574 
PEOFESSOE CAYLEY ON SKEW STTEFACES, OTHEEWISE SCEOLLS. 
46. The equations of a generating line (since it meets the directrix line #=0, y~0) 
may be taken to be 
x=oty, z=p(y—jy> 
the condition in order to the intersection of the generating line with the nodal conic is 
at once found to be 
ay 2 + 2hy + b -f 2/3( /+ go.) + c/3 2 — 0, 
and that for its intersection with the simple conic 
aa 2 +2/ia+5 + 20(m+/a)+«?0 2 =O ; 
and writing the equations of the generating line in the form 
the elimination of a, (3, 6 from these four equations gives the required equation of the 
scroll. Writing for a moment 
0 =« a 2 +2/w+/3, 
F =gcc +f 
M =la, -\-m , 
we find 
c/3 2 +2F/3 +9=0, 
(0y 2 + 2M yw + dw 2 )(. 3 2 -2 (Qyz + M wz)fi + 0z 2 = 0 ; 
© 
or, introducing at this place the condition c=0, the first equation gives j3= — , and 
we thence obtain 
0 ( 0^ 2 + 2 Myw + dvf) + 4F( Qyz + Mwz) + 4FV = 0 , 
or, what is the same thing, 
(0^+2Fz) 2 +2Mw(0^+2F*)+0<Zw 2 =O ; 
^_ ax' 2 -\-2hxy + by* ^ ^ ffx+fy , ^ ._ljc+wy. 
the equation of the scroll is 
(ax 2 + 2hxy + by 2 + 2gzx + 2fyz f 
+ 2(ax 2 +2hxy + by 2 +2gzx + 2 \fyz)(lx-\-my)w 
+ (ax 2 +2hxy+by 2 )dw 2 =Q. 
And we see from the equation that the surface contains the line (#=0, y= 0) as a double 
line, the conic 
w=0, ax 2 + 2hxy + by 2 + 2gzx + 2fyz = 0 
as a double curve, also the conic 
2 = 0, ax? + 2hxy + fo/ 2 + 2 Ixw + 2myw + dvf = 0 
