(5)

xo+ ( − 2 ô xi_xo+(i−1).0,
i=2, 3,... (n+1),

where nox, xo, the curve K is said to render I a limited weak [strong] minimum [maximum].

Geometrically these conditions mean that the curves compared to K must lie, in the case of a strong extremum (i.e., maximum or minimum), close to the curve K; in the case of the weak extremum, they must lie close to K and vary only a little from K in direction; in the case of a limited extremum, they must cut K at least once in every vertical strip of width ò. It is easy to show that if K is to render I an extremum (of any sort), (x) must satisfy the equation

which is known as Euler's (or less properly as Lagrange's) equation. For we have

(7) Ic=f*1ƒ{x, 4(x)+n(x), 4(x)+n'(x)}dx, which must be a minimum [maximum] for n(x) = =0. Replacing 7(x) by eλ(x), where (x) is a certain function and is a variable parameter, I will evidently be a function of the parameter & alone:

Integrating the second

(10) F'(0)=0=[d(x)iv(x, 4, 4)]*

+

d

that this proof, which implicitly assumes the existence of the second derivative of (x), does not involve any restrictions. (Cf. Bolza, 'Lectures, chap. i).

Assuming the further details without proof, it becomes evident that any curve K, y = f(x), which is to render I a minimum (of any sort) must satisfy the differential equation (6). Since f and its derivatives are known functions, (6) is an ordinary differential equation of the second order, linear in d2y/dx2(=y'). The coefficient of y' is d2f/dy'?(fv). If this coefficient fy' does not vanish, one and only one solution of (6) passes through a given point The general solution of (6) in a given direction. contains two arbitrary constants: y = f(x, α, B).

(12)

Any one of these solutions, i.e., any solution whatever of (6) is called an extremal. Hence the required curve K, if it exists, must be an extremal, and it is necessary to search for it only among the extremals. But K was to connect P. and P1. Usually, however, there is only one of the extremals (12) which passes through two given points, for the equations, (13) Yo = f(xo, α, 3), y1 = f(x,, α, }),

usually determine a and B, and hence also determine a single extremal joining P, and P1. If this is actually the case, either that extremal is the required solution K, or else there is no solution of the problem.

A large number of special cases lead to differential equations which can be solved directly.

f(x, y, y′) =√1+y'2, whence fa-fy=fav'=fvv = fxy=o, fv=y/(1 + y')', ƒy'y' = 1 / (1+y)3, and the equation (6) takes the form y' = o. The only solutions of this differential equation are the straight lines y-ax+b. It follows that if there is any curve of the class B in the plane along which the distance between two given fixed points is at a minimum, that curve is the straight line joining the two points. This result is independent of the Euclidean postulate, and depends only upon the definition of length by means of the preceding integral.

The problem of the brachistocrone, mentioned above, is to find the curve along which a particle with initial velocity v will descend most quickly from a given initial point P. to another given point P,. It is easy to show that the time of descent is given by the formula x1 √1+12 xo vo2-2g(x-xo)

t=

or since (x) evidently vanishes for x=x, and for x=x1,

(11) ƒ^*^'^(x)}{ƒ»(x, 4, 4′) − div(x, 4, 4′) } dx =0.

But (x) was itself any permissible function of x, and it is easy to show that the integral of such a product, of which one factor is arbitrary, can vanish only if the other factor vanishes. This gives precisely the equation (6). Certain further considerations are necessary to show

daf d

dx,

Vo-2g(x- -xo)

These extremals are cycloids on horizontal bases, the radius of the generating circle being (A+B)/2, and one cusp being at the point (A, C). Further investigation is necessary to decide just when a given pair of points can be connected by such a cycloid (cf. Bolza, 'Lectures,' p. 236). If such a cycloid can be drawn, we can infer that it is the solution if there is any solution. If no such cycloid can be drawn, we can infer that there is no solution in the region R.

