PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
121 
and so on, the successive terms a 2 , a! 2 , &c., 2 af \ 2 af, Sec. being derived by an obvious 
law from the first terms a 2 , 2 af, See . ; and these first terms are merely the coefficients of 
the terms X 4 , X 3 , Y, See. in the development of 
f , = {(a,b,e,d,f,g,h,l,m,oiJX,Y,Z,W) 2 } 2 ; 
viz. this is 
X 4 X 3 Y X 3 Z X 3 W X 2 Y 2 X 2 YZ X 2 YW X 2 Z 2 , X 2 ZW, X 2 W 2 , XY 3 , XY 2 Z, XY 2 W, XYZ 2 
a 2 2 af ‘lag 2al 2ab 2af 2am 2ac 2 an 2 ad 2 bh 2bg 2 Yl 2ch 
-\-h 2 +2 gh +2 M +g 2 +2 gl +l 2 +2 fh +2 hm +2 fg 
XYZW, XYW 2 , XZ 3 , XZ 2 W, XZW 2 , XW 3 , Y 4 , Y 3 Z, Y 3 W, Y 2 Z 2 , Y 2 ZW, Y 2 W 2 , YZ 3 , 
2 If 2dh 2cg 2cl 2 dg 2 dl, b 2 , 2bf, 2 bm, 2bc , 2^ 26d, 2 cf 
-\-2mg +2 Im -\~2>gn -J-2 In +/ 2 + 2fm -j-m 2 
-]-2wA 
YZ 2 W, YZW 2 , YW 3 , Z 4 , Z 3 W, Z 2 W 2 , ZW 3 , W 4 
2cm H-2^f 2dm, c 2 , 2cw, 2cd, 2dw, d 2 
+2/% -\-2mn -\-n 2 
and the values of the coefficients a, b, . . . which enter into the formulae are given by 
means of the following values of p, q, r ; viz. these are 
X 2 Y 2 Z 2 W 2 YZ ZX XY XW YW ZW 
p=/ AH, -F 2 ,-CF, -C 2 , -BF, -AF, BH, -AB, - B 2 , -BCJX, Y,Z,W) 2 , 
V-G 2 +CH + 2FG — 2CG +2CF 
q= ( — GH, BH, -CG, BC, -BG, -AG, AH, A 2 , AB, AC 
+CH +FG +FH +BG -BF +CF A ” h 
- F 2 -CH 
r = ( — H 2 , -FH, -F 2 , AC, FG, 2FH, GH, 2AG, -AF, -2BF 
— B 2 -G 2 +BH +CH - CG^ ” 
75. As an instance of the calculation of a single term, the coefficient of X 4 is 
(H, F, C, B, A-F, -G^AH-G 2 , -GH, -H 2 ) 2 ; 
viz. this is 
H(AH— G 2 ) 2 = A 2 H 3 — 2 AG 2 H 2 + G 4 H 2 
+FH 2 G 2 = FG 2 H 2 
+CH 4 = CH 4 
-f BGH 3 = BGH 3 
+ (A - F)( - AH 3 + G 2 H 2 )= -A 2 H 3 +AG 2 H 2 
+AFH 3 — FG 2 H 2 
— G( — AGH 2 + G 3 H) = AG 2 H 2 — G 4 H 2 
whole term is = (AF+BG-J-CH)H 3 , 
viz. there is the factor AF-j-BG-fCH as mentioned above. 
R 2 
