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

241 TITi PLANEIT5.
planets, it will be renadily understood, tljath if thle quantity of,
matter In the central bodly s is increased, whilc the distance of
the.revolving body remains1 the same), the velocity of thle revolving body mullst be incrleased'alo, inl o'rder to gecnerate a
suffiiett centrlifigal force. to counterbalan cell the increlased
force of attraction in the central body, arising firom the increase
of its mass; and that, were tlche force oft' attraction diminl slced
rey cnovinig the body:- to a glratel distance fiom. tlle cenltelr,
then t.e rate of its motion would also have to be diminislhed,
otlerwvisc tle centrifuigal force, would ovelpower the fiorce of
at.traction.: It is a. remark(able fatt, tlat thle members of the
solar system are so aljusted to each othler, it. respectt o to their
velocities, distances fiom the sun, periodic tilnes, andi g'ra\itation toward the central body, that if any one of these particu1ars is known, all the rest bccome lknown also.'1int1s, if it
mere foundl that a new —discovered planet moved with one-sixthl
thle velocity of the earth, we should know at once that its distant.ce fr'oll thle slln was tllirty-six times as great as the earthl's
(distance,) that its timne of revolution was two. hundred and1 sixteen years, and that its gravitation toward thle sulln was twelv
hundred and ninety-six times less thlan thlat'of the carthl' for
the distane is as tthe square of the numlber expressing thloe ceiprocal of its velocity, comparel wl ith that of thec cam'thl; its periodic
time as tle cube; and the reciprocal of gravity as. the fo1,rtil,
poWter of thle same number. All tllis follow.s fi'om Kepler's
third law-.... flt-that.. the squares of the.periodic timesl are as the
cuibes of the distances; and firom the law.:of: unll crsalii graitration --— t.lhat tthe force of attraction is inversely as the squarel of
tlhe distance. Tlhec four p)articuIlars named, therefore, constittte
a series of nlbcers in geometrical progression, of which the
first term is equal to tihe ratio. The trutlh of tlhis proposition
llay be demonstrated as follows:
Let 1) be the mIcan distance of a planet friom the sunl, Tr the
ratio of tle'diamet e r to: tile ciircumtfr'enco- of a cirtc, 1a0ndJ' P. the
time of. revolution airound the s\llnor per1iodic time;then tlhe
g lr]) 1)
expression for the velocity is'V —:- -. And Y2.c-.
P]t, bylc 1': 2)a; V o V) -^. Since
1311t) by Kcpler s law, or2 co:i~!i orVTM m!