PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
12 ; 
YW 3 is AC 2 - 3FC 2 +B 2 C, =AC 2 +B 2 C- 3C 2 F. Or, again, the coefficient of Y 3 Z being 
BFH— F 2 G, that of YZ 3 is — GFC+F 2 B, =BF 2 -CFG. 
77. But the equation may be written in the much more simple form 
X(-« 2 S + 3«/3y-2/3 3 ) (Kec. III.) 
+ Y ( — a/3^ + 2 ay 2 — /3 2 y) 
H-Z ( ay<$ — 2/3 2 S + /3y 2 ) 
+W( «S 2 -3/3yS+2y 3 )=0, 
or, what is the same thing, 
-i(3Xd s -Yd 7 + Zd fi -3Wd a )(ocT-6a/3y^~l-4ay 3 +4j3^-3j3y)=0, . (Bee. III.) 
as may be verified by actual substitution of the values of the coordinates. 
78. By what precedes, substituting for p, q, r their values in terms of a, /3, y, S, it 
appears that we have the remarkable identity 
(H, F, C, B, A-F, — GHQ3& — y 2 , /3y -a&, ccy-f 3 2 ) 2 
= (AF+BG + CH)X 
< X(-« 2 S + 3«/3y-2/3 3 ) ] 
I +Y ( — «j3^ + 2 ay 2 — /3 2 y) I 
H + m k 
+W( ah 2 -3j3yS+2y 3 ) 
79. In the case above considered of the tenth species, S(3 2 ), for which 
AF+BG+CH=|=0, the three forms of the reciprocal equation are of course absolutely 
equivalent to each other. The first form has the advantage of putting in evidence the 
fact that the reciprocal scroll is also of the tenth species ; the other two forms do not, 
at least obviously, put in evidence any special property of the reciprocal scroll. 
Reciprocals of Eighth Species , S(l, 3 2 ), and Ninth Species , S(l 3 ). 
80. If AF+BG+CH = 0, then the equation 
(H, F, C, B, A-F, -GJp, q, rf = 0 
is a scroll of the eighth species, S(l, 3 s ). The first form of the reciprocal equation 
becomes identically 0=0, on account of the evanescent factor AF+BG+CH, but the 
second and third forms continue to subsist, and either of them may be taken as the 
equation of the reciprocal scroll. Taking the third form, and calling to mind the sig- 
nifications of (a, (5 , y, l), viz. 
a=( . , H, -G, A£X, Y, Z, W), 
/3=(-H, ., F,BJ „ ), 
y-( G, -F, .,cx 
a=(-A, -B, -C, .£ 
), 
), 
