322 
ON THE QUADRIQUADRIC CURVE IN CONNEXION 
[843 
We satisfy the equation by 
x + iy + 6 z + 8iw = 0, 
• 1 1 • 
X — ly — Jj Z + -p. IW = 0, 
where 6 is an arbitrary parameter: hence these equations determine a generating line 
of the surface: and the coordinates of this line are 
a, b, c, f g, h 
viz. these values give a = —f, b= — g, c — — h. 
Similarly we satisfy the equation by 
x + iy + <f)Z — <f)iw — 0, 
. 1 1 . A 
x — iy —r z — — %w — 0, 
J 0 0 
where </> is an arbitrary parameter: hence these equations also determine a generating 
line of the surface: and the coordinates of this line are 
a, b, c, f g, h 
= ¿(,#.-1), 4,+t, iit + l), <¡> + 1, 
viz. these values give a =f, b = g, c=li. 
If for x, y, z, w we write x *Jp, y^q, z\!r, w \/s respectively, then we have the 
theorem that, for the quadric surface 
px 2 + qy' 2 + rz 2 + sw 2 = 0, 
the coordinates (a, b, c, f, g, h) of a generating line are such that 
%-±J^, h -=iJ2. 
f V qr g V rp h X pq 
the signs being all + or all —, according as the line belongs to one or other of 
the systems of generating lines. 
Take (a', b', c', f, g', h') for the coordinates of an arbitrary line, and write 
p, q, r, s = ag'h', b'h'f', c'f'g’, a'b'c'; the quadric surface is 
ag'h'x 2 + b'h'f y 2 + cfg'z 2 + a'b'c'vf = 0,