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Pierre Simon Laplace.

. (page 70 of 114)

easily proved by developing and reducing ; hence we have,

c'-^^ = i.{v/"^^^l + v/^^5^^r=i p. [8764e]

Taking the logarithms of both sides of this equation, and dividing by \/tng, we get t [8765].

t (4020) Adding Â±2 to both members of [8764], and extracting the square root, iqjqq ^
we get,

voLÂ» IV. 148

590 DESCENT OF BODIES FROM A GREAT HEIGHT. [Mec. Cel.

[8766]

at? =

The height h being given, and the time t determined by observation, we
may, from [8765], deduce the value of m. Then from- [8766] we can
ascertain the value of a.v', or the deviation of the body to the east of the
vertical. We can also determine m, by the figure and density of the body,

[8768] In a vacuum, or, in other words, when m [8740"] is infinitely small, we
shall have,*

[8769] aij' = -â€” â–  . \ / - . smJ.

S V g

There have been made, in Italy and Germany, several experiments upon

[8769] the fall of bodies, which agree with the preceding results. But these

experiments, which require very great care, ought to be repeated with still

greater accuracy.

[8766a'] v/2.v/^^^n^ ^^hWu, _^ ^ -hWm, ^ ^2.^^inc::r = c ^^'^"^ - ^-hW^g^

Dividing the second of these formulas by the first, we obtain,

[8766&]

c

[8768a]

c -\-C "I

Substituting this in the last terra of [8759], and the value of i [8765] in the first term of
its second member, we get [8766J.

* (4021) We have, by developing as in [55, 56] Int.,

substituting these in [8764], and rejecting 2 fi-om each member of the equation, we get

2mh-}-hc. = t^.mg-{-hc. Dividing by m, we get 2A = <2^ + ^^rms multiplied by 7Â».
[87686] y-oT . .

Now putting m = 0, we shall have 2h=:t^g, or t=\X _; substitutmg this in

[8760], after putting m = 0, we get the expression of,

[8768c ] aw' = i ngA sin.^ = ~ . \/ - . sin.d,

as in [8769].

X.v.'^^ie.] ON THE MOTION OF A BODY PROJECTED UPWARDS. 591

Case

16. We shall noiv consider the case in ivhich the body has a projectile ^^d'V^^

motion in space; and we shall resume the equations [8724, &c.], supposing, C^^'O]

, , . projected

/ "V \ I / upwards.

"â– " = '^\m) + """ ' [8771]

a..sin.Â« = Â£i.(|) + aâ€ž' [8737]; [8772]

50 that a-u' and au' will he the deviations of the body, from the vertical line
passing through the point ivhere the motion commences. The deviation o^u'
being in the direction of the meridian ; and the deviation a^v' being in the
direction of the parallel of latitude.* Then neglecting the resistance of the

air, we find that the equations [8724 â€” 8724"] will become of the following [8773']

forms ; f ^^1

differen-

Fun da-
mental
differen-
tial eqaa-

^ dds , _ . dv' ^ ds /'dy\ tions.

_ dds f dy\ , ddvl ^ cos.d ds /dy\ _ di/ /diA

= a. -â€” . -f- ) -f a. â€” - _2aw. -Tâ€” -. v â€¢( / )â€” 2aw.cos.^. â€” â€”g.( -/ ] ; [87751
dl^ \dd J ^ dt^ sin.d dt \dvij dt Â° \dsj' â– '

a dds /du\ , ddv' , ^ ds /dy\ , _, du'

= -râ€” .Ti.~)+cL. -â€” +2Â«.W.C0S.^. T- ( -77 ) +2an.cos.^. â€”

sm.6 dt^ \dTsJ ' dt^ dt \d6 J ' dt

2<x.n.sm.&. 4 ~ . ( ^

Multiplying [8774] by â€” ( "tt ) Â» and adding the product to [8775], we get

[8776]

dt sin.d ' \dvi

dy_
dd ^

Ino- frnm thft nrndiipt nf ci.n. , ^,,.

