It becomes
cos2(x, . . .
-H [5 i cos(x, v)cos(y, v)-5 -cos(x, v)-5; ]°wdy.
In forming the first term within the square brackets we then use the relations
6 I 6 I
5?eos(x, v) = -?cos(y, v),5~?cos(y, v) =l7cos(x, v), , 6 6 6 6 6 6
Q-9§ , T9-4; = -cos(y, u)§ c~é%-|-cos(x, 105, -9¥,
where p′ denotes the radius of curvature of the curve s′. The necessity of freeing the calculus of variations from dependence upon the notion of infinitely small quantities was realized by Lagrange, and the process of discarding such quantities was partially carried out by him in his Théorie des functions analytiques (1797). In accordance with the interpretation of differentials which he made in that treatise, he interpreted the variation of an integral, as expressed by means of his symbol δ, as the first term, or the sum of the terms of the first order, in the development in series of the complete expression for the change that is made in the value of the integral when small finite changes are made in the variables. The quantity which had been regarded as the variation of the inte ral came to be regarded as the first Zfcind variation, and the discrimination between maxima and Variation minima came to be regarded as requiring the investigation of the second variation. The first step in this theory had been taken by AIM. Legendre in 1786. In the case of an integral of the form ”;;F(x, y, y')dx
Legendre defined the second variation as the integral 1 au* a'»=F 6"F
~, @t(5y)'+2;);;W5y6y'+5;a(5y')” dx.-To this expression he added the term [§ a(6y)2];;, which vanishes identically because 6y vanishes at x=xo and 'at x=xl. He took a to satisfy the equation I
6“F 6“F da 6'F 2
wits = GW +“ ' .
and thus transformed the explgession for the second variation to 6
it-(W(5y'+may)2dx,
where
6'F 6'F
"'W=saV+°- I
From this investigation Legendre deduced a new condition for the existence of an extremum. It is necessary, not only that the variation should vanish, but also that the second variation should be one-signed. In the case of the First Problem Legendre concluded that this cannot happen unless 6'F[6y'2 has the same sign at all points of the stationary curve between the end points, and that the sign must be -I-for a minimum and
for a maximum. In the application of the g era metric method the function which has been denoted by f1 takes the place of ∂2F/∂y′2. The transformation of the second variations of integrals of various types into forms in which their signs can be determined by inspection subsequently became one of the leading problems of the calculus of variations. This result came about chiefly through the publication in 1837 of a memoir by C. G. J. Jacobi. He transformed Legendre's equation or the auxiliary function α into a linear differential equation of the second order by the substitution
er + ea rev J
ayay “" By” 'w dx
and he pointed out that Legendre's transformation of the second variation cannot be effected if the function 'w vanishes between the limits of integration. He pointed out further, that if the stationary curves of the integral are given by an equation of the form y=φ(x, a, b),
where a, b are arbitrary constants, the complete primitive of the equation for w is of the form
4> <15
U A6E+B ab
where A, B are new arbitrary constants. Iacobi stated these propositions without proof, and the proof of them, and the extension of the results to more general problems, became the object of numerous investigations. These investigations were, for the most part, and for a long time, occupied almost exclusively with analytical developments; and the geometrical interpretation which Jacobi had given, and which he afterwards emphasized in his Vorlesungen fiber Dynamik, was neglected until rather recent times. According to this interpretation, the stationary curves which start from a point (xo, yo) have an envelope; and the integral of F, taken along such a curve, cannot be an extremumif the point (20,110) where the curve touches the envelope lies on the arc between the end points. Pairs of points such as (x0, y0) and (Em no) were afterwards called conjugate points by Weierstrass. The proof .that the integral cannot be an extremum if the arc of the curve “flu” between the fixed end points contains a pair of conjugate g“ft points was first published by G. Erdmann (1878); W " 3
Fig. 2.
Examples of conjugate points are afforded by antipodal points on a sphere, the conjugate foci of geometrical optics, the kinetic foci of analytical dynamics. If- the terminal points are a pair of conjugate points, the integral is not in general an extremum; but there is an exceptional case, of which a suitably chosen arc of the equator of an oblate spheroid may serve as an example. In A the problem of the catenoid a pair of conjugate points on any of the, catenaries, which are the A stationary curves of the problem, is such that the tangents to the catenary at the two points A and A' meet on the axis of revolution (fig. 2). When both the end points of the required curve move on fixed guiding curves C0, C1, a stationary curve C, joining a point A0 of Co to a point A; of C1, cannot yield an extremum unless it is cut transversely by C0 at A0 and by C1 at A1. The envelope of stationary curves which B Do . set out from C0 towards C1, "and D are cut transversely by C0 at points near AO, meets C at a point Du; and the envelope of stationary curves which proceed from Co to C1, and are cut transversely by C1 at points near A1, meets C at a point 1, The curve C, drawn from An to A1, cannot yield an extremum A4 if Do or D, lies between A0 and A1, or if 'Do lies between 'Al and D1. These results are due to G. A. Bliss (1903). A simple example is afforded by the shortest line on a sphere drawn from one small circle to another. In fig. 3 D0 is that pole of the small circle AQBD which occurs first on great circles cutting AOB.) at right angles, and proceeding towards A1B1; D1 is that pole of the small circle A1131 which occurs first on great circles cuttin A1Bl at right angles, and drawn from points of A0B¢ towards A1B1. The arc AOA; is the required shortest line, and it is so A”
Fig. 3.
distinguished from B0B1 by the above criterion.
Jacobi's introduction of conjugate points is one of the germs from which the modern theory of the calculus of variations has sprung. Another is a remark made by Legendre (1786) in regard to the solution of Newton's problem of the Sources
solid of least resistance. This problem requires that a °'W°'f" curve be found for which the integral 522128 f >'>"“(I +y'”)"dy
should be a minimum. The stationary curves are given by the equatlon A 3'3"'(I+y )'”=<:0nst., V, a result equivalent to Newton's solution of the problem; but Legendre observed that, if the integral is taken along a broken line, consisting of two straight lines e ually inclined to the axis of x in opposite senses, the inte ral can (ine made as small as we please b sufficiently diminishing the angle of inclination. Legendre's remark amounts .to admitting a variation of Newton's curve, 'which is not a weak variation. Variations which are not weak are such that, while the points of a curve are but slightly displaced, the ta-ngents undergo, large changes of direction. They are distinguished as strong variations. A general theory of strong variations in connexion with the First Problem, and of the conditions which are sufficient to secure that the integral taken along a stationary curve may be an extremum, was given by Weierstrass in lectures. He delivered courses of lectures on the calculus of variations in several years between 1865 and 1889, and his chief discoveries in the subject seem to have been included in the course for 1879. Through these lectures his theory became known to some students and teachers in Europe and America, and there have been published a few treatise; and memoirs devoted to the exposition of his ideas. In the First Problem the following conditions are known to be necessary for an extremum. I. The path of integration must be a stationary curve. II. The expression 62F/6y'2, or the expression denoted by fl in the application of the parametric gms' nj method, must not change sign at any point of this curve digg; » between the end points. I II. The arc of the curve between A » the end points must not contain a pair of conjugate points.:All these results are obtained by using weak variations. Additional
