$1.QUARIS PENINSULAM.AMCNAM
UNIVERSITY OF MICHIGAN
LIBRARY VERITAS
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1837
ARTES
SCIENTIA
OF THE
TUEROR
CIRCUMSTIGE
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MISCELLANEOUS
T R A C T S
ON
Some curious, and very intereſting SUBJECTS
IN
MECHANICS, PHYSICAL-ASTRONOMY, and SPECULA-
TIVE MATHEMATICS;
:
WHERE IN
de la part
The Preceſfion of the Equinox, the Nutation of the
EARTH's Axis, and the Motion of the Moon in her
Orbit, are determined.
1710 - 176
By THOMAS SIMPSON, F. R. S.
And
Member of the ROYAL ACADEMY OF SCIENCES at STOCKHOLM.
---
L 0 N D 0 N,
Printed for J. Nourse over-againſt Katherine-Street in the Strand
MDCCLVII.
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TO THE
math,
Barya yang
10-14-23
RIGHT HONOURABLE
THE EARL OF
MACCLESFIELD, &c,
PRESIDENT of the ROYAL SOCIETY.
:
W
MY LORD,
HATEVER Luſtre, to the pub-
lic Eye, Works of Learning may
derive from the Patronage of the Great, it is
to your Lordſhip's perſonal Acquirements,
and extenſive Knowledge in the Mathemati-
cal Sciences, that my Ambition of deſiring
leave to prefix your Name to this Perform-
ance, is to be imputed: And indeed, My Lord,
an Author's natural Partiality permits me not
to hope, or wiſh, that any thing theſe ſheets
contain, will meet with a more general Ap-
probation, than what is due to the Propri-
ety of their being infcribed to the Earl of
MACCLESFIELD,
Were
A 2
DEDICATION.
.
>
Were your Character, My Lord, leſs
conſpicuous and diſtinguiſh'd, the Obligati-
ons I have to your Lordſhip’s Goodneſs,
would, alone, be Motives ſufficient to make
me gladly embrace this opportunity of pub-
licly expreſſing my warmeſt Gratitude, and
of teſtifying the perfect Eſteem, and pro-
foundeſt Deference, with which I am,
3
3
MY LORD,
-
.
:
Your Lordſhip's
moſt Obliged,
":
and ever Obedient
Humble Servant
Thomas Simpſon
bangsajt
nu.
PRE FACE.
THE TRACTS, or Papers compoſing the Work
here offered to
the Publick, were drawn up at ſeveral,, diſtant times, and upon
different occaſions ; either, with a view to clear up, or ſettle ſome dif-
ficult or controverted point in Aſtronomy, to shew the conformity of
Theory with Obſervations; or to extend and facilitate the analytic-
method of computation, by ſome improvements and applications, that
have not at all, or but ſlightly, been touched upon, at leaſt by any Eng-
liſh Author.
The firſt of theſe Papers, which is one of the moſt conſiderable in the
whole work, is concerned in determining the Preceſſion of the Equinox,
tation of the Earth's Axis, ariſing from the attraction of the fun and
moon ; wherein the late important diſcovery of Dr. Bradley, relating
to an apparent motion of the Fix'd Stars, unknown to former Aſtrono-
mers, is hewn to be intirely conſiſtent with the Theory of Gravitation.
This piece was drawn up about five years ago, in conſequence of
another on the ſame ſubject, by M. Silvabelle (a French Gentleman)
then delivered to me, for my opinion, fince printed in the Philoſophical
Tranſactions. -Tho I have particular reaſons for mentioning this cir-
cumſtance, I would not be thought to infinuate here, that my opinion had
any weight with Thoſe to whom the publication of that paper was owing:
I have, indeed, no reaſon to believe it.-Tho' the author thereof bad gone
through one part of the ſubject with ſucceſs and perſpicuity, and though
his concluſions were found perfe&tly conformable to Dr. Bradley's obſer-
vations, He nevertheleſs appeared (and ſtill appears) to me : to have
greatly failed in a very material, and indeed the only very difficult
part, that is, in the determination of the momentary alteration of
the poſition of the earth's axis, cauſed by the forces of the Sun and
Moon; of which forces, the quantities, but not the effects, are
truly inveſtigated.
The Second Paper, contains the inveſtigation of an eaſy, and very
exact method, or rule, for finding the place of a Planet in its Orbit,
from a correction of Dr. Ward's circular hypotheſis; by means of cer-
tain Equations applied to the motion about the upper focus of the ellip-
fis. From whence that table of Dr. Halley's, entitled, Tabula pro
expediendo calculo Æquationis centri Lunæ, may be very readily con-
ftru&cd.
.
PRE FACE.
The Simon
het
een en
ſtructed.-By,this method, the reſult, even in the orbit of Mercury, may
be found within a ſecond of the truth, without repeating the operation.
The Third, Shews the manner of transferring the motion of a Co-
met from a parabolic, to an elliptic Orbit; being of great uſè, when
the obſerved Places of a (new) Comet, are found to differ ſenſibly from
thoſe computed on the hypotheſis of a parabolic orbit.
The Fourth, is an attempt to hew, from mathematical principles,
the advantage ariſng by taking the mean of a number of obſervati-
ons, in practical Aſtronomy; wherein the odds that the reſult, this
way, is more exact, than from one ſingle obſervation, is evinced, and
the utility of the method in practice, clearly made to appear.--A
part of
this, and of the 7th paper, is inſerted in the XL IX volume of the Philoſo-
phical Tranſactions ; but the farther improvements here added, will (1
bope) be a fufficient apology for my printing the whole again, in this work.
The Fifth, contains the determination of certain Fluents, and the
reſolution of ſome very uſeful Equations, in the higher orders of fluxions,
by means of the meaſures of angles and ratios, and the right fines, and
verſed fines of circular arcs.
method of furd-divifors; wherein the grounds of that,
method, as laid
down by Sir Iſaac Newton, are inveſtigated and explained.
The Seventh, exhibits the inveſtigation of a general rule for the
reſolution of Iſoperimetrical Problems of all orders, together with
fome examples of the uſe and application of the ſaid rule.
The Eighth (and laſt) Part, comprehends the reſolution of fome ge-
neral, and very intereſting problems, in mechanics and phyſical Aſtro-
nomy; wherein, among other particulars, the principal parts of the
third, and ninth ſections of the firſt Book of Sir Iſaac Newton's Prin-
cipia,are demonſtrated, in a new, and very conciſe manner. But what,
I apprehend, may beſt recommend this part of the work, is the applica-
tion of the general equations therein derived, to the determination of the
lunar Orbit: In which I have exerted my utmoſt endeavours to render
the whole intelligible even to Thoſe who have arrived but to a tolerable
proficiency in the bigher geometry.
The
greater part of what is bere delivered on this
ſubject, was drawn
up in the year 1750, agreeably to what is intimated at the concluſion of
my Doctrine of Fluxions, where the general equations are alſo given.
The famous objection, about that time made to Sir Iſaac Newton's gene-
3
1
of
P R EF A CE.
of the Royal Academy of Sciences at Paris, was a motive ſufficient to
induce me (among many Others) to endeavour to diſcover, whether the
motion of the moon's apogee, on which that objection had its whole weight
and foundation, could not be truly accounted
for, without fuppofing a
change in the received law of gravitation, from the inverſe ratio of the
Squares of the diſtances. The ſucceſs was anſwerable to my hopes, and ſuch
as induced me to look
farther into other parts of the theory of the moon's
motion, than I firſt intended: but, before I had completed my dehgn, I
received the honour of a viſit from M. Clairaut (juſt then arrived in
England) of whom I learned, that he had a little before printed a piece
on that ſubject ; a copy of which I afterwards received, as a preſent at
his hands; wherein I found moſt of the ſame things demonſtrated, beſides
ſeveral others, to which I had not then extended my enquiry. Upon this,
I at that time defifted from a farther proſecution of the ſubject; being
chiefly diverted therefrom by a call then ſubſiſting for a new edition of
another work, in which ſome additions
ſeemed wanting. But I cannot
omit to obſerve here, in juſtice to M. Clairaut, that, tho' be indeed fell
into a miſtake, by too haſtily inferring a defect in the received law of at-
traction, from the inſufficiency of the known methods for determining the
effect of that attraction, in the motion of the moon's apogee, yet be was
bimſelf, the firſt who diſcovered the true fource, of that miſtake, and who
placed the matter in a proper light: Though there are ſome * wko
have, both before and fince, undertaken to give the true quantity of that
motion, from ſuch principles, only, as are laid down in the ninth ſection of
the firſt Book of the Principia : but that theſe Gentlemen, however they
may have made their numbers to agree, have
been greatly deceived in their
calculations, is very certain; ſince a conſiderable part of the ſaid motion
depends on that part of the ſolar force ačting in the direction perpendicu-
lar to the Radius-vector, which is by them, either intirely diſregarded,
or the effect thereof, not made one twentieth part of what it really ought
to be. There are others indeed, who have explained the matter, upon
true principles
, and with better ſucceſs
. Since M. Clairaut's piece firſt
made its appearance, the moſt eminent mathematicians, in different parts
of Europe, have turned their thoughts that way. But tho' what I now
offer on the ſame ſubject, may, perhaps, appear of leſs value, after what
has been already done by theſe great men, yet I am not very ſolicitous
upon that account, as it will be found, that I have neither copied from
* Vid. Walmſley's Theorie du mouvement des apſides (tranſlated into Engliſh) and
Vol. 47. N° 11. of the Philoſophical Tranſactions,
their
www
P R E F A C E.
.
:
their thoughts, nor detracted from their merit. The facility of the me
thod I have fallen upon, will, I flatter myſelf, be allowed by all, who are
appriz'd of the real difficulty of the ſubječt; and the extenſiveneſs thereof
will
, in ſome meaſure, appear from this, that it not only determines the
motion of the apogee in the ſame manner, and with the ſame eaſe, as the
other equations, but utterly excludes, at the ſame time, all terms of that
dangerous ſpecies (if I may to expreſs myſelf that have hitherto embar-
raſed the greateſt Mathematicians, and that would, after a great num-
ber of revolutions, intirely change the figure of the orbit. It thereby
appears, that all the terms, or equations in general, will be expreſſed
by fines and co-fines, barely, without any multiplication into the arcs
correſponding. From whence this important conſequence is derived,
that the mean motion, and the greateſt quantities of the ſeveral equations
will remain unchanged; unleſs diſturbed by the intervention of ſome
foreign, or accidental caule.
In treating of this ſubject, as well as in moſt of the other parts of the
enſuing work, I have chiefly adhered to the analytic method of Inveſ-
tigation, as being the moſt direct and extenhve, and beſt adapted to theſe
abſtruſe kinds of ſpeculations. Where a geometrical demonftration could
be introduced, and ſeemed
preferable, I have given one: but, tho' a pro-
blem, ſometimes, by this laſt method, acquires a degree of perſpicuity and
elegance, not eaſy to be arrived at any othcr way, yet I cannot be of the
opinion of Thoſe who affe&t to Shew a diſike to every thing performed by
means of fymbols and an algebraical Proceſs; fince, ſo far is the fyn-
thetic method from having the advantage in all caſės, that there are
innumerable enquiries into nature, as well as in abſtracted ſcience, where
it cannot be at all applied, to any purpoſe. Sir Iſaac Newton kimſelf
(who perhaps extended it as far as any man could) has even in the moſt
fimple caſe of the lunar orbit (Princip. B.3. prop.28) been obliged to call
in the aſiſtance of algebra; which he has alſo done, in treating of the
motion of bodies in renting mediums, and in various other places. And
it appears clear to me, that, it is by a diligent cultivation of the Mo-
dern Analyſis, that Foreign Mathematicians have, of late, been able to
puſh their Reſearches farther, in many particulars, than Sir Iſaac New-
ton and bis Followers bere, have done : tho’ it muſt be allowed, on the
other hand, that the fame Neatneſs, and Accuracy of Demonſtration, is
not every-where to be found in thoſe Authors; owing in ſome meaſure,
perhaps, to too great a diſregard for the Geometry of the Antients.
A D E
I
pppppTIAP Pppapa 636
papiromenamoro
DO
D E T E R MINATION
OF THE
PRECESSION OF THE EQUINOX,
And the different Motions of the EARTH's Axis,
Ariſing from the ATTRACTION of the Sun and Moon.
*HE PRECESSION of the Equinox, whereby the fix'd
ST ſtars appear to have changed their places by more
than a whole hign, ſince the time of the moſt ancient
Aſtronomers, is phyſically accounted for, from the at-
traction of the ſun and moon on the protuberant matter about
the earth's equator ; whereby the poſition of the ſaid equator
with reſpect to the plane of the ecliptic is ſubjected to a per-
petual variation. Were the earth to be perfectly ſpherical and
of an uniform denſity, no change in the poſition of the terre-
ſtrial equator could be produced, from the attraction of any re-
mote body; becauſe the force of each particle of matter in the
earth, to turn the whole earth about its center, in conſequence
of ſuch attraction, would then be exactly counterbalanced by
an equal, and contrary force. But as the earth, by reaſon of
the centrifugal force of the parts thereof, ariſing from the di-
urnal rotation, muſt, to preſerve an equilibrium, put on an ob-
late figure, and riſe higher about the equatoreal parts than at
the poles, the action of the fun on the ſaid equatoreal parts will
have an effect to make the plane of the terreſtrial equator to
coincide with that of the ecliptic: which would actually be
B
brought
::
2
Of the Preceffion of the Equinox,
$ ?" srt
on,
brought to paſs (neglecting other cauſes) was the ſun, or earth,
to remain fix'd in either of the folſtices, and the diurnal rotation
at the ſame time to ceaſe. But, though both the motions of
the earth contribute to prevent an effect of that fort, yet, in
conſequence of this action of the ſun, a new motion of rotati-
about that diameter of the equator lying in the circle of the
ſun's declination, is produced ; from which the preceſſion of
the equinox and the nutation of the earth's axis have their riſe.
The effect of the moon, as it is much more conſiderable than
that of the ſun, ſo is it likewiſe liable to ſome inequalities to
which that of the ſun is not ſubject. Were the inclination of
the lunar orbit to the plane of the equator to remain, always,
nearly the ſame, like that of the earth, the fame calculations
that anſwer'd in the one caſe would alſo anſwer in the other ;
but that inclination is continually varying, and, when the
aſcending node is in the beginning of Aries, is greater by above
{th part than the mean value; and therefore, as the force of
the moon to turn the earth about its center (other circum-
ftances remaining the ſame) is found, hereafter, to be as the
ſine of the double of the inclination, it is manifeſt, that, in the
ſaid poſition of the node, the motion of preceſſion will
much quicker than at the mean rate ; and conſequently that
an equation, depending on the place of the node, will neceſſarily
ariſe. The determination of which, as well as of the o her
motions of preceſſion and nutation ariſing from the attraction
both of the ſun and moon, I ſhall now proceed to ſhew: but
in order to pave the way thereto, it will be proper to begin.
with premiſing the ſubſequent Lemmasa
Here imprese con scrito yo he him for this page ease the site
om alebo na da se ne time to watch more than 4* * per **** RET MEER
go on
LEMMA F.
Suppoſing all the particles of a given ſpheroid A'Pápo to be
ſollicited parallel to the axis Pp, by forces proportional to the di-
fances from a plane PAOpa paſſing by the ſaid axis, in ſuch fort
that the two oppoſite ſemi-ſpheroids, A'Pp, ápp, may thereby be
equally urged in contrary directions; it is propoſed to determine
the whole effect of all the forces to turn the ſpheroid about its
center,
Let
and the different Motions of the Earth's Axis. 3
2
aa
a=ſemi-diam. OA' (perpend. to the plane PAOpa), Fig. 1.
A= area of the ellipſe PAOpa,
Let
g=force acting on a particle at the remoteſt point A',
ON, the diſt. of any ſection DONEQ from PAOpa:
Then, this ſection being alſo an ellipſe, ſimilar to Papa, we
ſhall have, by the property of the ellipſis, as A'O (aa) : A'O
ON*(aa—xx) :: PO’ : DN :: the area PAOpa (A) to
the area DONEC
(by the property of ſimilar
figures). Hence it is evident that Ax
***g will be
the ſum of all the forces whereby the particles in the ellipſe
DQEQ are urged parallel to the axis Pp of the ſpheroid;
which quantity, drawn into (x) the length of the lever ON,
will, conſequently, expreſs the effect of all the ſaid forces to
turn the ſpheroid about its center : and ſo the fluent of
AX xx
x xx xy, which is Ax xg (when x=a)
will truly expreſs one half of the quantity ſought.
=
Ах
aa
aa
2aa
aa **
ad
15
..
4аа
COROLLAR Y.
If the maſs, or content of the ſpheroid, which is A x 2ax
man
be denoted by S; then the force Ax xy, whereby the
ſpheroid tends to turn about its center, will be truly defined
by ķS xaxy, which therefore is juſt th part of what it
would be, if all the particles were to act at the diſtance of
the remoteſt point A'.
IS
LEMMA II.
Suppoſe a body to revolve in the circumference of a circle AFAF, Fig. 2.
whilſt the circle itſelf turns uniformly about one of its diameters
Aa, as an axis, with a very ſlow motion ; it is propoſed to deter-
mine the law of the force, ačting on the body in a direction per-
pendicular to the plane of the circle, neceſary to the continuation of
a motion thus compounded.
B 2
Let
4
of the Preceſſion of the Equinox,
Fig. 2.
Let AFaF and Afaf be two poſitions of the circle, indefinitely
near to each other, and let R and r be the two correſpond-
ing poſitions of the body; let alſo the planes RDn and mdc be
perpendicular to AF&F and to the axis AOa; in which planes
let there be drawn Rn and muc perpendicular to DR and dm,
meeting the plane Araf (produced out) in n and c; and let
there be drawn nv, parallel to the tangent Rtm, meeting mc in
V. If the velocity of the body along the circumference be ex-
preſſed by Rm, the velocity in the perpendicular direction Rn,
ariſing from the motion of the circle about the axis Aa, will
be repreſented by Rn. And, if the body were to be ſuffered
to purſue its own direction from the point R, it would, by the
compoſition of thoſe motions, arrive at the oppoſite angle v of
the parallelogram Rnum, in the ſame time that it might move
through Rm by the motion Rm alone ; and ſo would fall ſhort
of the plane by the diſtance cu*. It therefore appears that the
required force, neceſſary to keep the body in the plane, muſt
be ſuch as is ſufficient to cauſe a body to move over the diſtance
cu in the aforeſaid time; and that this force muſt, therefore,
be to the centrifugal force of the body in the circumference
(whoſe meaſure is et) as cv to et ; ſince the ſpaces deſcribed in
equal times, are directly as the accelerating forces.
