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

62

SPHERES

[CH. VII

homogeneous coordinates by putting x= X/ U, y= Y/U, z =Z/U,
where, U=0, is the equation of the infinite plane, the equation of
the family of concentric spheres is
(X + 2 + y Z) + U2= 0..................... (2).
Thus returning to the original coordinates, if (k (x, y, z) = 0, is the
equation of any sphere, centre at (xo, yo, zo), the equation of the family
of spheres with that centre is
a(" ~ ao ~ ao o _~ =.......(3),
XA (x, y, z) + U X0 o + go y + Zo = 0...... (3)
where, as usual, t is introduced to make the equation homogeneous
and is put equal to 1 after differentiation.
61. By recurring to equation (2) of ~ 60, we see that the plane
of yz, which is the plane perpendicular to the axis of x, is the plane
through the origin and through the common conjugate line of the axis
of x with respect to any of the spheres, centre the origin. Hence if,
( (x, y, z) = 0, is any sphere with centre Ao (xo, yo, zo), and Al is the
point (x,, y,, Zl), then the plane through Ao perpendicular to AoA1 is
(x o + Ax1) a+ (y + XY1) + (Z +Z) + (1 ) =..(1),
ax ay ax at
where
X=-24o/1^()j( +i- +(o a) h
/ o +'\ax / ayo +yo ' /at
and 0o, (~), etc. are the results of substituting the coordinates of
Ao in 0 (x, y, z), +, etc.
Let the left-hand side of (1) be written (Ao, Al, P),, where P is
the variable point (x, y, z). Thus the equation of the plane, perpendicular to the line AoAI and through the point Ao, is
(Ao, A,, P ) = 0...........................(2).
A quadratic surface of revolution round the axis of x is of the form
(cf. equation (2) of ~ 59)
b(y2+z2) + a + 2gx +c=0..................(3).
This can be written in the form
X {a (X2 + y:+ z2) + P} + a'x2 + gx + c'= 0.
Thus, if 4 (x,, z) = 0 is the equation of a sphere, centre Ao (xO, yo, Zo),