The system of equations (ii.) represents an operation of the group whatever the constants e1, e2, ..., e_r may be. Hence if e1t, e2t, ..., e_rt be replaced by a1, a2, ..., a_r the equations (ii.) represent a set of operations, depending on r parameters and belonging to the group. They must therefore be a form of the general equations for any operation of the group, and are equivalent to the equations (i.). The determination of the finite equations of a cyclical group, when the infinitesimal operation which generates it is given, will always depend on the integration of a set of simultaneous ordinary differential equations. As a very simple example we may consider the case in which the infinitesimal operation is given by X = x^2[Pd]/[Pd]x, so that there is only a single variable. The relation between x' and t is given by dx'/dt = x'^2, with the condition that x' = x when t = 0. This gives at once x' = x/(1 - tx), which might also be obtained by the direct use of (ii.). Relations between the infinitesimal operations of a finite continuous group. When the finite equations (i.) of a continuous group of order r are known, it has now been seen that the differential operator which defines the most general infinitesimal operation of the group can be directly constructed, and that it contains r arbitrary constants. This is equivalent to saying that the group contains r linearly independent infinitesimal operations; and that the most general infinitesimal operation is obtained by combining these linearly with constant coefficients. Moreover, when any r independent infinitesimal operations of the group are known, it has been seen how the general finite operation of the group may be calculated. This obviously suggests that it must be possible to define the group by means of its infinitesimal operations alone; and it is clear that such a definition would lend itself more readily to some applications (for instance, to the theory of differential equations) than the definition by means of the finite equations. On the other hand, r arbitrarily given linear differential operators will not, in general, give rise to a finite continuous group of order r; and the question arises as to what conditions such a set of operators must satisfy in order that they may, in fact, be the independent infinitesimal operations of such a group. If X, Y are two linear differential operators, XY - YX is also a linear differential operator. It is called the "combinant" of X and Y (Lie uses the expression _Klammerausdruck_) and is denoted by (XY). If X, Y, Z are any three linear differential operators the identity (known as Jacobi's) (X(YZ)) + (Y(ZX)) + (Z(XY)) = 0