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

52, 53]

AXES OF ROTATIONS

55

53. The infinitesimal rotation round the axis of y as axis is of the
form (cf. ~ 49, equations (1) and (2)),
dt +A3 (A +
d=AyiX + A (AiX+A3Z)J
dt= P21+P23-Y(P1X 3Z)...............(1),
dz
dt P31+/3Z - Z(/,lx +/3sz)
where Al + 33 =.2)
and n11A3: - A13/i1 > 0
Then, since (cf. ~ 45, axiom (1)) the motions form a group, by
combining this infinitesimal rotation with that round the axis of x,
another infinitesimal rotation of the group is found. Thus (cf. ~ 51,
equations (4)) an infinitesimal rotation of the group, assuming the
special axes and infinite plane of ~ 51, is of the form
dx
d =Al X + i3l3z- x (13x + ) Z)
dy = 21X + a22y + (323 + Ka23) Z- y (A 3 + 3z)........(3),
dt = A31 + Ka3y + (&3, 4~ Ka3) Z-Z (13x -+- 3) J
where K has any arbitrary value.
Hence (cf. ~ 52) we have
t31 0, 3t,
2,,, 23+ Ka =................. (4).
A31, Ka32, 332 + Ka33
But equation (4) holds for every value of K. Hence
A1 (a22 a3 - a23,a3) = 0.
Hence (cf. ~ 51, equations (5)),
= 0.................................(5).
Thence, again from equation (4), we find
1321 a32 -- 3l.31 a2: -........................ (6).
From equations (2) and (5) we find
Ps3 = O.................................(7).