PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
115 
with five given lines, or say simply the lines which belong to an involution*, I say that 
the locus of a line belonging to the involution, and meeting the skew cubic twice is the 
quartic scroll, tenth species , S(3 2 ). In the particular case where the line (instead of 
belonging to a proper involution) meets a given line, the locus is a quartic scroll, eighth 
species, S(l, 3 2 ). 
66. The analysis is almost identical with that given (Second Memoir, Nos. 47 to 50) 
in regard to the scroll S(l, 3 2 ). Considering a line defined by its “ six coordinates ” 
(a, b, c, f, g , h), the condition which expresses that the line shall belong to an involu- 
tion is 
(A, B, C, F, G, H ){a, b, c,f ; g, h)= 0, 
where (A, B, C, F, G, H) are arbitrary coefficients ; if they are the coordinates of a line, 
that is, if AF+BG+CH=0, then the condition expresses that the line (a, b, c,f, g, h), 
instead of belonging to a proper involution, meets the line (F, G, H, A, B, C). 
We have to determine the locus of the line ( a , b, c,f, g, h) the coordinates whereof 
satisfy the relation 
(A, B, C, F, G, H X«, b, c,f, g, h)= 0, 
and which besides meets the skew cubic yw—z 2 = 0, yz—xw— 0, xz — y 2 = 0. 
The equations of the skew cubic are satisfied by writing therein 
x : y : z : w = 1 : t : f : f; 
and hence taking 6, <p for the parameters of the points of intersection of the line 
(a, b, c,f, g, h) with the skew cubic, we have 
1, 0, 0 2 , Q 3 
l, <P, <P\ f 
as the coordinates of two points on the line in question ; whence forming the expressions 
of the six coordinates of the line, and omitting the common factor <p— Q, these are 
(a,b, c,f, g,h)=bq>, — (^-j-<p), 1, 0 2 -\ -0<p-\-<p 2 , Q(f>(0- |-<p), 0 2 <p 2 , 
and hence the condition of involution gives between the parameters Q, <p the equation 
(A, B, C, F, G, — 0 — <p, 1, b 2 -\-Q<p-\ -<p 2 , Q$(0-\-<p), 0 2 <p 2 ). 
* The theory is explained in my memoir “ On the Six Coordinates of a Line,” Camb. Phil. Trans, t. xi. 1868. 
In explanation of the subsequent analytical investigations of the present memoir, it is convenient to remark 
that if on a given line we have the two points (a, /3, y, 8) and (a', /3', y , 8'), and through the given line two 
planes Aa;+Ih/+Cz+I)w = 0 and A'x+IS'y+C'z+D'w— 0 • then we have 
@y' — /3 V : ya! —y'a, : a/3' — a'/3 : a.8' —a'S : /3 8' —(Z'8 : y8' —y'8 
=AD'— A'D : BD'-B'D : CD'-C'D : BC'-B'C : CA'-CA : AB'-A'B ; 
and denoting either of these sets of equal ratios by 
a : b : c ; f : g : h, 
then (a, b, c,f, g, h) satisfy identically the relation af-{-bg-\-ch—0, and are said to be the six coordinates of 
the line. 
