326 A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. [410 
scroll; that is, the reciprocal scroll is a scroll of the ninth species, S (I s ). Or stating 
the theorem more completely: For the scroll, eighth species, >5(1, 3 2 ), 
(H, F, C, B, A-F, - 6%, q, r)> = 0, 
generated by a line meeting the line (F, G, H, A, B, G), and the skew cubic p = 0 
q = 0, r = 0 twice, the reciprocal scroll is of the ninth species, S (l 3 ), 
X (- a-8 + 3a(3ry - 2/3 3 ) 
4- F (— a/38 4- 2ay 2 — /3 2r y) 
+ Z ( cvy8 — 2/3 2 S 4- /3y 2 ) 
4- IF ( a.8- — 3/3y8 4- 27 3 ), 
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 
(*#X, F) 4 Z (* \X, Y) 3 +W(* r %X, F) 3 = 0 
of a scroll >5(1 3 ), to obtain the equation of the reciprocal scroll >5(1, 3 a ). But I 
content myself with a very particular case. I consider the equation 
Y-Z 2 - Y 3 W- Z 3 X = 0, 
which belongs to a scroll >5(1 3 ) having the line F=0, Z = 0 for its triple line. To 
find the equation of the reciprocal scroll, write 
— Z 3 +\x = 0, 
2 YZ 2 — 3 F 2 TF 4- = 0, 
2 Y-Z — SZ 2 X +\z = 0, 
— F 3 4- Xw = 0, 
we find without difficulty, reducing by means of the equation of the scroll, 
X 2 (;jw-z 3 ) = - 3£ 2 {F 4 + 3XZ (XZ - F 2 )}, 
X 2 (aw — yz) — 3 Y-Z- {YZ - 3 WX\, 
X 2 (xz - y-) = - 3 F 2 {Z* 4- 3 YW (YW - Z% 
Hence writing for a moment 
n = {F 4 4- 3X£(XZ - F 2 )} {Z 4 4- 3YW (YW - Z 2 )} - Y 3 Z 3 (YZ-3 WXf, 
we have 
n= Y*Z 4 + 3Y 5 W(YW—Z 3 )4- 3Z 5 X(XZ — Y*) + 9XYZW(Y S Z* - Y 3 W- Z 3 X + XYZW) 
- F 4 F 4 4- 6 Y 3 Z 3 X IF - 9 Y-Z-X 2 IF 2 ,