Weierstrass, in 1879, gave a fourth necessary condition. He defines a new function, E(x, y P)=i (x, Y')—f (x, 3 P) -(y-p) (., y, P)• Then Weierstrass's (fourth) necessary condi tion for a minimum [maximum] is E(x, y, y, > 0 < 0] x
It is possible to show (cf. Hedrick, (Bull. A. M. S.,) IX, 1) that for a limited minimum the conditions remain the same except that Jacobi's condition may be omitted. The condi tions in the various cases may be summarized in the following scheme : It is seen on glancing at the table that from the simple conditions (Euler's and Legendre's) for limited weak variation we proceed to any other case by adding Weierstrass's conditions in the case of a strong minimum, and Jacobi's in case of an unlimited mimimum, only. A fifth necessary condition for a strong extremum, in dependent of all the others, has been discovered by Bolza.
In special problems the irksomeness of these conditions can sometimes be circumvented. For • a instance, given a problem in which >0 for all values of x, y, y', then the necessary and I sufficient condition for a limited strong minimum is the possibility o finding a solution of Euler's joining t e two given end points. Such is the case in the geodetic problem and also in the integral which leads to Hamilton's principle; and in each of these cases, fortunately, a limited strong minimum is all that is desired. Similar Mf simplification occurs in every case when -,-- >0 oY'' for all values of x, y, y'. For then Legendre's and Weierstrass's conditions are always satis fied, and may be abstracted from the above table. For this reason Hilbert has called a problem in which >0 for all x, y, y' con tained in a singly connected region R, in which the given end points lie, a °regular)) problem of the Calculus of Variations.
Considering the examplei ; 1/1-Fyidx, we see that for all finite values of x, y, and y' whatever. Since E(x. Y. y',2 , P) l.' (x. Y. 0 o f ?C' P Pr —fil it follows that such an example surely satisfies Weierstrass's sufficient condition, provided that a field exist in the manner specified above. But in this case, since the extremals are all straight lines in the plane, it is obvious that all other conditions are satisfied. Hence the straight line joining any two points actually minimizes the given integral, i.e., the straight i line is the °shortest° line between any two of its points if the preceding integral be the definition of length.
In the problem of the brachistochrone,. men tioned above, it is shown that the extremals found (cycloids) actually render the integral of the problem a minimum provided no cusp lies between the end points. (Cf. Bolza, tures,' Chap. 4, pp. 126, 136, 146).
Returning to the integral which defines length, it is evident that sonic other integral might as well have been selected as the defini tion of length, if we are not to assume an intui tive knowledge of it. The variety of choice is
limited only by the selection of those properties which we desire to have hold. This leads very naturally to The Inverse Problem of the Cal culus'of Variations: Given a set of curves which form a two-parameter family. What is the con dition that they be the extremals of a prob lem of the Calculus of Variations? What are the conditions that they actually render the integral thus, discovered a. minimum? Let a, b) be the given family. Then (cf. Bolza, 'Lectures,' p. 31) the integrand of any integral for which these are extremals must satisfy the equation of ay ay Olf _ _ Oy Oy'Ox Oy'Oy' y • alay'' where y" G(x, y, y') is the differential equa tion of the given family. This equation for f(x, y, y') always has an infinite number of solutions, of which only those are actually solu tions of the given inverse problem which satisfy the relation f >0, and these are solutions in any region free from envelopes of one-para meter families of the given extremals. Some interesting conclusions for particular forms are to be found in a paper by Stromquist, 'Trans actions of American Mathematical Society' (1905).
Another interesting class of problems are the so-called isoperimetnc problems. These are problems in which a further restriction is placed upon the solution by that it shall give a second (given) integral a given value. Such is, for example, the problem of finding the curve of maximum area with a given perimeter. The problem is treated by means of the so-called method of multipliers, which is too long for presentation here. Consult Bolza, 'Lectures,' Chap. 6. • This article is too short to give any account of the details of the work for double integrals. Suffice it to say that the known methods follow closely those given above for simple integrals. In the other possible problems mentioned above the same holds true. An interesting appli cation of these other problems occurs in the well-known Problem of Dirichlet, which is fun damental in mathematical work. Another is the important problem of Minimum Surfaces. Another is the well-known theory of mechanics based upon Hamilton's Principle or one of the analogous mechanical principles. The modern methods have made these theories more rigorous.
Bibliography.— The following is a list of the more Important works and articles published in America concerning the Calculus of Varia tions: Bliss, 'Thesis' (Chicago 1901); and various papers, 'Annals of Mathematics' and 'Transactions of the American Mathematical Society' ; Bolza, various papers, 'Bulletin Amer ican Mathematical Society' ; 'Transactions American Mathematical Society,' etc. (1901 06) ; brochures published in the Chicago Decen nial publications, including the Lectures on the Calculus of Variations mentioned above (Chicago 1904) ; Carl!, 'Calculus of Variations' (New York 1885) ; Hancock, various papers in 'Annals of Mathematics' and 'Calculus of Variations' (Cincinnati 1894) ; Hedrick, articles in 'Bulletin American Mathematical Society' (1901-05) ; Osgood, 'Annals of Mathematics' (II, 3) and 'Transactions American Mathemati cal Society' (II) ; Whittemore, 'Annals Of Mathematics' (II, 3).
The foreign literature is well collected for reference in the footnotes to Bolza's lectures and in the following books and articles: Kneser, 'Variationsrechnung' (Braunschweig 1900) ; 'Encv. d. Math. Wiss.,) (H, A 8) (Leip zig 1904) ; Moigno-Lindeloff, 'Calcul des Varia tions' (Paris 1861); Pascal, 'Calcolo delle Variazioni' (Milano 1897, German trans., Leipzig 1899) • Todhunter, 'History of the Calculus of Variations' (Cambridge 1861) Zermelo u. Hahn, 'Envy. d. Math. Wiss.,) (II A 8a) (Leipzig 1904).
The literature is 'altogether extremely ex tensive, covering, as it does, a period of over 200 years. It is evident that the more important papers for present use are those of recent date.
An important phase of the subject which has necessarily been overlooked is the general proof by Hilbert (1900) that at least an improper minimum always exists. Consult Bolza, 'Lec tures,' (chap. 7).