v 
340] A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 
207 
LLS. [340 
3 corresponding- 
regard to the 
: the plane of 
the coincident 
i the line of 
1 (that is, the 
through a be 
plane by the 
id the foot of 
le of approach. 
Le signification 
by the plane 
merally, meets 
>ear that the 
nes, and that 
lis is not the 
ne 1' must be 
is a line not 
If, however, 
nt—that is, if 
3urve in such 
curve on in 
— a — 1. And 
;ent, then the 
secutive point 
y the branch, 
,s the a-tuple 
fold tangent; 
say a united 
of the scroll 
ind a /3-tuple 
tiple point in 
inches having 
') tuple point 
iut the order 
e case where 
(1, 1, on) are 
= 2 to, is = m. 
That is, we may suppose that the curve m has a united a (+ /3) tuple point (a + /3 = on), 
and may take the directrix line to pass through this point, and the line of approach 
to be the common tangent of the /3 branches; and this being so, the order of the 
scroll will be 2m — a — /3, = m. It may be added that if the curve on has, besides the 
a (+ /3) tuple point, a y-tuple point, then the scroll will have a 7-tuple generating line, 
and so for the other multiple points of the curve on. 
16. We may, in the same way as for the scroll 8 (1, 1, on), consider the different 
sections of the scroll >3(1, 1, on) of the order on. The general section is a curve of the 
order 001, having an a(-f/3)tuple point at the intersection with the directrix line, and 
a 7-tuple point, &c. corresponding to the multiple generating lines, if any. A section 
through the directrix line is in general made up of this line counting a times, and 
of /3 generating lines through the point which corresponds to the plane of the section; 
if the section pass also through a 7-tuple generating line (7 ;(> /3, in the same way 
as for the scroll >3(1, 1, m)), then, of the /3 generating lines, 7 unite together in the 
7-tuple generating line. The general section through a 7-tuple generating line breaks 
up into this line counting 7 times, and a curve of the order on — 7, which has on 
the directrix line an a — 7 (+ /3 — 7) tuple point and a S-tuple point, &c. at its inter 
sections with the other multiple generating lines, if any. 
Equation of the Scroll >3(1, 1, on) of the order on, Article Nos. 17 and 18. 
17. Taking for the equations of the directrix lines (x = 0, y — 0) and (2 = 0, w = 0), 
and supposing that these are respectively an a-tuple line and a /3-tuple line on the 
scroll a + /3 = on, it is obvious that the equation of the scroll is 
(*$#, y) a (z, wf = 0. 
In fact starting with this equation, if we consider the section by a plane through the 
line («=0, y— 0), say the plane y = \x, then the equation gives 
x a ( *][1, X) a (z, ovf = 0 ; 
that is, the section is made up of the line (x = 0, y = 0) reckoned a times, and of 
/3 other lines in the plane y = \x\ and the like for the section by any plane through 
the line (z — 0, w = 0), say the plane z = vou. Hence the assumed equation represents 
a scroll of the order on, having the two lines for an a-tuple line and a /3-tuple line 
respectively, and conversely such scroll has an equation of the assumed form. 
Case of a y-tuple geoierating line. 
18. The multiple generating line meets each of the lines (x = 0, y — 0) and (z= 0, w=0); 
and we may take for the equations of the multiple generating line x + y = 0, z + w = 0. 
This being so, the foregoing equation of the scroll may be expressed in the form 
(+$>> yY{z, z + wf = 0, 
( U, V, IF,..) (z, z + wf = 0, 
or say