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Also by the theorem of Leibnitz, and that given in (119),

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do-1.c"n(-1)
dxo-1

+

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dra-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,

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that is, dн 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

d index of dx tions of x.

which affects it; and in which A, A1,

...

A, are func

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

It is manifest that ▲, is (−)”

(

d2 v

dy()); but the other coefficients, viz., ▲。, ▲1..., are of a form so complicated that it is useless to calculate them in the general case.

Thus substituting (123) in (112),

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dr. Ann(n)

+

+...+

dx

dx2

dxn

In dx,

En dx, (124)

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dy, and A, A1, are determinate functions of x.

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

η

the form of the right-hand member of (123), dнn dr is an exact differential when 7 is replaced in dн 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. OH is manifestly a differential expression containing and its derived-functions up to the 2nth order inclusive.

η

Let z be a value of 7 satisfying the equation dн= 0; that is, suppose z to be a function of a which substituted for η renders dн = 0. Such a value is given in (114), Art. 352: so that we have

A02+

d.A, z d2. Aq z"

+

dx

dx2

+ +

d".A, z(n)

dx"

= 0. (125)

In dн let nz be substituted for 7, and let the result be multiplied by z, and be subsequently represented by u for convenience of notation so that

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Then the following investigations will prove

(1) that ude is an exact differential, whatever is the value of ŋ:

(2) that u de will have the same form as U, except that the

d dx

index of will in each term be diminished by unity: in other

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Multiplying (125) by ŋz, and subtracting it from u as given

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which series consists of pairs of terms; of each of which the

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to these let the theorem given in (119), and that of Leibnitz be respectively applied: then

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(-)k

m dm-1.A (m) (12)
dxm-1

1

m(m-1) dm-2.A, 2(m) (172)′′
dxm-2

+

1.2

(~)m-1 m d.A,,(m) (12) (m−1)

1

dx

m

(-) Am (m) (n); (130)

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 ude 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

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where м is a constant, and A", z(*) are functions of x, and in
which therefore the order of the derived of ʼn is the same as the

d

index of which affects the whole subordinate subject; and

dx'

(2) that there are other terms, the general type of which is
plainly,

m(m-1)...(m-k+1) m (m-1)... (m-k'+1) dm-k.Az(*)(*') ŋ(m—k')

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m2

dxm-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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m(m—1)... (m—k' + 1) m (m −1)... (m—k+1) dTM ̄k ́.Am2(k') z(k)n(m−k)

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so that there are pairs of terms of the form

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where c = Am() z(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

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d.b1n d2.b2n"
dx
dx2

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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.B1n d2.B2n"

d".B, n(")

u = Bon+

+

+ +

;

dx

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dx2

but B1 = 0, because it has been proved in the preceding Article Bo= that uda is an exact differential: and consequently

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where B1, B2, B3, ... B, are functions of a: 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 determine them, if necessary, in each particular case. It is plain

however that the only term in (128) which will give d".A, (nz)()

2

dx"

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

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;

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

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

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and from this we infer as a corollary, that as dнdy or noй when expressed in the form (133) is a particular case of (134), so will the integral of dнndæ be of the form (135): but as dн and (134) contain n, n,... n("), and n≈, (nz)', (nz)", ... (nz)(") in corresponding places, so in the integral of dнndx, when expressed in the

η

form (135), ŋ, n ́‚....... »(") must be replaced by 2, (2), (2)";

and therefore

η,

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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 ŋ, will satisfy dн = 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. dнndx, 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),

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