dt ' dt

-J-j or (j-)' In like manner, multiplying [8774] by ^â€¢(^)Â» ^"^ [8777]

adding the product to [8776], we get [8779], neglecting terms arising from

[8778], neglecting terms arising from the product of aw. j , aw. â€” , by

[8773a]

fr:

* (4022) The deviation, in the direction of the parallel of latitude, is au' [8740,8739] ;

and it is shown, in [8736, 8736'], that if aw = as.fj-j , the deviation of the body from

the parallel of latitude, or in the direction of the meridian, v?ill vanish. The difference
between this and the real value of azt [8771], is aw', which represents the actual deviation 1.87736]
in the direction of the meridian, as in [8773].

t (4023) Neglecting the resistance of the air, we shall have, as in [8705, 8707],
S=:0. Substituting this value of S, and those of aw, av [8771, 8772], we find that the [8774a]
equations [8724, 8724', 8724"] become respectively as in [8774, 8775, 8776].

592 ON THE MOTION OF A BODY [Mec. Cel.

the products of an.|, an. J, by (|) or (g). Finally, by

Approxi
mate

Ta'ptT neglecting similar terms in [8774], it changes into [8780].

jectile.

[8778] == a. ^ â€” 2ayi.cos.^. ^ ;

dt^ dt '

[8779] = a. ^' + 2a7i.cos.^. ^-f â€” 2aw.sin.^. ^ ;

dt^ dt dt'

[8780] = a. ^ + 2aw. sin.^. ^ -g.

dt^ ^ dt ^

Now integrating these equations, and fixing the origin of the co-ordinates
[8781] a5, aw', at?', at the point where the motion commences, and the origin of the
time t at the commencement of the motion, we shall have,*

* (4024) If we take the diiFerential of [8779], and substitute in it the values of
idu' dds

a.-â€” , a.â€” , deduced from [8778, 8780], we get, after dividing by dt,

[8781a] 0= a. ^' -f 4a7i2.(cos.24-f sin.^d). -^ â€” S^n.sinA

Substituting cos.^^-{-sin.^^= 1, multiplying by dt, integrating and adding the constant
quantity â€” 4C'n^, we obtain,

[8781&] == a. -I 4- 4a.n^.v'â€”2gnt.s\n.6â€”4C'n^.

This linear equation of the second degree, is the same as that which is solved in
[8781c] [865a, Sic], putting y = a.v', a=2n, a.Q^ = â€”2gnt.sm.6â€”4Cn^, b=C, (p=s,

in order to conform to the present notation ; and the integrations being performed in the

manner pointed out in [8656], we get, by reduction,
[8781rf] av' = -â€” C.sin.(2n^+s) + ^^^ . f + C'.

Without going through the labor of these reductions, we may more easily verify this value

of au', by substituting it in the differential equation [87816], which will vanish by this

[8781e] substitution. This value of at>' contains the three arbitrary constant quantities C, C, s,

but C, C, depend on each other. For av' vanishes when ^ = [8781], and then the

value of au' [8781^] becomes = â€” C.sin.s + C, or C' = C.sin.Â£. Substituting
[8781/]

this in au' [8781<Z], it becomes as in [8783], with the two arbitrary constant quantities

C, e, which are required for the complete integral. IVIultiplying [8778] by dt, integrating,

[8781/'] adding the constant quantity B.s'm.d â€” SCn.cos.^.sin.s, and using a.v' [8783], we get

successively.

X. V. ^ 16.] PROJECTED VERTICALLY. 593

Devia-
tions,

[8782]

olm' = Bt.smJ-{- Igt^.smJ.cosJ -{- C.cosJ.lcos.(2nt-\- s) â€” cos.s} ;

ci-i?' = "^ ' .t â€” C.{sm.(2nt-{- s)â€” sin.s} ; [8783]

a5 = BUosJ + ^gt\cos.^&â€”C.sinJ.{cos.(2nt-^s)â€”cos.s], [8784]

J5, C, Â£, being three arbitrary quantities, which depend on the initial
velocity of the body in the direction of the three co-ordinates.