Let now the ratio of the angular celerity of the circle about
its axis to that of the body in the circumference, be ſuppoſed as
r to unity; then, the latter of theſe celerities being repreſented
by Rm, the former will be defined by r x Rm; and conſe-
quently the celerity (Rn) in the direction Rn, by rx Rmx
OF
Moreover, becauſe of the ſimilarity of the triangles DRn and
dmc, it will be, as DR : Rn (rx Rmx DF) :: dm (DR+sm)
Rmx DR
tox Rmxsm: from whence, taking away
Rm sm
DR
*}. I am not sur
: mc
Ү Х
OF
OF
Rm X SM
the value of my or Rn, we get cv=rX
: which is
OF
* The lineola cr, lying in the plane of the circle, muſt be anſwered by
a force tending to the center of the circle; with which we have nothing to
do in the preſent conſideration.
in
}
-": :*****... Fw:-
>
i
and the different Motions of the Earth's Axis.
5
in proportion to the meaſure of the centrifugal force et, or it's
Rml"
equal as r x sm to Rm, or, becauſe of the ſimilar tri-
2OR
angles ORD and Rsm, as ar X OD to OR or OA.
Hence it is evident that the body, to continue in the plane
of the circle, muſt be conſtantly acted on, in a direction per-
pendicular to the plane, by a force varying according to the co-
fine of the diſtance AR of the body from the extremity of the
axis; whoſe greateſt value, at A, is to the centrifugal force in
the circle, as ar to unity. 2. E. I.
COROLLARY I.
If, inſtead of one, a great number of bodies or corpuſcles,
ſo as to touch one another and thereby form a continued ring
AFaF, were to revolve at the ſame time, and to be acted on
in the ſame manner (tut is to ſay, by forces in the ratio of
the diſtances from the diameter FF perpendicular to the axis
Aa), it is evident that they would all continue in the ſame
plane. And this will alſo be the caſe, when a number of con- Fig. 3.
centric rings ERGEG, &c. are ſuppoſed to perform their revo-
lutions together about the common axis AEea. For, aſſuming
B to denote the centrifugal force of a corpuſcle in the outer-
moſt ring AR'FaF, the centrifugal force of an equal corpuſcle
(R') in the ring ERPEG, will be equal to BXOA : whence, by
the foregoing proportions, 2r XBXOA will be the force act-
ing perpendicular to the plane at E: and 2r x B x
OA" OR
(=2r.xBxD) will be the true meaſure of the force acting
on a corpuſcle at R'; which, as r, B, and OA are all of them
conſtant, is evidently as the diſtance from the diameter FF.
Whence it follows, becauſe the diſtance below FF becomes
negative, that the forces above and below that diameter muſt
have contrary directions.
COROLLARY II,
Whatever hath been ſaid in the preceding Corollary holds
equally,
OE
OE
1
OE OD
X
+ c • • • • • trữ » * *
6
of the Preceſſion of the Equinox,
equally, when the line or axis Aa, about which the plane is
ſuppoſed to turn, hath a progreſſive motion, or is carried uni-
formly forward, parallel to itſelf; provided the angular celerity
about that axis continues the ſame; as is evident from the re-
Fig. 4. folution of forces. Hence it follows, that, if a circle E'ECée,
conſidered as compoſed of an indefinite number of concentric
rings, be ſuppoſed to revolve uniformly about its center C,
whilſt the center itſelf and the right-line OC (which, to help
the imagination, may be taken as the axis of a cone E'Oé,
whoſe baſe is E’Eeé) move uniformly in the plane PápA' about
the point (); I ſay, it follows that the forces neceſſary to keep
the particles in the plane, under ſuch a compound motion, will
be the very fame as if the circle was to turn about the line Ee
(perpendicular to the plane PápA') at reſt, with an angular
celerity equal to that of the center Cabout the point O: becauſe,
the angle OCE' being always a right oile, the angular celerity
of the moveable circle about the line ECe (which remains
every-where parallel to itſelf) will, evidently, be equal to the
angular celerity of the center of the circle about the point O.
From whence and the preceding Corollary it is manifeft, that
the forces which, acting parallel to PCO, are neceſſary to retain
the particles in the plane E'Eée, will be, every-where, as the
diſtances from the diameter E'Cé, or the plane PápA', let the
diſtance of the plane E Eee from the center O be what it will.
}
*
COROLLARY III.
Conceive now OĄPapa'á to be an homogenous Auid, re-
volving uniformly about the axis POp, under the form of an
oblate ſpheroid *; whilſt the axis itſelf is ſuppoſed to turn
about the center O, in the manner explained above: then it will
appear, from what is there delivered, that the particles of the
fluid, to continue in equilibrio among themſelves, muſt be ſo-
licited parallel to the axis, by forces that are as the diſtances
from the plane PápA'; ſuch, that the force acting at the re-
moteſt point A may be defined by 2rß; where B (by Corol. I.)
* That the particles will remain in equilibrio, under the form of an ob-
late ſpheroid (when the axis is at reſt), is demonſtrated in Part II. Sect. 9.
'of my Doctrine and Application of Fluxions.
repreſents
A
and the different Motions of the Earth's Axis. 7.
repreſents the centrifugal force in the circumference AdaA' of
the greateſt circle, and r the meaſure of the angular motion
of the axis itſelf, that of the rotation, about the axis, being
denoted by unity. But it appears further, from Lemma 1, that
the efficacy of all the ſaid forces to turn the ſpheroid about its
center (making here=2rB) is truly defined by 2rßxSxOA.
Whence it is plain, that all the particles of the body will remain
in equilibrio among themſelves, under the two different moti-
ons above explained, when the whole force producing the mo- Fig. 4.
tion of the axis, is expreſſed by 2rß x S x OA. And, when
the forces reſpecting the ſeveral particles are ſuppoſed to act ac-
cording to a different law, the effect produced by them will be
the ſame, provided their joint efficacy, to turn the body about
its center, be the ſame : ſince the ſame force must be anſwer-
ed, or ſatisfied with the ſame kind and degree of motion in
the whole body; if we except only, the exceeding ſmall diffe-
rence that will ariſe from the alteration of the figure; which figure
will not be accurately a ſpheroid, in this caſe, but nearly luch,
as the motion of the axis and, conſequently, the forces pro-
ducing it, are ſuppoſed very ſmall. Neither will the axis con-
tinue to move in the ſame plane, when the direction of the
forces is not every-where parallel to the axis ; the motion pro-
duced in the body being always about that diameter (Aa)
wherein the whole perturbating force may be conceived to act,
as by a lever, to turn the body about its center. Laſtly, it may
be obſerved here, that the time of revolution about the axis
will not, in this caſe, continue accurately the ſame ; ſince a
change of the figure muſt neceſſarily be attended with a change
in the time of revolution. But this change of motion about the
axis, when we regard the effect of the perturbating forces of
the ſun and moon upon the earth, is ſo extremely ſmall, as to
be quite inconſiderable, even in compariſon of the very
flow
motion of the axis above ſpoken of.
LEMMA III.
Suppofing all the particles of a given ellipſe MFNf to be urged
from a right-line GG coinciding with a given diameter MN, by
forces
8
Of the Preceſſion of the Equinox,
!
forces proportional to the diſtances from the ſaid line, ſuch that the
force acting at a given diſtance a, may be expreſſed by a given
quantity x; it is required to find the whole efficacy of all theſë
forces, to turn the ellipſe about its center 0.
If BC be ſuppoſed parallel to GG, interſecting OT, perpen-
dicular thereto, in D, then the force with which a particle, at
any place V in that line, is urged in the direction wV parallel
to OD, will be expreſſed by < x Vw, or 7 x OD; and con-
ſequently it's efficacy to turn the ellipſe about its center by
2 x D x Ow, or 7X OD X DV. Let there be taken Cu=
DV; and the efficacy of a particle at v will, in like manner,
be had equal to ? x OD x Dv: which, added to that of the
Fig. 5. former particle at V, gives 2x OD x DC. Therefore, ſeeing
the joint action of any two particles in DC, equally diſtant
from the middle one I, is expreſſed by the ſame quantity
OD DC, the efficacy of all the particles muſt conſe-
quently be equal to that quantity drawn into half the number
of the particles; and ſo is truly expounded by 2x ODxDC*.
By the ſame argument, the force of all the particles in the line
BD to turn the ellipſe about its center, the contrary way, will
be * '
OD x BD’. Therefore the difference of theſe two
values, * * ;0D BD —CD”, is the whole force of all the
particles in the line BC, to turn the ellipfe about its center
(downwards), which expreſſion, if Ff the conjugate dia-
meter to MN be drawn, biſecting BC in E, will become
zxz0D x BD + CD x BD-CD = * OD BC DE.
Put, now, OF=1, OM=d, FH (perpendicular to MN)=f,
8; and let OE and OD, conſidered as variable; be de-
noted by x and y, reſpectively. Then, by the property of
the
al
OH
and the different Motions of the Earth's Axis.
9
dd XCC
CC
ya
c
a
Eiga
X CC
ex
$
х
the ellipſis, it will be, cc : dd :: cc-- XX : BEI'=
2d V cc **
and conſequently BC =
Alſo (by fimilar trian-
gles) c:f::*:y=f; and c:8::* : DE Hence
our expreſſion 2xOD DE BC, derived above, by ſubſtitut-
infg theſe values, becomes 2 x 2dje xcc— xx]xx*: and there-
fore the whole fluent of 2 x 20 xxl*xxxy, or of its equal
2dfg
xic— xx/xx* xfi, will be the force of all the parti-
cles in the ſemi-ellipſe MFN. In order to the finding of this
fluent, let A be taken to denote the area of the ſemi-ellipſe,
or, which is the ſame, the fluent of 2d xcc xx}" x få; then,
by compariſon, the whole fluent of 20 xcc— *x*xx* xf, when
x=0, will be found to be Ax 1c*: whence that of our given
xcc— xx*xx*xri, muſt conſequently be
XEXc*A=*x ifgA = ** FH X OH A; the
double of which, or FHxOH x area MFNfM, is there-
fore the true meaſure of the whole force whereby the ellipſe
tends to move about its center. 2. E. I.
C
C
$
TE
expreſſion, 2x2df
*
COROLLARY I.
If the fame value be required by means of the angle AOM
included between the diameter MN and the principal axis AOa
(ſuppoſed to be given); then let Pop and FR be drawn per-
pendicular to Ox, and TF to OT, meeting OA produced, in
Qį ſuppoſe L to be the interſection of AO and FH; and let Fig. 6.
the fine and co-ſine of the ſaid given angle AOM (to the ra-
dius 1) be denoted by m and n, reſpectively. Becauſe FL is
perpendicular to the tangent TQ, we have, by the property
of
C
the
e
IO
Of the Preceſſion of the Equinox,
;
the ellipſis, as AO’: AOʻ-OP:: OR: OL :: OR ⓇOQ
(AO): OLXOQ; and conſequently AO-OP=OLXOQ.
But, I : 1:; OL : OH
and, 1:m:: OQ: OT (FH);
whence, by compoſition, 1 : mn:: OL ⓇOO(=AOʻ— OP”)
: FHXOH = mn x AO - OPP: and ſo, by ſubſtituting this
value above, we get
xmax AO-OPx area of the ellipſe,
for another expreſſion of the required force.
a
4
COROLLARY II.
Hence may be eaſily deduced the force by which the fphe-
rojd, generated by the rotation of the ellipfe about its lefſer axe
Pp, tends to turn about its center, when all the particles are
urged from a plane GG paſſing through the center, by forces
proportional to the diſtances from the ſaid plane. For, as any
ſection of the ſpheroid, parallel to the middle one Apap, is alſo
an ellipſe, ſimilar to it, the area of that ſection will be in pro-
portion to the area of ApaP (which I ſhall denote by 2.)
as the ſquare of its greater ſemi-axe, to the ſquare of the
greater ſemi-axe OA of the given ellipſe PApa: ſo that, if
OA be denoted by a, PO by b, and the diſtance of the ſaid
ſection from the center of the ſpheroid by t, we ſhall have,
aa : aa- uu (= fq. greater ſemi-axis of that ſection, by the
property of the circle) :: 2:2x the area of the fec-
tion. Moreover, by reaſon of the ſimilar figures, we have
x aa - Uu, the difference of
the ſquares of the greater and lefſer femi-axes of the ſection.
Therefore, by ſubſtituting theſe values in the above general
expreſſion, we get
ии х , x
= 2X x2x
a* — 2a*u* + **) for the force of all the
particles in that ſection to turn the body about the common axis
of motion ſtanding at right-angles to the plane PApa. This quan-
tity, drawn into i, will, therefore, be the fluxion of force of the
ſemia
aatu
aa
ca : aa
bb :: aa
аа bb
44:
aa
aa
uu
aa
7
inn aa bb
х
Х
хаа
4
ad
mn na
bb
X
4
na
aa
and the different Motions of the Earth's Axis.
II
bb
8a3
aa
X
d 4
15
15
40
X
3
1
ſemi-ſpheroid in which that ſection is; whoſe fluent, when u=a,
will be found 2x
x x : the double where.
of, or 2 x mn x aa
bbx 44Q, muſt conſequently be the re-
quired force of the whole ſpheroid : which force, as is
known to expreſs the content, or maſs of the ſpheroid, will
alſo be truly defined by Ž x 1 x aa — bbxS; S being put (as
in the preceding Lemmas) to repreſent the ſaid content or
maſs.
PROBLEM I.
To determine the efficacy of the ſun's attračtion, on a corpuſcle,
any where in the body of the earth, to turn the earth about it's
center.
SC
;
Let CDHE repreſent the earth, C the center thereof, S that Fig. 7.
of the ſun, and GCG a plane perpendicular to the line CS
joining the centers of the earth and fun; let D be the place of
the corpuſcle, and upon the diagonal SD let the parallelogram
QCSD be conſtituted; producing QD to meet GCG in K.
If F be taken to denote the ſun's abſolute force on a particle
at the center C, his force on a particle at D will be FX SDA
which may be reſolved into two others, the one in the di-
rection DC (which has no effect at all to turn the earth about
its ter); and the other in the direction DQ, expreſſed by
SC XDQ: from which the force F, in the parallel di-
SD
rection CS, being deducted, the remainder Fx
will
be the true meaſure of that part of the force in the direction
DQ, whereby the particle at D tends to change its poſition
with reſpect to the plane GCG. But this value is reducible
to FX
SC-SD X SC + SC X SD SD2
; which, as SC SD
(by reaſon of the great diſtance of the ſun) is nearly equal to
DK,
Fx
SD
SC_SD3
SD3
ز
SD3
C 2
1 2
Of the Preceſſion of the Equinox,
DK X 3SD?
DK, will become = FX
3F x €, nearly:
which, drawn into CK, will be as the required efficacy of
that force to turn the body about its center. 2. E. I.
SD3
****!--:; **.
a
ť
DK
COROLLARY J.
A= (SC) the diſtance of the ſun and earth,
the ſemi-equatoreal diameter of the earth,
If there be T=the time of the annual revolution,
taken
the time of the diurnal revolution, and
B the centrifugal force of a particle at the equa-
tor, ariſing from the diurnal revolution;
A
ABtt
then, ſince: 11 ::B: F, or F= (by the known laws
ATT
of central forces) it is evident that the force 3Fx with
SC"
which a particle at D tends fro. the plane GCG, will alſo be
truly defined by B x 3 x Hence it appears that the ſaid
force is directly as the diſtance from the plane ; and that the
value thereof, at the diſtance of the earth's ſemi-equatoreal
diameter, is truly defined by Bx; being in proportion to the
centrifugal force at the equator, ariſing from the diurnal rota-
tion, as thrice the ſquare of the time of the diurnal revoluti-
on, to once the ſquare of the time of the annual revolution,
1
DK
Fig. 7.
COROLLARY II.
Moreover, from hence the perturbating force of the moon,
or any other planet, will be given : for, fuppoſing S to repre-
ſent the planet, it's abſolute force (F) at the center C, will be
as its quantity of matter, applied to the ſquare of the diſtance
SC: and ſo our general expreſſion 3Fx Ds will here become
3DK XS
; which, becauſe the quantities of matter in bodies
SC
of the ſame denſity, are as the cubes of their ſemi-diameters,
will alſo (ſuppoſing the poſition of D to remain the fame)
be
and the different Motions of the Earth's Axis.
13
:.'. W
be as the cube of the ſemi-diameter of the planet directly, and
the cube of its diſtance SC, inverlly; or, which is the ſame,
as the cube of the fine of the apparent ſemi-diameter directly,
and the cube of the radius inverlly. Hence it is manifeſt, that
the perturbation forces of planets, of the fame denſity, are in
proportion directly as the cui es of the fines of their apparent
ſemi-diameters, or as the cibes of the ſemi-diameters, them-
ſelves, very near Therefore the ſun and moon, appearing
under cqual vin -diameters, have their perturbating forces in
the ſame ratio with their denſities.
PROBLEM II.
To determine the change of the poſition of the terreſtrial equator,
ariſing from the action of the fun on the whole maſs of the earth,
during any very ſmall interval of time.
Let OAPap be the earth, under the form of an oblate Fig. 8.
ſpheroid ; let AlaL be the plane of its equator, and HICL a
plane paſſing through S'the center of the fun, and making
right-angles with the plane of the meridian HAPCp.
It appears, by Corol. I. to Prop. I. that the relative force
whereby a particle of matter, any where in the earth, tends,
through the action of the fun, to recede from the plane GG,
perpendicular to HICL, is directly as the diſtance from the ſaid
plane ; and that the ſaid force, at the diſtance (a) of the ſemi-
equátoreal diameter from the plane, is truly defined by B x *.
