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