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SECTION 3.- Investigation of the critical values of a definite inte

gral, whose element-function involves derived-functions. ; 349.] In all the preceding problems of maxima and minima, the differentials contained in the element have been taken in their most general forms; no supposition has been made as to one or more being equicrescent or as subject to no variation, and they have not been put in the forms of derived-functions; and the solutions, it will be observed, have been deduced from first principles, and without the intervention of general formulæ : the results arrived at are left in their symmetrical forms; and hereby have we been able to infer geometrical properties, which are frequently the only available definition of the function which satisfies the critical property that is required. Now for elegance and symmetry nothing more can be desired : but we have not investigated any critical function whose element contains differentials above the second order; the simplest cases only have been considered, and a slight inspection of the general results of Art. 316 will shew that the complexity of the formulæ rapidly increases if higher differentials enter into the calculation : in this latter case then, it is desirable to simplify the formulæ as far as is possible, ere they become the subjects of inquiry; and as such a simplification is obtained by making one of the variables equicrescent, and by using derived-functions instead of differentials, although it is with the loss of symmetry, it is necessary to consider the conditions under which a definite integral, whose element-function involves derived-functions of different orders, may have a critical value. And there is also another reason why the subject must be investigated from this point of view : it is only when the elementfunction is of this form that criteria for discriminating maxima and minima have been constructed. We proceed then at once to the investigation.

Let the definite integral, whose maximum or minimum is to be determined, be

= 'vdx ; where

v = f(z, 8, 9, y,... yo), .. . (102) using the notation of derived functions.

Now the variation of u, where v is of the given form, has already been investigated in Art. 303; and for an abridgement

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of notation of the results of that Article which are given in equation (56) of it, let Su dr" dy" w dn-ly(n)

:-)* dan 1 sa

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du = a + Hw dx ;

(105) L do Jo and as u is a maximum or minimum, du = 0; and to satisfy this condition it is manifest that

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); H = 0;

(106) of which expressions the former depends on the values of certain variables and their derived functions at the limits; the latter by integration gives the general functional relation, and thereby the form whence the required critical value may be found. Now since v contains y(") or , , which contains

d "y()

dx ' will generally contain , and therefore will be a differential equation of the (2n)th order: the solution of this equation will therefore contain an arbitrary constants; and the determination of these depends on the values which a = 0 assumes at its limite ing values; the process however of finding these, being similar to that explained in Art. 318, it is unnecessary to repeat; but it is desirable to investigate one or two cases in which the equation H = 0 assumes particular forms analogous to those of Art. 319, and thereby admits of immediate integration.

350.] First, suppose v not to contain the first m of the quantities y, y, y", ...; then the equation 1 = 0 becomes

dmy(m) ďm+ly(m+1)

dom - dam+1 +... = 0; and this admits of m successive integrations; and thus we have

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which is a differential equation of an order not higher than
2n-1; and therefore whenever v does not contain x, the equa-
tion 1 = 0 always admits of being integrated at least once.
Thirdly, let v = f(y); then, by the preceding equation,

v=c+Y'y';
but as v and y' contain ý' only, this may be put into the form

= 4* = P(c);
.: y = f(c)x+G;

(109)
and thence we infer that a linear function, as (109), is such that
the variation of any functio deduced from it vanishes.
Lastly, if v=f(y, money,
V=c+Y'y'.

(110)

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351.] Let us apply the preceding method to the solution of one or two problems.

Ex. 1. Let u= by die
Here v = you; and therefore equation (108) is applicable.

r=over ) = ; . romana ngayon ay 1".

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which is the equation to a straight line; and we have the same result as that arrived at in Art. 322. · Ex. 3. Let the element-function be of the form given in Art. 308; viz. v=f(x, y, y', y", ...2, 2', z", z", ...); and let us also take a simple case, and suppose that

u = l {1+y^2 +z2;& dx ; then, by reasoning similar to that which has been frequently employed, both the terms under the integral signs in equation (69), Art. 308, must vanish : and therefore, as v involves only 'y and :', we have

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which are manifestly the equations to a straight line in space. A straight line therefore is the shortest distance.

Ex. 4. To find the plane curve of given length enclosing the greatest area between itself, two given extreme ordinates, and the axis of x.

This is a problem of relative critical value. Let a be a constant multiplier; then,

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so that

v = y +à{1+y'?}; and because v does not involve X, (108) is applicable; and

1 ay

{1+ y2} as y+d{1+y2}} = 1 +

+ 142

{1+ y2'}" whence

(y-C1)? + (x — c)2 = 1?, which is the equation to a circle, whose radius is equal to 1; and 1 may be expressed in terms of the known length of the curve by a process similar to that of Art. 330.

A comparison of the two methods by which problems have been solved plainly shews that, although the former immediately involves first principles and from them is directly deduced; yet, as the results assume complicated forms when all the differentials are retained, it is convenient to make one of the variables equi. crescent, and to express the element-function in terms of derived-functions, and then to apply the process of these latter articles.

SECTION 4.-The discriminating conditions of Maxima and

Minima. 352.] The process which has been developed in the preceding articles of this chapter, and which has been applied to the solution of problems involving maxima and minima of definite integrals, although necessary, is yet insufficient for the object

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