It appears moreover, from Corol. II
. to Lem. III. that a ſphe-
roid, acted on in this manner, tends, through the joint force
of all the particles, to turn about its center with a force ex-
preſſed by 2x
XS; y being the force wherewith a
particle, at the diſtance a, is urged from the plane GG. There-
fore, expounding y by Box
XS,
for the true meaſure of the force whereby the whole earth
tends to turn about its center, through the ſun's attraction.
mn X aa
bb
5
3tt
TT: We have BX
x 34
3tt mn X aa_bb
х
TT
5a
But
14
Of the Preceſſion of the Equinox,
3
But it is proved, in Corol. III. to Lem. II. that, if a ſphe-
roid OAPap revolving about its axis Pp, be at the ſame time
acted on by forces tending to generate a new motion, at right
angles to the former, it will, in conſequence of ſuch action,
have another motion, about the line Aa; whereof the celerity
will be in proportion to that of the former motion about the
axis (Pp), as r to l; the whole force with which the ſpheroid
tends to turn about its center, whereby this motion is produ-
ced, being expreſſed by zrßxSxa. Let this force, there-
fore, be made equal to that found above, by which the earth
tends to turn about its center : by which means we have
and therefore r =
TT
From whence it appears that the earth, in conſequence of the
ſun's attraction, has a motion about the line Aa (lying in the
plane of the ſun's declination) whereof the celerity will be in
proportion to that of the diurnal motion about the axis Pp, as
to unity: where t and T expreſs the re-
2TT
ſpective times of the diurnal, and annual revolutions, a and b
the greateſt, and leaſt ſemi-diameters of the earth, and and
n the fines of the ſun's declination and polar diſtance.
ara
311
та хаа
ble
x
3tt mn X aa - bb
x
2TT
aa
4
5
*5a
3tt
bb
mn X aa
X
аа
PROBLEM III.
Fig. 9.
To determine the preceſſion of the equinox, and the nutation of
the earth’s axis, cauſed by the ſun, during any very ſmall interval
of time ; on the ſuppoßtion of an uniform denſity of all the parts of
the earth.
Let Do be the ecliptic, on the ſurface of the ſphere ;
and let ACQP be the poſition of the equator, when S is the
fun's place in the ecliptic, and SA his declination.
It is evident, from the laſt propoſition, that the angle Aa,
or pAb, deſcribed by the equator about the point Ā, in any
very ſmall time ť, by means of the ſun's attraction, is in pro-
portion to (x 360°) the angle deſcribed in the ſame time, by
meana
and the different Motions of the Earth's Axis.
15
mn X aan -bb
х
is to U-
aa
пn Xaa - bb
X
2TT
378'
bb
as
3tt
means of the diurnal motion, as
3tt
2TT
nity; and therefore is truly defined by 360°x_386
or 360° x x kmn; fuppofing (for the ſake of bre-
2TT
vity) to be denoted by k. But it will be (p. Spherics)
fine of a (or): ſin. A (:: ſin. Aa : ſin. a) :: angle
Aa : 0 360° * x kmn x the quantity of
2TT
the preceſſion required.
Again, if aD and ac be taken as arcs of 90 degrees each ;
ſo that Dc may be the meaſure of the angle a; we ſhall alſo
have (p. Spherics) as Rad. ; ſin. AC (co-lin. A) :: ſin. CAC
3tt
(Aa): fin.Cc:: angle Aa:Cc=360°x xkmnx
Rad.
the required nutation, or the decreaſe of the inclination of the
equator to the ecliptic. & E. I.
fin. A
fin.
co-f. A
2ТТ Хkmnx.
COROLLARY.
If we be made perpendicular to aA, it will be, as we : Cc
(:: ſin. A: fin.CA):: tang. A: Radius ;
alſo, a : :: Rad. : fin. a: whence, by compound-
ing theſe proportions, we have a: Cc :: tang. A:
fin. a. From which it appears, that the quantity of the pre-
ceſſion, for any very ſmall time, is to that of the nutation
correſponding, as the tangent of the ſun's right aſcenſion to
the fine of the inclination of the equator to the ecliptic.
;܀
PROBLEM IV.
To determine the preceſion of the equinox, and the nutation of the
earth's axis, cauſed by the fun, from the time of his appearing in
the equinoctial point, to his arrival at any given diſtance therefrom.
Every thing being ſuppoſed as in the preceding Problem, put Fig. 9.
the fine of the angle =p, its co-fine =9, the arch S=2,
its
16
Of the Preceffion of the Equinox,
1:1), per a ****
}
px
Rad.
3tt'
2TT
܀
XX, and
its fine =x, its coſine = y, and the length of the ſemi-
periphery DP=e: then, p. ſpherics, it will be, co-fin. .
AS X co-fin..A(=yx Rad.) =y; and fin. AS (= )
px : whence, by multiplying theſe two equations together, we
have fin. AS x co-fin. AS X Co-fin. -A(= mn x co-ſin. A)
=pxy: and ſo, by ſubſtituting this value for its equal, our
expreſſion for the nutation, during the time ť (given by the
laſt Problem) will here be reduced to 360° x x kpxy.
But the time wherein the ſun's longitude -S is augmented
by the particle ż will be Tx; which being wrote in the
room of t, we thence have 360°x x x xyż: and this, by
ſubſtituting ✓ 1
inſtead of their equals y
and ż, will be farther transformed to 360° x
x kp x xx.
Whoſe Auent, 360°x**
kpxs, is conſequently the true quan-
tity of the nutation that was to be determined.
Again, with regard to the preceſſion of the equinox, the in-
creaſe thereof (by the Corol. to the precedent) being in proporti-
on to the correſponding decrement of the inclination of the
equator to the ecliptic, as the tangent of A to the fine of 2,
or, in ſpecies, as to p, it therefore appears (by multi-
9x
plying the fluxion of the nutation by
that 360° x
pv 1- xx
3t x kqx will be the fluxion of the quantity under con-
VI
kq
fideration ; whoſe fluent, which is 360°x x
is therefore the preceſſion itſelf. 2. E. I.
NI
37
4T
e
2e
.....................
qx
NI
4T
2
XX
-XV 1- **
Х
2
COROL.
현
​and the different Motions of the Earth's Axis.
17
kq, whoſe quadruple,
X
4
;
47
COROLLARY I.
When the ſun arrives at the ſolſtitial point D, the value of
x being =1, and that of z=e, the quantity of the preceffi-
on becomes barely equal to 360° x
3t
360°xampfx kq, will be the whole of the annual preceffion,
depending on the fun; which, in numbers (by making t=1,
T= 366 9
3665, q=.91723 = co-line of 23° 28', a = 231,
b=230, and k (= aa--b6
comes out 21" 7'". But
it will appear from what follows hereafter, that this quantity,
derived on the hypotheſis of an uniform denſity of all the
parts of the earth, ought to be reduced to about 143", to agree
with obſervations.
aa
230
1
:.
>
ett
ką
4
3t
X
4T
COROLLARY II.
Since the preceſſion during 4th of the annual revolution is
found to be 360° x
kq
e next we have as je: z:: 360°x I
: (360°xi xxxthe mean preceſſion during the time
of deſcribing the arch : which being taken from the true
preceſſion, 360° x hxz—XVI—xx, the remainder,
360° x x—*
kqx1
I -- xx, will conſequently be the equa-
tion of the preceſſion ; which therefore is to the mean pre-
ceſſion, as
- XV i
2xv 1 - xx : 22;
that is, as -
fin. 22 : 22. But the mean preceſſion, in the
time of deſcribing the arch 2, is 21" 21" (or rather 14;"
by Corol. I. Therefore the equation correſponding will be
x fin. 22 = 1" 42" x ſin. 22, when the denſity is
taken as uniform; but when taken to correſpond with the obfer-
vation, it will be x ſine 27 =-1"10" x fine 2z. Hence
XX : %, or as
Z
2€
2e
21"7"
4e
143
40
D
it
18
Of the Preceſion of the Equinox,
***FUME
it appears, that the greateſt equation of the preceſſion (when
the ſun is in the mid-way between the equinox and ſolſtice) is
1" 10"; and that the general equation (which is ſubtractive
in the firſt and third quadrants of the ecliptic) will be in pro-
portion to the ſaid greateſt equation, as the fine of twice the
fun's diſtance from the equinoctial point is to the radius.
7
ve
3t
Х
2e
*
That it was the then
you would like to be a comment
px²
as
COROLLARY III.
Furthermore, becauſe the quantity of the nutation is, univer-
ſally, equal to 360°x kpx?, it will therefore be the greateſt
4T
poſſible, when the ſun is in the ſolſtice and x is the greateſt poſ-
fible : after which it will decreaſe, according to the ſame law
whereby it before increaſed ; 'till, on the ſun's arrival at the
other equinoctial point, it intirely vaniſhes, and the inclination
is thereby reſtored to its firſt quantity. It is alſo evident that
the quantity of the nutation will, in all circumſtances, be in
proportion to the preceſſion, during 4th of the ſun's revolution,
2 px" to, that is, as the product under the
ſquare of the fine of the ſun's longitude and the tangent of the
inclination of the two planes of the equator and ecliptic, is to
the length of an arch of 90 degrees. According to which pro-
portion (taking the ſaid preceſſion th of 141") the greateſt
nutation comes out one ſecond, very near; the inclination of the
two planes decreaſing from the time of the ſun's leaving the
equinoctial points, to his arrival at the ſolſtices, and that in the
duplicate ratio of the fine of his diſtance from the ſaid equi-
noctial points.
It may be obſerved that, in order to avoid trouble, the quantities
p and q are taken as conſtant ; the error, or difference thence
ariſing ſcarcely amounting to co booth part of the whole value.
to
9
2
or as
*THE W
* 9.: :... I ...
SCHOLIU M.
Sir ISAAC Newton, in finding the preceſſion of the equi-
nox, conſiders the protuberant matter about the earth's equator,
as a ring of moons, revolving uniformly round the center of
the
and the different Motions of the Earth's Axis.
19
en
is
ve
O
le
may be
it
C
V
e
n
t
n
the earth in 24 hours ; and by virtue of that aſſumption, from
the motion of the lunar nodes, before determined, he infers
the motion of the nodes of the ſaid ring, and from thence the
preceſſion of the equinox. We have proceeded upon other
principles, and by a very different method ; and it
worth while to remark here, that, as the preceſſion of the
equinox is deducible from the motion of the nodes of a ſatel-
lite, ſo, on the contrary, the motion of the nodes of a fatellite
may be very eaſily deduced, as a Corollary, from our general
formula, for the preceſſion of the equinox.
Thus if the value of b be ſuppoſed indefinitely ſmall (fo
that the figure of the earth, or ſpheroid, may be conceived as
flat as poſſible) we ſhall have k (= abb) )=; and the ex-
preſſion in Corol. I. will then become 360 x 31 xq, exhibit-
ing the motion of the node of a ring, or of a number of con-
centric rings, during the time (T) of one whole revolution of
the body about the fun (vid. Corol. I. to Lem. II.) But it will
alſo appear, from the articles here referred to, that the place
of a ſatellite, moving in a circular orbit, will always be found
in a ring or plane revolving in the manner there ſpecified.
Hence 360°xx 1 xq will likewiſe expreſs the motion of the node
of a ſingle moon, or ſatellite, in the time (T) of one whole revo-
lution of the primary planet: which value, when the inclination
to the ecliptic is but ſmall, will be equal to 360° x1 nearly.
Hence the mean motion of the node of a ſatellite, in a circular or-
bit, is in proportion to the mean motion of the primary planet about
the ſun, as ths of the periodic time of the ſatellite, is to the whole
periodic time of the primary planet. It follows moreover, that,
let a planet have ever ſo many ſatellites, the mean motions of
the nodes of them all will be in proportion, directly as the times of
revolution of the ſatellites themſelves ; and, conſequently
, the periodic
times of the nodes, inverſly, as the periodic times of the reſpective
fatellites.
D 2
The
2
*
20
Of the Preceſſion of the Equinox,
......
The proportions uſed by Sir Isaac Newton, in inferring
the preceition of the equinox from the motion of the lunar
node, agree exactly with thoſe above determined; it may,
therefore, ſeem the more ſtrange that there ſhould be ſo wide
a difference between the conclufion derived, in Corol. I. of the
laſt Problem, and that brought out by that celebrated Author ;
who makes the quantity of the annual preceſſion, depending
on the ſun, to be no more than 6" 7" 20^: which is not the
half of what it is here found to be. But to give the Readet
what ſatisfaction I can in this particular, and to diſcover the
error (if any ſuch ſhould have crept into my calculations) I
ſhall now attempt the ſolution by a different method: in order
to which it will be requiſite to premiſe the two following
Lemmas.
L EMMA IV.
If every particle in the circumference Alal of a given circle
tends to turn the circle about one of its diameters IL, as an axis,
by a force proportional to the ſquare of the diſtance therefrom ; it
is propoſed to find the whole force of all the particles, to turn the
circle about that diameter.
If, from any point E in the circumference, there be drawn
EH perpendicular to the given diameter IL, the force of a
Fig. 10. particle at E will, by hypotheſis, be defined by EH'; which
quantity, drawn into (Em) the Auxion of the arch IE, will
therefore be the fluxion of the quantity to be determined. But
Em, becauſe of the ſimilar triangles Emn, EOH, will be equal
; and therefore the fluxion fought=OAXEHxEn.
But EH x En is the fluxion of the area IHE: whence it is
evident, that the force of all the particles in the whole cir-
cumference, will be truly defined by OA x area of the circle, or
OAXOA x the ſemi-circumference, or OA’x the number of para
ticles. 2. E. I.
COROLLAR Y.
Since the force of a particle at A is expreſſed by OA', it fol-
lows that the force of all the particles in the whole circumfe-
rence will be equal to half the force of an equal number of
particles acting at the diſtance of the higheſt point A.
LEMMA
SEK tomu
OE x En
to
EH
Forma
and the different Motions of the Earth's Axis.
21
i
g
ir
7,
le
te
000
e
I
r
$
L EMMA V.
To determine the momentum of rotation of a given Spheroid
APaOp, revolving uniformly about its axis Pp, with a given
angular celerity.
Let ENF be an ordinate to the generating ellipſis APap,
parallel to the axis of rotation Pp: make AO (perpendicular
to Pp)=a; OP (=Op)=b; ON=x; EN=y; and let
p denote the ſemi-periphery of the circle whoſe radius is unity.
Then it will be, as 1 : 2p :: x : 2px, the periphery of the Fig. 11.
circle generated by the point N. Therefore 2px x 2y will be
the meaſure of the ſurface generated by the ordinate EF, in the
revolution of the ellipſis about its axis Pp: which, drawn into
the ſquare of the diſtance ON, gives 4pyx3 for the momentum
of rotation of all the particles in the ſaid ſurface : ſo that
the fluent of 4pyx3x will be the true meaſure of the force
to be determined.
Now, by the property of the ellipfis, we have x*=*x bb-yy:
and conſequently va =
-aayi: whence, by ſubſtituting
De * b*yj — ysj in the room of its equal -- *3x, our fluxion is
transformed to 4 g * x b*y*j—ytj: whoſe fluent, when y=b,
8pa+b
is found equal to
15
COROLLAR Y.
is known to expreſs the maſs or content of the
ſpheroid, the momentum of rotation of any ſpheroid about
its axis appears, therefore, to be juſt the ſame as would ariſe,
if şths of the whole maſs was to revolve at the diſtance of the
higheſt point (A) from the axis of motion.
K
bb
at
Since 4pa
3
PROBLEM V.
To determine the alteration of the poſition of the terreſtrial
equator, ariſing from the action of the fun on the whole maſs of the
earth, during an inſtant of time.
Let
22
Of the Preceſſion of the Equinox,
}
Fig. 12.
Let OAPap be the earth, under the form of an oblate ſphe-
roid ; let AlaL be the plane of its equator, and HICL a plane
paſſing thro' S the center of the ſun, making right-angles with
the plane of the meridian HAPCp and with the plane GG. It
is found, in Prob. I. that the force whereby a particle, at any
point E in the equator AELa, tends from the plane GG, is in
proportion to that reſpecting the higheſt point A, as the di-
Itance EF to the diſtance AK, or as ED to AO (ſuppoſing EF
parallel to AK, and ED to AO); whence it is evident that the
force on the particle at E, in a direction perpendicular to the
plane of the equator, muſt be to the force on a particle at A,
in the like direction, in the very fame ratio of ED to AO, that
is, in the ratio of the coſine of the arch AE to the radius.
But this, by Corol. I. Lem. II. appears to be the law of the
forces under which a ring of particles AELal, detached from
the earth, may continue in equilibrio, in the ſame plane, under
a twofold motion about the center 0, and about the line Aa as
an axis.
Imagine now this ring to be exceeding denſe, ſo that its
momentum of rotation about its center 0, may be equal to that
of the earth itſelf, or ſo that the two bodies may equally en-
deavour to perfevere in the ſame ſtate and direction of motion,
in oppoſition to any new force impreſſed. Then it is evident,
that, were the forces whereby the two bodies tend to turn
about the line LI, through the ſun's attraction, to be alſo equal,
the ſame effect, or alteration of motion, would be produced in
both; and conſequently, that the effects produced, when the
forces applied are unequal, will be in proportion directly as the
forces. Now the force whereby a particle at A is urged from
the plane GG, is found to be B x
3tt
( by Prop. I. Co-
rol. I.); which, in a direction perpendicular to the plane of the
3tt
equator or ring, will be B x
- B x
TT^AO^AO
T'T
Therefore, the force acting on a particle at E, in a like di-
rection, being expreſſed by B x 3 xmn x the effect
thereof to turn the ring about the line IL will be expreſſed
by
AK
X
TT AO
AK OK
Х Х
3tt
X mn.
ED
AO
and the different Motions of the Earth's Axis. .
23
3tt
ED2
X mn X
TT
AO;
1
t
L
3tt
TT
.
3tt
2TT
Tix mn be-
2TT
B
which being as the ſquare of the di-
ftance ED, it follows (from the Corol. to Lem. IV.) that, if Mbe
taken to denote the maſs of the ring, the whole force by which
the ring tends to move about the line LI, as an axe, through
the action of the ſun on all the particles, will be truly defined
by B x
3tt
X mna x M. Again, becauſe Bx
TT
X mn is the
force acting on a particle at A, in a direction perpendicular to
the plane of the ring, it is evident, from Corol. I. to Lem. II.
that the ring will, in conſequence of that force, have a motion
about the line Aa as an axe; whoſe celerity will be to the ce-
lerity of the other motion about the center, in the proportion
of r to 1, or of 301 X mn to I; becauſe, BX
ing=Bx2r, r will here be =
3tt
x mn. Therefore, if N
be aſſumed to denote the ſun's force to turn the earth about
its center, we ſhall (from the above obſervation) have,
x mna x M (the force of the ring):N::
3tt
xmn (the
2TT
motion of the ring)
the required motion of the
BAM
earth itſelf, about the ſame line Aa.
But it appears, by the Corol. to Lem.V. that the maſs (M) of the
ring (to have an equal momentum) muſt be juſt ths of the maſs
(S) of the earth: theref.our laſtexpreſſion is equal to which
by ſubſtituting B x
x S, in the room of its equal
N, at laſt becomes _ 31t
į being exactly the ſame as
2 TT
was before found in Prop. II. The aſcertaining of which is the
only real difficulty in the ſubject; ſince, that being once known,
every thing that follows after is purely mathematical ; nothing
more being required than to take the ſum, or fluent of thoſe
inſtantaneous alterations, in order to have the whole alteration,
for any finite time propoſed ; as is actually done in Prop. IV.
which therefore it will be needleſs to repeat.
