and to each other in 
the same manner that the 16 lines of the quartic surface are related to the nodal