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will give the value of

it will be sufficient to write

d2 y d2 z
d12

dy, d z, and a alone are made to vary,

d R d R

dx'

dy

and the resulting equation,

d V = da,

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da; or, the same result will be obtained if

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An instance will illustrate the principle of the method; if we multiply the equations of p. 376, by dx, dy, dz, respectively, add them and integrate the equation so formed, there will result this integral equation,

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in which, a is an arbitrary quantity introduced by integration. If we compare this with V = a, (see p. 378.)

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But if R be a function of x, y, z, the left-hand side of the equation is (see Prin. Anal. Calc. pp. 78, &c.) the complete differential of R, which is usually thus expressed dR; consequently,

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This result has been obtained by means of the symbol d, and of the process indicated by it; that is, (see p. 378.) by an abridgment of the direct and plainer method. This latter, however, in the present case, is easily instituted; thus, since

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substitute these values in d V, and there will result,

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* If dx, dy, dz, only are made to vary and not r, the last term in dV (1. 12.) may be suppressed as well as the second terms in II. 14, 15, 16. and the same result will still subsist; which is the first part of the rule of p. 379.

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- (-), was suppressed, since ♪ x, dy, dz, are equal nothing;

for, the differential equations of the first order are the same both in the undisturbed and disturbed system: and since the differentials of x, y, z, when the disturbing force acts, have been expressed by dx + dx, dy + dy, dz + dz,

we must necessarily have

dx= O, dy = 0, 8% = 0,

which, in fact, are three equations of condition between the variations of the arbitrary quantities; for instance, if x contain two arbitrary quantities a and b, then

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which equation determines the relative values of da and ♪ b.

Hence, the abridged method, under its most simple form, is made to depend on equations such as these

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We have used the symmetrical equations of p. 376. and the equations derived from them, chiefly for illustration: but the method applies equally to any equations derived from the preceding. In deducing the variations of the other elements, it is not intended to use the symmetrical equations; because, as it has been observed, their integration affords not simply the elements of a planet's orbit, but functions of those elements.

In the instance used for exemplifying the equation = a (see p. 379.) it so happens that the quantity a therein introduced, is the mean distance, one of the elements. But this circumstance is peculiar to that differential equation. The five other equa

tions may be constructed so as to involve each only one arbitrary quantity; but, then, such arbitrary quantity will not be an element.

The three equations of p. 376. being of the second order admit, each, of two integrations: each integration introduces an arbitrary quantity, therefore, there will be six differential equations of the first order (like the one of p. 378.) and six arbitrary quantities.

Any equation such as V = a, of the first order belongs equally (see p. 381.) to the disturbed and undisturbed system. Hence, to the same systems the integral of that equation must also belong. The finite expressions, therefore, of x, y, z are the same but there is this distinction to be noted in their values; in the latter, which is the elliptical system, the arbitrary quantities are constant, whilst in the former they are variable, their variations being determined by formulæ similar to that already obtained. Now, since the expressions for x, y, and z are the same, the curves of which these are the co-ordinates must be similar, or of the same kind; but, in the undisturbed system, the curve is an ellipse: the curve therefore to which x, y, z are the co-ordinates, when the arbitrary quantities a, b, c, &c. are variable, must be also an ellipse: this, however, is not the curve described by the body it is merely the ellipse that would be described were the disturbing forces to cease at that point of time for which the arbitrary quantities a, b, c, &c. were determined. At the next instant a, b, c, &c. have different values, and the ellipse is of different dimensions: so that, as it is easy to see, the successive ellipses form a series of ellipses of curvature to the real curve.

:

It is easy to see that the method which has been described rests on the same principle as that which is technically denominated the Variation of the Parameters, and which was employed in pp. 96, &c.

We will now proceed to deduce the variations of the elements

*These are not independent, the one of the other; the equation which connects them reduces their number to five.

on the principles already laid down, but by the aid of those equations which involve the projection of the radius vector, the longitude and the tangent of the body's latitude: these equations are (see pp. 92, 66.)

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dR dR dR

which

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dv

ds

`p2. (1 + s2)}

and, if we rescind from these the terms

depend on the disturbing force, there will remain three equations for determining the elliptical laws of the body's motion.

If we appropriate, as in p. 378. the symbol to represent the effects of the disturbing force, then when that force acts

d2 v = d (d v + ô v) = d2 v + dò v,

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Substitute these values in the two first equations, and there will result, by virtue of the equations of condition,

δρ = 0, δυ= 0,

and of the elliptical equations mentioned in 1. 9.

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or, if we suppose the latitude s to equal nothing, and r to be the

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