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ON SKEW SURFACES, OTHERWISE SCROLLS. 
[339 
symbols, used (according to the context) to denote geometrical forms, or else the orders 
of these forms. Thus G (on, n, p, q) denotes either the lines meeting the curves 
on, n, p, q each of them once, or else it denotes the order of such system of lines, 
that is, the number of lines. And so S (on, ox, p) denotes the Skew Surface or Scroll 
generated by a line which meets the curves on, n, p each once, or else it denotes the 
order of such surface. 
4. G (on, n, p, q): the signification is explained above. 
5. S (to, n, p): the signification has just been explained; but as the surfaces 
8(m, ox, p), S (to 2 , oi), S(m 3 ) are in fact the subject of the present memoir, I give the 
explanation in full for each of them, viz. S (on, n, p) is the surface generated by a 
line which meets the curves to, ox, p each once; S (on 2 , ox) is the surface generated by 
a line which meets the curve on twice and the curve n once; S (on 3 ) the surface 
generated by the line which meets the curve on thrice. As already mentioned, these 
surfaces and their orders are represented by the same symbols respectively. 
6. ND (on, oi, p). The directrix curves on, n, p of the scroll S (on, ox, p) are nodal 
(multiple) curves on the surface, viz. on is an wp-tuple curve, and so for n and p. 
Reckoning each curve according to its multiplicity, viz. the curve on being reckoned 
i [np]' 1 times, or as of the order on. \ [oxp] 2 , and so for the curves oi and p, the 
aggregate, or sum of the orders, gives the Nodal Director ND (on, n, p). 
7. NG(nx, oi, p). The scroll S (on, n, p) has the nodal generating lines G (on-, oi, p), 
G (on, v?, p), G (on, ox, p 2 ). Each of these is a mere double line, to be reckoned once 
only, and we have thus the Nodal Generator 
JS r G (on, n, p)= G (on 2 , n, p) + G (on, oi 2 , p)+ G (on, n, p 2 ). 
But to take another example, the scroll S (on 2 , n) has the nodal generating lines 
G (on 3 , oi), each of which is a triple line to be reckoned \ [3] 2 , that is, three times, 
and also the nodal generating lines G (on 2 , oi 2 ), each of them a mere double line 
to be reckoned once only; whence here NG(oox 2 , ?i) = 3G(on 3 , n)+G(on 2 , n 2 ). And so 
for the scroll S (on 3 ), this has the nodal generating lines G (to 4 ), each of them a 
quadruple line to be reckoned \ [4] 2 , that is, six times; or we have NG (on 3 ) = 6G (on 3 ). 
8. NR (on, ox, p). The scroll S (on, n, p) has besides the directrix curves on, ox, p 
or Nodal Director, and the nodal generating lines or Nodal Generator, a remaining 
nodal curve or Nodal Residue, the locus of the intersections of two non-coincident 
generating lines meeting in a point not situate on any one of the directrix curves. 
This Nodal Residue, as well for the scroll S (oox, ox, p) as for the scrolls S(oox 2 , ox) and 
8'(to 3 ) respectively, is a mere double curve to be reckoned once only; and such curve 
or its order is denoted by NR, viz. for the scroll S(on, n, p), the Nodal Residue is 
NR (on, n, p). 
9. NT (on, n, p). The Nodal Director, Nodal Generator, and Nodal Residue of the 
scroll S (on, ox, p) form together the Nodal Total NT (on, n, p), that is, we have 
NT (on, n, p) = N1) (on, ox, p) + NG (to, n, p) + NR (on, ox, p); 
and similarly for the scrolls 8 (on 1 , ox) and S (to 3 ).