SCHO
Bx. 3tt
TT
N
N
Baxşšiwhich,
bb
3t+
mn Xaa
x
TT
5a
bb
mn X ad-
хаа
x
aa
* mingi
24
Of the Preceſſion of the Equinox,
N
** martwy
SCHOLIU M.
Fig. 12. From this laſt method it will not be very difficult to deter-
mine what the reſult ought to be, when the denſity, inſtead of
being every-where the ſame, is ſuppoſed to increaſe or decreaſe
from the ſurface to the center, according to a given law. For,
let N, as above, be taken to denote the force whereby the
earth tends to turn about its center, by the action of the ſun
(the determination of which will be given by-and-by); and let
M, alſo as before, expreſs the quantity of matter in an exceed-
ing denſe ring, at the equator, having the ſame time of revo-
lution, and momentum of rotation, with the earth itſelf; then it
will appear, from the laſt Problem, that, let the figure and
denſity of the earth be what they will, the celerity of the an-
gular motion about the aforeſaid line Aa, will be to that about
the axis, in the ratio of to 1. Now, if the momentum
BAM
of the earth about its axis (which I ſhall denote by R) be com-
puted (by taking the ſum of the products of all the particles
by the ſquares of their, reſpective, diſtances from the axis) the
value of M will be known; becauſe the momentum of the ring
(by the fame rule) being Mxa', we have Mxaº =R;
and conſequently
ſo that, the general proportion
BAM
between the celerities of the two motions, about the line Aa, and the
axe Pp, will be that of
to unity.
BR
But now, in order to find theſe values of N and R, accord-
ing to any hypotheſis of denſity, we muſt look back to the third
and fifth Lemmas; from the latter of which it appears, that the
value of the momentum (R), when the denſity is uniform, will
8pab
be truly defined by ; a being the ſemi-equatoreal diame-
ter, and b the ſemi-axis of the ſpheroid, and p the meaſure
of the periphery of the circle whoſe radius is unity.
Now let us ſuppoſe the ſpheroid, inſtead of being every-
where of the ſame denſiiy, to be compoſed of elliptical ſtrata,
whoſe denſities
vary according to any given law of the diſtances
of ſuch ſtrata from the center of the ſpheroid.
Then,
N
aN
;
BR
aN
#
>*
15
and the different Motions of the Earth's Axis.
25
a
1 banyo
15
I
15
Then, putting z = the ſemi-equatoreal diameter of any ſuch
ftratum, and ſuppoſing the correſponding ſemi-axis to be in pro-
portion thereto, as w to i which proportion may be aſſumed,
either, as conſtant or variable), we ſhall, by writing in the
room of a, and wz in the room of b, have 8 x wzb; whoſe
Auxion, or indefinitely ſmall increment, will, therefore be the
momentum of the ſtratum or ſhell, whereof the ſemi-equatoreal
diameter is z, and thickneſs (at that diameter) z; on the for-
mer ſuppoſition, that the ſpheroid is every-where of the ſame
denſity. But in the preſent caſe this momentum muſt be drawn
into the quantity, which, according to hypotheſis, expreſſeth the
meaſure of the denſity anſwering to the value of 2; which
quantity we will repreſent by D; then will the product
8pD
x flux. wx5 be the general fluxion of the momentum,
when the denſity is variable: and therefore the fluent of this laſt
expreſſion, when z = a, and w= =, will be the true value
of the required momentum (R) in the preſent caſe.
After the ſame manner, the correſponding value of N (from
Lem. III.) may be determined : for, retaining the above no-
tation and ſuppoſitions, it is evident (from thence) that the ſaid
value of N (which is there expreſſed by xa– x s,
xa" — b* x 48ąb) will here, by ſubſtitution, become
W375; and conſequently that Х XDX,
flux. wzs — W3z5 will be the required fluxion of the value of
N: where is a conſtant quantity (by hypotheſis), the force
being proportional to the diſtance, and the meaſure thereof (y)
at the given diſtance a, equal to Bx :
as has been before
ſhewn, in Prop. II.
Now from the whole of what is above laid down, it will
appear evident, that the angular celerity of the motion about
the line Aa (ſuppoſing that about the axis to be denoted by
E
unity)
7
mn
Х X
5
2x
mn
or х
5
3
ay
4mnp
X Xwzi - W325
15
y
4mnp
15
a
a
1
a
3tt
.
2
26
Of the Preceſſion of the Equinox,
2TT
Auent of D x flux. wzs
A
unity) will be truly defined by zmntf x Auent of D.x Aux.wz:—w23
which, therefore, will be known, when the relation of %, w,
and D is aſſigned.
If w be ſuppoſed conſtant, or, which is the ſame, if all the
ſtrata are conceived to be ſimilar to one another, then our ex-
preſſion will become 3mntt
w
X
2TT
W3 x fluent of D X Aux. x5
w x fluent of D x Aux. 5
II
3mntt x
W
W3
bb
3tt mn X aa
Х
2TT
аа
2TT
w
b
3mntt
XiW2
(becauſe
2 TT
w - ): which concluſion appears to be the very fame with
that found when the denſity was ſuppoſed uniform. From
whence it is evident that an increaſe or decreaſe of denſity, in
going towards the center, makes no ſort of difference bere ';
provided the ſurfaces of the ſeveral ſtrata are all ſimilar to one
another and to the ſurface of the earth. If indeed the ſtrata
are diſſimilar, the caſe will be otherwiſe ; as will be ſeen by
the following example : which ought not be looked upon as a
matter of mere ſpeculation ; ſince it will appear, in the ſequel,
that the preceſſion of the equinox cannot be accounted for, fo
as to agree with the phenomenon, upon the ſuppoſition of an
uniform denſity of all the parts of the earth; the reſult, this
way, coming out about d part greater than the real quantity,
determined by obſervation.
Let then, as before, the greateſt ſemi-diameter of
tum be denoted by %, and let the leaſt ſemi-diameter (lying in
the axe of the earth) be in proportion thereto as I -- 22° to
unity; alſo let the denſity be ſuppoſed to increaſe, in approach-
zo
ing the center of the earth, in the ratio of 7-5-IX ; fo
as to vary according to ſome power (zu) of the diſtance, and
that the meaſure thereof at the center, may be to that at the ſur-.
face, in any given ratio of 7 to 1. Then, by taking , Q, and v, as
conſtant quantities, and writing i -12°, and 7—7IX
inſtead of their equals w and D, we ſhall here have
D
any ſtra-
4
***
.
a
4
and the different Motions of the Earth's Axis.
27
125
ixen
W,
the
X-
vn +0+5 x 2narts.
TE X 2124ti.
uſe
z
-1 X
ETTT-IX
momentiralaman
mente
ith
a
7
om
in
re;
ne
zta
by
a.
D x flux. wz5 - W135 = 7 - - 1***+5* 2929+4ż,
nearly (becauſe, to render the calculus leſs laborious, the terms
involving a’ and 23 may be here neglected as inconſiderable);
the fluent of which expreſſion, when z = a, will be found
7-1X0+5 x 200
4+5
oto+5
0 +9+ 5
Moreover we have, in this caſe, D x fluxion of wz5
* 5242 — 5+ 0x22p+uz
x 524ź, nearly (becauſe 5+ 0 xazott, as the earth is nearly
ſpherical, is inconſiderable in reſpect of 52+); whereof the
fluent, when z=a, will be had = 405 --
7. -1x52_v2 + 5xas.
ut 5
v + 5
Now let theſe two values be ſubſtituted in the general expreſſi-
3mntt fluent of D x flux. wz5w3zs
; by which means it be-
fluent of D x flux. wzs
comes
3mntt
Un tp + 5x 0-to 5
x 21am. But, when % = 2,
2TT
v+o+5XUF + 5
=> ) will be = 1 - sa*, and w= 1 — 28a*, nearly ;
whence we have 220
w? - 1
ſo, by ſubſtitution, our laſt formula becomes
3mntt x VA + 0 + 5xv + 5
2TT
Whence it appears that the
v to + 5 x UT + 5
required motion, in this caſe, is to the motion, when the den-
ſity is ſuppoſed uniform, in the proportion of
Uno+5xv+5
oto +5X Un + 5
to unity.–From this proportion a great number of Corollaries
may be drawn; but theſe will be, more properly, conſidered
hereafter.
on
2TT
el,
X
fo
an
w
nis.
ys
bb
aa
bb
I
; and
aa
aa
aa
bb
X
аа
a-
in
to
h-
fo
nd
Ý
as
).
+
D
E 2
PRO-
Bahagia
28
Of the Preceſſion of the Equinox,
Fig. 13.
1
4
}
PROBLEM VI.
To find the quantity of the preceſſion of the equinox, and alſo
that of the nutation of the earth's axis, cauſed by the moon, dur-
ing the time of half a revolution in ber orbit.
Let fFNeЕ be the orbit of the moon (on the ſurface of
the ſphere) interſecting the ecliptic vs p in N; and ſuppoſe
F_DEP to be the poſition of the equator, on the moon's
paſſing it at F, and fbDea the poſition thereof when ſhe re-
pafſeth it again at e: let moreover the quantity of the annuat
preceſſion ariſing from the ſun (given by Prob. IV.) be denoted
by A; and let the ratio of the denſities of the moon and fun
be expreſſed by that of m to unity: then, taking to repreſent
the given time in which the moon is moving from F to e, the
mean quantity of the preceſſion, ariſing from the fun, in that
time, will be x A: and therefore, ſince the perturbating
forces of the ſun and moon are as the denſities (by
. Prob. I.
Corol. II.) it is evident that the preceffion (Ee) caufed by the
moon, in the ſame time, with reſpect to the plane of her own
orbit, would be truly expreſſed by mxx 4, were the or-
bit's inclination to the equator to be always the ſame as that of
the ecliptic to the equator : but, ſince the magnitude, as well
as the poſition, of the angle E varies, with the place of the
node, the ſaid quantity'm x 5 x A muſt therefore be dimi-
niſhed in the ratio of the co-fine of E to the co-fine of q (as ap-
co-fin. E
pears by the ſaid Prob. IV.) and then we ſhall
get
r?
for the true value of the preceſſion Ee, cauſed by the moon,
with reſpect to her own orbit.
But now, in order to refer this to the ecliptic, it will be re-
quiſite to obſerve firſt of all, that, as the inclination of the
earth's axis, at the end of every half revolution, on the return
of the ſun or moon again into the plane of the equator, is re-
ſtored to its former quantity (by Corol
. III. to Prob. IV.) it fol-
lows, ſeeing the angles E, e, F, f are thus equal, that the
triangles
MTA
T
Х
co-fin.gr
and the different Motions of the Earth's Axis.
29
allo
ur
co-fin. E
Х
X
T
;
co-f, g x rad.
triangles DEe and DfF will alſo be equal and alike, in all
reſpects ; and fo, DE + De being = DE + DF = a ſemi-
circle, both DE and De may be taken as quadrantal, or arcs
of 90° each : whence, if ve R, the meaſure of the angler,
be ſuppoſed to meet aED in r, it will be, as fin. ED (radius)
MTA ſin. E Xco-f. E
: fin. e (E) :: Ee (MFA
)
: EDe
co-fin.r
alſo, as fin. a (p): fin. PD (co-fin. PE) :: EDe : pa
the required quantity of the
preceſſion :
And, as fin.r (radius) : fin. DR (PE) :: EDe (RDr): Rp
т
the correſponding quantity of
co-fin. r x rad. .]
the nutation, or the decreaſe of the inclination of the equator
to the ecliptic. 2. E. I.
MTA
T
X Х
of
oſë
n's
e-
lat
ed
in
nt
he
at
fin. E x co-fin. E x co-fin. rE
fin. r x co-fin. or x rad.
MTA fin. E x co-fin. E x ſin. E
X
ig
1.
ne
n
rad.
fin. r
COROLLAR Y.
It is evident from hence that the quantity of the nutation is
fin. E co-fin. rE
to that of the preceſſion, as to
or as fin. p to
rad. x co-fin.nE
; that is, as the fine of op to the co-tangent of
fin. rE
PE. It appears moreover (becauſe fin. E : fin.ON :: fin. N
: fin. PE, P. ſpherics) that the former of theſe quantities is
MTA fin. N x fin. NX co-fin. E
alſo truly explicable by т
: which
co-fin. r x rad.)
expreſſion will be of uſe in the following Problem.
of
11
e е
Х
***
PROBLEM VII.
To determine the preceſſion of the equinox, and the quantity of
the nutation of the earth's axis, cauſed by the moon, during the
time of half a revolution of the node of the moon's orbit.
Things being ſuppoſed as in the preceding Problem, let the Fig. 14.
diſtance PN of the node from the equinoctial point be denot-
ed by %, its fine by x, and its co-fine by. y; let alſo the fine
of the angle NepE be put = a, its coſine :b, the fine of
N
1
3
30
Of the Preceſſion of the Equinox,
N=1, its co-fine
its co-fine =d, the ſemi-periphery Y 2 =, and
the time of half a revolution of the node = R.
су
b
I
hb-
ha + bb
b
у
>
d
If QQ be ſuppoſed perpendicular to NE, it will be, p-fphe-
rics, as co-fin. Nop (v) : radius (1) : : co-tang. N () : tang.
NYQ= ; let this be denoted by h, then the ſecant of the
ſame angle will be = Vi+bb, its fine = and its
vithh
co-fine = : whence, by the known rules for finding
Vithh
the fine and co-line of the difference of two angles, the fine
of EQQ will alſo be had = and its cofine
Vithh
VI+ bb
Whence, again, p. ſpherics, we have, as fin. NopQ: fin. EpQ
:: co-fin. N:co-fin. E
hb — axd bd — ad=bd_acy;
alfo, as co-in. NQ: co-fin. EQ:: co-tang. Nm (2)
: cotang. PE =b+tax
ad + bcy, becauſe h
But, by the Corol. to the laſt Problem, the quantity of the nu-
tation for the time t, is expreſſed by
co-fin. NrEx rad.
which, in algebraic terms (by ſubſtituting the above values),
will become
But the time t, during which
the longitude (2) of the node is increaſed by ż, being to R the
time of half a revolution of the node, as ž to e, its value will
therefore be expounded by Rxč, or its equal Rx
and ſo by ſubſtituting this value, and writing v 1 - xx in the
room of
y, our laſt expreſſion will be reduced to
T
á *
acxä; whoſe fluent mAR
x bd - bdV 1 - XX - acxx
(=
су
ſin. Nxfin.r Nxco-f. E
Bu
expreſſed by mTA
Х
T
cx xhd
MTA
Х
T
acy
b
i
EV I
e
MAR
X
Х
bdxx
X
VI-**
T
eb
and the different Motions of the Earth's Axis.
31
nd
be-
g.
he
its
ig
де
b
асх
oh
Q
I
abd *
+ bb
v;
abe
.)
(=maRx x bd x verſed fine 2 — ac X verſed ſine 2%) is.
conſequently the meaſure of the nutation, or the decreaſe of the
inclination of the equator to the ecliptic, cauſed by the moon,
from the time of the node's coinciding with the equinoctial
point P, to its arrival at the poſition N.
Again, with regard to the preceſſion of the equinox, the in-
creaſe thereof being in proportion to the decrement of the in-
clination, as the co-tangent of PE to the fine of NPE (by
ad + bcy
the Corollary to the precedent) or, in ſpecies, as (or
ad + bcV 1 --X*
to a, its fluxion will therefore be had by mul-
tiplying that of the nutation, given above, into ad + bcv T-
and fo is found to be
MAR
Х X
—aa x cdi — abc áv 1 — XX;
T
vi
whoſe fluent, which is
x abdéz+bb-aax cdx - abc+z- abcʻxv 1
(= arex dd – cc xabz+bb – aaxcdxf.z-abccxſ.2%),
muſt confequently be the preceſſion itſelf.
But, at the end of half a revolution, when the node N ar-
rives at the other equinoctial, point, both this, and the ex-
preſſion for the nutation, will become much more ſimple, x
being then = 0, and z=e; whence the nutation will be =
MAR
X
X
and the precefſion equal to
T
x dd - =
MAR
XI - C (becauſe cc + dd =1).
T
Q. E. I.
COROLLA RY I.
It appears from hence, that the mean preceſſion of the equi-
nox, ariſing from the action of the moon, is in proportion to
what it would otherwiſe be, if the moon's orbit was to coin-
cide with the ecliptic, as I
zac to unity : whence the true
value
XX
T
MAR
I
Х
abe
MAR
Х
T
-
Boy
;
),
h
2cd
MAR
т
MAR
j
e
Il
MAR
T
:
e
ܟ
32
Of the Preceſſion of the Equinox,
"
1
2cd
to dd
d
C
1
value thereof is to that depending on the ſun, in a 'ratio com-
pounded of the ratio of the denſity of the moon to the denſity
of the fun, and the aforeſaid ratio of 1-{cc (or 0,988) to
unity.
CORO L L AR Y II.
It
appears likewiſe, that the whole quantity of the nutation,
in half a revolution of the node, is to the correſponding quan-
tity of the preceſſion, as
cc, or as unity to
The one that is, as the radius to the exceſs of the co-
tangent, above half the tangent of the orbit's inclination, drawn
into (1.5708) the meaſure of half the periphery of the circle
whoſe diameter is unity. This proportion, in numbers, ſup-
poſing the mean inclination of the orbit to be 5° 8', will be
found to be as 10 to 174, very near.
COROLLARY III.
Moreover, ſeeing the preceſſion in half a revolution of the
node is xdd — cc, we have, as e : 7::
x dd -CC
MAR
Ex xdd — cc, the quantity of the mean preceſſion
during the time in which the node moves over the arch , or
PN. This being ſubtracted from the true preceſſion, found
above, the remainder
- aa x cdx
abc+xv 1 — XX will conſe-
quently be the equation of the preceſſion, or the exceſs of the
true above the mean:
which equation or exceſs, if we neglect
the term
abcºx/ T — xx (whoſe value, by reaſon of the
ſmallneſs of c, never amounts to th of a ſecond) will evi-
dently be at its greateſt value at the end of th of a revolu-
tion, on the node's arrival at the folſtice ; when it becomes
MAR
x bb - aa x cd; and, is therefore, in proportion to
Τ'
MAR
the whole, or greateſt quantity of the nutation,
during half a revolution of the node, as bb aa : 2ab, or
MAR
T
MAR
T T
:
T
I
MAR
X
T
x bb
abe
3
I
X
abe
2cd
Х
T
as
and the different Motions of the Earth's Axis.
