But t't, or the difference between the times of describing the half periphery of the ellipse, and of describing the arc YPX is known, it being the difference between half the anomalistic year and the given interval, in time, between the observations which were made when the sun was at Y and at x : let this difference be represented by D; then

whence t may be found. Hence the time of being at Y, and the longitude of Y, as well as the velocity at that place being known, the time of arriving at P, and also the longitude of P may be found.

201. The excentricity of the earth's orbit, considered as an ellipse, may be conceived to be known approximatively from the relative values of the aphelion and perihelion distances, for these being represented by a and p respectively, a + p will represent the major axis of the orbit, and (a—p) will be the distance of the centre from the focus of the ellipse; a-p will be an expression for that distance when the semitransverse axis of the ellipse is supposed to be unity: this excentricity is usually represented by e; and a more accurate method of determining it will be presently explained (art. 207.).

also

2 a

202. Having determined the periodical time of a tropical revolution of the earth about the sun, astronomers divide 360 degrees by the number of days or hours in the length of the year, and consider the quotient as the mean daily or hourly motion of the sun; then, dividing the longitude of the perihelion point, found as above, for a given time by the mean daily motion of the sun, the result will be the number of days from that time since the sun's mean longitude was zero, or since the sun was in the mean equinoctial point. The mean longitude of the sun for any given time may then be found on multiplying the number of days, hours, &c., which have elapsed since the sun had no mean longitude by his daily motion in longitude; and the difference between this mean longitude of the sun and the longitude of the perihelion point for the given time constitutes what is called the mean anomaly at that time. The true anomaly is the difference between the true longitude of the earth and the longitude of the perihelion point. A third anomaly, which is called excentric, is used when it is required to find the relation between the true and mean anomalies.

203. In order to investigate this relation, let ADP be half the periphery of the elliptical orbit, c its centre, s one of the foci, or the place of the sun, P the perihelion point, and let E be the place of the earth. With CP as a radius, describe the semicircumference AQP, and draw the radius vector SE; also through E draw QR perpendicular to CP, join s and Q, C and Q, and let fall SM perpendicularly on co. Then the angle PSE is the true, and PC Q the excentric anomaly.

By the natures of the circle and ellipse,

D

RQ RE segment QPR seg. EPR,

M

с S R

and (Euc. 1. 6.) RQ: RE: triangle QSR triangle ESR; therefore, since by conic sections

RQ: RE: area of circle: area of ellipse,

by equality of ratios and composition,

P

sector SQP: sector SEP :: area of circle: area of ellipse, or ellipse sector SEP: circle sector SQP.

But the areas described by the radii vectores being proportional to the times of describing them (art. 194.) if T represent the time in which E describes the periphery of the ellipse, and t the time in which it describes the arc PE,

Tt area of ellipse sector SEP:

therefore, by equality of ratios,

Tt area of circle: sector SQP;

and if CP1, so that the area of the circle is represented by

π

π(=3.1416) the sector sQP is equal tot. To this sector

T

adding the area of the triangle scQ or CQ.SM which (if cQ or CP=1, CS=e, and the angle scq=u) is equal to 1⁄2 e sin. u, we have the area of the sector CQP equal

π

to

T

t + e sin. u.

Now the same sector is equal to CP.PQ, or u; therefore,

But again,

2π

T

expresses the mean angular motion of E about

s, being equal to the quotient arising from the division of the circumference of a circle whose radius is unity by the time of

a complete revolution in that circumference. Let this be represented by n; then

ntu - e sin. u. . . . . (A).

Since t is reckoned from the instant that E was at the perihelion point P, nt expresses the mean anomaly; therefore, from this equation we have the mean, in terms of the true anomaly.

204. Next, by conic sections, SE CP-CS.CR, which, since CP or CQ=1, cse, and the angle QCP = u, also, representing SE by r,

e cos. u. (B);

but, by conic sections, representing the angle PSE by 0, we have

and, in the second member, substituting the value of cos. 0, we get, after reduction,

This is a convenient expression for the true, in terms of the excentric anomaly.

205. A convenient expression for the radius vector is found in the following manner. We have above,

r = 1 -e cos. u,

which (Pl. Trigon., art. 35.) becomes r = sin.2u); or, putting cos.2u + sin.2 to which it is equivalent, we get,

1e (cos.21 u u for 1 (= rad.2),

r = cos.2 1 u — e cos.2 1 u + sin.2 1 u + e sin.2 u :

or, dividing the preceding equation for r by (1 + e) sin.2 1 u,

206. The difference between the true and mean anomaly at any given time is that which is called the equation of the centre at that time; and the determination of the value of this element when a maximum, is of great importance, since, by means of that greatest value, the value of the excentricity can be obtained more correctly than by the method stated above.

A

Since the velocity of the earth, when in aphelion, is less than when in perihelion, the mean velocity must be greater than the former and less than the latter; and since, from the symmetry of the ellipse, the increase of velocity in moving from aphelion to perihelion follows the same law as the decrease from perihelion to aphelion; there must be a point as p and p' on each side of AP, the line of the apsides, at equal distances from either of its extremities, at which the true is exactly equal to the mean velocity. Now, if a fictitious planet E were to be in conjunction with the earth p at the perihelion point P, and both were to set out E from thence at the same time, the radius vector SE of the former turning about s with a uniform angular motion equal to the mean angular velocity, while the radius vector sp of

p'

P

C

E

S

the earth turns about the same point with the elliptical

angular movement, it is evident that while the latter exceeds the mean movement, the angular distance between sp and SE will go on increasing, and that when the elliptical movement becomes less rapid than the mean movement, that angular distance will begin to diminish. The angle Esp, or the difference between the mean and true angular movements of the earth, that is, the equation of the centre, is evidently equal to zero when the earth is at the perihelion, and also when it is at the aphelion point of the orbit; and it follows, from what has been said, that it attains its maximum value on either side of the line AP at the instant when the elliptical movement becomes exactly equal to the mean

movement.

Hence, if the longitudes of the sun be taken from a table of such as have been computed from the observed right ascensions and declinations, for the two times when the daily differences of longitude are equal to the mean daily difference, that is, when the elliptical velocity of the sun or earth is equal to the mean velocity, the difference between those longitudes being represented by the angle psp, or twice asp, while the mean angular movement of the earth or sun, during the time elapsed between those observations, may be represented by ESE' or twice ASE; the difference between Asp and ASE, or between PSE and Psp will be the maximum equation of the centre, which may therefore be so determined. It may not be possible to obtain from the observations two longitudes of the sun at the precise moments when the true velocities of the sun or earth were equal to the mean velocity; but, since the equation of the centre varies very slowly, it is evident that if two longitudes of the sun were taken when the true velocities are equal to one another, and nearly equal to the mean velocity, the error in the computed value of Esp be very small, and it may afterwards be corrected by other means. It should be remarked that the longitudes obtained from the observations need not be corrected, on account of the movement of the perihelion in the interval of time between the instants at which the earth is in p and in p', since this movement affects equally the true and the mean longitudes.

2 π T

will

207. It has been shown above that the mean angular velocity is expressed by which, if T be given in seconds, will be the value of a circular arc (whose radius is unity) described about s by uniform motion in one second of time. Now, if v represent the angular velocity described by the