The problem of finding the geodetic lines on a given surface is that of minimizing the integral,

1 =

(E+2Fy' +Gy's) hdx,

fy'y'(x, 4(x), 4'(x))>o[ <o] for xox < be satisfied along the supposed solution y=&(x) between the end points. We shall prove this, and we shall see that the same condition is actually a sufficient condition for a weak limited minimum if the sign be removed. Jacobi then showed, by means of the second variation of the given integral, that a third necessary condition for a minimum [maximum] is that the quantity

(x, xo)=1(x)2(Xo) — 72(X)71(x0) should not vanish for any value of r in the interval x< x < x, where 4 (x, x) is a solution of the equation

1=0,

dfy'y' (dfyy' ƒx'y'7" + dx + -fyy dx which vanishes for x=x.. The proof, which is omitted, can be found in Bolza, 'Lectures,' chap. ii. A beautiful geometrical interpretation of this condition exists: if we consider the one parameter family of extremals through (o, yo), and call their envelope E, the extremal which joins ro to any point beyond its point of tangency with E cannot possibly render the integral a minimum [maximum] between those two points, ie., the envelope of the extremals through (ro, yo) bounds all the points which can possibly be reached by a minimizing extremal from r.

It was long believed that Jacobi's condition, together with the previous two, was a sufficient condition. That such is not the case was first

pointed out by Weierstrass, who also showed that Jacobi's condition, while not sufficient for a minimum in general, is sufficient for a weak minimum (if the point (x, y) lies inside the envelope of the extremals through (ro, Yo). (Cf. Bolza, 'Lectures, chap. iii.)

That the preceding conditions are not sufficient is most readily seen by giving an actual example in which the extremals, though all the above conditions are satisfied, do not minimize the integral. Such is the example (see Bolza, 'Lectures, p. 73),

f(x, y, y)=y'y' + 1)2. Here the extremals are straight lines, but it is easy to join two points for which all the preceding conditions are satisfied by a simple broken line for which the value of the integral is less than that along the straight line extremal. Of course, the comparison line used varies considerably from the straight line extremal in direction, though not in position.

Weierstrass, in 1879, gave a fourth necessary condition. He defines a new function, E(x, y, y', p) = f(x, y, y) -f(x, y, p)

-(y' - p)fp (x, y, p). Then Weierstrass's (fourth) necessary condition for a minimum [maximum] is

E(x, y, y', po Sol x ≤x≤x1, where x, y, are the values of x, y, dy/dx along the extremal between the end points, and where y is any finite number whatever. (Cf. Bolza, 'Lectures, chap. iii.) Since we have Limit E(x, y, y', p)] y' = p \ (y' - p)2

nit [

=

a2f

it follows that it is also necessary that fy'yo, which is precisely the second (Legendre's) necessary condition mentioned above. It is easy to show that if (a) the end points can be joined by an extremal K, (b) a one parameter family of extremals [y=(x, a)] can be found, one of which is K itself, and one and only one of which passes through each point of the plane near K, so that y=p(x, y) can be found, i.e., a function which gives the slope of the extremal of the family at any point (x, y) near K, then the integral

= { '*, (f(x, y, p) + (y' − p) fp(x, y, p)]dx

J =

ΧΟ

for all x and y near K and for the function p(x, y) just mentioned and for any finite value of y' whatever, is a sufficient condition for a strong minimum, if the sign of equality holds only for p= y. (Cf. Osgood, Annals of Mathematics, II, 3; Bolza, 'Lectures,' chap. iii.)

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 conditions in the various cases may be summarized in the following scheme:

Sufficient.

Necessary. 1

Limited Variations.

Unlimited Variations.

Weak. Strong. Euler's, Euler's, Legendre's Legendre's

necessary, Jacobi's necessary.

necessary.

necessary, Jacobi's necessary, Weierstrass's necessary.

Euler's Legendre's sufficient, Jacobi's necessary, Weierstrass's sufficinet.

It is seen on glancing at the table that fron 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 minimum, only. The above table represents substantially the present known conditions.

