PROFESSOR CAYLEY ON SKEW STJEEACES, OTHERWISE SCROLLS. 567 
which is of the order a+2|3, =2 m—u, so that the a(-f-/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 5=0 ; the factor y 3 divides 
out, and the equation is 
%{ayw — cyz - dxzf( y y+P-w— 
which is of the order «+/3, =m, so that here the reduction caused by the a(-f/3)tuple 
point is =a+/3. We may without loss of generality substitute ax for cy+dx , and 
then, putting also a=l, we find that when the equation of the curve m is as before 
X(ywY(*Xx, yY+^'= 0, 
but the plane through the directrix line (#=0, y= 0), and the point on this line, are 
respectively given by the equations x—Xy, z=xw, the equation of the scroll is 
2(yw-xzf'(*X^ y) a+p ~^= 0. 
23. The result maybe verified by considering the section by any plane y—Xx through 
the directrix line. Substituting for y this value, we find 
X^x^'ikw - zf(*X 1 , x )«+^=0, 
which is of the form 
x«(fXx, 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, . . .XI, yw-xzf= 0, 
where U, V, W, . . . are functions of x , y of the forms 
(*)>> y) m > (* y) m ~\ (*0O> y) m ~\ • • • 
Assuming that these contain respectively the factors 
(y—zxy, (y-zxy~\ (y—zxy - 2 . . ., 
where then the equation takes the form 
(U 7 , V', W' . . -Xy—xx, w(y—zx)-\-x(zw—z)) y = 0, 
where the coefficients TJ', Y', W', . . . are functions of x, y, z, w of the orders m — y, 
m— y— 1, m— y— 2, . . . ; or, what is the same thing, the equation is 
(U", V", W", . .. Jg-xx, zio-zy= 0, 
where U", Y", W", . . . are functions of x, y, z, w of the order m — y. The scroll has 
thus the y-tuple generating line 
y — zx=0. zw—z=0. 
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