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follows the limits of the multiple integral (1) to be determined by the condition (a)*.

In like manner it is clear that when

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the expansion of V in powers of u will contain none but the even powers of this variable.

Again, it is quite evident from the form of the function V that when any one of the s+1 independent variables therein contained becomes infinite, this function will vanish of itself.

3. The three foregoing properties of V combined with the equation (2) will furnish some useful results. In fact, let us consider the quantity

fdæ,d.,...da,duu". {(dr.)2 + (dr.)".

fdxdx....dx.duuTM ̄'.

dx

+

+...+!

+

dx.

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where the multiple integral comprises all the real values whether positive or negative of x, x,,...,, with all the real and positive values of u which satisfy the condition

X2
1 + + +

...

x u2
+ <........... .(6),
h2

ar, a,,... a, and h being positive constant quantities; and such that we may have generally

a, > a,.

In this case the multiple integral (5) will have two extreme limits, viz. one in which the conditions

u2 h2

+ + + =

...

1 and u= a positive quantity... (7)

* The necessity of this first property does not explicitly appear in what follows, but it must be understood in order to place the application of the method of integration by parts, in Nos. 3, 4, and 5, beyond the reach of objection. In fact, when V possesses this property, the theorems demonstrated in these Nos. are certainly correct but they are not necessarily so for every form of the function V, as will be evident from what has been shewn in the third article of my Essay On the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. [See pp. 23-27.]

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Morcover, for greater distinctness, we shall mark the quantities belonging to the former with two accents, and those belonging to the latter with one only.

Let us now suppose that V" is completely given, and likewise V or that portion of V' in which the condition (3) is satisfied; then if we regard V or the rest of V' as quite arbitrary, and afterwards endeavour to make the quantity (5) a minimum, we shall get in the usual way, by applying the Calculus of Variations,

d' V d2 V n-s dv

0=-fdx, dx,... dx, du u** 81 ** dx+

— fdxdx..... dx uTM• §v, dv

du

du

+

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seeing that &V=0 and 8V,'=0, because the quantities V" and I'' are supposed given.

The first line of the expression immediately preceding gives generally

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n-s dv

+

น du

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.(2'),

which is identical with the equation (2) No. 1, and the second

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From the nature of the question de minimo just resolved, there can be little doubt but that the equations (2′) and (9) will suffice for the complete determination of V, where V" and V are both given. But as the truth of this will be of consequence in what follows, we will, before proceeding farther, give a demonstration of it; and the more willingly because it is simple and very general.

4. Now since in the expression (, u is always positive,

0

every one of the elements of this expression will therefore be positive; and as moreover V" and V are given, there must necessarily exist a function V, which will render the quantity (5) & proper minimum. But it follows, from the principles of the Calculus of Variations, that this function Vo, whatever it may be, must moreover satisfy the equations (2) and (9). If then there exists any other function V, which satisfies the last-named equations, and the given values of V" and V, it is easy to perceive that the function

V=V2+A (V1 ~ V)

will do so likewise, whatever the value of the arbitrary constant quantity A may be. Suppose therefore that A originally equal to zero is augmented successively by the infinitely small increments SA, then the corresponding increment of V will be

SV=(V1- V) SA,

and the quantity (5) will remain constantly equal to its minimum value, however great A may become, seeing that by what precedes the variation of this quantity must be equal to zero whatever the variation of V may be, provided the foregoing conditions are all satisfied. If then, besides V, there exists another function satisfying them all, we might give to the partial differentials of V, any values however great, by augmenting the quantity 1 sufficiently, and thus cause the quantity (5) to exceed any finite positive one, contrary to what has just been proved. Hence no such value as V, exists.

We thus see that when " and V are both given, there is one and only one way of satisfying simultaneously the partial differential equation (2), and the condition (9).

5. Again, it is clear that the condition (4) is satisfied for the whole of V; and it has before been observed (No. 2) that when is determined by the formula (1), it may always be expanded in a series of the form

V = A + Bu2 + Cu' + &c.

Hence the right side of the equation (9) is a quantity of the order u; and u being evanescent. this equation will then

evidently be satisfied, provided we suppose, as we shall in what follows, that

ns+1 is positive.

If now we could by any means determine the values of " and V belonging to the expression (1), the value of V would be had without integration by simply satisfying (2′) and (9), as is evident from what precedes. But by supposing all the constant quantities a,, a,, a,...... a, and h infinite, it is clear that we shall have

0 = V",

and then we have only to find V, and thence deduce the general value of V.

6. For this purpose let us consider the quantity

[dxdx..........dx ̧duuTM- {dV dʊ_ dV dū

+

dx, dx, dx, dx,

+

+.

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the limits of the multiple integral being the same as those of the expression (5), and U being a function of x, x,, ...... x, and u, satisfying the condition 0 U" when a, a,,......α, and h are infinite.

=

But the method of integration by parts reduces the quan

tity (10) to

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since 0 V"; and as we have likewise 0 ==

==

tity (10) may also be put under the form

U", the same quan

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Supposing therefore that U like V also satisfies the equation (2'), each of the expressions (11) and (12) will be reduced to its upper line, and we shall get by equating these two forms of the same quantity:

dU'
du

dV'

fdxdx... dx. u'” ̄1 y' = ƒdx ̧dî ̧.......dx, du u'"-'U' ;

the quantities bearing an accent belonging, as was before explained, to one of the extreme limits.

Because satisfies the condition (9), the equation immediately preceding may be written

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If now we give to the general function U the particular

value

U = {(x, — x,')2 + (x ̧ — x,')' + ... + (x, − x,')" + u2)}T,

which is admissible, since it satisfies for to the equation (2), and gives U" 0, the last formula will become

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in which expression u' must be regarded as an evanescent posi

tive quantity.

In order now to effect the integrations indicated in the second member of this equation, let us make

x-x" = up cos1; x-x"=u'p sin 0, cos 0,;

x-x"=u'p sin e, sin e, cos 0,, &c.

ρ

1

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