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= 2
do-1.0"n(0-1) do-2.0"'"'(0-2)

dwo-1 T dxo-2 which terms are of the form given in (116); and by a similar process the other terms of (122) may be transformed into equivalents of the same form ; so that ultimately,

d.Ay n d2.A, y" du.A n'" d".A, n(n) OH = Aon+

- +...+ '

;(123) dx dx2 T dx3


. that is, o h consists of a series of terms in each of which the order of the derived-function of n or dy is equal to the order of the index of which affects it; and in which Ag, Ay, ... An are functions of x.

A process similar to that pursued above is also applicable if r-8 is an odd number.

It is manifest that an is (-)" ( 2); but the other coefficients, viz., Ag, Ay..., are of a form so complicated that it is useless to calculate them in the general case. Thus substituting (123) in (112), is .d.a, ń d.A, "

du T d.x2 where n = dy, and An, Ag, ... are determinate functions of x.

354.] Now we proceed to shew that, when &H is expressed in

.. to

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the form of the right-hand member of (123), ohndx is an exact differential when n is replaced in dh by a certain value; and that consequently the second variation of the required definite integral admits of integration : hereby we shall be led to a reduction of the result to such a form as will immediately indicate its sign.

dh is manifestly a differential expression containing n and its derived-functions up to the 2nth order inclusive.

Let z be a value of n satisfying the equation 8h = 0; that is, suppose z to be a function of x which substituted for n renders OH = 0. Such a value is given in (114), Art. 352: so that we have d.Az d2. Asa

d”.A, z(n) 10" 7 dxdx2t.. In dh let nz be substituted for n, and let the result be multiplied by z, and be subsequently represented by u for convenience of notation : so that v=x{ 4 d.AZ (12) d®.A()" d".An (12)(m)*

= z{40 92 +222 + +...+ dm}. (126) Then the following investigations will prove

(1) that u dx is an exact differential, whatever is the value of n:


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(2) that / u dx will have the same form as u, except that the index of will in each term be diminished by unity: in other words, that we shall have

d.B, u" d?.B, U'" u dx = B, u' +

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dm.Amz(m) 92 m dm-1. Amz(m) (nm)' m(m-1) dm-2.Amz(m) (12)"

- dxn-1 + 1.2 dxm-2

m-1 m d.Amez(m) (nz) (m-1) (-)m-1"

-(-)" Am2(m) (n2)" ;(130)

1 dx
as the last term is the same in both expressions they disappear
in the subtraction; and as these are the only integral terms,
(128) consists of a series of derived-functions; and therefore u,
which is made up of a system of terms satisfying the distributive
law, is also a derived-function, and u dx is integrable immediately
by virtue of its form.

355.] Upon an examination of the series (129) it appears (1)
that there are terms of the form

dm-k.Amz(k) z(k)(m-k)

M. dam-k
where m is a constant, and a", z(k) are functions of x, and in
which therefore the order of the derived of n is the same as the
index of î, which affects the whole subordinate subject; and
(2) that there are other terms, the general type of which is
m(m-1)...(m-k+1) m(m-1)...(m_k' +1) dm-k.Ama(k)z[k”) (

mk) 1.2.3...k


d xm-k and of these, if k and k' correspond to any particular term, so must there also be another to which k' and k correspond, and which is therefore

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Izm (m-1)...(m-K' +1) m(m-1)...(m-k+1) dm-k.Amalk) 2(k)n(n-k) 1.2.3...k

1.2.3...k so that there are pairs of terms of the form sam-k.cnm-K), ?

(131) dxm

ks where c = Am2(b)2(k), and is therefore a function of x.

Now in Art. 353 it has been shewn that a pair of terms, such as (131), can be expressed in a series of terms of the form

d.6, n d2.6, n" dm.bong(m)

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and we shall suppose all the terms of (129) to be so expressed;
and by a similar process all the terms of (130); so that ultimately
by addition

. d.B, n , do. B, n, d”.B, n®)


+ dan but B= 0, because it has been proved in the preceding Article that u dx is an exact differential : and consequently

d.B, ņ d?.B, n" d".Bm7(9).

V = de + dra +...+ dion and therefore, finally, , , d.B, n" d2.B.

dn-1.B. (H) x = Bin +

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where By, B2, B3, ... B, are functions of x: the general form of
these may be found, but in the general case it is too complicated
to be available for any useful purpose; and it is better to deter-
mine them, if necessary, in each particular case. It is plain
however that the only term in (128) which will give“ - is
„d".A, (92)(*)

; and that the only term of this latter expression · when expanded as in (129), which is of the required form, is

d". Anza ()

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and that if z is a function of x for which, when substituted for n in the expression for ồn, the whole vanishes; then

d.A, (nz), d®.42 (12)” ... +d".An (2)"}dt (134)

dx d x a dx is an exact differential by virtue of its form, and independently of the value of n; and that its integral is of the form

d.B.n" d2.B, n' dn-1.B. 7(n)

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and from this we infer as a corollary, that as 8h dy or n dẢ when expressed in the form (133) is a particular case of (134), so will the integral of ohndx be of the form (135): but as oh and (134) contain n, ñ, ... 7("), and nz, (nz)', (nz)”, ... (92)(n) in corresponding places, so in the integral of dh ndx, when expressed in the form (135), n, ñ, ...qm) must be replaced by ., (),... (3.)"; and therefore

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Now the process to be pursued is as follows; we must find a value of z; that is, we must investigate a certain expression, which, when substituted for n, will satisfy ôh = 0: this is given in Art. 352, and by (115); hereby we shall be able to integrate by parts the infinitesimal element-function of the second variation, viz. Ohndx, and to express it in the form (136): and in the general case, by the repetition of a similar process we shall ultimately arrive at an expression consisting of two factors, of which one will be a complete square, and the other, which is easily determined, will by its sign determine the sign of the second variation of the definite integral, and hereby give the required criterion of the critical values.

357.] For a first application of these criteria let us take the case wherein v = f(x,y,y'); so that by (104),

H = Y - am = 0;

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