Suppose^ for example, that the body is projected vertically upwards, with a
velocity equal to K. The positive values of s being here counted \.^^^^'\

ds
downwards, we shall have, at the origin of the time t,* a. â€” = â€” K. We [8787]

[8785]

Vertical
projec-

[8781/1]

du'
a.â€”- =2n.cosJ.(iv'4-B.sm.6 â€” 2Cn.cos.6.sin.s [8781g-]

at

= 2n.cos.5. } ^' ' .t â€” C.sin.{2ni-{-s)-{-C.sm.s ^ -\-B.smJâ€”2Cn.cos.6.s\n.s

= ^.sin.^ 4~^^-s'n'^Â«cos.4 â€” 2Cn.co5.6.sin.{2nt-\-s).

IVIultiplying this last expression by dt, and integrating, we get aw' [8782] ; the constant
quantity â€” C.cos.lcos.s being added, so as to make aw' vanish when i=0, as in [8781].

The second differential of [8783], divided by dt^, gives a. â€” =4Cn^.sin.(2n<+ s) ; [878H]

substituting this in [8779], and then dividing by 27i, we get [8781/]. Substituting in its
second member the expression [8781A], connecting the terras multiplied by C, and
reducing by putting 1 â€” cos.^^ = sin.^4, we get [8781w]. Dividing this by sin.^, we
obtain [8781o], whose integral gives as [8784]; the constant quantity -{-^'Sin.^.cos.s â€¢â–  â– '
being added to the integral, so as to make as vanish when <s=sO, as in [8781].

a.sin.^. ^ =2Cn.sin.(2n^ + Â£)+a.cos.^. ^ [8^8"]

==2Cn.sm.{2nt-\-s)-{-cos.L\B.sinJ-{'gt.sin.LcosJ â€” 2Cn.cos.^.sin.(2n^-|-Â£)| [8781m]

= 2Cn.smJ^6.sin.{2nt-]-s)-\-B.sinJ.cos.6-}-gt.sinJ.cos.^6 ; [878]Â»]

ds
a. â€” z=2Cn.sm.6.s\n.(2ni-\-s)-\-B.cos.6-\-gt.cos.^6. [8781o]

* (4025) If the body be projected upwards, in the direction of the radius r, or r â€” as,
with the velocity K, it will pass over the space â€” arfs = Kdt, in the first moment of
time dt, as in [8787], without altering the values of aw' or av'; so that at the

du dv

commencement of the motion we shall have â€” = ; â€” = 0, as in [8787']. Now

dt dt [87876]

taking the differentials of aw', au', as [8782, 8783, 8784], dividing them by dt, and
then substituting < = 0, and the values [8787, 8787'], we obtain the equations
[8788, 8789, 8790] respectively.

VOL. IV. 149

594 ON THE MOTION OF A BODY [Mec. Gel.

dii dr/

[8787'] shall also have at that origin, â€” - = ; â€” = ; therefore,

[8788]

= ij.sin.fl â€” zCw.cos.^.sin.s;

[8789]

= ;^.sinJâ€” 2Cw.cos.Â£;
2n

[8790]

Hence we deduce,*

â€” K= B.C0S.6 +2Cn.sin.(5.sin.s.

[8791]

^ . K.sm.6
C.sm..= ^^ ;

[8792]
[8793]

B = â€” JT.cosJ.

Devia- which give,!

tions with "

a vertical ^ Jf^ p. %

[8794] aw' == â€”sin.^.cos.^. } - .(2ntâ€”sm.2nt) + ^ .{lâ€”2n^t^-^cos,2nt) i ;

projec- ^ J

tion.

[8795] ^v' == ^-^ . ^ -^.(2ntâ€”sin.2nt)â€”K.(lâ€”cos.2nt) \ ;

[8796] a5 = ^Kt+igf-\- ^^^ . { l^2nH''â€”cos.2nt] + ^^^ . (2nt^sm,2nt).

