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proposed, because no discriminating conditions of maxima and minima have been investigated. For the existence indeed of such critical values it is necessary that the first variation should vanish; but at the same time such vanishing is consistent with the definite integral being either a maximum or a minimum or a constant, and with being none of these : the truth of this statement is evident from the ordinary theory of maxima and minima. For a critical value it is necessary that the first variation of the definite integral should not only vanish, but also change its sign: and I know of no process immediately applicable by which to determine whether a function deduced from the differential equation h= 0, see Art. 349, and involving an arbitrary constants, will or will not cause the required change of sign of du. In accordance then with the theory explained in Art. 149, Vol. I, we are obliged to have recourse to the second variation of the definite integral, with the object of determining its sign, and hereby to obtain the discriminating condition; so that when du = 0, and if déu does not vanish, and does not become infinite or discontinuous, and does not change its sign within the limits of integration, u is a maximum or minimum according as 8u is negative or positive. We proceed to the further development of these conditions.

But, to narrow the investigation as far as possible, I will take the case which has last been considered; that, namely, in which the infinitesimal element-function involves x, y and the derived functions of y, and in which also x is not only equicrescent but undergoes no variation; that is, ox is not one of the subjects of calculation, but the variation is due to a variation of y only: or, geometrically viewed, the displacement of the point on the curve is in a direction parallel to the axis of y only: for it is to this case that Jacobi*, the discoverer of the criteria, has confined himself. And first let the object of the research be clearly understood.

If the infinitesimal element-function contains a derived function of the nth order, the differential equation 1 = 0 will generally be of the (2n)th order, and therefore the value of y deduced from it is of the form y = f (x, C1, C2, ... C2n),

(111) and contains an arbitrary constants which have been introduced

* Zur theorie der Variations-Rechnung und der Differential-Gleichungen, von C. G. J. Jacobi; Crelle, vol. XVII, p. 68.

in the process of integration : and therefore, if u is the given definite integral, it is plain that, after the substitution of y by means of the above equation, u will depend partly on the form of the function f, and partly on the arbitrary constants. It may seem then that the critical value of u will depend on both these quantities : as to the constants, however, it has been shewn that all their values may be determined by means of the given limiting values of the variables and of the derived functions; and hence, as these are determinate constants, the value of the definite integral cannot be made critical by any change of them : and even more than this, did u depend on such quantities it would become an integral (not differential) function of many variables, and would have its critical value determined by the ordinary rules of the differential calculus.

It is then the other question which we have to discuss; namely, whether the form of the function deduced from the equation H= 0 is such as to render the definite integral a maximum or minimum. For this purpose we must, as in Art. 310, calculate oʻu, and determine its sign, subject to the conditions that when ou = 0,

(1) 8-u has the same sign for all values of the variables and their derived functions between the limits;

(2) 82u does not become infinite for any values between the limits;

(3) 8-u does not vanish: for if so, we must, in accordance with the theory of maxima and minima, proceed to the investigation of 8ou and 84u, and so on; a work beyond our present purpose. Let the definite integral, which is the subject of inquiry, be

u = ('vdx ; where

v = f (x, y, y', ', ... Y(")); then, by equation (105), since w = dy, because dx = 0,

71 fi

a | + | HỒg ô ; and as h = 0, u = | SH 3g da;

(112) the sign of which is to be determined ; and we have moreover to examine whether it can change its sign or not. PRICE, VOL. II.

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352.] For this purpose it is necessary that du should not vanish when du = 0; that is, du must not vanish, when h = 0. Now the complete integral of h = 0 is y = f(x, C1, C2,...C2n),

(113) the right-hand member of which contains 2n arbitrary constants. Let each of these constants vary, and let oy be the consequent variation of y; so that

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and becomes 1 +ðh: then, as the varied value of y differs from the original value only in the arbitrary constants, it must also satisfy 1 +; and as the variations of the arbitrary constants are arbitrary, we may replace them by new constants C1, C2,...Can, so that the equation Òh = 0 becomes satisfied by

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and as y contains 2n arbitrary constants, so will dy also contain them: but oh = 0) cannot involve derived functions of an order higher than the 2nth; and therefore the above value of sy is the complete integral of the equation dh=0); consequently, if dy is of the form given in (115), h and du vanish simultaneously; and u cannot thus far have a critical value. Hence, if u = 0 is satisfied by an equation of the form (113), the first thing to be done in the examination of the character of that result is to inquire whether by satisfies the equation (115); if so, for that value of y, u is neither a maximum nor a minimum, and it is unnecessary to pursue the inquiry.

353.] If however the form of y given in (113), and which satisfies h=0, is not such as to satisfy (115), we must return to (112), and examine the sign of 8 h.

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it appears that the preceding expression for 8h contains terms of the form _d dv d .öy ) .

(116) : ? dy(+)2 dwa s wherein the order of the derived of dy is the same as the index of so which affects the whole of its subordinate subject; and it appears also that the other terms may be grouped in pairs of the form, d" (cd".@y) (– jo-o de Component);


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where c = 0,(P) 4,6)

(118) the connecting sign in (117) being + or -, according as s–y is even or odd.

Now all the terms of which such a series as (117) is composed can be put into the form (116); and consequently dh admits of being expressed in a series of terms, the type of each one of which will be

dk 5 dk.dy?

- dock {Ar dak } where a, is a determinable function of x.

By the theorem given in (57), Art. 426, Vol. I, if p and Q are two functions of x, whose derived functions are denoted by P', p", ... P("), (', e", ... QC"),

d"Q d".PQ nd"-1.pq n(n − 1) dn-2.p"Q dwn = dan -ī dxn-1 + 1.2 dx*–2

- n d.p(n-1)

...(-)"-1"da"( - )"p() Q; (119) which theorem we shall apply to the subordinate subjects of dif

ferentiation in (117), when some convenient substitutions have been made.

Let oy =n; so that employing the ordinary notation of derived-functions


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