THEORETICAL ASTRONOMY. 99 
* be the vernal equinox. The arrows give the direction of the motions of 
the earth and planet, as also the order of the signs of the ecliptic. The 
planet P will now be seen from the sun, 8, in the direction Sp; the 
angle vSp, or the arc Yp, will consequently be the heliocentric longitude 
of the planet, and the angle ‘ST, or the arc v¢, the heliocentric longitude 
of the earth. From the earth T, the planet P will evidently be seen in the 
direction Tp’. It is further allowable, on account of the almost infinite 
distance of the fixed stars, to consider the line TA parallel to St as meet- 
ing in the'same point of the celestial sphere, and consequently A and Y as 
coinciding. The angle ATp’ then, or the arc vp’, is the geocentric longi- 
tude of the planet ; and since the plane of the planet at P has a certain incli- 
nation to the common plane of the inner and outer circles, the angle PSQ, 
or‘the arc PQ, will represent the heliocentric latitude of the planet, if the 
arc PQ be supposed drawn from the planet P, perpendicular to the plane of 
the earth’s orbit. Finally, the angle PTQ will be the geocentric latitude 
of the planet. The angle PTS, at the earth, will be the elongation of the 
planet, or its apparent angular distance from the sun, this elongation. being 
(pl. 6, fig. 14) equal to the heliocentric longitude of the earth, + 180°, 
diminished by the geocentric longitude of the planet. From the preceding 
it follows that p will be the helvocentric, and p' the geocentric place of the 
planet, with respect to the ecliptic. We can speak in the same manner of 
the heliocentric and geocentric position of a planet with respect to the 
equator, and consequently of the heliocentric and geocentric right ascension 
and declination. 
We have still to speak of the commutation and the annual parallax of a 
planet. The commutation is the angle PST (jig. 14), at the sun, S, 
obtained by deducting the heliocentric longitude of the earth from the helio- 
centric longitude of the planet. The annual parallax is the angle TPS, 
at the planet P, obtained by deducting the heliocentric longitude of the 
planet from its geocentric longitude. It is very evident that we can never 
speak of the geocentric place of the earth. 
Some other Important Elements in the Theory of the Planetary Motions. 
21. The mean anomaly, the true anomaly, the perthelion distance, and the 
aphelion distance, are far more important to the theory of the planetary 
motions than the commutation and the annual parallax. If, for instance, m 
( fig. 15) be the mean place of a planet, p the true place, then rays, Sm, Sp, 
from the sun to these two positions will form, with respect to the major axis 
AP, of the elliptical planetary orbit, the angles mSP’, pSP, which are respect- 
ively called the mean and the true anomaly of the planet. Their great import- 
ance consists in this, that the mean anomaly expresses the uniform or the mean 
motion in the circle, while the true anomaly expresses the true motion in the 
ellipse. The true anomaly, or the angle PSp,eonsists of two parts, angles 
mSP’ and mSp. This latter angle, mSp, however, which evidently is the 
difference between the true and mean motions of the planei, is called the 
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