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Calculating in the same way the parts of the integral due to the remaining pairs of faces, we get for the final result

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Hence (17) simply expresses the fact that the surface-integral (18), taken over the boundary of S, is equal to the sum of the similar integrals taken over the boundaries of the elementary spaces of which we have supposed S built up.

We may interpret (17) by regarding u, v, w as the component velocities of a continuous system of points filling the region S, and supposing to represent some property (estimated per unit volume) which they carry with them in their motion. The surfaceintegral on the left-hand side of (17) expresses then the amount of which enters S in unit time across its boundary; whilst the above investigation shews that (19) expresses the rate at which the property is being accumulated in the elementary space dxdydz. The theorem then asserts that the total increase of o within the region is equal to the influx across the boundary. A particular case is where u, v, w are the component velocities of a fluid filling the region, and is put = p, the density. See Art. 12. Corollary 1. Let = 1; the theorem becomes

dv

- SSS (du + dy + du) dx dy dz.

[] (lu+mv + nw) dS = −

If u, v, w be the component velocities of a liquid filling the region S, the right-hand side of this equation vanishes, by the equation of continuity; so that

[[ (lu+mv+nw)dS0(20),

which expresses that as much fluid leaves the region as enters it.

dy dy dy

Corollary 2. Let u, v, w=

respectively, where

・dx' dy' dz'

is a function which, with its first differential coefficients, is finite and continuous throughout S. Then

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where dn is an element of the inwardly-directed normal to the surface of S. Substituting in (17) and performing the differentiations indicated, we find

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By simply interchanging and we obtain (provided & be single

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Equations (21) and (22) together constitute Green's theorem.

(22).

Corollary 3. In (21) let & be the velocity-potential of a liquid, and let 1; we find, since v2 = 0,

=

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which is in fact what (20) becomes for the case of irrotational motion. Compare Art. 44.

Corollary 4. In (21) let y=4, and let & be the velocitypotential of a liquid. We obtain

2

2

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dz

[[] { (db)* + (do)2 + (db)} dx dy dz = - [] odds.

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аф

ds......(24).

dx

dn

If we multiply this equation by p it becomes susceptible of a

simple dynamical interpretation. On the right-hand side

do dedn

notes the normal velocity of the fluid inwards, whilst pp is, by Art. 26, the impulsive pressure necessary to generate the actual

motion. It is a proposition in Dynamics* that the work done by an impulse is measured by the product of the impulse into half the sum of the initial and final velocities, resolved in the direction of the impulse, of the point to which it is applied. Hence the right-hand side of (24) when modified as described, expresses the work done by the system of impulsive pressures which, applied to to the surface of S, generate the actual motion; whilst the lefthand side gives the kinetic energy of this motion. The equation (24) asserts that these two quantities are equal, thus verifying for our particular case the principle of energy.

have

Hence if T denote the total kinetic energy of the liquid, we

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Corollary 5. In (24) instead of & let us write

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course satisfy (10) as & does; and let us apply the resulting theorem to the region included within a spherical surface of radius r having any point (x, y, z) as centre. With the same notation as in Art. 46, we have

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an essentially positive quantity. Hence, writing q2 = u2 +v2 + w3, we see that

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is positive; i.e. the mean value of q3, taken over a sphere having any point as centre, increases with the radius of the sphere. Hence q cannot be a maximum at any point of the fluid, as was otherwise proved in Art. 45.

65. We shall require to know, hereafter, the form assumed by the expression (25) for the kinetic energy when the fluid extends

* Thomson and Tait, Natural Philosophy, Art. 308.

to infinity and is at rest there, being limited internally by one or more closed surfaces S. Let us suppose a large closed surface Σ described so as to enclose the whole of S. The energy of the fluid included between S and Σ is

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where the integration in the first term extends over S, that in the second over Σ. Since we have by (11)

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where C may be any constant, but is here supposed to be the constant value to which was shewn in Art. 51 to tend at an infinite distance from S. Now the whole region occupied by the fluid may be supposed made up of tubes of flow, each of which must pass either from one point of the internal boundary to another, or from that boundary to infinity. Hence the value of the integral

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taken over any surface, open or closed, finite or infinite, drawn within the region, must be finite. Hence ultimately, when Σ is taken infinitely large and infinitely distant all round from S, the second term of (27) vanishes, and we have

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where the integration extends over the internal boundary only. If the total flux across the internal boundary be zero, we have

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Thomson's Extension of Green's Theorem.

66. It was assumed in the proof of Green's Theorem that and were both single-valued functions. If either be a cyclic function, as may be the case when the region to which the integrations in (21) and (22) refer is multiply-connected, the statement of the theorem must be modified. Let us suppose, for instance, that is cyclic; the surface-integral on the left-hand side of (21), and the second volume-integral on the right-hand side, are then indeterminate, on account of the ambiguity in the value of & itself. To remove this ambiguity, let the barriers necessary to reduce the region to a simply-connected one be drawn, as explained in Art. 54. We may suppose such values assigned to that it shall be continuous and single-valued throughout the region thus modified (Art. 56); and equation (21) will then hold, provided the two sides of each barrier be reckoned as part of the boundary of the region, and therefore included in the surface-integral on the left-hand side. Let do, be an element of one of the barriers, K1 the cyclic constant corresponding to that barrier, dn the rate of variation of in the positive direction of the normal to do. Since, in the parts of the surface-integral due to the two sides of

dy

dy

do is to be taken with opposite signs, whilst the value of 1' dn on the negative side exceeds that on the positive side by ê, we get finally for the element of the integral due to do,, the value

dy

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SS

dn

do

Hence (21) becomes, in the altered circumstances,

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where the surface-integrations indicated on the left-hand side extend, the first over the original boundary of the region only,

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