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will consider only one simple instance: viz. that in which it is

1

required to find 82u, when u =

=['v dx,

and

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dv

dv

but 8v = (dv) &x + (dy) by + (dry) by " + (dy )ży" +.

82v = ô (dv) 8x + 8 8 (dv) by + 8 (dv) by2 +

dy

dy

dv

+ (dv) 32x + (dv) 83y + (dy ) 8y +

dx

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dy

(dv)...

dy'

8x

(73)

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are functions of x, y, y', y”.....;

8. (dr) = (day) &x + (

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(day) by + (dvd) by +
(dy'dy

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+ 2 { (dry) 8x by + ( 127 ) 8.x by + ... + (d) by by +... }; (75)

2{(

d2v dx

and therefore

dx dy'

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By similar processes may d3u, d'u,... be calculated.

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311.] We come now to the investigation of the variation of a multiple integral; but instead of taking the problem in its most general form, I will consider the case of a double integral only; for the principle on which the inquiry is founded is the same in all cases; and the number of terms in the result increases so rapidly with each new integral sign, that by taking any higher order the formulæ are so complicated as to require new symbols and new modes of abbreviation, and no result useful for our present purpose is arrived at.

And I shall consider only a simple case of a double integral: that, viz. in which the element-function involves x, y, z, (z being an undetermined function of x and y), and the partial derived functions of z of the first and second orders; and in which also the limits of integration are given by an inequality in accordance with the principles explained in Art. 214: so that the range of integration includes the values of the variables corresponding to all points within a given closed surface. This case will suffice for all the examples to which I shall at present have occasion to apply the calculus; and the student who desires further information will find an investigation of the general case in those memoirs of Ostrogradsky, Sarrus, Delaunay, and Lindelöf, which are mentioned in the foot-note of page 411. In those by Sarrus and Lindelöf especially the difficulties of the variations of double definite integrals are elucidated, and to them I am under great obligation for the following investigation. A peculiar symbol, which they call the symbol of substitution, has been largely employed by them; I however have found the symbols already employed in this work sufficient for the purpose.

Let the double definite integral which is the subject of vari

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and let us suppose v to be a function of x,y,z, and of

d2 z

dx2

x and y.

d2z

dx dy

(77)

dz

dy

-), (d), where z is an undetermined function of

dy2

For the sake of abbreviation let the following symbols

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du=

the upper and lower accents referring to the x- and y- partial
derivations respectively. Also let us take the limits in their most
general form; that is, as the integral stands in (77), let Y, and Y, be
Yo
functions of a, and let X1, X, be independent of x and y; although,
if the order of integration is changed, x, and x, are functions of
y, and Y, and Y are independent of x and y. Now, taking the
total variation of u as given in (77),

Υ

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substituting for 8.dy dx from equation (23). And integrating
by parts the last terms, and remembering that the order of
integration is indifferent, provided that attention is paid to the
value of the limits,

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Χρ

Yo

=

Yo

Χρ

XoYo

Now let us assume that in the general variation of the ele-
ment, dx dy = 0, because we shall thereby shorten an expres-
sion which is under any circumstances long, and (see Art. 301)
shall not abridge the generality; and let us also replace dz by w,
so that is an arbitrary function of
x and y; then

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where w', w,, w", ... are partial derived functions of w.

(83)

Let z, z, z,, z",... denote the partial differential coefficients of v with respect to z, z, z,, z",... respectively; then

so that

du =

dv = zdz +z′dz′+z,dz,+z′′dz′′ + z ̧ ́dz, +z,„„dz,„,
= zw + z w2 + z,w, + z" w " + z, w, + Z,,,,;

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Χρ

+

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Το

ΤΟ

Χρ

(84)

(z w + z 'w' + z,w,+z′′ w′′ +z;w; +z,w)dy dx. (85)

Now the last part of this equivalent of du involves not only w, which is an arbitrary function of x and y, but also its x- and y-partial derived functions of the first and second orders; and as it is to be integrated with respect to the same variable for which it is differentiated, several of these terms are capable of considerable simplification by means of integration by parts, and of the theorems for the variation of definite double integrals which are given in Art. 217.

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Thus taking the x-integral of this equivalence with the limits

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Again, taking the last three terms in the latter part of the second member of (85), as they involve second derived functions of w, two integrations will be required for their simplification. Let us first take the term whose element-function is z"w"; then, by the theorem expressed by (87),

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The first term of this second member does not admit of further reduction, because the element-function of a y-integral involves two x-derived functions. The second and third terms admit of the following simplification.

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so that substituting these equivalents in (89), we have

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Hence, substituting all these equivalents in (85), we have

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