* (4026) Dividing [8789] by 2n, we get [8792]. IVIultiplying [8788] by sin J,

and [8790] by cos.^, then adding the two productsj we get,
[8791a] â€” Z.cos.^ = B.{sm,^6 +^03.^^} =: B,

as in [8793]. Substituting this value of B in [8788], we get,
[879161 ^ "^ â€” ir.sin.^.cos.5 â€” 2Cw.cos.^.sin.s.

Dividing this by â€” 2?i.cos.5, we obtain [8791].

t (4027) Substituting,
sm.{2nt-\-s) = sin.27i^cos.s-f cos.2n^.sin.s ; cos.(2Â»^4~^) ^^ cos.2n^cos.s â€” sin.2n^.sin.s,
[22, 24] Int., in [8782, 8783, 8784], together with the values [8791â€”8793], we obtain,

/ Tr . - - â–  1 o . Â« .1 . ^ e-.sin.^ _, , X'.sin.^ . _ ^ fi-.sin.^ )
[8794a] OLti' = â€” X^.sin.^.cos.d+i^r.sni.^.cos.d-fcos.5. < .cos.2wH 5 â€” â€¢sin.2n^ â€” ^"i" C '*

[87946 ] av' = ^â€” â€” . < + ^ â€” ^â€”:r â€¢ sm.2nt + â€” â€” . cos.2n< â€” [ ;

2n ' ( 4n^ ' 2n 2n )

[8794c] cw = â€”Kt.cos.^6-{-igt^.cos.^6-\-sin.6. j â€” ^^^ .cos.2n^ '^'^^^'^^^"^^^^ \ '

By a different arrangement of the terms, we may change [8794a] into [8794] ; [87946]
[8794rf] into [8795] ; and [8794c] into [8796] ; observing that in [8794c] we must change cos.^d
into 1 â€” sin.^i).

X. V. Â§ 16.] PROJECTED VERTICALLY. 595

Reducing these expressions to series, and neglecting quantities of the order rg^ggq

n^ we obtain,*

aw' == ; [8797]

av' = ^ntK(gt^3ir).sin.d ; [8798]

a5 =â€”Kt-\-igt\ [8799]

These expressions show that the deviation of the body, in the direction of the

meridian aw', is very small; and that it is only sensible in the direction

of the parallel of latitude a^v'. If we suppose ^=0, in [8798], we shall

have the same expression of the deviation as in [8768c]. If we suppose

K to be given, and we wish to find the point where the body will strike the t^^^^]

earth, we must put as = 0, whence we get gt = 2K; consequently, f [8801']

, 4n.^^.sin.^
^^ = 3^ â€¢ [8802]

To reduce this formula to numbers, we shall observe that n is the angle ""^^"j^

[6800]

â€¢arallel of
atitude.

described by the rotation of the earth in a centesimal second of time [8727'],

40"
and this anele is equal to -r-z-â€”â€”, ; because the duration of the sideral day [8803]
Â° ^ 0,99727 -^

is 99727" ; we must reduce it to parts of the radius, or, in other words,

divide it by the radius in seconds 636620". g is double the space which [8804]

gravity causes a heavy body to describe in the first centesimal second of its

* (4028) We have, by developing as in [43, 44] Int.,

sin .2ni = 2Â«<â€” | n^ 1 3 +&c. ; cos.2n^ = lâ€”2n^ t^-{-^nU ^â€”hc. ; [8796a]

hence we deduce, ^

-1 . i2niâ€”sm.2nt\ = %n^t^^kc. ; -^.nâ€”2Tv^t^^cos.2nt) == ^^n^t^-i-hc. ; [87966]

SO that if we neglect terras of the order n^, as in [8796'], we shall find that both
expressions in [8796i] will vanish ; and then the equations [8794, 8796] will become
as in [8797, 8799] respectively; moreover [8795] changes into [8798].

t (4029) Putting as = in [8799], we get gt == 2K; substituting this in [8798]

we obtain a.v'=â€”-^nt^.K.s'm.6; and by using i= â€” [8801'], it becomes as in [8802]. [8802a]

Now the earth, by its diurnal motion, describes 4000000" in one sideral day of 99727" of
the centesimal division ; hence the arc n, corresponding to one second [8727'], is

40" 40"

0//^99727 ' ^^ ^"= 0^^,99727x636620 ' i" parts of the radius. Substituting this in [8802], [88026]

together with the values of g, K, 6 [8804 â€” 8308], we get the expression of au' [8807], j-gg^g ,
or by reduction as in [8808].

