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

38

INFINITESIMAL TRANSFORMATIONS

[CH. IV

For brevity put
(aM^i) n (x__, y, ) n, (-1, r) )
(= (x, y, z~, al...a r)
(a2 (x' Y z'a,"a,) a = a (x,y,z)=qn, (n=1, 2,...r) (3).
z al, _ a, = yt,, (=, 2,....)
( an ) al.~,... ar
Now any transformation (T) of the group can be expressed in the
form
Tx = 02 (x, y, z, al~ + elt, a2~ + e2t,..., ar~ + ert).
Ty = 02 ( y, z, al~ + elt, a, ar + e t.........(4).
Tz = 03 (x, y, z, a~ + elt, a20 + e2t,..., a,P + ert)
Hence, since the functions 01, 02, 43 are analytic, if t is not too
large, we find, remembering equations (2) and (3),
Tx = x + t (el $ + e2 2 +... + er) + terms involving t2, t3, etc. )
Ty = y + t (el1 + e2vq +... + ervr) + terms involving t2, t3, etc. I (5).
Tz = z + t (el 4 + e2.2 +... + e,.t) + terms involving t2, t3, etc.)
Hence in the limit when t diminishes indefinitely, writing
dx
Tx = x + - t, etc.,
dt- el l + e.^2 + * + err
= el l + e2 2 + + err.....................(6)
dt = el, + e2, +...+ err
These equations define the infinitesimal transformations of the
group, every value-system of ratios of el, e2,...er defining one Infinitesimal Transformation.
36. Conversely by integrating equations (6) of ~ 35, it can be
proved that the form of any finite transformation of the group can be
recovered. Assume that we have found in this way
x=w (t, C2, C2, ), y =a2 (t, Clt ce, 3), =A (t, C, C2, C ),
where C1, C, Cs are the constants introduced by the integration. Let