33
1
n-
ty
to
as I :
that is, as the radius to the tangent of double
the inclination of the equator to the ecliptic.
2ab
bb
aa
n,
n-
to
O-
ed
X
- -V1-
in
le
D-
be
COROLLARY IV.
Furthermore, ſince the value of c (the fine of the orbit's
inclination) is but ſmall, the laſt term of the general ex-
preſſion for the nutation, as well as that for the exceſs of the
true preceſſion above the mean, may be rejected, without
producing any conſiderable error ; whence the nutation is re-
MAR
duced to
XI -
xx, and the preceſſion to
T
MAR
x ma x bb—aa x cdx. Hence it appears that the de-
creaſe of the inclination, from the time of the node's leav-
ing the equinoctial point P, will be as the verſed ſine
( (i-Vi=xx) of the node's true longitude ; and that the
exceſs of the true preceſſion above the mean, will be always as
the-line (x) of the ſame longitude.
T
ne
CC
in
or
d
le
t
le
I.
SCHOLIU M.
The quantity of the annual preceſſion of the equinox arif-
ing from the force of the ſun, is found in Prob. IV, to be
21"7""; upon the ſuppoſition of all the parts of the earth be-
ing homogenous, and in a ſtate of fluidity. If, therefore, this
quantity be taken from (50%) the whole, obſerved, annual pre-
ceſfion, ariſing from the ſun and moon conjunctly, the remain-
der 28" 53"" will conſequently be the mean annual preceſſion
depending on the moon; which being increaſed in the ratio of
1000 to 988 (according to Prob.VII. Corol. I.) gives 2914", for
the quartity of the preceſſion, if the orbit of the moon were
to coincide with the plane of the ecliptic. Hence it will be
(by the fame Corol.) as 21"7"' is to 29"14", ſo is the denſity of the
fun to the denſity of the moon, according to this hypotheſis
. But
it is evident from experience (whether we regard the proporti-
on of the tides, or the accurate obſervations of Dr. BRADLEY)
that the denſity of the moon in reſpect to that of the ſun,
cannot be ſo ſmall as it is here aſſigned.
F
It
c T
o
1,
S
34
Of the Preceſſion of the Equinox,
had to
It is true, there is no way of knowing the exat ratio of the
denſities of the two luminaries ; ſince theory, for want of fuffi-
cient data, fails us here. And as to the method, by obſerving
and comparing the ſpring and neap tides * (whether we regard
the quantities or times of them) it cannot be otherwiſe than very
precarious ; conſidering the many obſtacles and intervening
cauſes by which they are perpetually, more or leſs, influenced
and diſturbed. Upon the whole it therefore ſeems to me, that
the beſt method to ſettle this point (as far as the nature of the
ſubject will allow of) is from the obſerved quantity of the nu-
tation itſelf; agreeable to what has been hinted on this head
by that celebrated Aſtronomer, to whoſe accurate obſervations
we owe this important diſcovery.
x
at xg
X g is the
IO
93
31
X
Joco
to
31
31
Let us, therefore, take g to denote the greateſt nutation of the
carth’s axis, as given by obſervation ; and then, 'if f be taken to
repreſent the mean annual preceſſion, given in like manner, it
will
appear (by Prob. VII. Corol. II.) that 174
978
part of the ſaid annual preceſſion depending on the moon ;
whence the remaining part, owing to the ſun, muſt neceffarily
be f -- 74 x8 (=31f7-589). Therefore we have (by Pro-
blem VII. Corol. I.) as
58g
58g
58g
988
3if - 588
to 1, fo is the denſity of the moon to the denſity of
the fun; which, in numbers (making g= 18"), will come
out as 2,09 to I. But if the value of g be ſuppoſed only a
ſecond or two greater or leſs than 18", the reſult will be ſenſi-
bly different, as may be ſeen in the annexed Table ; wherein,
beſides the ratio of the denſities, are alſo exhibited the mean
quantities of the annual preceffion, depending on the forces of
the ſun and moon, reſpectively; together with the greateſt
equation of the ſaid precefſion, as given by Problem VII. Co-
rollary III.
1000
988
}
* Sir Isaac Newton, by this method, makes the proportion to be as 45
to l; and M. DANIEL BERNOULLI, only as 2 to 1.
Greateſt
and the different Motions of the Earth's Axis.
35
he
ffi-
ing
ird
Ratio of the Annual preceſ- Mean annual Greateſt equa-
Greateſt denfities of fion cauſed by preceffion cauſ- tion of the pre-
outation. the fun and the fun. ed by the moon.cellion cauſed
by the moon.
moon,
ery
Seconds
16
I : 1,51
I : 1,77
I : 2,09
1 : 2,50
1 : 3,01
ng
ed
jat
he
U-
ad
16 19
Seconds Tliirds Seconds Thirds Seconds Thirds
1
1
1
20 3 29 57
14 58
18 II
31 49 15 54
33 41
14 27 35 33
1746
I 2 35 37 25
18
4.2
17
18
19
20
16 50
ns
he
to
it
he
1;
lý
James
g
of
Were I to deliver my opinion which of the different caſes
here put down anſwers beſt to the phenomenon, and the gene-
ral law of gravitation, I ſhould, without heſitating, fix upon
that preceding the laſt; which, upon the whole, will be found
to agree better with Dr. BRADLEY's obſervations than any
of the others : beſides, though the obſervations on the
tides cannot be relied on to any great degree of exactneſs, yet,
by them, it is ſufficiently evident, that the perturbating force
of the moon cannot be to that of the ſun in a leſs proporti-
on than of about 2. to 1.
From the greateſt nutation, and the greateſt equation of the
preceſſion, given above, the quantity of the nutation and the
equation of the preceſſion, correſponding to any given poſition
of the lunar node, may be very eaſily determined : for, firſt, it
will be, by Corol. IV.
As the radius is to the fine of the nodė's diſtance from the neareſt
equinoctial point, ſo is the greateſt equation of the preceffion to the
equation fought.
Which muſt be added to the mean preceſſion when the
node (viz. the aſcending one) is in any of the fix ſouthern
ſigns ; but ſubtracted, when in any of the fix northern ones.
Secondly, it will appear, by the ſame Corollary, that the de-
creaſe of the inclination of the equator to the ecliptic, from the
time of the node's coinciding with the equinoctial point qp
is
proportional to the verſed ſine' of the node's preſent diſtance
from that point : whence it follows, that the ſaid inclination
will be at its mean value when the node is in the folſtice ; and
conſequently, that the difference between the mean; and true
values,
e
f
F 2
.
36
Of the Preceffion of the Equinox,
5
Ecliptic.
D's co
Sii, o
Sig. 1
sig. di
Aud
Add
:
O
8,2
4,7
values, will be as the difference between the verſed ſine of the
node's preſent diſtance from P, and the verſed fine of go de-
grees, that is, as the co-line of the node's diſtance from p.
Therefore, to find the nutation at any given time, it will be,
As the radius is to the co-fine of the node's diſtance from the neareſt
equinoctial point, ſo is the greateſt nutation to the nutation fought
Which, to have the true obliquity of the equator to the
ecliptic, muſt be added, when the node is in any of the fix
aſcending ſigns 18, H, 9,8, II; but, otherwiſe, ſubtracted.
The following Table, ſhewing by inſpection, as well the
equation of the preceſſion, as that of the obliquity of the eclip-
tic, is computed from the proportions here laid down ; upon
ſuppoſition that the greateſt quantity of the nutation is 19 ſeconds.
The Equation of the Precellion of the 'The Equation of the Obliquity of the
Equinox.
D's 8 Siq, o Sig. I Sig. II | Subtr.
from v Sig. V? Sig VII Sig. VII
from or Sig. Vi Sig. Vo Sig. VII Subtr.
Deg. Seconds Seconds Seconds Deg. veg. Seconds Seconds Seconds Deg.
0,0
8,8
15,3 30
5 1,5 10,1 16,1 25
IO
3,0
1,1,4
16,7 20
7,3
15 4,5
12,5
17,2 15
20 6,0 13,6 17,5
6,1
25 7,4 14-5 17,7
25
5,5
0,9 5
30 8,8
15,3 17,7
30
8,2 4,7 0,0
Subtr. Sig. V Sig. IV|Sig. IJ!D's 8 Subtr. Sig. V Sig. IV Sig. 111 's 88
Add
sig. Xi] Sig. X Sig. IX | from n Add Sig. XI Sig. X Sig IX from a
Fig. 15.
To place what has been delivered above in another view,
ſuppoſe PE to be equal to the mean diſtance of the pole of the
equator from the pole E of the ecliptic; in which (produced)
let there be taken PA and PB equal, each, to half the greateſt
nutation ; and about AB, as an axis, conceive an ellipſe ACBF
to be deſcribed; whoſe other axe CF is to AB in the ratio of
the co-fine of 2EP to the co-fine of EP (that is, in numbers,
as 7444 to 10000, or as 3 to 4, nearly); then, if the point P
repreſents the mean place of the pole of the equator, the true
place will always be found in the circumference of the ſaid
ellipſe. And if, on the diameter AB, a circle ADBG be alſo
deſcribed, and the angle APS be made equal to the diſtance of
the node from the equinoctial point ; then, I ſay, a perpen-
dicular SRy, falling from the point S upon the diameter AB,
SR
will
ono
7,8
4,0
IO
9,5
9,4
9,3
9,2
9,0
3,3
30
25
20
15
IO
6,7
15
20
2,4
1,7
IO
on õr
8,7
>
and the different Motions of the Earth's Axis.
37
he
le-
P.
rest
t.
he
ſix
d.
he
the co-
bb--aa
2
P-
on
is.
bb-
аа
ene
bb aa
:
2
id
tr.
g.
5
will interſect the circumference of the ellipſe in the point where
the pole of the equator (p) is, at that time, polited. For,
firſt, it is clear, from what has been already remarked, that
AE and BE will be the greateſt and leaſt diſtances of the two
poles, as being equivalent to the reſpective inclinations of the two
planes, the equator and the ecliptic : from whence and Corol.
ĪV. it is manifeſt, that ER or Ep will be the true diſtance of the
ſaid poles, when the verſed fine of the node's diſtance (APS)
from pis AR. Moreover, by conſtruction, CP : AB :: |
ſine 2EP : co-fin. EP; that is, in ſpecies, CP : AB :
: b. And, p. ſpherics, tang. PEC : tang. PC (:: PEC : PC,
nearly) :: rad. (1): a. Therefore, by compounding theſe two
proportions, we have PEC : AB :: : ab :: bb
2ab: which proportion, for finding the angle PEC, is the very
fame with that determining the greateſt difference of the mean,
and true longitudes, as given by Corol. III. Whence it eaſily
follows, that the angle ŘEp will expreſs the difference of the
mean and true longitudes, at the given poſition of the node;
ſince, as the radius : fine APS (:: PD : RS :: PF : Rp) :: the
angle PEC : the angle REp, as it ought to be, by Corol. IV.
The ratio of CF to AB is here determined to a geometrical
exactneſs, as no-ways depending, either, on the denſity of the
moon, or on any other phyſical hypotheſis.
Having now laid down the general proportions for the nu-
tation of the earth's axis, and the preceffion of the equinox, I
ſhall here ſubjoin the neceſſary rules for determining how
much the declinations and right-aſcenſions of the ſtars are af-
fected by thoſe inequalities.
1°. For the alteration of a ſtar's declination, and right-aſcen-
fion, ariſing from the nutation of the earth's axis ; it will be
As the radius is to the fine of the ſtar's right-aſcenſion, ſo is
the nutation (or the given alteration of the equator's inclination to
the ecliptic) to the alteration of the ſtar's declination, cauſed by
the nutation ;
And, as the co-tangent of the ſtar's declination is to the co-kne
of its right-afcenfon, ſo is the nutation to the alteration of the
ftar's right-aſcenſion, correſponding.
2°. For
.
8
ท
V,
le
1)
ſt
F
of
ܕܕ
P
e
d
0
f
1
38
Of the Preceſſion of the Equinox,
2°. For the alteration of the ſtar's declination and right-af-
cenſion, ariſing from the preceſſion of the equinox; it will be
As the co-fecant of the obliquity of the ecliptic is to the co-fine
of the ſtar's right-aſcenſion, ſo is the preceſſion of the equinox (or
the alteration of the ſtar's longitude) to the alteration of the ſtar's
declination, cauſed by the preceſſion ;
And as the co-fine of the ſtar's declination is to the co-tangent of
its angle of poſition, ſo is the alteration of declination, found by the
laſt proportion, to the alteration of right-afcenfion, anſwering thereto.
Any one, but little acquainted with the ſphere, will eaſily
ſee when theſe equations are additive, and when ſubtractive :
nor will it be at all difficult to comprehend the reaſons upon
which they are founded ; they being nothing more than fo
many particular caſes of the general relation ſubſiſting between
the fluxions of the ſides and angles of a ſpherical triangle *. It
will not, however, be improper to remark here, that, when
the quantity of the preceſſion, in the ſecond of the preceding
caſes, amounts to ſome minutes, it will be neceſſary, in order
to have the conclufion ſufficiently exact, to make uſe of the
mean right-aſcenſion, at the middle of the given interval; which,
from the given right-aſcenſion at the beginning of the interval,
may be eſtimated near enough for the purpoſe, in moſt caſes,
without the trouble of a calculation: but in other caſes, and when
the utmoſt exactneſs is required, it will be neceſſary to repeat
the operation.
It may not be improper to obſerve likewiſe, that, beſides the
equations depending on the poſition of the lunar nodes, com-
puted above, there is a ſmall motion of nutation and preceſſion
ariſing from the moon's declination; whereof the greateſt quan-
tity is to the greateſt quantity of that depending on the ſun, in
a ratio compounded of the ratio of the denſities of the two
bodies, that of their periodic times, and that of the fines of the
inclinations of their reſpective orbits to the plane of the equa-
tor, nearly (as appears by Prob. IV. and VI.) Whence it is
evident, that this part of the nutation, depending on the moon's
declination, cannot, in any circumſtance, amount to more than
about th of a ſecond; a quantity too ſmall to merit attention
in the practice of Aſtronomy.
* See my Doctrine of Fluxions, Part II. Se&. I.
Remarks
and the different Motions of the Earth's Axis.
39
s
IT
.
Remarks on fome Particulars in the preceding Theory
and Calculations; in order to explain and obviate
certain difficulties and objections that may thence
arife.
T may be obſerved, in the firſt place, that we have, all
along, conſidered the effects of the ſun and moon ſepa-
rately; and, conſequently, have ſuppoſed them to be no-ways
influenced or diſturbed by each other. This may ſeem too .
bold an affumption ; eſpecially, as it is known that the tides,
which are produced by the very ſame forces, depend upon,
and are greatly varied by, the different poſitions of the two lu-
minaries.
To remove this objection, let PPSM repreſent the plane Fig. 16.
of the earth's equator po-, its interſection with the plane of
the ecliptic, PS the right-afcenſion of the ſun, and PM the
right-aſcenſion of the moon; and let the forces of the two
bodies to turn the earth about its center, in thoſe poſitions, be
repreſented by f and F, reſpectively.
Theſe forces may be conſidered as acting perpendicular to
the plane of the equator in the points S and M, and will be
equivalent to, and have the ſame effect with, one ſingle force,
equal to them both, acting in their center of gravity N. But,
by mechanics, the force f+F, acting at N, will (if the radius
OP be drawn through N) be equivalent to another force, act-
ing at P, expreffed by f+Fx Op, or f+Fx MR (ſuppor-
ing NQ, PR, as alſo SB and MC, to be perpendicular to
po).
But the quantity of the preceſſion, during a given moment
of time, is known to be as the force, and as the fine of the
right-aſcenfion, conjunctly (by Prob. III.); from whence the
two quantities ariſing from the ſun and moon, conſidered ſepa-
rately, are expounded by fx SB, and Fx MC, reſpectively.
But, fuppofing both bodies to act together, or, which is the
fame, fuppofing one ſingle force, expreſſed by F+Fx NQ
PR
to
40
Of the Preceffion of the Equinox,
1
PR
to act at P, the quantity of the preceſſion will then (by the
very ſame rule) be truly defined by f+Fx NO X PR, or its
equal f+Fx NQ; which quantity, by the property of the
center of gravity, is known to be equal to f x SB +Fx MC.
Hence it is manifeſt that, whether the forces of the lumina-
ries be joined together, or treated apart, the reſult will be the
ſame.
The next difficulty, relates to the excentricity of the lunar
orbit, and the inequality of the motion in that orbit; which
may be thought ſufficient to occaſion a ſenſible deviation from
rules founded on a ſuppoſition that pays no regard to them.
Fig. 17.
In order to clear up this point alſo, imagine ADBE to be an
ellipfe, in which the moon is ſuppoſed to revolve, about the
center of the earth, placed in the lower focus F. of the ellipſe :
let AB be the tranſverſe axis of the ellipſe, perpendicular to
which, through F, draw the ordinate IH; moreover let there
be drawn any two other lines DE, de, through the focus F, to
make a very ſmall (given) angle DFd with each other.
The perturbating force of the moon, at the diſtance DF,
will (by Prop. I. Corol. II.) be, inverſly, as the cube of that di-
ſtance; and the time of deſcribing the given angle DFd will,
it is well known, be directly as the ſquare of the ſame diſtance.
Therefore, by compoſition, the quantity of the moon's action,
during the time of deſcribing this angle, will be in the ſimple
ratio of the ſaid diſtance, inverſly
. Hence it appears, that the
ſum of the forces employed, during the times of deſcribing the
oppoſite angles DFd, EFe, will be truly defined by FO + Fe
FE + FD
or its equal FEXFD
Upon AB let fall the perpendiculars DN and EM; fo Thall
FE - FH : FI (FH) - FD :: FM: FN (p. ellipſe) :: FE :
FD (p. fim. triang.) : conſequently FE XFD-FHXFD
FH X FE FDX FE, or 2FE XFD = FHX FE + FD:
FE + FD
therefore, as it appears from hence that
the meaſure
FEXFDS
of the ſaid forces, is, every-where, equal to the conſtant quan-
tity
and the different Motions of the Earth's Axis.
4.I
e
ES
e е
1
e
r
2
2
tity F it is evident that the excentricity of the orbit and the
poſition of the apogee have no effect on the motion of the
earth's axis.
