The axioms of descriptive geometry, by A.N. Whitehead.

60-62]

FAMILIES OF CONCENTRIC SPHERES

63

the equation of a quadric surface of revolution round the line joining
Ao to Al (xi, yl, za) is
X (x,, z) + K (Ao, Al, P)2 + v (A, A1, P)4 o ( o + O~ + ) a,+
ax ay a a,/, at,,,
+ P o + yo + + /) =o............... (4).
The family of quadric surfaces of revolution round any line must
include every family of concentric spheres with its common centre at a
point on the line. Accordingly taking Ao to be the origin, and < (x, y, z)
to be x1 + y2 + z2+ 1, the family of spheres at any point (xi, y, z1) is
included in the family
X (x2 + y2 + z2 + 1) + X (X1X + yiy + zJz)2 + 2r (x1x + yy + z1z) + p = 0,
that is, in the family
X (x2 + ~y2 + z2) + M (XIX + y'y + zzl)2 + 2v (xix + yly + zaz) + a = 0.. (5).
For this is the family of quadrics of revolution round the line joining
the origin to the point (xi, y,, z1).
62. Consider any two infinitesimal projective transformations in
the plane of xy. One transformation is defined by
dx
dt = alx + a12Y + a3 - x (ax + acy)
dt = a21x + a2y + a - y (alx + a2y)
The other is defined by
dx- = bllX + b2y + b13 - x (bxb + b2y)
dty = b2x + b2by + b23 - y (b6l + b2y)
Now each of these transformations leaves a family of curves latent,
the locus of points, which either are the points of contact of members
of the respective families, or are points on a curve common to the two
families, is given by
allx + a,1y + a13 - x (a x + a2y) a2x + a22Y + a,3 - (a1x + a2y)
b1 + bl2y + b13 - x (bx + by) b2 + b2y + b23- y (bx + b2y)
This locus is a cubic curve.
Now consider two rotations belonging to the congruence group
under consideration. Let one be about the point (0, 0, 0), and the