An introduction to astronomy: designed as a text-book for the use of students in college. By Denison Olmsted ...

E,slEME1rNT'i OF' THIE PI,ANETARY OlrrITS'. 233
axes gives tle radius of a circle of trhe same area as the
ellipse. Tltherefore, with the center S, at the distance SB:...:
V/lKx-() -:ik, (describe the circle CEi:I, thelln ill the: area of
this circle be equal to that of the ellipse. At the samle tilme
that a body departs firom A. the a phelion, lot a body b)egin to
Imove with a uniform motion'rofi(n tlhroughl thl e periphery
CE(';F', and performin a whole revolution iin tlhe same period
that tlhe planet (lesribes the ellipse; the motion of thlis body
will repl esent the equable or metan motion of the planet, llnd
it will describe around S areas or sector' of circles which are
proportional to the times, anld equal to tile elliptic areas dcscribed in tlhe stame time by thle planct. Suppose tile body
dcscribing thwe circle to be at MN1:; then takitng the sector AS]P
- CSS., I) is t; e ttrue place of tlc 1)planct. l The aingle CSN[ is
is thle mealn anomlaly,x CS"I) the true, and I:)SM the equation
of the centerc. iut in a circle, sectors;ary as their acs;
therefore, thle sector D1)SANM may be used for tle equationl; b)ut
taking' CS1) from tile equals CSM.\[ and ASSP,.1)SMI:.::::: ACI.R,
which therefore measures thle equation of tlhe center. ]N'ow
ACJ)IP obviously i'ner ese till it becwnrcs AEf], thlat is, when
tle planet lhas reachted the point where the two orbits intersect.
it mlay be shown, thltt after passing EIt, the equation diriminisis. TIhe hlalf lorbits AE:13, CEG, are described in tile saml
time; and tIle mean pl)ace, therefore, remains ill advance of
tlce true, till they'Ca} G anld 3B togctller. JLet V' be the
mean place, and ]. tile trute plaee, at a certtailn moment; then
thel angle (SV is tile mealllanomaly, and GCS? tihe true, and
\VSm tlle equation. The sectors AS1, O(SV', are equal, being
delscribed in the same time. Takillg C1T.RS from each, A.CE.
];w.2;1t. +'VSi m;.'. VSz AC:- Et]a;lt; t}hat is, the equlation
rSm has dilninisiled by In'EmR since the l)anet was at Ei.
Therlfore:'tE is the pl)ace of greatest equation of tile center.:lBtt 1E is also the plhace wlcere the mean and true angular
motions tare equal, becautse thel equal sectors of the two orbits
describlcd iln ach instlant, htave the same lcngtht at tl at point,
and1 tlherefore thle saine aglgic. I c lncc, the greatest equaltio
occurns w/e'e tzie meZan (a'ut' fmotion is equal to the t.rue.
Anoloialy is now r1ckoltned om oi perildetio (Art. 200); but that change idoes
not afict th1e correctles s O(f this reasonil1ng.