An objection may, perhaps, ariſe, with regard to the addi-
tion of the forces employed by the moon in oppoſite parts of
her orbit ; which ſtep may be looked upon as arbitrary: but
the reaſon upon which it is founded will be clear, by conſider-
ing that the moon's inclination to the plane of the equator,
in oppoſite points of her orbit, is always the ſame ; and that,
therefore, the very fame effect in the alteration of the poſition
of the equator will be produced, whether the whole force
employed during the deſcription of the correſponding oppoſite
angles, be equally, or unequally, divided, with reſpect to the
ſaid angles; ſince the ſaid force acts with the ſame advantage,
or under the ſame circumſtance of declination, in both caſes.
Another difficulty that may ariſe, is in relation to our having
made the effect of the ſun's force to be about part leſs than
the quantity reſulting from calculations founded on hydroſtatical
principles and the hypotheſis of an uniform denſity of all the parts
of the earth. But, that the phenomenon cannot be truly accounted
for, upon this hypotheſis, appears from the concurrence of all
experiments in general: for, whether we regard the menfura-
tion of the degrees of the earth, the accurate obſervations of Dr.
• Bradley, or the proportions and times of the tides, the caſe is
the ſame, and requires a much leſs effect from the action of the
ſun than reſults from, or can conſiſt with, the ſaid hypotheſis.
But if the denſity of the earth, inſtead of being uniform, is
ſuppoſed to increaſe from the ſurface to the center (as there is
the greateſt reaſon to imagine it does), then the phænomenon may
be eaſily made to quadrate with the principles of gravitation ;
and that according to innumerable ſuppoſitions, reſpecting the
law whereby the denſity may be conceived to increaſe.
Thus, conformable to the hypotheſis laid down in the Scho-
lium after Prob. V. the motion of the equinoctial points will
be in proportion to the motion of the ſame points, when the
denſity is ſuppoſed uniform, as
VT +®+ 5*v + to r, that is;
v tot 5 XUT + 5
G
*
as
42
Of the Preceſſion of the Equinox,
υφΧ π -1
as I
I
3
VO XI
I
1
VO
I
3
V
1
ll
near-
3
is to 1 : therefore, by making 1 -
v+o+ 5X VA + 5
υφΧ π -
I (agreeable to what has been above
uto+ 5 Xun+ 5
obſerved), we ſhall have
: by means
v++ 5 X VA + 5 3
of which equation, the relation of T, Q, and v may be ſo af-
ſigned, as to give the true quantity of the preceſſion, and that
innumerable ways. As one inſtance hereof, let us ſuppoſe
a=2, or that the denſity at the center is juſt double to that
at the ſurface; and let the value of Q be ſuppoſed very great,
or, which comes to the fame, let the ſtrata in the lower parts
of the earth, be ſuppoſed very nearly ſpherical, or orbicular :
then our equation will become
; which,
uto+ 5 x 20 + 5
becauſe o is ſuppoſed very great, will be
20 + 5
ly; whence v is given = 5: ſo that, according to this hypo-
theſis, the decreaſe of denſity, in going from the center of the
earth to the ſurface, will be in the quintuplicate ratio of the
diſtances from the center.
No One can imagine that we pretend here to aſcertain the
ſtructure and denſity of the interior parts of the earth : all that
is attempted, is to Thew (which indeed is all that can be done).
that the preceffion of the equinox may be truly accounted for
upon the principles of gravitation, though not in the hypotheſis
of an uniform denſity of all the parts of the earth, unleſs
by aſſuming the difference of the leaſt and greateſt diameters
much ſmaller than it is found to be, either, from hydroſtatical
principles, or by an actual menſuration of the degrees of the
earth's meridian.
There remains yet another particular that I cannot avoid tak-
ing ſome further notice of; which is the wide difference to be
found between our concluſion, in Prob. IV. Corol. I. and that
brought out by Sir Isaac Newton (in Prop. 39. Book III. of
kis Principia) from the very fame data.
A
I am
3
and the different Motions of the Earth's Axis. .
43
upon; but
ve
ns
af-
nat
oſe
at
at,
rts
I am ſenſible that this is a delicate point to touch
then I know likewiſe, that I might leave my Readers diſſatisfied,
were not I to endeavour to point out the cauſes of the ſaid dif-
ference. At firſt I had, myſelf, a ſtrong ſuſpicion that I had,
ſomewhere, fallen into an error ; which put me upon attempt-
ing the ſolution by different methods, as the moſt proper way
to arrive at certainty, and to diſcover the miſtake, if any ſuch
had crept into my calculations. Two of theſe methods í have
given; the others ſeemed unneceſſary. The exact concurrence
of them all, firſt made me think, that it was not impoſſible but
there might be a fault in that Author's ſolution; and occaſion-
ed my looking into his method with a more particular attenti-
on than I had before regarded it with.
What, at firſt, ſeemed moſt doubtful to me was his hypo-
theſis, that the motion of the nodes of a ring would be the ſame,
whether the ring were fluid, or whether it conſiſted of a hard rigid
matter * : this, I ſay, did not ſeem at all clear, at firſt; but
upon recollecting the demonſtration of my ſecond Lemma
(wherein this point is fully, though not directly, proved) I was
foon convinced that the fault (if ſuch there was) muſt be
owing to ſomething elſe.
In the next place, his third Lemma did not appear to me
ſo well grounded as the two preceding ones. In this Lemma
r:
h,
ro
O-
ne
ie
le
at
-).
20
is
[s.
'S
11
e
* The celebrated mathematician M. D'ALEMBERT, who has with great
fubtlety expatiated on Sir ISAAC NEWTON's ſolution of this Problem, repre-
ſents the above hypotheſis, as ill founded ; and ſays, that, when the ring is in a
fluid ſtate, the particles, or detached moons will not have their centers in one
and the ſame plane (il eſt certain que des lunes iſolées n'auroient pas toujours leurs
centres placés dans un même plan). Now if, by this, we are to underſtand, that
the deviation from a plane is fomething ſenſible in compariſon of the nutation in
queſtion, what is advanced is repugnant to what is demonſtrated in our ſecond
Lemma. But if an exceeding ſmall deviation (depending on the ſecond term of
a ſeries) be only intended (and ſuch it muſt be, if any thing at all), ſuch a ſup-
poſition will make nothing againſt our Author's aſſumption; as, in phyſical
ſubjects, a perfect accuracy is not to be expected. This learned gentleman him-
ſelf allows, that, the conſidering of all the particles (or the ring of moons) as
being in the ſame plane, produces no error in the conclufion : from whence
it might, with ſome reaſon, be imagined, that the hypotheſis itſelf, could not be
otherwiſe than true. And it ſeems farther plain to me, that, whatever lights that
Author's overſights in the ſolution of this problem are capable of being placed,
his real miſtakes are two only,
he
t
f
G2
44
Of the Preceſſion of the Equinox,
This propor-
--
he determines, that the motion of the whole earth about its axe,
ariſing from the motion of all the particles, will be to the motion
of a ring about the ſame axe, in a proportion compounded of the
proportion of the matter in the earth to the matter in the ring, and
of the number 925725 to the number 1000000.
tion is, indiſputably, true, in the ſenſe of the Author : but
there is a difference between the quantity of motion, fo con-
fidered, and the momentum whereby a body, revolving round
an axis, endeavours to perſevere in its preſent ſtate of motion,
in oppoſition to any new force impreſſed. Now it ſeems clear
to me, that it is this laſt kind of momentum that ought to be re-
garded, in computing the alteration of the body's motion, in
conſequence of any ſuch force. And here every particle is to
be conſidered as acting by a lever terminating in the axis of
motion: ſo that, to have the whole momentum ſought, the
moving force of each particle muſt be multiplied into the length
of the lever by which it ſuppoſed to act: whence the momen-
tum of each particle will be proportional to the ſquare of the
diſtance from the axis of motion; as it is known to be in find-
ing the centers of percuſſion of bodies, which depend on the
very ſame principles.
Now, according to this way of proceeding, it will be found,
that the momentum of the whole earth (taken as a ſphere) will
be to the momentum of a very ſlender ring, of the ſame dia-
meter, revolving in the ſame time, about the ſame axe, in a
proportion compounded of the proportion of the matter in the
earth to the matter in the ring, and of the number 800ooo to
the number 1000000. Which proportion, therefore, differs
from that of Sir Isaac Newton, given above, in the ratio of
800000 to 925725: ſo that, if his reſult, which is 9" 7'"20,
be increaſed in this ratio, we ſhall then have 10" 33", for the
quantity of the annual preceſſion of the equinox, ariſing from
the force of the ſun; allowing for the above-mentioned dif-
ference.
It appears further, by peruſing his 39th Propoſition, that he
there aſſumes it as a principle, that, if a ring, encompaſſing the
earth, at its equator AlaL (but detached therefrom) was to tend,
or begin, to move about its diameter LI with the ſame accelera-
tive
and the different Motions of the Earth's Axis.
45
e,
22
be
ed
r-
it
I-
d
),
ir
it.
n
0
f
e
tive force, or angular celerity, as that whereby the earth itſelf
tends to move about the fame diameter, through the action of
the ſun (at S); that then the motion of the nodes of the ring
and of the equator would be exactly the ſame. Now this Fig. 8.
would indeed be the caſe, were not the effects of theſe forces
whereby the two bodies tend to move about the diameter LI,
to be influenced and interrupted by the other motions about
the axe of rotation Pp, and that according to a different ratio,
depending on the different figures of the earth and ring.
A ſphere, let the direction of its rotation be which way
will, that is, let it move about what diameter it will, has al-
ways
the ſame momentum, provided it has the fame angular ce-
lerity: but the momentum of a very flender ring, revolving
about one of its diameters, appears (by Lem. IV.) to be only
the half of what it would be, if the revolution was to be
per-
formed in a plane, about the center of the ring. Whence it is
evident, that the ring Alal, to acquire the ſame motion of
preceſſion and nutation with the earth's equator, ought to
tend to move about the diameter LI with an accelerative force
double to that whereby the earth itſelf tends to move about the
ſame diameter, through the action of the ſun: fince, in this
caſe, the quantities of motion, or the momenta, generated in
the two bodies, during any very ſmall particle of time, would
be exactly proportional to the reſpective momenta of rotation,
whereby the bodies endeavour to perſevere in their preſent ſtate
and direction of motion, in oppoſition to any new force im-
preſſed. Hence it follows that all concluſions, relating to the
change of the poſition of the earth’s axe, drawn from the prin-
ciple above ſpecified, muſt be too little by juſt one half; and
conſequently that the quantity of the annual preceſſion of the
equinox, ariſing from the action of the ſun, ought to be the
double of 10" 33"; which is 21" 6", and agrees, to a third,
with what we before found it to be, by two different methods.
1
e
:
A
*
1
C
T
A very exact Method for finding the Place of a
Planet in its orbit, froin a Correction of Ward's
hypotheſis, by means of One, or more Equations,
applied to the motion about the
upper
focus.
T
1
t
Fig. 18.
L
1
one per
Sa sh sh
(
2.
a
t
le
fi
***ET ABPC be the ellipſes in which the planet revolves
about the ſun in the lower focus S; let F be the
up-
per focus, and M and m any two places of the pla-
net indefinitely near to each other; and let FM, Fm,
SM, Sm be drawn, as likewiſe MN, perpendicular to the greater
axis AP: from the center F, with the radius FD=1, let the
circumference of a circle DEe be deſcribed, and from its inter-
ſection with FM draw EH perpendicular to AP: put AO
(=OP)=a, OB(=OC)=b, OF (=OS)=1, FM=u,
FH=%, EH=y, DE = %, and Ee=, then SM being
(= AP-FM) = 2a -- u, by the nature of the ellipſes, and
FM
FN (=FH *
FE
xu, we have, by a known property of
triangles, SM+FMXSM-FM (2ax 2a-2u)=2SFX ON
(4cxc + xu): from which equation u (FM) is found
(becauſe bb = aa--cc): whence SM (= 2a
atex a to cx
aa + 16 + 2acx
- u) is alſo had in terms of x, being
a + 6x
Now the area Efe being expreſſed by ż (=FEX Ee),
we have FE (1): FM’ :: : the area MFm
zŻ X FM".
Therefore, the angle SMm being equal to FMA,.or FmA
(by the nature of the curve) we alſo have FMxMm : SmxmM
(or FM : SM) :: the area ( 3 x FM') of the triangle FMm :
the area of the triangle SMm= 2 x FM x SM (Elem. 23. 6.)
16?ż X aa tocc+20cm
162cż X I
= 6*+
bż t
a to cx
atal
1 b*c*yaż
the fluxion of the area ASM. In order to find the
2 + cz]"
fluent
a?
bb
f
f
V
a
t
t
A very exact Method for finding, &c.
47
1
20*
boste
a 3
a²
az;
6²c²
- ;
2a?
a3
6²c²
2a?
3a3
323
2
4aa
2
6
bedp
b²dz
62cxy
as
fluent hereof, let be reſolved into the ſeries
at call
&c. (whereof the two firſt terms will, here, be ſufficient) by
b2c3xży?
which means our fluxion will become bʻz + *f*c*y*z
which, becauſe yż=-*, and xż=ý, will be reduced to
bcy?;
*b*ż +
x - Ky
whoſe fluent will therefore be
6%c3y8
truly expreſſed by béz + x area DHE
-, or bz
bºc%y3
+ x xy (becauſe the area DHE = : DE
X DF
- HF ⓇEH = 2 exy). This expreſſion, when
M coincides with P, and z is the ſemi-circumference DEK
b²p
b2cp b_dp
(=p), will become +
(ſuppoſing d =it
c) = the area of the ſemi-ellipfis ABP. Therefore we have,
b2c3y3
(= the area ASM) :: p=
208y3
the length of an arch of 180°) : %-
the
3a3d
length of an arch (A) expreſſing the planet's mean anomaly :
с*xy
2cy3
from which equation, z = A +; +
4a2d
x
ſin. 2) (becauſe xy, or co-lin. 2 x ſin. 2 =
3ad
; ſin.2z): where the two laſt terms being very ſmall in com-
pariſon of the others (and, therefore, z nearly = A), we may,
inſtead of fin. Z and ſin. 22, ſubſtitute ſin. A and fin. 2A; by
which means we have z=A+ d * ſin. 2A+
x ſin. A.
From whence it appears, that, in order to have the angle AFM
upper focus, the mean anomaly (A) of the planet at
the time given, muſt be increaſed by the quantity, or correc-
tion
x ſin. 2A +
x fin. A.
But to expreſs the value of this correction in ſeconds of
a degree (which in practice is the moſt commodious) it
will
2
2
4aa
303
(²x)
2a’d
c?
=A+
· X
2ad
3a'd
203
fin. 2x+
203
3a3d
at the
cc
203
4ad
3a'd
48
A very exact Method for finding
he
1
rec]
the
rect
CC
203
3a3d
c²
ad
40%d
or
137513 xx fin. Al
will be, as 3,1416 (the length of an arch of 180 degrees)
is to 648000 (the number of ſeconds in that arch) ſo is
x ſin. 2A +
x ſin. Al to 51567 x x ſin. 2A +
the number of ſeconds in the ſaid
correction : the logarithm of the latter term of which will,
therefore, be = 5,1383 — log.d + 3 log 5 + 3 log. fin. A ;
and that of the former = 4,7123 — log.d + 2 log.. + log.
fin. 2A. But, to render theſe expreſſions ſtill more convenient
for practice, the log. of d, by reaſon of its ſmallneſs, may be,
either, intirely neglected, or elſe fo aſſumed, to be nearly a
mean of what it is known to be in the planetary orbits. Ac-
cordingly, by aſſuming the excentricity c=
of the mean
diſtance (which is a ſmall matter leſs than the excentricity of
Mars, but ſomething greater than thoſe of the Moon, Saturn,
and Jupiter), the value of d (=it and
) will be = 1,0032,
and its logarithm = 0,0014.
= 0,0014. Whence the log. of the former
part of our correction, by ſubſtituting this value, will be 5,1369
+ 3 log. 5 + 3 log. fin. A = 3 X 1,7123+log. +log.f. A;
and that of the latter = 4,7109 + 2 log.. + log. fin. 2A :
which, expreſſed in words at length, give the following
proj
heli
thu:
A
let
ſign
mea
The
Vs
8
1
100
CC
f
F
ſu
2aa
th
ac
fa
who
quir
T
be,
The
lo
fi
H
Practical Rules.
1°. To the ſum of the conſtant logarithm 1,7123 and the
log. of the excentricity in parts of the mean diſtance, add
the log. ſine of the mean anomaly; the ſum (rejecting the ra-
dius) being tripled, will give the log. of the firſt equation in
ſeconds) to be added to the mean anomaly.
2°. To the ſum of the conſtant log. 4,7109 and twice the
log. of the excentricity, add the log. ſine of twice the mean
anomaly; the fum (rejecting the radius) will be the log. of the
ſecond equation ; to be added or ſubtracted, according as the
mean anomaly is leſs, or greater than go degrees.
Here
anor
T
the Place of a Planet in its Orbit.
49
Here, and in what follows, the anomaly is to be always
reckoned from, or to the aphelion, the neareſt way ; in which
the ſeconds may be omitted, in computing the propoſed cor-
rections. Which corrections being made, the true anomaly,
or angle at the lower focus, will be had from the common
proportion, by ſaying, as the aphelion-diſtance is to the peri-
helion-diſtance, ſo is the tangent of half the mean anomaly,
thus corrected, to the tangent of half the true anomaly ſought.
As an example of the uſe of the method here laid down,
let the excentricity be ſuppoſed = 0,048219 (being that af-
ſigned, by Dr. HALLEY, to the orbit of Jupiter) and let the
mean anomaly be 45°.
Then, log. excent. . . 2,6832
2 log. excent. 3,3664
+ conſt. log. 1,7 123
+ conſt. log. 4,7109
log. for iſt equ. 0,3955
log. for ad cqu. 2,0773
log. ſin. anom. 9,8494 log. fin. 2 anom. 10,0000
0,2449
2d equ. =
1191" 2,0773
firſt
51" 0,7347
From 7,978537 log. of perihelion-diſt. 0,951781,
ſubtr.
0,020452 log. of the aphelion-diſt. 1,048219;
the rem. 1,958085, will be a (3d) conſt. log. for this orbit: to which
add 9,617596 = log, tang.
log. tang. cor. anom. 22° 31' 2",
ſo ſhall 9,575681 = log. tang true anomaly 20° 37' 40":
whoſe double, 41° 15' 20", is therefore the true anomaly re-
quired.
The ſame excentricity being retained ; let the mean anomaly
be, now, ſuppoſed 120 degrees.
