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