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(1.0). tan. sin (0'—0)+(1. 2). tan p′′. sin (0′-0′′)+&c.

=(2.0). tan. sin (0"−0) + (2. 1). tan p'. sin (0"− 0')+&c.

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As these quantities do not contain the mean longitude, nor its sines or cosines, they depend on the configuration of the orbits only.

Approximate Values of the Secular Variations in these four Elements in Series, ascending according to the powers of the Time.

480. The annual variations in the elements are readily obtained from these formule; but as the secular inequalities vary so slowly that they may be assumed to vary as the time for a great many centuries without sensible error, series may be formed, whence very accurate values of the elements may be computed for at least a thousand years before and after the epoch. Let the eccentricity be taken as an example. With the given values of the masses and mean longide tudes of the perihelia determined by observation, let a value of

9 dt the variation in the eccentricity, be computed from the preceding

equation for the epoch, say 1750, and another for 1950. If the lat

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the quantities

'de, being relative to the year 1750. Hence, ë

dt dt

being the eccentricity of any orbit at that epoch, the eccentricity e at any other assumed time t, may be found from

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with sufficient accuracy for 1000 or 1200 years before and after 1750.

In the same manner all the other elements may be computed from

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For as 7 and are given by observation, 7 and II, which are functions of them, may be found. All the quantities in these equations are relative to the epoch.

These expressions are sufficient for astronomical purposes; but as very important results may be deduced from the finite values of the secular variations, the integrals of the preceding differential equations must be determined for any given time.

Finite Values of the Differential Equations relative to the eccentricities and longitudes of the Perihelia.

481. Direct integration is impossible in the present state of analysis, but the differential equations in question may be changed into

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and substituting the differentials in article 477, the result will be

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dt

= − { (1.0) + (1 . 2) + &c. } h' + 1.0 h +

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1.2 h"

It is obvious that there must be twice as many such equations, and

as many terms in each, as there are bodies in the system.

482. The integrals of these equations will be obtained by making

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It is easy to see why these quantities take this form, for if

0, h' 0, &c., l = 0; l' = 0, &c., then

h'

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And by article 214 h = N sin (gt + 6), N and 6 being arbitrary constant quantities. In the same manner = N cos (gt + 6).

483. If the preceding values of h, h', h'', &c., l, l', l'"', &c., and their differentials be substituted in equations (131), the sines and cosines vanish, and there will result a number of equations,

Ng={(0.1)+(0.2)+(0.3)+&c.}N - [0.1 N 0.2 N"-&c.

N'g={(1.0)+(1.2)+(1.3)+&c.}N' – 1.0 N-1.2 N"-&c.(132)

&c.

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&c.

equal to the number of quantities N, N', N', &c., consequently equal to the number of bodies in the system; hence, if N', N", N'", &c., be eliminated, N will vanish, and will therefore remain indeterminate, and there will result an equation in g only, the degree of which will be equal to the number of bodies m, m', m", &c. The roots of this equation may be represented by g, g1, 52, &c., which are the mean secular motions of the perihelia of the orbits of m, m', m", &c., and are functions of the known quantities (0.1), 0.1, (1.0), 1.0, &c., only. When successively substituted in equations (132), these equations will only contain the indeterminate quantities N, N', N", &c.; but it is clear, that for each root of g, N, N', N", &c., will have different values. Therefore let N, N', N", &c., be their values corresponding to the root g; N1, N, N", &c., those corresponding to the root g; N2, N, N", &c., those arising from the substitution of g, &c. &c. ; and as the complete integral of a differential linear equation is the sum of the particular equations, the integrals of (131) are

h = N sin (gt+6) + N1 sin (g1t+61) + N ̧ sin (g,t+62)+&c. h'N' sin (gt+6) + N,' sin (g1t + 61) + Ng' sin (g2t+6)+&c. (133)

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1= N cos (gt+6) + N, cos (g1t +61) + N2 cos (g,t +62) + &c.,

l'= N' cos (gt + 6) + N' cos (g1t + 61) + N2 cos (g2t +62) + &c.

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for each term contains two arbitrary quantities N, 6; N1, 61, &c. 484. Since each term of the equations (132) has one of the quantities N, N', &c., for coefficient, these equations will only give values N' N" ; &c., N N'

of the ratios

so that for each of the roots g, g1, ga, &c., one of the quantities N, N1, N2, &c., will remain indeterminate.

To show how these are determined, it must be observed that in the expression

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of article 474, S and S' are the coefficients of the first and second terms of the development of

(a2-2aa' cos B + a'2}},

which remains the same when a' is put for a; and the contrary, that is to say, whether the action of m' on m be considered, or that of m Hence if m, n', and a', be put for m', n, and a,

on m'.

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But if the mass of the planet be omitted in comparison of that of the sun considered as the unit,

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