596 ON THE MOTION OF A BODY PROJECTED VERTICALLY. [M6c. C61.

fall [8761'] ; and this space, in the latitude of Paris, is equal to T^^^'%32214i,
[8805] If we suppose, for an example, that the velocity K is 500 metres per

second, we shall have for Paris, whose latitude is 54Â° ,2636, d equal to the
[8806] complement of this latitude, or 6 = 45^,7364. Hence we get,

[8807] (x.v' = â€” 4..500'"'*'''. ( ) . . sm.45Â°,7364 :

^ \ 7- ,32214 y 0,99727X636620" "*^"''' '

whence we deduce,

[8808] ^v' = â€” 128"'''^*S9.

Deviation TMs exprcsscs the distance of the place, where the body falls upon the earth,
piSectL. ^0 ^^^ ^^^^ of ^he place of projection. For the rotatory motion of the earth,
being from the west towards the east, the negative values of ^v' are to be
taken in the opposite direction, or from the east towards the west.

[8809]

X. vi. <^ 17.] MOTIONS OF ATTRACTING BODIES. 597

CHAPTER VI.

ON SOME CASES WHERE WE CAN RIGOROUSLY ASCERTAIN THE MOTIONS OF SEVERAL BODIES WHICH

MUTUALLY ATTRACT EACH OTHER.

17. The problem of the motions of two bodies, mutually attracting
each other, can be accurately solved, as we have seen in the second book,
[531 â€” 534] ; but when the system is composed of three or a greater
number of bodies, the problem, in the present state of analysis, can be
solved only by approximation. The following cases are however susceptible
of a rigorous solution.

If we suppose the different bodies to be situated in the same plane, so that
the resultant of the forces, acting upon each one of them, may pass through
the centre of gravity of the system, and that the different resultants may be theorem
proportional to the respective distances of the bodies from this centre, then it [881 1]
is evident that ifive impress upon the system an angular rotatory motion about
its centre of gravity, so that the centrifugal force of each body may be equal
to the force ivhich attracts it towards that centre,* all the bodies will continue [88ir]
to move in circles about that point, retaining, in relation to each other, the
same relative positions, so that they will appear to describe circles about
each other.

[8810]

[881 0'l

First

[8812]

* (4030) If we suppose all the bodies to have the same angular rotatory motion about j-gg-, .
the common centre of gravity, the centrifugal force of any one of them will be proportional
to its distance from that centre, as in [54'] ; and by hypothesis [8810', &c.] the whole
action of the bodies, upon any one of them, is reduced to a simple attraction towards the
centre of gravity, with a force which is also proportional to the distance of the body from
that centre. Now as both these forces are proportional to the distance from the centre of
gravity, it is evident that we can adjust the rotatory motion, so that they may exactly pgg.. .
balance each other, as in [8811'].

VOL. IV. 150

theorem

[8814J

[8814"]

MOTIONS OF ATTRACTING BODIES [Mec. Cel.

