Hence
| F′ = F + t | dF | + | t2 | + | d2F | + ... |
| dt | 1·2 | dt2 |
| = F + tX·F + | t2 | X·X·F + ... |
| 1·2 |
It must, of course, be understood that in this analytical representation of the effect of the finite operation on F it is implied that t is taken sufficiently small to ensure the convergence of the (in general) infinite series.
When x1, x2, ... are written in turn for F, the system of equations
| x′s = (1 + tX + | t2 | X·X + ...)xs, (s = 1, 2, ..., n) |
| 1·2 |
(ii.)
represent the finite operation completely. If t is here regarded as a parameter, this set of operations must in themselves constitute a group, since they arise by the repetition of a single infinitesimal operation. That this is really the case results immediately from noticing that the result of eliminating F′ between
| F′ = F + tX·F + | t2 | X·X·F + ... |
| 1·2 |
and