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which gives the complete value of ou.
Now this expression consists of three distinct groups and classes of terms:
(1) The first two terms in the first row of (95); these are independent of w, and only involve the value of 8x and dy at the limits; these consequently express the variation of u which is due to the deformation of the functions which assign the limits. This fact is also otherwise evident inasmuch as these terms have arisen from the variation of dy dx and not from that of v. Those which arise from the variation of v are contained in the following groups.
(2) The last term in the first row, and the terms contained in the next three rows of (95): these do not depend on the general value of w; but only on the values which it and its partial derivedfunctions w' and w have at the limits; and consequently on the functions which assign the limits of integration.
(3) The definite double integral which is in the last row of (95). This depends on the arbitrary function w, and cannot be determined unless that function is given.
312.] In the calculation of the value of du which is given in (95), the variations at the limits have been taken in the most general form, and have not been subjected to any conditions. In the application however of this expression, the limits are frequently determined by certain equations or inequalities, by reason of which the preceding value is much reduced.
Now one of the most common cases is that in which the range of integration is determined by a certain closed function or surface, of which the equation is given, so that values of the vari
ables corresponding to all points within this surface are included within the integral, and all values outside are excluded ; and thus if f(x,y,z) = 0 = L is the equation to the limiting surface, the integral includes those values of the variables for which i is negative. This, it will be observed, is the case which I have considered in Art. 214, and have applied in Ex. 7, Art. 258. In a more general case the form of this surface may vary, but I shall assume it now not to be subject to any deformation. Moreover, I shall also assume that L = 0 gives only two real values of z; that if z is eliminated by means of L=0, an
), we have only two real values of y, which are y, and yo ; and that if z and y are eliminated by means of L=0,
= 0, the result gives only two real values of X, which are x, and xo. Under these circumstances of limits Y,=y, when x=xy, and also
'Y 7x1 when x=Xo; so that all terms of the form a dy vanish, because the y-limits are equal ; and consequently the second term of the second group of (95) vanishes.
If the limits of x and y are constants, their variations vanish; and consequently the first group of terms in (95) disappears.
313.] If the element-function of the double definite integral which is the subject of variation, is a function of x,y,z, (4), (0), where as heretofore z is an undetermined function of x and y, so that u =) v dy dx ;
(96) xo Yog where v is a function of x, y, z, z', z,, according to the notation of Art. 311; and where the limits of integration are taken in the most general form; then, taking the results of that article, and putting z"=,=z,, = 0; so that also z" = =2,= 0; from (95) we have
which is the general value of ou. The several groups of terms admit of explanation similar to that given in Art. 304.
If the district of integration is bounded by a closed surface, then, as in the preceding article, the first term of the second group in (97) vanishes.
314.] The preceding examples of variation are sufficient both to illustrate the theory and for the solution of special problems to which we shall apply the calculus. It is good however to consider a difficult subject, such as that under discussion, from another point of view. We have conceived the quantity involving the unknown function to be resolved into its elements, and the definite integral of these elements to be the finite quantity which is the subject of inquiry : and the limits have been taken to be values whose symbols have subscripts 1 and 0. Now imagine the definite integral to represent some property of a plane curve, and between the values X, and xo; this restriction is convenient to fix our thoughts; and let the quantity X7-X, be resolved into n elements, and $1,$,, &z, ... Én_1 be the values of x corresponding to the points of division, and the corresponding values of y be 91, 92, ... Yn-1: then, as the definite integral is the sum of a series of quantities, of each of which the element is a type; so if we replace the definite integral by its equivalent series, it will be a function of xo, $1, 62, ... Én_1, Xm, that is, of n+1 variables; and when the elements are infinitesimal, of an infinite number of variables. This then is a distinguishing mark of the calculus of variations; its immediate subjects of inquiry are functions of an infinite number of variables generally independent of each other; but as these functions consist of a series of terms, all of which are of the same form, the differential, or variation of the sum of them, is equal to the sum of the differentials or variations of the separate terms: hence the cause of 8 and / being subject to the commutative law. The principles of the calculus of variations therefore are only different from those of the differential calculus, because its subject is a function of an infinite number instead of a finite number of variables.
It will also be observed, that if for the definite integral the equivalent series of terms involving intermediate variables is substituted, the number of variables that enter into each term will depend on the order of the highest differential which enters into the element-function ; thus if the element-function involves dạy, three consecutive values of y will enter into each term; and so for other forms of the element-function.
I will not however enter on further inquiry into this method of the calculus of variations, because the process is much longer than, and ultimately leads to the same results as, the preceding. But because the principles of the calculus become hereby resolved into their most simple elements; nay rather, because the processes of perhaps the most transcendental analysis hereby become capable of geometrical interpretation and construction, I shall take an opportunity, in the next Chapter, of solving a simple problem by this method ; and the mode of application will thereby be evident.
APPLICATION OF THE CALCULUS OF VARIATIONS TO PROBLEMS
OF MAXIMA AND MINIMA.
SECTION 1.-Determination of the critical values of a definite
integral whose element-function involves variables and their differentials.
315.] We proceed to apply the principles of the preceding Chapter to a large class of problems of maxima and minima involving unknown functions.
At this part of our treatise it is superfluous to repeat the conditions and the criteria for determining maxima and minima values of known functions, which depend on particular values of the subject-variables of these functions; for the whole question has been fully discussed in Chapter VII of Vol. I, and the reader is supposed to be familiar with it. Suppose however that the problem is to determine the form of a curve or curved surface between certain limits, so that a property of it, such as its length or the area inclosed by it, may have a maximum or minimum value; the principles of Vol. I are plainly insufficient, because the form of the function is unknown; and we have recourse to the following mode of solution : let the property, whose value is critical, be resolved into its elements, the element being a known function of the variables and their differentials, and this being independent of the relation between the variables ; then the sum of all these, or, in other words, their definite integral, is the quantity whose critical value is to be found, and by which means the form of the function is to be determined. The definite integral therefore is the subject of inquiry, and is such as those whose variations have been calculated in the preceding Chapter.
316.] Let u represent the definite integral, of which the critical value is to be determined; and first suppose that the variables and their differentials of which it is a function are independent of each other; that is, that there is no equation of relation amongst them: a maximum or minimum of such a kind is