X
-,S. [340'
340] A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 209
)r /3), if the
f y) y ~\ the
which equation, substituting therein for B its value in terms of X, gives the parameter
p which enters into the equations of the generating line ; or, what is the same thing,
the equation of the scroll is obtained by eliminating X, B, p from the equation just
mentioned and the equations
hing, x, y, z, w,
3, z + w — 0 as
y = Xx, z — Bw — px, 6 = ^.
* 1 cX + d
22. These last three equations give
to 24.
^ V g _ ay+ bx Bw-z (ay + bx) w — (cy + dx) z
x’ cy + dx’P x x
directrix line,
section of the
0), or foot of
r the equation
tead of being
; line and the
4 the form
and substituting these values, we find for the equation of the scroll
2 (ay + bxy~P'y p ' [(ay + bx)w — (cy + dx) zf (*\x, yy+P-W = 0,
which is of the order a + 2¡3, = 2m — a, so that the a. (+ /3) tuple point, in the case
actually under consideration, produces only a reduction = a. If however the line of
approach coincides with the tangent of the /3 branches, then 6 = 0; the factor y 3 divides
out, and the equation is
2 (ayw — cyz — dxzY (*][x, yy+P-W = 0,
. inclusive.
which is of the order a + ¡3, = m, so that here the reduction caused by the a (+ /3) tuple
point is =a + /3. We may without loss of generality substitute ax for cy + dx, and
itrix line, then
of this line
then, putting also a = 1, we find that when the equation of the curve m is as before
2 (yw) P (*][x, y) a+p ~ 2/3< = 0,
rectrix line is
but the plane through the directrix line (x = 0, y = 0), and the point on this line, are
f approach is
respectively given by the equations x = Xy, z = Xw, the equation of the scroll is
the common
2 (yw — xzY (*\x, yy+P-W = 0.
vever, for the
23. The result may be verified by considering the section by any plane y ~ Xx
through the directrix line. Substituting for y this value, we find
of intersection
x a Xx^' (Xw — zY (*]£!> Xy +p ~P' = 0,
which is of the form
x a (•!*$#, Xw — zf = 0;
so that the section is made up of the directrix line (x = 0, y = 0) reckoned a. times and
of /3 lines in the plane y — Xx = 0, the intersections of the plane y — Xx = 0 by planes
such as z = Xw —px.
Case of a <y-tuple generating line.
24. The equation of the scroll may be written
(U, V, W, yw — xzy = 0,
C. V.
27
