Chap. IV.]

ELLIPTICAL MOTION OF THE PLANETS.

accomplished by Maclaurin's Theorem, for if u' be the value of u

But when e 0, unt + e sin u, becomes unt; and from

9

and 6 sin nt — 9 sin3 nt = − sin nt + sin 3nt; hence

+ &c. &c. &c.

This series converges rapidly in most of the planetary orbits on account of the small value of the fraction which e expresses.

387. Having thus determined u for any instant, corresponding values of v and r may be obtained from the equations ra(1‐e cos u)

and

tan v =

1+e tanu;

1-e

but it is better to expand these also into series ascending according to the powers of e; and in functions of the sines or cosines of the mean anomaly.

Determination of the Radius Vector in functions of the Mean Anomaly.

but as r is a function of e by the equation r = a (1 e cos u); and u is a function of e by u = nt + e sin u, therefore,

The differential of the latter expression according to e is

This gives a value of the radius vector in functions of the time.

*

Kepler's Problem. To find a Value of the true Anomaly in functions of the Mean Anomaly.

388. The determination of v in terms of nt is Kepler's problem of finding the true anomaly in terms of the mean anomaly; or, to divide the area of a semicircle in a given ratio by a line drawn from a given point in the diameter-in order to accomplish this, a value of v in functions of u must be obtained from

c being the number whose logarithm is unity; hence the equation in question becomes

The true anomaly may now be found in terms of the mean anomaly.

389. In order to have v in terms of the mean anomaly and of the powers of e, values of u, sin u, sin 2u, must be found in terms of the sines of nt and its multiples; and X, X2, &c. must be developed into series according to the powers of e. Both may be accomplished by La Grange's Theorem, for if

If i be successively assumed to be 1, 2, 3, &c., this equation will give all the powers of A in series, ascending according to the powers of e.

Again. If we assume unt + e sin u, is a function of u which is a function of e;

Values of u, sin u, sin 2u, &c., may be determined from this ex

pression by making

successively equal to nt, e. sin nt, &c. The substitution of these, and of the powers of λ, will complete the development of v, but the same may be effected very easily from the h.dt expression dv=

of article 372, or rather from