In special problems the irksomeness of these conditions can sometimes be circumvented. For a2f instance, given a problem in which ayo for all values of x, y, y, then the necessary and sufficient condition for a limited strong minimum is the possibility of finding a solution of Euler's quation joining the 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 92f simplification occurs in every case when for all x, y, y'. For then Legendre's and Weierstrass's conditions are always satisfied, and may be abstracted from the above table. For this reason Hilbert has called a problem in 02f which > for all x, y, contained in a singly connected region R, in which the given end points lie, a "regular" problem of the Cal

culus of Variations.

2

we

y,

Considering the example Vi+y2 dx, see that f>o for all finite values of x, and y whatever. Since E(x, y, y', p) = ——ƒv'v·(x, y, §), yp, 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, ie., the straight 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, mentioned 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, Lectures,' chap. iv., pp. 126, 136, 146).

Returning to the integral which defines length, it is evident that some 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 Calculus 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 y= F(x, 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

where y=G(x, y, y') is the differential equation of the given family. This equation for f(x, y, y′) always has an infinite number of solutions, of which only those are actually solutions of the given inverse problem which satisfy the relation f>o, and these are solutions in any region free from envelopes of one-parameter families of the given extremals. Some interesting conclusions for particular forms are to be found in a paper by Stromquist, Transactions of American Mathematical Society' (1905).

Another interesting class of problems are the so-called isoperimetric problems. These are problems in which a further restriction is placed upon the solution by requiring that it shall give Such a second (given) integral a given value. 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. (See Bolza, Lectures, chap. vi.) 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 application of these other problems occurs in the well-known Problem of Dirichlet, which is fundamental in mathematical work. Another is Another is the well-known theory of mechanics the important problem of Minimum Surfaces. 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 Variations: Carll, Calculus of Variations' (New York, 1885); Osgood, Annals of Mathematics) (II, 3. and Trans. A. M. S., II); Whittemore, Annals of Mathematics' (II, 3): Hancock, (Various papers in Annals of Mathematics and Calculus of Variations' (Cincinnati, 1894); Bliss, 'Thesis' (Chicago, 1901); and various papers, Annals of Mathematics and Transactions American Hedrick, Mathematical Society); 'Bulletin American Mathematical Society,' IX (1901-5); American Bolza, various papers, Bulletin Mathematical Society,' (Transactions American Mathematical Society, etc. (1901-5): Brochures published in the Chicago Decennial Publications,

including the Lectures on the Calculus of Variations mentioned above (Chicago, 1904).

The foreign literature is well collected for reference in the foot-notes to Bolza's 'Lectures,' and in the following books and articles: Todhunter, 'History of the Calculus of Variations' (Cambridge, 1861); Moigno-Lindeloff, 'Calcul des Variations? (Paris, 1861); Pascal, 'Calcolo delle variazioni' (Milan, 1897, German Trans., Leipzig, 1899); Kneser, Variationsrechnung) Braunschweig, 1900); Kneser, 'Ency. der Math. Wiss., II, A 8' (Leipzig, 1904); Zermelo u. Hahn, Ency. der Math. Wiss., II, A Sa' (Leipzig, 1904).

The literature is altogether extremely extensive, covering, as it does, a period of over two hundred 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. (See Bolza, 'Lectures,' chap. vii.)

EARLE RAYMOND HEDRICK, Professor of Mathematics, University of Missouri.

