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

52

LATENCY OF POINTS ON AXIS

[CH. VI

49. The transformation of ~ 46 (1) on the latent axis of x
(i.e. y=O, z= ) is given by
dx 2
d = a11ax - a X2.
If an $ 0, the solution is
X Xo eallt.
all -a1 all-a1Xo
If al = 0, the solution is
1 1
o at.
x XQ
But (cf. ~ 48) when t 27r/v, we find X=-X for every value of X0.
Hence a1 = 0, a, = 0.
Thus every point on any line is latent for a rotation round it with one
point of it latent. This fundamental theorem will be cited by the
shortened statement, that 'every point on an axis of rotation is
latent.';
Thus equations (1) of ~ 46 for the infinitesimal rotation round the
axis of x, reduce to
dt = a12 + a1 - X (a2y + a3Z)
dy= ay + a2z-y (a2y+ az)...............(),
dt
dt = a2y + a33z- z (a2y + a3z)
where a22 + a3 0
and a22a33- a3a32 > O )
50. The condition that
lx+my+nz=O, (1*0)....................(1),
should be a latent plane for the rotation (1) of ~ 49 is that
dx d dy dz
I + m nz + n t = 0........................ (2),
dt dt dt
whenever (1) is satisfied. Hence substituting for
dx dy dz
dt' dt' dt'
and using (1), we find
al2 + a22m + a,2n = (3)
al31 + a2m + a33n =...............