substituting for sin p its value sin H sin (z+p) = sin H sin Z (equa. 10, page 43), and reducing,

and as i is very small, sin i sin (A + a) does not differ sensibly from i sin A, and we thus have in seconds (For. 47, page 343),

Solution of Kepler's Problem, by which a Body's Place is found in an Elliptical Orbit.

Let APB be an ellipse, E the focus occupied by the sun, round which P the earth or any other planet is supposed to revolve. Let the time and planet's motion be dated from the apside or aphelion A. The condition given is the time elapsed from the planet's quitting A; the result sought is the place P; to be determined either by finding the value of the angle A EP, or by cutting off, from the whole ellipse, an area A EP bearing the same proportion to the area of the ellipse which the given time bears to the periodic time.

There are some technical terms used in this problem which we will now explain.

Let a circle A M B be described on A B as its diameter, and suppose a point to describe this circle uniformly, and the whole of it, in the same time as the planet describes the ellipse; let also t denote the time elapsed during P's motion from A to P; then if A M

t

=

× 2 A MB, M will be the place of the point that moves period uniformly, whilst P is that of the planet's; the angle ACM is called the Mean Anomaly, and the angle A EP is called the True Anomaly.

Hence, since the time (t) being given, the angle ACM can always be immediately found (see Art. 243, p. 101), we may vary the enunciation of Kepler's problem, and state its object to be the finding of the true anomaly in terms of the mean.

Besides the mean and true anomalies, there is a third called the Eccentric Anomaly, which is expounded by the angle D C A,

and which is always to be found (geometrically) by producing the ordinate N P of the ellipse to the circumference of the circle. This eccentric anomaly has been devised by mathematicians for the purposes of expediting calculation. It holds a mean place between the two other anomalies, and mathematically connects them. There is one equation by which the mean anomaly is expressed in terms of the eccentric; and another equation by which the true anomaly is expressed in terms of the eccentric.

We will now deduce the two equations by which the eccentric is expressed, respectively, in terms of the true and mean anomalies.

Let t = time of describing, A P,

P = periodic time in the ellipse,

a = CA,

ae= E C,

v = < PEA,

u = DCA; (whence, E T, perpendicular to DT, = EC

x sin u),

P = PE,

T = 3.14159, &c.;

then, by Kepler's law of the equable description of areas,

an equation connecting the mean anomaly nt, and the eccentric u.

In order to find the other equation, that subsists between the true and eccentric anomaly, we must investigate, and equate, two values of the radius vector P: or E P.

First value of p, in terms of v the true anomaly,

Second, in terms of u the eccentric anomaly,

=

(a e + a cos u)2 + a2 sin2 u (1 —— e2)

= a2 {e2 + 2 e cos u + cos2 u} + a2(1 — e2) sin2 u

=

=

2

a2 { 1 + 2 e cos u + e2 cos3 u}.

Hence, extracting the square root,

p = a (1 + e cos u).

but, in order to obtain a formula fitted to logarithmic computa

analytically resolve the problem, and, from such expressions, by certain formulæ belonging to the higher branches of analysis, may v be expressed in the terms of a series involving n t.

Instead, however, of this exact but operose and abstruse method of solution, we shall now give an approximate method of expressing the true anomaly in terms of the mean.

MO is drawn parallel to DC. (1.) Find the half difference of the angles at the base of the triangle ECM, from this expression,

-e

tan | (CE M — CME) = tan ↓ (CEM+CME) × 1 + e

in which, CEM, + CME ACM, the mean anomaly.

=

(2.) Find C E M by adding (CEM+CME) and

(CEM

- CM E) and use this angle as an approximate value to the eccentric anomaly DCA, from which, however, it really differs by LEMO.

=

(3.) Use this approximate value of DCA ECT in computing ET which equals the arc D M; for, since (see p. 368), × DEA, and (the body being supposed to revolve

t =

P

area

in the circle ADM)=.

P

area

ХАСМ, area A E D

= area A C M, or, the area DEC + area A CD = area D C M + area ACD; consequently the area DEC = the area DCM; and, expressing their values,

Having then computed ET = D M, find the sine of the resulting arc DM, which sine OT; the difference of the arc and sine (ET-OT) gives E O.

(4.) Use E O in computing the angle E MO, the real difference, between the eccentric anomaly DCA, and the < MEC; add the computed EMO to Z ME C, in order to obtain DCA. The result, however, is not the exact value of 4 DCA, since ZEMO has been computed only approximately; that is, by a process which commenced by assuming MEC, for the value of the DCA.

For the purpose of finding the eccentric anomaly, this is the entire description of the process; which, if greater accuracy be required, must be repeated; that is, from the last found value of < DCA = 4 ECT, ET, EO, and EMO must be again computed.

NOTE TO PROBLEM XIV, (Page 277.)

Rules for finding the Moon's Longitude, Latitude, Hourly Motions, Equatorial Parallax, and Semi-diameter, for a given time, from the Nautical Almanac.

Reduce the given time to mean time at Greenwich; then,

For the Longitude.

Take from the Nautical Almanac the calculated longitudes answering to the noon and midnight, or midnight and noon, next preceding and next following the given time. Commencing with the longitude answering to the first noon or midnight, subtract each longitude from the next following one: the three remainders will be the first differences. Also subtract each first difference from the following for the second differences, which will have the plus or minus sign, according as the first differences increase or decrease.

Find the quantity to be added to the second longitude by reason of the first differences, by the proportion, 12h. : excess of given time above time of second longitude :: second first difference: fourth term.

With the given time from noon or midnight at the side, take from Table XCIII the quantities corresponding to the minutes, tens of seconds, and seconds of the mean or half sum of the two second differences, at the top: the sum of these will be the correction for second differences, which must have the same sign as the mean.

The sum of the second longitude, the fourth term, and the correction for second differences, will be the longitude required.

For the Latitude.

Prefix to north latitudes the positive sign, but to south latitudes the negative sign, and proceed according to the rules for the longitude, only that attention must now be paid to the signs of the first differences, which may either be plus or minus.

The sign of the resulting latitude will ascertain whether it is north or south.