Calcut'ta ("the ghaut or landing-place of Kali" from a famous_shrine of this goddess), India, the capital of British India, and of the presidency and province of Bengal, is situated on the left bank of the Hooghly (Húghli), a branch of the Ganges, about 80 miles from the Bay of Bengal. The Hooghly is navigable up to the city for vessels of 4,000 tons or drawing 26 feet; the navigation, however, on account of sand-banks which are continually changing their size and position, is dangerous. The river opposite the city varies in breadth from rather more than a quarter to three quarters of a mile. The city may be said to occupy an area extending along the river for about five miles from north to south, and stretching eastward to a distance of nearly two miles in the south, narrowing in the north to about half a mile. The eastern boundary is nominally formed by what is known as the Circular Road, the Lower Circular Road forming part of the southern boundary. Another eastern boundary on the north is the Circular Canal, which runs for some distance parallel to the Circular Road. The southwestern portion of the area thus spoken of is formed by the Maidan, a great park stretching along the river bank for about one and three quarter miles, with a breadth in the south of one and a half miles. This grassy and tree-studded area is one of the ornaments of Calcutta; it is intersected by fine drives, and is partly occupied by public gardens, a cricket ground, race-course, etc., and partly by Fort William, which rises from the iver bank. It was built in 1757-73, being begun by Clive after the battle of Plassey, and is said to have cost about $10,000,000. Along the river bank there is a promenade, and drive known as the Strand Road, which has for the most part been reclaimed from the river by successive embankments. Along the east side of the Maidan runs Chauringhi Road, which is lined with magnificent residences, and forms the front of European fashionable residential quarter. Along the north side of the Maidan runs a road or street known as the Esplanade, on the north side of which are Government House and other public buildings. The European commer

They

cial quarter lies north of the Esplanade, between it and another street called Canning Street, having the river on the west. The centre of this area is occupied by Dalhousie Square (enciosing a large tank or reservoir), and here there are a number of public buildings, including the post-office, telegraph office, custom house, Bengal secretariat, etc. The European retail tradthe quarter occupies a small area to the east of ing above area. Everywhere outside of the European quarters Calcutta is interspersed with bastis, or native hamlets of mud huts, which form great outlying suburbs. "The growth of the European quarters, and the municipal clearings demanded by improved sanitation, are pushing these mud hamlets outward in all directions, but especially toward the east. have given rise to the reproach that Calcutta, while a city of palaces in front, is one of pigstyes in the rear. First among the public buildings is the Government House, the viceregal residence, situated, as already mentioned, on the Esplanade. It was built in 1799-1804, and with its grounds occupies six acres. Four wings extend toward the four points of the compass from a central mass which is crowned with a dome and approached from the north by a splendid flight of steps. Besides accommodating the viceroy and his staff it contains the council chamber in which the supreme legislature holds its sittings. The high court, the town hall, the bank of Bengal, the currency office, post-office, etc., are among the other public buildings in this locality, while further to the north stands the mint, near the bank of the Hooghly. The chief of the Anglican churches in Calcutta is the cathedral of St. Paul's, at the southeastern corner of the Maidan, a building in the "IndoGothic style, with a tower and spire 201 feet high, consecrated in 1847. St. John's Church, or the old cathedral, is another important church, in the graveyard surrounding which is the tomb of Job Charnock, founder of Calcutta. The chief Presbyterian church is St. Andrew's or the Scotch Kirk, a handsome Grecian building with a spire. The Roman Catholics have a cathedral and several other churches; and there are also places of worship for Greeks, Parsees, and Hebrews. Hindu temples are numerous, but uninteresting; among the Mohammedan mosques the only one of note is that which was built and endowed by Prince Ghulam Mohammed, son of Tippoo Sultan. The religions, edu cational, and benevolent institutions are numerous. Various missionary and other religious bodies, British, European, and American, are well represented. There are four government colleges the Presidency College, the Sanskrit College, the Mohammedan College, and the Bethune Girls' School. There are five colleges mainly supported by missionary efforts; besides several others, some of them under native man◄ agement. Other educational institutions include Calcutta Medical College, a government school of art, Campbell Vernacular Medical School, and a school of engineering at Howrah, on the western side of the river. Besides these there is the Calcutta University, an examining and degree-conferring institution. Among the hos pitals are the Medical College Hospital, the General Hospital, the Mayo Hospital (for natives), and the Eden Hospital for women and children. The Martinière (so named from its founder, Gen. Martin, a Frenchman in the Company's ser