PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
119 
and we have therefore 
or omitting the first equation, we have (independent of §) a system which is clear must 
be equivalent to a single equation. 
71. I take any one of these equations, for instance the equation 
or, what is the same thing, 
qrZ-r 2 Y+(pr-q 2 )W=0, 
and I proceed to reduce it so as to obtain the result in a symmetrical form. For this 
purpose I observe that from the values of a, (3, y, £, if only AF-J-BG+CH=|=0, we 
have 
X : Y : Z : W= ( . , 
-c, 
B, 
-F* 
:( C, 
. , 
-A, 
-GX 
:(-B, 
A, 
• 5 
-HI 
:( F, 
G, 
H, 
•I 
and substituting these values, the equation in question becomes 
qr(— Bes+A/3— H§) 
— r 2 ( C a — Ay — ) 
+(pr-q 2 )( F« + Gj3+H y )=0. 
This becomes 
Ar(q/3+r y) 
— Bqm 
— Cr 2 a 
+F(pr— q 2 )« 
+ G{r$+(pr— q 2 )/3 
+H{— qr^+(pr-q 2 ) 7 } 
=Ar( — pa) =0 
— Bqra 
— Cr 2 « 
+F(pr-q> 
-}- Gpqcs 
— Hp 2 a, 
viz. the whole equation divides by a ; and, omitting this factor, the equation is 
Apr + Bqr + Cr 2 + F(q 2 - pr) - Gpq + Hp 2 = 0, 
or, what is the same thing, it is 
(H, F, C, B, A-F, -G5Cp, q, r)*=0, 
where I recall that we have 
P> q, r=/3^ — y\ Py—och, ay— /3 2 , 
MDCCCLXIX. E 