The bodies being in the preceding position, if we suppose that the polygon,

[8813] at ivhose angles the bodies may be imagined to be placed, varies in any

Second manner, but always retaining a similar figure, it is evident that the laiv of

attraction, being supposed to be proportional to any power whatever of the

distance, the resultants of the forces which act upon the bodies will be to each

[8814] other, at all times, as the distances of the bodies from the centre of gravity

of the system.* This being premised, we shall now suppose that all the

bodies, when in a state of rest, are innpressed at the same instant with

velocities proportional to their distances from this centre, and in directions

equally inclined to the radii drawn from this point to each of the bodies ;

* (4031) For the purpose of illustrallon. we shall suppose the attraction of the bodies

r8fti4 1 "PÂ°" G^c\i other to be as the power n of the distance; so that if, at the commencement
of the time t, we represent the distances of the bodies w', m", &;c. from m, by s, s', &c.,

[88146] the action of the bodies m', m", &,c. upon m, will be expressed by w'.s", tw^.s'", &c.,

[8814c] in the direction of the lines s, s', Stc. respectively. IMoreover if we represent by r, r', f,
&;c. the radii, drawn from the centre of gravity of the system to the bodies w, m', m", he.
respectively, we shall have, according to the hypothesis assumed in [8814, Stc],

[8814rf] Kr, Kr', Kr", &;c. for the resultants of all the forces of attraction acting upon these bodies
respectively ; K being of the same magnitude for all the bodies at any one moment

[8814e] of time whatever. In the hypothesis [8811, 8812], where the polygon does not vary, the
value of K is also, at all times, invariable ; but when the polygon varies in magnitude,
but not in its figure, as in [8813, Stc], the value of K may also vary in successive
moments of time, but at any particular instant it must, by hypothesis, have the same value
for all the bodies m, w', w". Sec. Now if we suppose the figure of the polygon to vary as
in [8813], so that every linear measure s, s', s", &tc., r, r', r", &;c., corresponding to
the time ^ = 0, may be increased in the ratio of a to 1 ; and we then represent the new
values, corresponding to the time t, by the Italic letters s, s', s", Stc, r, r', r" , &c.
respectively ; we shall have,

[8814^] s = as, /=as', /^ffs", &tc. ; r=Â«r, '/ = ax' r^' = av'\ &c.

In this case the action of the bodies m', ml' , hjc. upon m [88146], will be changed into
m'.s% m"./'*, &c. ; or into the equivalent values m'.a^.s", m".a". s'", &,c. ; each of them
having increased from its original value [88146] in the ratio of a" to 1 ; and as the
resultants [8814c?] must also increase in the same ratio, they will become Ka^-r,
Ka~.r', Ka".r", Sec. respectively ; so that if we put for symmetry K=. Ka"~^, and use

[8814Z] the values of r, /, r", Sic. [8814A], these resultants will become Kr, Z/, Kr" , &c.
respectively ; or, in other words, they will be prop-rtional to the augmented or new

[8814m] distances r, r', r" , Stc. of the bodies from the centre of gravity of the system ; it being
evident that the position of this centre does not vary in consequence of these changes.

'- "^ These results are in conformity with the remarks of the author in [8814', &;c.].

[8814/]
[8814^]

X. vi. <Â§. 17.]

RIGOROUSLY ASCERTAINED.

599

then the polygons, formed at each moment by the right lines which connect
these bodies, will be similar ; the bodies will describe similar curves,* both
about the centre of gravity of the system and about each other, and these
curves will be of the same nature as that which a body attracted towards a
fixed point would describe.

To apply these theorems to an example, we shall consider three bodies,
whose masses are m, m', m", which attract each other, according to the

[8815]

[3815']

[8816]

"^ (4032) The velocity of projection of any one of the bodies, as for example that of
the body m, is by hypothesis proportional to its distance r, from the common centre of
gravity of the system, at the time of the commencement of the motion ; and as the angle of
inclination of the line of projection with the radius, is the same for each of the bodies, it is
evident that the area described in the first instant of time dt^ will be proportional to r^ ;
so that we may represent the double of this area by cr^.dt ; c being the same for all the
bodies m, m', rrl\ &;c. Comparing this with the expression of the same area cdt [366],
we get c = cr^. Moreover the force 9 [373], acting on the body m at the time i^ is
represented by Kr [8814/]. Substituting these values of c, 9, in the expression of dv
[376], we get the value of dv [8815e] ; and by accenting the letters we get the similar
expression of dv\ corresponding to the motion of the body m' ; observing that K is the
same for both bodies [8814Z] ;

