566 
PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
Equation of the Scroll S(l, 1, m) of the Order m, Article Nos. 19 to 24. 
19. We may take x=0, y=0 for the equations of the twofold directrix line, z = 0 
for the equation of the plane of the curve m (an arbitrary plane section of the scroll). 
•Then (a-f-j3 =m), if the curve m have at the point (#=0, y= 0), or foot of the directrix 
line, an a(+(3)tuple point, and if moreover we have y=0 for the equation of the 
common tangent of the (3 branches (viz. if the plane y=0, instead of being an arbitrary 
plane through the directrix line, be the plane through this line and the common tangent 
of the (3 branches), the equation of the curve m will be of the form 
yr^'= o, 
where the summation extends to all integer values of |3' from 0 to (3 , both inclusive. 
20. Taking y—\x for the equation of any plane through the directrix line, then the 
corresponding point on the directrix line will be the intersection of this line (#=Q, 
y=0) by the plane z=dw, where 0= ; the foot of the directrix line is given by the 
value 0=0, or X= — -? and the equation of the line of approach is therefore y=—^x; 
this should coincide with the line y— 0, which is the common tangent of the (3 branches; 
that is, we must have 5=0; I retain, however, for the moment the general value of 5. 
21. The equations of a generating line will be 
y=\x, z=0w—px; 
and then taking X, Y, (Z=0) and W for the coordinates of the point of intersection 
with the curve m, we have 
Y=xX, O = 0W-^X, 
2(YWf(*XX, Y)“ + ^'=0, 
and thence 
s(^)%ii^r'- 2 "=o, 
or, what is the same thing, 
(*X 1 , *.)“ + ^'=0 ; 
which equation, substituting therein for 0 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, 0, p from the equation just mentioned 
and the equations 
. . a\ -)- b 
y=™’ 6= ~i ■ 
22. These last three equations give 
. y a __ay + bx 6w—s (ay + bx)w — {cy + dx)s . 
x cy-\- ax x x 
and substituting these values, we find for the equation of the scroll 
1,{ay + bxf~ p f[(ay + bx)w — (cy+dx)z~f\*Jx, y) a+p ~ 2p '=0, 
