122 
PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
76. Throwing out the factor in question, AF+BG+CH, the equation of the reci- 
procal scroll is found to be 
0= X 4 .H 3 (Rec. II.) 
+X 3 Y . — 2GH 2 
+X 3 Z . AIT 2 — 3FIT 2 + G 2 H 
+X 3 W . — 3AGH— 3BH 2 +G 3 
+X 2 Y 2 . 2FH 2 +G 2 H 
+X 2 YZ . — AGH + BH 2 + 2FGH — G 3 
+X 2 YW. A 2 H + 3AFH+ AG 2 +BGH— 2CH 2 — 3FG 2 
+X 2 Z 2 . — 2 AFH + AG 2 -f CH 2 — FG 2 + 3F 2 H 
+X 2 Z W . - 2 A 2 G - 2 ABH + 3 AFG + 6BFH - BG 2 - CGH 
+X 2 W 2 . A 3 +3ABG-3ACH + 3B 2 H+3CG 2 
-fXY 3 -2FGH 
-f X Y 2 Z AFH - BGH - 3F 2 H + 2FG 2 
+ XY 2 W 2 ABH - 2 AFG - BFH + 2 CGH + 3F 2 G 
+XYZ 2 — AFG — 2BFH + BG 2 — CGH 
+XYZW A 2 F-3AF 2 -3ABG+ACH-2B 2 H+BFG+5CFH-2CG 2 
+XYW 2 2A 2 B-3ABF-F2ACG+B 2 G+BCH-6CFG 
+XZ 3 i\.F 2 — 2 CFH -F CG 2 — F 3 
-F XZ 2 W 2 ABF - 2 ACG - 2BCH - 3BF 2 + CFG 
+XZW 2 A 2 C + AB 2 + 3 ACF — 3B 2 F+ BCG — 2C 2 H 
+XW 3 3 ABC — B s + 3C 2 G 
-F Y 4 F 2 H 
+Y 3 Z BFH— F 2 G 
+ Y 3 W AF 2 + B 2 IT — 2 CFH - F 3 
+Y 2 Z 2 -BFG+CFH+F 3 
+Y 2 ZW ABF — B 2 G + BCH + 2CFG 
+Y 2 W 2 AB 2 — 2 ACF — B 2 F + C 2 H + 3CF 2 
+YZ 3 BF 2 — CFG 
+ YZ 2 W ACF + 2B 2 F — BCG — 3CF 2 
+ YZW 2 ABC + B 3 — 2BCF — C 2 G 
+YW 3 AC 2 +B 2 C-3C 2 F 
+Z 4 CF 2 
+ Z 3 W 2BCF 
+Z 2 W 2 B 2 C + 2C 2 F 
+ZW 3 2BC 2 
+W 4 C 3 , 
where, in regard to the symmetry of this equation, it is to be observed that we may 
interchange X and W, and Y and Z, leaving A, F unaltered but interchanging B and 
— G, and also C and H ; thus the coefficient of X 3 Z being AH 2 — 3FH 2 + G 2 IT, that of 
