218 
A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 
[340 
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48. The equations of the cubic curve may be taken to be 
1 X, 
V, z 
i y> 
z, w 
or, what is the same thing, 
xz — y 2 = 0, xw — yz = 0, yw —z 2 = 0 ; 
those of the directrix line may be represented by 
ax + j3y + r yz+8w = 0 , 
ax + /3'y 4- 7'z 4- 8'w = 0 ; 
or, what is the same thing, if 
/3y' - fi'y = a, a8' - a'S =f 
ry a ' - ry'a = b, /38' - /3'8 = g, 
a/3' — a'/3 = c, 7 8' — 7'8 = h, 
or 
th 
wl 
he 
or, 
wl 
ref 
(and therefore identically af+bg + ch = 0), the line is defined by means of its “six 
coordinates ” (a, b, c, f, g, h). 
49. The equations of the cubic curve are satisfied by writing therein 
x : y : z : w — 1 : t : t 2 : t 3 , 
and therefore the coordinates of any two points on the curve may be represented by 
(1, 6, 6 2 , 6 3 ) and (1, </>, </> 2 , cf> 3 ); hence, if x, y, z, w are the coordinates of a point in 
the line joining the last mentioned two points, we have 
x : y : z : w = l+ m : W + m(f> : W 2 + m(f> 2 : 16 3 + m^ 3 , 
which equations, treating therein l, m as indeterminate parameters, give the equations 
of the line in question. And putting moreover 
p = yw — z 2 , y — yz — xw, r = xz — y 2 , 
we have identically 
p : q : r = 6(f) : — (6 + (f>) : 1. 
50. In order that the line in question may meet the directrix line, we must have 
l (a + /3 6 + 7 6 2 + 8 6 3 ) + m (a + /3 </> + 7 </> 2 + 8 4> 3 ) = 0, 
l (a' + /3'6 + <y'6 2 + 8'6 3 ) 4- m (a + /3'cf) 4- 7'(f) 2 + 8'(f> 3 ) = 0 ; 
that is, eliminating l and m, we must have 
a 4- /3 6 4- 7 0 2 4- 8 6 3 , a 4- /3 4- 7 (j> 2 4- 8 4> 3 =0, 
a 4- /3'6 4- 7'0 2 4- 8' 6 3 , a 4- /3'<£ 4- 7'(f> 2 4- 8'(f> 3 
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