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

96 UNIVYJCESAT GRlAYI? nt'Xrl Io.
Nowt, it is shownv in collnie setionss, that INFO:: by
similar triangles, INFO_-).(, NI. k Substituting. this for INF
g ) 2( ]') j
ill tle eqlluation ftr 5:It'I- Fabove, we have NI.I: = N. i enco,`
in thi equation foir NM, wte i]hve
IN'I N[. I,
INow the sector.13IM N is measure d l)y - iM. NKA;.' NK.1(= —
1 _.I..N 4147AMN12:II"lIN
2I tFMN; an d N', I M it ut, as equal
areas alre described by tle radius vector il eqtual timles, ]'MIN
is constant. Therefore
INtl (:::: fJ) - -ti;
tllat is the centripetal force varies inversely as the square of
the distance.
$80. it is tlhus proved, that in any elliptical orbit (cescriled about the focus as tile center of attraction, the intenlsity of that attraction varies inver-sely as the square iof tile iadilus vector. As, there is nothing in thle foregoing, demonstration to limit tile conclusion to tile orbits whici are nearl y circular, like thtose of th1e planets, we are at liberty to apply it to
orlbits of extreme eccentricity, as those of thle comets. \d it
is proved by Newto n in his Priillia, that tlhe satin la\ of
force is necessary, in order that a body Ilmay describe anyi olne
of the conie sections about its fotcus as twhe center of attraction.
18 t1. A1nd not only does this law prevail in all parts of anly
* Jackson's Clonli Sections, The sale may be derived ftrom Coffil's (on.
See.. Prt. V., (Curvature. it' or MO'::::: b- a and b beting tihe semi-axes;.
MO. X(l t FM.MV)" (. AMultiply by ((b'~)', aml divide by its eyqual (FP.VI,)';
th en t (M ~- 1 - )'I since FMP and V!f, r e similar! But
2:':::P P being tile paranmcter;,. t:-' J'(Vt...'i.?'/
p 1,2iF c'