So far, I have generally mentioned problems as definite and special as possible, in the opinion that it is just such definite and special problems that attract us the most and from which the most lasting influence is often exerted upon science. Nevertheless, I should like to close with a general problem, namely with the indication of a branch of mathematics repeatedly mentioned in this lecture—which, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, is its due—I mean the calculus of variations.[51] The lack of interest in this is perhaps due in part to the need of reliable modern text books. So much the more praiseworthy is it that A. Kneser in a very recently published work has treated the calculus of variations from the modern points of view and with regard to the modern demand for rigor.[52] The calculus of variations is, in the widest sense, the theory of the variation of functions, and as such appears as a necessary extension of the differential and integral calculus. In this sense, Poincaré's [Pg 37] investigations on the problem of three bodies, for example, form a chapter in the calculus of variations, in so far as Poincaré derives from known orbits by the principle of variation new orbits of similar character. I add here a short justification of the general remarks upon the calculus of variations made at the beginning of my lecture. The simplest problem in the calculus of variations proper is known to consist in finding a function of a variable such that the definite integral assumes a minimum value as compared with the values it takes when is replaced by other functions of with the same initial and final values. The vanishing of the first variation in the usual sense gives for the desired function the well-known differential equation In order to investigate more closely the necessary and sufficient criteria for the occurrence of the required minimum, we consider the integral Now we inquire how is to be chosen at function of , in order that the value of this integral shall be independent of the path of integration, i. e., of the choice of the function of the variable . The integral has the form [Pg 38] where and do not contain and the vanishing of the first variation in the sense which the new question requires gives the equation i.e. we obtain for the function of the two variables , the partial differential equation of the first order The ordinary differential equation of the second order (1) and the partial differential equation (1*) stand in the closest relation to each other. This relation becomes immediately clear to us by the following simple transformation We derive from this, namely, the following facts: If we construct any simple family of integral curves of the ordinary differential equation (1) of the second order and then form an ordinary differential equation of the first order