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 Hence dF t^2 d^2F F' = F + t-- + --- ---- + ... dt 1.2 dt^2 t^2 = F + tX.F + --- 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 t^2 x'_s = (1 + tX + --- X.X + ...)x_s, (s = 1, 2, ..., n) (ii.) 1.2 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 t^2 F' = F + tX.F + --- X.X.F + ... 1.2 and t'^2 F" = F' + t'X.F' + ---- X.X.F' + ... 1.2 is (t + t')^2 F" = F + (t + t') X.F + ---------- X.X.F + ... 1.2 The group thus generated by the repetition of an infinitesimal operation is called a _cyclical_ group; so that a continuous group contains a cyclical subgroup corresponding to each of its infinitesimal operations.