ON SKEW SURFACES, OTHERWISE SCROLLS.
169
[339
339] ON SKEW SURFACES, OTHERWISE SCROLLS. 169
CLIII. (for
863.]
I will also, though it is less closely connected with the subject of the present memoir,
refer to a paper by M. Chasles, “Description des courbes à double courbure de tous
les ordres sur les surfaces réglées du troisième et du quatrième ordre’X 1 ).
The present memoir (in the composition of which I have been assisted by a
correspondence with Dr Salmon) contains a further development of the theory of the
skew surfaces generated by a line which meets a given curve or curves : viz. I con
sider, 1st, the surface generated by a line which meets each of three given curves of
the orders on, n, p respectively ; 2nd, the surface generated by a line which meets a
given curve of the order on twice, and a given curve of the order n once; 3rd, the
surface which meets a given curve of the order on three times ; or, as it is very
convenient to express it, I consider the skew surfaces, or say the “ Scrolls,” S (on, oi, p),
S (on-, oi), S (on 3 ). The chief results are embodied in the Table given after this intro
duction, at the commencement of the memoir. It is to be noticed that I attend
throughout to the general theory, not considering otherwise than incidentally the effect
of any singularity in the system of the given curves, or in the given curves separately :
the memoir contains however some remarks as to what are the singularities material
to a complete theory ; and, in particular as regards the surface S (on 3 ), I am thus led
to mention an entirely new kind of singularity of a curve in space—viz. such a curve
has in general a determinate number of “ lines through four points ” (lines which
: “ On the
bhe order n
-2) points,
(, the class
9n a Class
line which
e shown to
ain number
ese, it was
l right line
is found to
the curve,
the Degree
ven for the
meet the curve in four points) ; it may happen that, of the lines through three points
which can be drawn through any point whatever of the curve, a certain number will
unite together and form a line through four (or more) points, the number of the lines
through four points (or through a greater number of points) so becoming infinite.
Notation aoid Table of Results, Articles 1 to 10.
1. In the present memoir a letter such as on denotes the order of a curve in
space. It is for the most part assumed that the curve has no actual double points
or stationary points, and the corresponding letter M denotes the class of the curve
taken negatively and divided by 2 ; that is, if h be the number of apparent double
points, then M=—^ [inf + h : here and elsewhere [on] 2 , &c. denote factorials, viz.
[m] 2 = on (on — 1), [onf = m (on — 1) (on — 2), &c. It is to be noticed that for the system
of two curves on, on', if h, h' represent the number of apparent double points of the
two curves respectively, then for the system the number of apparent double points is
Qg a given
> generated
;. And in
r line the
any linear
ne theories
tensions ”( 4 ).
= mom! + h + In!, and the corresponding value of M is therefore — \ [m + on] 2 + mon + It + h!,
which is = — % [on] 2 + h — f [on'] 3 + h', which is =M + M'.
2. The use of the combinations (on, oi, p, q), (on 3 , n, p), &c. hardly requires ex
planation ; it may however be noticed that G (on, oi, p, q) denoting the lines which
meet the curves on, n, p, q (that is, curves of these orders) each of them once,
G (on 3 , oi, p) will denote the lines which meet the curve on twice and the curves n
and p each of them once ; and so in all similar cases.
3. The letters G, S, ND, NG, NR, NT (read Generators, Scroll, Nodal Director,
Nodal Generator, Nodal Residue, and Nodal Total) are in the nature of functional
1 Comptes Rendus, t. lui. (1861, 2 e Sem.), pp. 884—889.
C, Y.
22