Then, log. ſin. anom. 9,9375 log. ſin. 2 anom. 9,9375
log. for firſt equat. 0,3955 log. for ad equation 2,0773
0,3330 ed equat. = 103
1031" 2,0148
equ. =
N
firſt equat.
equat. = 10"
10" 0,9990
Hence 120° + 10" - 1'43" = 119° 58' 26" = the cor.
anomaly.
Therf. log. tang. cor. anom. 59° 59' 13" 10,238331
+ third conſt. log. for this orbit 1,958085
log. tang: 1 true anom. 57° 32' 10" 10,196416
H
Whence
50
A very exaćt Method for finding
4
fin. 2
alfo
1
I
ftitut
203 >
4
and,
e²x
4
It
2
ſuch
difre:
Whence the anomaly itſelf is given
= 3* 25° 4' 20".
If the excentricity be aſſumed 0,066777 (being the
greateſt that Dr. HALLEY gives to the lunar orbit) the two
conſtant logarithms, to be added to the fines of the mean ano-
maly and of its double, will be 0,5369 and 2,3601: whence if
the ſaid anomaly be taken = 50° (at which, according to the
Doctor's Table, pro expediendo calculo æquationis centri luna, the
whole correction is a maximum), the former part of the ſaid
correction will be found = 18", and the latter part = 225;":
therefore the ſum of both is 244", or 4' 4"; agreeing, exactly,
with the quantity given in the Table. And in the
And in the very ſame
manner, the proper corrections correſponding to other anoma-
lies and excentricities may be computed ; the error never
amounting to above a ſingle ſecond in any of the planets, ex-
cept Mars and Mercury : in the place of Mars, the greateſt
error will be two or three ſeconds; and in that of Mercury,
about as many minutes. As to the Earth and Venus, the fe-
cond equation, alone, will be fufficient to give their places to
leſs than a ſecond.
To obtain a farther correction, which will be necef-
ſary when the orbit is very eccentrical, we may (inſtead of
the two firft only) make uſe of a greater number of terms of
the ſeries - Home
36**2
+
1b2c%y?
which means the fluxion (byż +
of the area MSA
a + cx
deſcribed about the focus S, which is proportional to the time,
will be here repreſented by
ib*z tib* xe*y*2e3y* x2 + 3ety*x*3- 4eSyaxiž &c. (fup-
poſing e' = á). But it is well known that yü (fin. Zl")
co-fin. 22'; whence yºx (= y* Xco-fin. z) co-
co-fin. 22.X co-fin. % co-fin. % - co-fin. %
co-fin. 32*
I co-fin. z - co-fin. 32;, and therefore y***
* This, and all that follows to the ſame purpoſe, is nothing more than the
application of the THEOREM, That the rectangle of the co-fines of any two angles
(the radius being unity) is equal to half the ſum of the co-fines of the ſum and
difference of thoſe angles.
cum:
bap
463x3 30.
=
at "); by
at
as
of ti
АВР
(area
b
-X
arch
ſubfi
fe"
1
1
2
I
fin. z
-
I
7,
I
4
4
5%.
N
the
affur
it
whi
the Place of a Planet in its Orbit.
51
4
fin.2%
)
1
8
I
f
co-fin.3%
T6
1
8
fine 3%
.
3e4
42 +
32
20
co-fin.z Xco-fin.x-co-fin. 32 x co-fin. z= + co-
ģ co-ſin.2x-co-fin.42=;- co-fin. 43: whence
alſo y*x3 = 'co-fin. co-ſin. 4Xco-ſin.x = co-fin.z
o co-ſin.5%. Which values being now ſub-
ftituted for their equals, our fluxion, above given, will become
b*to be into x
x 2g CO-fin. 22 -
2e3 xżco-l.z-ix 3žco-f.32+364 x - x 43c0-1.42
40x200-1.2
n's x 3žco-f.32 X 5200-1.5% t&c.
and, conſequently, the fluent thereof = bʻz tibi into
e*xx-fin. 22.-243 x 1f.z-tſ32+3e4xx- fin.4%
445 x ffinez
As
5. ſine 52 & C. =into
item + e4 x 2 e3 + Les x ſin.z e* fin. 2% +
Les + zes x ſin. 32 fin. thfin. 52 (ſuppoſing al
all
ſuch terms wherein e riſes to more than five dimenſions, to be
diſregarded). This expreſſion, when z=p=the ſemi-cir-
cumference AEK) becomes b'xit fe + fet &c. xp
•
26² p *рха
b*px Tee?
abp = the area
✓
W a² (²
of the ſemi-ellipſe ABP. Hence it will be, as abp (area
ABP) : abz-16?x23 + les x ſin.x—16" x te* fin. 22 &c.
(area ASM) :: P (the length of an arch of 180°) : %
xestes x ſin.z
bxéfin. zz &c. = the length of the
arch A, expreſſing the planet's mean anomaly : whence, by
ſubſtituting f= ba we have A=%-ite*x fesſin.z.
a
fe* fin. 2z+ite x+fe3fin. 32
3fe* x fin.42 + x fine
5%.
Now, to find from hence the value of z, in terms of A and
the fines of its multiples, we may, for a firſt approximation,
aſſume ite x fe3 ſin. % + fe* fin. 2z &c. as equal to
itex fe3fin. A + +fe'fin. 2A -Ite* x_fe3 ſin. 3A &c.
which laſt quantity being denoted by 2, we ſhall have A =
Q, and conſequently x= Q
H2
But
CC
I
aa
b
a
4
32
20
At
1.
2
52
A very exact Method for finding
pro
+
180
wil
ano
oft
ded
]
by
hal
hal
ſqu
ܐ
But ſin. % (=ſin. A +Q)=ſin. A x co-ſin.Q+ ſin. Qx
co-fin. A =lin. A + Qx co-fin. A, nearly (becauſe Q being
very ſmall, its co-line will differ inſenſibly from the radius,
and the fine very little from the arch itſelf). In the ſame
manner, f. 22(=f.2A +2Q)=f.2A +2Q x co-f. 2A, ſin. 32
f.3A+3Qxco-f.3A, &c. Which values being therefore ſub-
ſtituted in the given equation, it will become A=%–2–2x
i tex+fe:co-f.A+ fe co-f. 2A itex fe3co-f.3A&c.
But the firſt term of the ſeries, ite xfe3 co-ſin. A, drawn
into Q (if all terms having more than five dimenſions
of e be neglected) will be barely 2fe3 x co-ſin. A x
afe' fin. 2A = fes x co-ſin. A x fin. 2 A afles x
fin. 3A + ſin. A.
In the ſame manner fe co-ſin. 2A X-Q=-fe*co-fin.
2A xfe3 fin. A +fe ſin. 2A 5fe3 ſin. 3A f%e5 x
fin. 3A — fin. A-tófe4x ſin.4A +fres x ſin.5A+ſin. A:
And, laſtly, it be x fe co-ſin. 3A X Q_equal to
fe3 co-fin. 3A XQ fe3 x co-fin. 3A x istė“ ſin. 2A
of es x ſin. 5A
fin. A. The ſum of all which will
be fºesſin. A - Mofes fin.3A Folf etſin.4A + sf*es
ſin. 5A ; which added to %- Q (or its equal, % - i te* x
fez fin. A - fe' ſin. 2A, &c.) gives A=% cite-fe*
xe3f ſin. A heffin. 2A +ite - fe*f x ze3ffin. 3A
35+26* * -fin. 4A ++of+mif*es x fin. 5A. From whence
we have x = A + i + 12x5e3ffin. A+effin. 2A
xte?fſin.3A + 5e** fin. 4A — 37effin.5A, very near ;
becauſe in all terms having more than three dimenſions of e,
the quantities J and f may be uſed indifferently, for each other,
without producing any errors but ſuch as conſiſt of more than
five dimenſions of the converging quantity e.
But, fince it is known that ſin. 3A = 3S
S3, and ſin.
5A = 5S -- 2053 + 1655 (S being the fine of A) we ſhall,
by ſubſtituting theſe values in the 2d, 4th, and 6th terms (after
proper
the
I
pli
I
24
0
32
co
ſu
21
5ee
8
32
240
;
g
la
ir
F
ti
1
ta
0
the Place of a Planet in its Orbit.
53
15
팅
​32
180 x 60 x 60
15
32
upper focus
Q
proper reduction) have x= A +i+ 4eexe3f53 - 370'Lgs
5ef?
+ fx ſin. 2A +
x fin.4A. Hence it appears that
xi+ 4eex;cfS:_370495 ++eff.2A+ 50*f*f. 4A
3,1416
will expreſs the number of ſeconds, to be added to the mean
anomaly, in order to have the angle AFM at the
of the ellipſis, correſponding to that anomaly. From whence is
deduced the following method of calculation.
Let F denote the logarithm of half the leſſer axis divided
by half the greater, E the log. of the eccentricity divided by
half the greater axis, G the log. of the ſum of the ſquares of
half the greater axis and twice the eccentricity divided by the
ſquare of half the greater axis.
Take P
1,71277 + E +F+G,
= 1,14130 + E + F,
R = 4,71236 + 2E + F,
S = 4,50824 + 4E + 2F;
then the logarithms of four equations in ſeconds), to be ap-
plied to the mean anomaly (A), will be
3 x P + log. fin. A - log. rad.
5xQ + log. fin. A - log. .
R + log. ſin. 2A - log. rad.
S + log. fin. 4.A - log. rad.
Of which equations the firſt is always to be added, and the ſe-
cond always ſubtracted; the other two being to be added, or
ſubtracted, according as the fines of their reſpective arguments,
2A and 4A, are poſitive, or negative.
The two principal of theſe equations agree with thoſe before
given ; and are the ſame, in effect, with the two equations
laid down (without demonſtration) by Sir Isaac Newton,
in the Scholium to the 31ſt Propoſition of the firſt book of his
Principia. The latter of which, in the Laws of the Moon's Mo-
tion, prefixed to that Work, ſeems to be repreſented, as defec-
tive ; it being there aſſerted, that, the inequality in the motion
of a planet about the upper focus, conſiſts of three parts; as if the
nature of the ſubject admitted of juſt that number, and no
more
A
54
A very exact Method for finding
whi
WH
Of
61
ing
for
wh
Х
more; whereas the parts, or equations ariſing in the confide-
ration, are without number, being the terms of an infinite
feries, wherein the eccentricity is the converging quantity.
Sir ISAAC NEWTON has given two terms of this ſeries, which
are right; but the new equation added by his Commentator, is
not fo; the ſign thereof, the coefficient, and the law by which
it increaſes and decreaſes, being all different from what they
ought to be.
This equation (expreſſed according to the above notation,
where i repreſents half the greater axis, f half the leſſer axis,
e the eccentricity, and A the mean anomaly) he makes to be
Lift x ſin. Al'x co-fin. A. But it ought to be +
3ft2f²
32
et ſin.4A (=
5e4
x fin.4A, nearly), as is ſhewn above; this
32
being the 3d term of the general ſeries, and the next in order
after thoſe given by Sir ISAAC NEWTON ; who appears, more
than once*, to have been diſadvantageouſly (I might ſay, un-
fairly) repreſented, and that, under the covers of his own book:
a circumſtance that cannot be attended to without ſome con-
cern and diſlike, by thoſe who entertain a due regard for the
merit of an Author to whom the mathematical world is ſo
much indebted.
I ſhall now put down one ſingle example of the uſe of the
equations above derived; wherein I ſhall ſuppoſe the eccentri-
city to be 100% 8% parts of the ſemi-tranſverſe axis (the ſame
as is aſſigned by Dr. Halley to the orbit of Mercury). Here,
then, we have E= 1,313635; F (= log. V 1-ee =
log. I-ee)=-0,009406; G(=log. It fee)=0,068024;
whence P=1,046; Q=0,453; R = 3,3302;
3,3302 ; S= 1,744:
W
any
thi
fin
th
uſo
tri
an
fir
th
I
ga
fu
ас
w
if
a
li
td
* In the 28th Propoſition of his third book, it is found that the moon's
diſtance from the earth in the fyzigies is to the diſtance in the quadratures (ſet-
ting aſide the confideration of eccentricity) as 69 to 70; which is confirmed by
what is demonſtrated in a ſubſequent part of this our Work, as well as by the
calculations of others; nevertheleſs the truth of this proportion is called in que-
ſtion, and a new one is laid down, which makes the faid diſtances to be in
the proportion of 59 to 60. See Laws of the Moon's Motion, p. II and 12.
which
V
the Place of a Planet in its Orbit.
55
upper focus
which values will ſerve in all caſes belonging to this orbit.
Whence, ſuppoſing the given anomaly to be 50°, we have
1,0460
0,4530 3,3302 1,744
9,8842 9,8842 9,9933 9,534
0,9302 0,3372 3,3235
1,278.
3
5
2,7906 1,6860
Of which reſulting logarithms, the numbers correſponding are
617, 48, 2106, and 19; whereof the firſt and third be-
ing added, and the other two ſubtracted, we have 50° 44' 16"
for the corrected anomaly, or the angle at the
whereof the half is 25° 22' 8":
Therefore log. tang. 25°22' 8"
9,6759338
to log. of the ratio of the gr. and leaſt diſt. 1,8185730
log. tang. 1 true anomaly 17° 20' 28" 9,4945068.
Which concluſion is true to a ſecond. Nor will the error, in
any part of this orbit, amount to more than about two or
three ſeconds. - If you would have the reſult depended on to a
fingle ſecond, or if the orbit be ſuppoſed ſtill more eccentrical
than that of Mercury, then the following method may be of
uſe.
Say, as half the greater axis of the ellipfis is to the eccen-
tricity, fo is the ſine of the mean anomaly to the fine of an
angle; which ſubtract from the mean anomaly, and to the log.
fine of the remainder (which I call the eccentric anomaly) add
the ſum of the log. of the eccentricity and the conſtant log.
1,758123: the aggregate (rejecting the radius) will be the lo-
garithm of an angle, in degrees and decimal parts ; which,
fubtracted from the angle firſt found, leaves a correction to be
added (under its proper ſign) to the mean anomaly: with
which corrected anomaly, let the whole operation be repeated,
if needful, by always adding the laſt correction to the mean
anomaly. Then it will be, as the greater ſemi-axis of the el-
lipfis is to the lefſer, fo is the tangent of the corrected anomaly
to the tangent of the angle at the upper focus of the ellipſis :
whence the angle at the lower focus, or the true anomaly, may
alfa
56
M
have
ty
9,2866211;
A very exact Method for finding
alſo be known by the common proportion, and that to any
aſſigned degree of exactneſs.
This method, in all the planets, except Mars and Mercury,
anſwers to a ſecond, at one operation. In the former of theſe
two, the error, when greateſt, will amount to about three or
four ſeconds; and in the latter, to nearly as many minutes ;
in which caſe, three operations will be neceſſary: but in order
to avoid that trouble, the following calculation may be uſed;
which is ſo exact as to anſwer, even in the orbit of Mercury,
to leſs than half a ſecond, without repeating the operation.
Add together twice the log. of the eccentricity, the log. ſecant
of the angle firſt found (as above), and the log. co-line of the
ſum of the eccentric anomaly and mean anomaly once cor-
rected (in all of which angles the ſeconds may be neglected).
The
aggregate (ſubtracting twice the radius) will be the log. of
a fraction to be added to unity, when the ſaid ſum of anoma-
lies is between 90 and 270°; but otherwiſe, ſubtracted there-
from: then the log. of this ſum, or remainder being ſubtracted
from the log. of the firſt correction, you will have the log. of
the true correction to be added (under its proper fign) to the
mean anomaly given.
Thus, for example, let the mean anomaly be 70; and let
the eccentricity be = 0,20589 (as in the preceding exam-
ple).
Here, fin. mean anom. 70°
9,9729858
log. eccent.
1,3136353
whence 58° 50',67 = the eccentric anomaly :
whoſe fine
9,9323552
+ the log. proper for the orbit . 1,0717583
= log. of 10°,0952
1,0041135:
which angle, or its equal, 10° 5',71, being ſubtracted from
11°9',33, leaves the firſt correction 1° 3,62 = 63,62.
Moreover,
Ai
+
who
quir
the Place of a Planet in its Orbit.
57
Moreover, by adding together 70°, 1° 3', and 58° go', we
have
129° 53', whoſe co-fine
9,8070
+ ſec. 11° 9' ang. firſt found 10,0083
+ twice log. eccent.
log. of 0,0272
2,4425
Log. firſt cor. 63',62 1,80359
· log. 1,0272
0,01187
log. true cor. 61,9 1,79172
2,6272.
Now the mean anom. +true cor.=71°1',9,
01:9;}10,4638087
whoſe log. tang.
+ log. of the ſemi-conjugate axis
1,9905940
log. tang, of 70° 38',82
10,4544027
And the log.tang. of the half hereof, 35° 19,41 9,8504350
+ log. of the ratio of the gr. and leaſt diſtances 1,8185730
log. tang. of 25° 1',02
9,6690080
whoſe double 50° 2',04, or 50° 2' 2", is the true anomaly re-
quired.
UI
sic
.....:
I
A DE-
telustersdecreencestostcardులుతcracted
that
ତିଙ୍କ
ಧರ್ಮಾಜ್ಯದಾದ್ಯ
is the
А
or CO
D Ε Τ Ε R ΜΙ Ν Α Τ Ι Ο Ν
c?
OF THE
whic
Difference between the Motion of a Comet in an
Elliptic, and a Parabolic Orbit.
th
L
LE
meet
equa
ſeque
(to tl
I: 2
Whi
II
C.
- 4ac - 400
video
ac - CC X ax **
019
аа
I -
radit
Fig. 19. ***ET PNG be a parabola, and PBH a very excentrica
ellipſis, having the ſame focus S, and vertex P with
* the parabola ; let moreover N and n be conſidered as
cotemporary poſitions of two bodies, in theſe orbits,
moving from the perihelion P at the ſame time, about the ſun
in the focus S. Make NBC perpendicular to PSO, and call
PC, X; PS, c; and the greater axis of the ellipfis,, a : then
the lefſer axis will be=2Vixã; the parameter
and the ordinate BC = 2
21
(by the proper-
ty of the ellipſis) = 2c*x} x 1-
2c*?
nearly. This laſt taken from CN (= 2c*x*,
by the nature of the parabola) leaves c*x* x. +
which being multiplied by i and the fluent found, we thence
have cm x
trener for the meaſure of the area NPB, or
NPv, very near : which ſubtracted from the area NSP
CNX CS = +
20*x* x x — 2c*x* x*-
cix}x+x), leaves c*x* xc+ *-
= the area:
VPS. Moreover, the parameter of the parabola being 46, and
that
min
C
I
}
1
2a
2a
it is
BN;
ang
SN
20%
2x²
gor
3a
*
CNX PC
20*
2**
3a
50
wi
܀
Of the Difference of Motion of a Comet, &c.
