If a differential invariant of a continuous group of the plane be equated to zero, the resulting differential equation remains unaltered when the variables undergo any transformation of the group. Conversely, if an ordinary, differential equation [f](x, y, y1, y2, ...) = 0 admits the transformations of a continuous group, i.e. if the equation is unaltered when x and y undergo any transformation of the group, then [f](x, y, y1, y2, ...) or some multiple of it must be a differential invariant of the group. Hence it must be possible to find two independent differential invariants [alpha], [beta] of the group, such that when these are taken as variables the differential equation takes the form F([alpha], [beta], d[beta]/d[alpha], d^2[beta]/d[alpha]^2, ...) = 0. This equation in [alpha], [beta] will be of lower order than the original equation, and in general simpler to deal with. Supposing it solved in the form [beta] = [phi]([alpha]), where for [alpha], [beta] their values in terms of x, y, y1, y2, ... are written, this new equation, containing arbitrary constants, is necessarily again of lower order than the original equation. The integration of the original equation is thus divided into two steps. This will show how, in the case of an ordinary differential equation, the fact that the equation admits a continuous group of transformations may be taken advantage of for its integration. The most important of the applications of continuous groups are to the theory of systems of differential equations, both ordinary and partial; in fact, Lie states that it was with a view to systematizing and advancing the general theory of differential equations that he was led to the development of the theory of continuous groups. It is quite impossible here to give any account of all that Lie and his followers have done in this direction. An entirely new mode of regarding the problem of the integration of a differential equation has been opened up, and in the classification that arises from it all those apparently isolated types of equations which in the older sense are said to be integrable take their proper place. It may, for instance, be mentioned that the question as to whether Monge's method will apply to the integration of a partial differential equation of the second order is shown to depend on whether or not a contact-transformation can be found which will reduce the equation to either [Pd]^2z/[Pd]x^2 = 0 or [Pd]^2z/[Pd]x[Pd]y = 0. It is in this direction that further advance in the theory of partial differential equations must be looked for. Lastly, it may be remarked that one of the most thorough discussions of the axioms of geometry hitherto undertaken is founded entirely upon the theory of continuous groups. _Discontinuous Groups._