The need for series which converge for all values of the time; Poincaré's series.

We have already observed (§ 32) that the differential equations of motion of a dynamical system can be solved in terms of series of ascending powers of the time measured from some fixed epoch; these series converge in general for values of t within some definite circle of convergence in the t-plane, and consequently will not furnish the values of the coordinates except for a limited interval of time. By means of the process of analytic continuation it would be possible to derive from these series successive sets of other power-series, which would converge for values of the time outside this interval; but the process of continuation is too cumbrous to be of much use in practice, and the series thus derived give no insight into the general character of the motion, or indication of the remote future of the system. The efforts of investigators have therefore been directed to the problem of expressing the coordinates of a dynamical system by means of expansions which converge for all values of the time. One method of achieving this result is to apply a transformation to the t-plane.