498]
67
498.
ON THE INVERSION OF A QUADRIC SURFACE.
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xi. (1871),
pp. 283—288.]
The inversion intended to be considered is that by reciprocal radius vectors, viz.
if as, y, z are rectangular coordinates, and r 2 = x 2 + y 2 + z 2 , then x, y, z are to be
changed into ~ 2 , - 2 . But it is convenient to introduce for homogeneity a fourth
ry*Z yZ yZ,
coordinate w, = 1; and the change then is x, y, z into
xw 2 yw 2 zw 2
y2 ^ y 2 ^ y 2
Starting from the quadric surface
(a, h, c, d, f g, h, l, to, n\x, y, z, w) 2 = 0,
or, what is the same thing,
(a, h, c, f g, K$x, y, z) 2
+ 2w {lx + my + nz)
+ dw 2 —
the equation of the inverse surface is
w 2 {a, b, c, f, g, hjx, y, z) 2
+ 2w {lx + my + nz) r 2
+ dP =
where r 2 = x 2 + y 2 + z 2 . The inverse surface is thus a quartic having the nodal conic
w = 0, x 2 + y 2 + z 2 = 0 (circle at infinity); and having the node x = 0, y = 0, z = 0 (the
centre of inversion); or say it is a nodal bicircular quartic surface, or nodal anallagmatic.
9—2
