[Pd][f]1 [Pd] [Pd][f]2 [Pd] [Pd][f]_n [Pd]
-------- ------ + -------- ------ + ... + --------- -------
[Pd]a_i [Pd]x1 [Pd]a_i [Pd]x2 [Pd]a_i [Pd]x_n
be represented by X_i, (i = 1, 2, ..., r), then the increment of F is
given by
(e1X1 + e2X2 + ... + e_rX_r)F[delta]t.
When the equations (i.) defining the general operation of the group
are given, the coefficients [Pd][f]_s/[Pd]a_i, which enter in these
differential operators are functions of the variables which can be
directly calculated.
The differential operator e1X1 + e2X2 + ... + e_rX_r may then be
regarded as defining the most general infinitesimal operation of the
group. In fact, if it be for a moment represented by X, then (1 +
[delta]tX)F is the result of carrying out the infinitesimal operation
on F; and by putting x1, x2, ..., x_n in turn for F, the actual
infinitesimal operation is reproduced. By a very convenient, though
perhaps hardly justifiable, phraseology this differential operator is
itself spoken of as the general infinitesimal operation of the group.
The sense in which this phraseology is to be understood will be made
clear by the foregoing explanations.
We suppose now that the constants e1, e2, ..., e_r have assigned
values. Then the result of repeating the particular infinitesimal
operation e1X1 + e2X2 + ... + e_rX_r or X an infinite number of times
is some finite operation of the group. The effect of this finite
operation on F may be directly calculated. In fact, if [delta]t is the
infinitesimal already introduced, then
dF d^2F
-- = X.F, ---- = X.X.F, ...
dt dt^2