428 
[859 
859. 
ON THE COMPLEX OF LINES WHICH MEET A UNICURSAL 
QUAPTIC CURVE. 
[From the Proceedings of the London Mathematical Society, vol. xvn. (1886), 
pp. 232—238.] 
The curve is taken to be that determined by the equations 
x : y : z : w = 1 : 6 : d 3 : d 4 , 
viz. it is the common intersection of the quadric surface © = 0, and the cubic surfaces 
P = 0, Q = 0, R = 0, where 
© = xiu — yz, P = x 2 z — y 3 , Q = xz 3 — y 3 w, R = z 3 — yiv 3 . 
Writing (a, b, c, f g, h) as the six coordinates of a line, viz. 
(a, b, c, f g, h) = (J3z — yy, yx — az, ay — ¡3x, aiu — 8x, ¡3w — 8y, yw — Sz), 
if (a, /3, y, 8), (x, y, z, w) are the coordinates of any two points on the line ; then, if 
the line meet the curve, we have 
. hd - g6 3 + ad 4 = 0, 
— h . +fd 3 +bd 4 = 0, 
9-/0 • + cd 4 = 0, 
— a — bd — cd 3 . = 0, 
from which four equations (equivalent, in virtue of the identity af-\- bg + ch = 0, to two 
independent equations), eliminating d, we have the equation of the complex. The form 
may, of course, be modified at pleasure by means of the identity just referred to, but one 
form is 
il, = a 4 — b 3 li + bf 2 g + eg 3 — acfh + 2c 2 h 2 — 4 a 2 ch + af 3 — a 3 f= 0, 
as may be verified by substituting therein the values a — — bd— cd 3 , g =fd — cd 4 . 
h = fd 3 + bd 4 . The last-mentioned equation is thus the equation of the complex in 
question, in terms of the six coordinates (a, b, c, f g, h).