124 PROFESS OK CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
it is to be observed that a= 0, (3=0, y=0, o=0 are the equations of four planes passing 
through a common line, viz. the line whose coordinates are (A, B, C, F, G, H), and the 
equation thus puts in evidence that this line is a triple line on the reciprocal scroll ; 
that is, the reciprocal scroll is a scroll of the ninth species, S(l 3 ). Or stating the theorem 
more completely: For the scroll, eighth species, S(l, 3 2 ), 
(H, F, C, B, A-F, -GXp, q, r) 2 =0, 
generated by a line meeting the line (F, G, H, A, B, C), and the skew cubic p— 0, 
q=0, r=0 twice, the reciprocal scroll is of the ninth species, S(l 3 ), 
X ( — a 2 & +3aj3y — 2/3 3 ) 
-J-Y (■— e4/3§-j-2ay 2 — /3 2 y) 
+ Z ( ay&-2/3«& + (3y 2 ) 
+ W( a § 2 -3/3y>+2/ ), 
having for its triple line the reciprocal line (A, B, C, F, G, H). 
81. It should of course be possible, starting from the equation 
(*XX, Y) 4 +Z(*£X, Y) 3 +W(#'XX, Y) 3 =Q 
of a scroll S(l 3 ), to obtain the equation of the reciprocal scroll S(l, 3 2 ). But I content 
myself with a very particular case. I consider the equation 
Y 2 Z 2 — Y 3 W — Z 3 X = 0 , 
which belongs to a scroll S(l 3 ) having the line Y=0, Z=0 for its triple line. To find 
the equation of the reciprocal scroll, write 
— ’Zi 1 -\-'kX'= 0, 
2YZ 2 — 3Y*W+ty=0, 
2Y 2 Z— 3Z 2 X+ta=:0, 
— Y 3 +Aw, 
we find without difficulty, reducing by means of the equation of the scroll, 
7, 2 (yw-z 2 )=- 3Z 2 {Y 4 +3XZ(XZ-Y 2 )}, 
X\xw-yz)= 3Y 2 Z 2 { YZ- 3WX}, 
\\xz — y 2 ) = — 3 Y 2 { Z 4 + 3 YW ( YW — Z 2 ) } . 
Hence writing for a moment 
{Y 4 -f3XZ(XZ— Y 2 )} {Z 4 +3YW(YW— Z 2 )} — Y 2 Z 2 . (YZ — 3WX) 2 , 
we have 
G = Y 4 Z 4 + 3Y 5 W (YW — Z 2 ) -j- 3Z 5 X(XZ — Y 2 ) +■ 9XYZW(Y 2 Z 2 — Y 3 W — Z 3 X+XYZW) 
— Y 4 Z 4 +6Y 3 Z 3 PW- 9Y 2 Z 2 X 2 W 2 , 