cr2.t/r , . cr'S.rfr'

dv =

dv' ==

We shall now suppose that at the commencement of the motion, when < = 0, we have
if=br, and that the angles v, v', commence at that time, so that their places of origin
must be on the lines r, r', respectively ; and we shall then compute, by means of the
formulas [8815e], the relation of the arcs v, v\ when the general value of r' is expressed
by / = br. In this case we find, by substituting 1/ = 5r, / = br, in the expression
of dv' [8815c], that the numerator and denominator can be divided by the constant
quantity b^; so that this value of dv' will become identically the same as that of c?y,
[881 5e], and we shall then have dv = dv'-f or by integration v==v; supposing, as in
[8815e'], that both angles commence at the origin of the time t. Hence we see that when
/ = br, we shall have v' = v', therefore the figures described by the bodies m, mf, will
be similar; consequently the areas described by the radii vectores will be as r^ to r'^, or
as I to ZÂ»2 ; and as the areas described in the time dt, at the origin of the motion, are in
the same ratio, the times of describing the equal angles v, v', by the bodies m, mf, must
be equal. This is conformable to the remarks in [8815, &;c.]. What we have here stated
relative to the paths of the bodies about the common centre of gravity being similar to
each other, may evidently be applied to the relative motions of the bodies about any one
of them, considered as at rest, since they must also be similar.

[8815a]

[88156]
[8815c]

[8815<?]

[8815e ]
[8815e']

[8815/]
[8815g-]

[8815/i]

[8815i ]

[8815A]

[8815/ ]
[8815ni]

[8819]

600 MOTIONS OF ATTRACTING BODIES [M^c.C^l.

[8816'] function (^{r) of the distance r. We shall put ar, y, for the co-ordinates of
m, referred to the plane which connects these bodies, and to the centre of gravity

[8817] of the system ; also ar', y', for the co-ordinates of m\ and x'\ y, for those
of m". Then the force acting upon m, parallel to the axis of x, and
drawing towards the centre of gravity, is,*

[8818] m'. ^ . (x-x') + m". ^ . (x-x") ; [^;'a7a.?efto^:! '^ ]

s being the distance of m from m', and s' that of m from m". The
force acting upon m parallel to the axis of y, is,

[8820] m'. ^ . (2/-2/)+^"- ^ â€¢ {yâ€”lf')' [^pafa.r?o^;? " ]

S S

Likewise the force acting upon m' parallel to the axis of x, is,

[8821] m. ^-^ . (a/â€” a;)+m". ^^ . (x'â€”x") ; CTrlA'^l

s s

s" being the distance of m' from m". The force acting upon m' parallel to

the axis of y, is,

[88231 m. ^\ (y'^y)+m". ^-^ . (y'-f). [^TaraniiToT. "1

O

Lastly, the forces acting upon m", parallel to the axes of xf', if, are
respectively,

[8824] m. ^ . (a:"-a:) +m'. ^^ . (a/'â€” a/) ; [^%'a^.Â£r..'""]

Now, in order that the resultant of the two forces which act upon m.

[8822]

* (4033) The action of m' upon m, in the direction s, is m'.9(s) [8816']. Resolving
it, in a direction parallel to the axis of a?, it becomes rd, â€” .{od â€” x) in a direction
opposite to the origin of the co-ordinates [393'], or to conform to the present hypothesis
[8833], ni.â€” .{x â€” a/), towards the origin. In like manner the force of m" upon m,

is m". â€”-. (a; â€” x") ; the sum is as in [8818], which represents the whole force of m'
and m" upon m, resolved in a direction parallel to x. The other forces [8820 â€” 8825]
[8818c] are found in a similar manner ; all these forces being supposed to tend towards the origin
of the co-ordinates, as in [8833].

[8818a]

[88186]

X. vi. -^ 17.] RIGOROUSLY ASCERTAINED. 601

parallel to the axes of x and y, may pass through the centre of gravity of rggoei
the system, it is necessary that these forces should be in the ratio of x to