The various classic devices for the integration of differential equations may be explained simply from a single standpoint—that of infinitesimal transformations leaving the equations invariant. What is still more important than this unification of diverse known methods, infinitesimal transformations furnish us a new tool, likely to succeed when the ordinary methods fail, since they enable us to take into account vital information ignored by the ordinary methods. In fact, the new method does not confine attention to the differential equation and ignore the data of the problem of which the equation is an analytic formulation, but makes use of the data itself in order to obtain one or more infinitesimal transformations leaving the equation invariant. Accordingly the new method is most readily applied successfully to differential equations arising in geometry or mechanics. Why bother with a dead equation whose origin is unknown or has been concealed?
Although no previous acquaintance with differential equations is presupposed, the paper is not proposed as a substitute for the usual introductory course, but rather to provide a satisfactory review ab initio and at the same time to present the unifying and effective method based on groups.
The important topic of differential invariants is given considerable attention at appropriate places throughout the paper. Application is made in § 54 to the congruence of plane curves and their intrinsic equations.
Finally, we obtain in § 55 a complete set of functionally independent covariants and invariants of the general binary form and deduce the facts that every polynomial invariant of the binary quadratic or cubic form is a polynomial function of its discriminant, while every polynomial invariant of a quartic form is a polynomial function of two specified invariants. This method of attack provides an easy introduction to the complicated algebraic theory of invariants as well as the relation between that subject and the topic of functionally independent invariants.
The writer is greatly indebted to the founder of the theory of continuous groups, Sophus Lie, whose lectures he attended in 1896. Numerous valuable suggestions on the manuscript were received from Professor Bliss. It has been used in classes by Dr. Barnett and the writer.
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