C. IX. 
32 
[595 
595] 
or, what is the same thing, 
ON A SENATE-HOUSE PROBLEM. 
249 
x = 2 (a -1) 2 (a - 2) (l - 
: y : a ... («-2}>(l-A) 
: z 
2a (a - 1) 
: w 
a 2 (a — 1) (a — 2) 
: e 
«(«-l)(<x-2) (l-£-), 
where, for the sake of homogeneity, I have introduced the factors ^1 - — ^ and ^1 — — ^ 
viz. we have x, y, z, w, 6 proportional to quartic functions of the arbitrary parameter 
a, or the curve is a unicursal quartic. Writing in the equations a = 0, 1, 2, oo successively, 
we see that this quartic curve passes through the four points 123, 234, 341, 412 (inter 
secting at these points the lines 13 and 24 respectively); and writing also a=l ±i we 
see that the curve passes through the points P, Q, the coordinates of which now are 
l the points 
iting in the 
') — 0, wx = 0, 
z — 0) and 
x = y = z = w = (1 + i) 9. 
It should admit of being proved by general considerations that, in 4-dimensional 
geometry when 4 quadric surfaces partially intersect in two lines, the residual inter 
section consists of 2 points; and that, when they intersect in the two lines and in a 
unicursal quartic met twice by each of the lines, there is no residual intersection—but 
this theory has not yet been developed.