59
4ac
4CC
4ac
400
fo
>
a
that of the ellipſis
we have v 40 to
✓
is the parabolic area NSP (c*x* xc + x) to the correſponding,
or cotemporary elliptical area nSP = 1-xcxxc+ x
c*x* x1-xct** (nearly) = c*x* xc+;
which fubducted from the area vPS, leaves cix? x
CC
2a
ба
8
CC
CX
2**
2a
2a
52
the area usn.
cm.
c²z²
26224
6374
CC
CZX
24
2a
ga
2a
I
NI
3z
X
2a
"...siin...........
I
2²
w
Let now NM be ſuppoſed perpendicular to the tangent NA,
meeting the axis in M; then SM, SN, and SA will be all
equal to each other, by the nature of the parabola ; and con-
ſequently the angle PMN equal to half PSN; whoſe tangent
(to the radius. I) let be denoted by % : whence it will be, as
1: 2 :: MC(20) : CN (2c***); and conſequently x =
Which value being ſubſtituted in the area vSn, it will become
XIX 24; and this di-
vided by 1 Svº (=ixc+ x c* X1 + zz ), gives
for the meaſure of the angle vSn, in parts of the
I + ZZ
radius (1). Therefore, if m be put - 3437 = the number of
minutes in an arch equal to the radius, and u =
I + 2z
it is evident that X mu will expreſs the meaſure of the faid
angle in minutes of a degree. To find now the ratio Sn to
SN, which ſtill remains to be determined, we have (by Tri-
gonometry) No
ſin.ByF
co-fin. M
c+*
xcix?
x BN = tang. M ~ BN
ct czz
Xitzz. Alſo (ſuppoſing nb perpendicular to Sv) it
will be, as 1 (radius): (the arch meaſuring the angle nSo)
?
m.in. . ...
ZXI-Z
5
· *...*..
· W. WAV. V...
fin.B
ſin. M
X BN
ſin. AFC
fin. AFS
X BN =
2 X CX2 X
un
X X X X
a
CCZZ
60
Of the Difference of Motion of a Comet
ccu
fun
::cxi+ zz (=Sn, nearly): nb xi+zz: whence,
again, by ſimilar triangles, 1: 2 :: CM : CN):: xitzz
сси
Cezu
CCZZ
(nb): bv =
xit zz: this added to No
XI+38,
a
ing
log
thir
will
beit
lon
49
fun
CCZ
gives Nb =
xu +2x1 + xz: which therefore is to SN
a
(cx1 + xz) as cm xut x to unity; and conſequently the re-
quired proportion of Sn (or Sb) to SN, as I - xu + z to
unity.
cz
duc
par
ſon
ſcri
ſibl
of
From this laſt concluſion, and that derived above, exhibit-
ing the value of the angle NSn, the following Table is com-
puted ; whoſe uſe is thus : Find, in the firſt column, the co-
met's longitude from the perihelion, as given from the hypo-
theſis of a parabolic orbit (either by Dr. HALLEY's Table, or
any other of the like kind); againſt which, in the third co-
lumn, you have the logarithm of a number of minutes (ex-
preſſed in the ſecond column); from which fubtracting the
logarithm of the ratio of the greater axe of the ellipſe divided
by the perihelion diſtance, the remainder will be the logarithm
of a number of minutes to be added to, or ſubtracted from the
aforeſaid longitude, as the Table directs: whereby the comet's
longitude, for the ſame time, in the elliptic orbit will be given.
And if, from the logarithm found in the fourth column, the
logarithm of the ſame ratio be alſo ſubtracted, the remainder,
abating 10, will be the logarithm of a quantity to be taken
from the logarithm of the comet's diſtance from the ſun, com-
puted according to the aforeſaid hypotheſis.
Thus, for example, let the greater axis of the ellipſis be
ſuppoſed = 35,727, and the perihelion diſtance — 0,5825
(anſwering to the orbit of the comet of the year eighty-two);
and let the longitude from the perihelion, according to Dr.
HALLEY's Table, computed for a parabolic orbit, be 44 3' 20";
correſponding to which, the logarithm of the diſtance from the
fun
in an elliptic and a parabolic Orbit.
61
ſun is given = 0,065838. Here, then, the log. of 35,727 be-
0,5825
ing = 1,7877, this value is to be ſubtracted from both the
logarithms 2,9228 and 9,0555, ſtanding againſt 44°, in the
third and fourth columns of the annexed Table : from whence
will be found the two quantities 13',65 and 0,001853 ; which
being ſubtracted from 44° 3' 20", and 0,065838, the required
longitude from the perihelion is given from thence = 43°
49' 41", and the logarithm of the comet's diſtance from the
fun 0,063985.
The ſame Table, not only furniſhes an eaſy way, for de-
ducing the motion in an elliptic orbit, from the motion in a
parabolic one, but may be farther uſeful in determining, in
ſome degree, the ſpecies of the ellipſis which a new comet de-
ſcribes, when the obſervations thereon are found to differ fen-
fibly from the places computed according to the hypotheſis
of a parabolic orbit.
To
.....
1
62
Of the Difference of Motion of a Comet
ſubtr.
fubtract
ſubtract
fubtr.
fubtract
ſubtract
2
19 30 1,4770 5,8200 ||4198362,9220 9,0006
60 1,7770|6,4220||428362,92299,0195
31 901,9533 6,773543 836 2,92329,0378
4 1192,0778 7,0242 44 836 2,9228 9,0555
5.149 2,1740 7,2178 45 835 2,9217 9,0727
6179 2,2521 7,2759 46 832 2,9200 9,0894
7 208 2,31787,5095 || 47 827 2,91769,1056
8 237 2,3745 7,6250 ||488212,91459,12 15
9266 2,4242 7,7269 49 814 2,9107 9,1369
10294 2,468317,8178 50 806 2,9061 9,1520
11 322 2,50747,8998 51 796 2,9007 9,1666
12 349 2,5431 7,9747 52 784 2,8945 9,1808
133762,57608,0437 53 772 2,88749,1947
14 4032,6058 8,1074 ||54 758 2,8795 9,2083
15 430 2,63318,1666 55 743 2,87089,2215
164551 2,6583 8,2217 156 726 2,8610 9,2344
17 480 2,68168,2735571708/2,8500 9,2469
18 505 2,7032 8,3222 58 6882,83789,2592
195292,7233 8,3682 159 6572,8243 9,2712
20 552 2,74208,4116160 645 2,8095 9,2829
21 575 2,7594 8,4528 || 61 | 621 2,79329,2943
22 596 2,7755 8,4921|62 596 2,7754 9,3055
23 617 2,79068,5295 1163 570 2,75579,3163
24 637 2,80468,5652 ||64 542 2,7338 9,3270
25 1657 2,817618,599465 512 2,70959.3374
26 676 2,8297 8,632066 482 2,68279,3477
27 16942,8410 8,6634 67 450 2,65269,3576
28 710 2,8514 8,6934 68 416 2,61879,3674
29 1726 2,8610 8,7224 ||693812,58059,3769
30 741 2,8698 8,7502 ||70344 2,5363 9,3862
31 755 2,87798,7770 1171 30612,4857 9,3954
32 17682,8853 8,8030 172 267 2,42589,4044
33 780 2,8920 8,8280 173 2262,35379,4132
34 7912,89808,8521 174 184 2,26399,4218
3580112,9034 8,8755 175 1402,1458 9,4303
36809 2,9081 8,8980761 95 1,97679,4386
378172,9122 8,9198 77 481,68409,4468
388232,9156 8,9409178 add add
39 8292,91848,96141179| 491,6910 9,4624
40183312,920518,9813 180/100 2,0006
}
:
9,4546
in an elliptic and a parabolic Orbit.
63
add
add
ſubtract
add
ſubtraét
I 22
9,7602
+
9,8866
add
810 152 2,182719,4777 121°
3547 3,5499
9,7516
82 206 2,3139 9,4851 3675 3,5653
83 261 2,41709,4924 || 123 3806 3,5805 9,7690
84) 318 2,50219,4996 || 124 3941 3,5956 9,7781
85 376 2,5750 9,5067 || 125 4078 3,6105 9,7874
86
435 2,63869,5138 126 4220 3,62539,7971
871 496 2,69519,5207 || 127 4364 3,6399 9,8070
88 558 2,74709,5275
128 4513 3,6545 9,8172
89 622 2,7936995342 || 129 46673,66909,8278
90
688 2,8373 9,5409 | 130 4825 3,6835 9,8387
91 754 2,87769,5475 || 131 4989 3,6980 9,8500
92 8232,91529,5540 || 132 5158 3,7125
9,8618
93 892 2,95069,5605 || 133 5332 3,7269 9,8740
94 963 2,98379,5670 || 134 55123,7413
95 1036 3,0154 9,5734|| 135 5698 3,7557 9,8997
96 11103,04559,5797 || 136 5890 3,7701 9,9132
97 11863,0742 9,5860 || 137
6088 3,7845 9,9273
9812643,1016 9,5923 | 138 6297 3,7991 9,9419
99 1343 3,12799,5986 || 139 6513 3,8138 9,9571
100 14233,1531 9,6050 || 140 67393,82869,9729
101 1505 3,1774 9,6113 || 141 6974 3,8435 9,9893
102 1588 3,2008 9,6176 || 142 7219 3,8585 10,0064
103 1673 3522369,6239 || 143 7475 3,8736 10,0242
104 1761 3,24579,6302144 7743 3,8889 10,0427
105 18:50 3,2671 | 9,6366145 80243,9044 10,0620
106. 1941 3,2879 9,6430 || 146 8316 3,9199 10,0822
107 2033 3,3081 9,6495 || 147 86303,9360 10,1032
108 2127 3,3278 9,6560148 8962 3,9524 10,1251
10922233,3469 9,6627 || 149 9312 3,9690 10,1481
110 2321 | 3,3656 9,66941509681 3,9859 10,1720
2421 3,3840 9,6763 | 151 10072 4,0031 10,1970
112 2523 3,4019 9,6831 | 152 10491 4,0208 10,2231
113 2627 3,4195 9,6901 153 10940 4,0390 10,2504
114 2734 3,4368 9,6972 || 154 11416 4,0575 10,2792
115 2843 3,453819,7044 155 11929 | 4,0766 10,3094
116 29543,4704 9,7119 156 12483 4,0963 10,3412
117 3068 3,4868 9,7195 | 157 18086 4,1168 10,3746
118, 3184 3,5030 9,7272 || 1.58 13740 4,1380 10,4098
119 3302 3,5188 9,7352 || 159 14455 4,1600 10,4469
120 3422 13,5345 19,7433 160 | 15223 I 4,1825 | 10,4861
III
N
An
*
An Attempt to Thew the Advantage ariſing by
Taking the Mean of a Number of Obſervations,
in practical Aſtronomy.
T
2
IR SIL
SR
1
HOUGH the method practiſed by Aſtronomers, in
order to diminiſh the errors ariſing from the imper-
fection of inſtruments and of the organs of ſenſe, by
taking the mean of ſeveral obſervations, is of
very
great utility, and almoſt univerſally followed, yet has it not,
that I know of, been hitherto ſubjected to any kind of demon-
ſtration.
In this Eſſay, ſome light is attempted to be thrown on the
ſubject, from mathematical principles : in order to the appli-
cation of which, it ſeemed neceſſary to lay down the following
fuppoſitions.
1. That there is nothing in the conſtruction, or poſition of
the inſtrument whereby the errors are conſtantly made to tend
the ſame way, but that the reſpective chances for their hap-
pening in exceſs, and in defect, are either accurately, or nearly,
the ſame.
2. That there are certain aſſignable limits between which all
theſe errors may be ſuppoſed to fall; which limits depend on
the goodneſs of the inſtrument and the ſkill of the obſerver.
Theſe particulars being premiſed, I ſhall deliver what I have
to offer on the ſubject, in the following Propoſitions.
goo ;
PROPOSITION I.
Suppoſing that the ſeveral chances for the different errors that
any ſingle obſervation can admit of, are expreſſed by the terms of
the ſeries yan.....4-3, 7-2, rus,
pi, gº, go, goz, ps3
where the exponents denote the quantities and qualities of the re-
Spective errors, and the terms themſelves, the reſpective chances for
their happening ; it is propoſed to determine the probability, or
odds,
Of the Advantage ariſing, &c.
65
m
n
gov
I
(by the
X I
NI
n ņI
n
NU
you are
nrw.
I
2
3
2
N N
nt 2
2
2
3
I
2
3
odds, that the error, by taking the mean of a given number (n) of
obfervations, exceeds not a given quantity (
It is evident from the laws of chance, that, if the given ſeries
+7-379-2+r-stritrityl tr} .+ gius
expreſſing all the chances in one obſervation, be raiſed to the
nth power, the terms of the ſeries thence ariſing will truly ex-
hibit all the different chances in all the propoſed (n) obſerva-
tions. But in order to raiſe this power, with the greateſt faci-
g 2011
lity, our given ſeries may be reduced to y-vx
known rule for ſumming up the terms of a geometrical pro-
greſſion); whereof the nth power (making w = 20 +1) will
be gav gezo)" x 1 -r1-*; which expanded, becomes
+
- Ign2w-
22 300-
j320-*+&c.
x into it nr +
n+In+ 2 nntun n+ 2
g3 +
n+374 + &c.
Now, to find from hence the ſum of all the chances, where-
by the exceſs of the poſitive errors above the negative ones, can
amount to a given number m preciſely, it will be ſufficient (in-
ſtead of multiplying the former ſeries by the whole of the lat-
ter) to multiply by ſuch terms of the latter only, as are ne-
ceſſary to the production of the given exponent m, in queſtion.
Thus the firſt term (rv) of the former ſeries, is to be mul-
tiplied by that term of the ſecond whoſe exponent is nv +.m,
in order that the power of r, in the product, may be gom : but
it is plain, from the law of the ſeries, that the coefficient of
this term (putting nv +m=9) will be ".n+1.nt2.n+3 (9),
q being the number of factors; and, conſequently, that the pro-
duct under conſideration will be monts.n+2.n+3 (9) x gott.
Again, the ſecond term of the former ſeries being
the exponent of the correſponding term of the latter muſt
therefore be --w+nutm9-w), and the term it-
K
ſelf,
4
I
2
3
4
I
2
3
4
*
Nr-no
66
Of the Advantage ariſing by Taking the Mean
I
2
N
- gon.
I
2
3
I
2
n+in+2.n+3 (9) x yama
I
2
3
4
1
2
3
4
N
N
+
I
yum
2
3
4
I
2
n
nI n2
gon
2 3
1
2
3
4
1
felf
n n+in+2
(9
w) x 79 ; which, drawn into
3
--- ---<, gives ********? (9—2v)
nu.+2 (9—W) x npm for the ſecond
term required.
In like manner the third term of the product whoſe expo-
nent is m, will be found, Fn2(9–2w)*
And the ſum of all the terms, having the ſame, given expo-
nent, will conſequently be
n
+
guma
n ntin+2.n+3 (9-W) x nur
n+in+2.n+3 (9
2w)
n ntin+2.n+3 (9–3w)
&c.
&c.
From which general expreſſion, by expounding m by 0, +1,
1, +2, 2 &c. ſucceſſively, the ſum of the ſeveral
chances whereby the difference of the poſitive and negative
errors can fall within the propoſed limits (+ m, – m) will
be found: which divided by the total of all the chances, or
gou no x 1 — gozel" x 17", will be the true meaſure of the
probability fought. From whence the advantage, by taking
the mean of ſeveral obſervations, might be made to appear :
but this will be ſhewn more properly in the next Propoſition;
which is better adapted, and to which this is premiſed, as a
Lemma.
REMAR K.
If r be taken =1, or the chances for the poſitive, and the
negative errors be ſuppoſed accurately the ſame; then our ex-
preſſion, by expunging the
r, will be the
with that ſhewing the chances for throwing n t 9 points,
preciſely, with n dice, each die having as many faces. (w) as
the reſult of any one ſingle obſervation can come out different
ways. Which may be made to appear, independent of aniy
kind
i
powers of
very fame
of a Number of Obſervations in practical Affronomy. 67
V...
2,
I,
kind of calculation, from the bare confideration, that the
chances for throwing, preciſely, the number m, with n dice,
whereof the faces, of each, are numbered
32
-0, +1, +2, +3.... tv, muſt be the
very fame as the chances whereby the poſitive errors can ex-
ceed the negative ones by that preciſe number: but the former
are, evidently, the ſame as the chances for throwing preciſely
the number 0 + 1X1 + m (or n +9) with the ſame n dice,
when they are numbered in the common way, with the terms
of the natural progreſſion 1, 2, 3, 4, 5, and ſo on; becauſe the
number upon each face being, bere, increaſed by vt1, the
whole increaſe upon all the (n) faces will be expreſſed by
V+Ixn; ſo that there will be, now, the very fame chances
for the number otixntm, as there was before for the
number m; ſince the chances for throwing any faces aſſigned
will continue the ſame, however thoſe faces are numbered.
3ru-
PROPOSITION II.
Suppoſing the reſpective chances for the different errors, which
any fingle obſervation can admit of, to be exprefſed by the terms of
the ſeries go + 2r?~ + 3r2
to +1.00
+ 2q + goo (Whereof the coefficients, from the middle one (0+1)
decreaſe both ways, according to the terms of an arithmetical pro-
greffion); it is propoſed to find the probability, or odds, that the
error, by taking the mean of a given number' (t) of obſervations,
exceeds not a given quantity (75).
Following the method laid down in the preceding propoſiti-
on, the ſum, or value of the ſeries here propoſed will appear
to be
gut
(being the ſame with the ſquare of the
geometrical progreſſion ratoxitr+r+r3..... tro). And
the power thereof whoſe exponent is t (by making n = 2t,
and w =°+) will therefore be you to X 7 - foru" x 1g
ngo-to +
n.7**
to
&c. into i + nr +
XIT
I
ri
ܕ݁ܬܳܟ
***
p2ro intro
I
2
I
K 2
.
Of the Advantage ariſing by Taking the Mean
68
n+Iga t
2
2
3
nin +1.n+243 +&c. Which two ſeries's being the
fame with thoſe in the preceding Problem (excepting only,
that the exponents in the former of them are expreſſed in terms
of t, inſtead of n) it is plain, that, if q be here put = tvtm
(inſtead of nv + m) the concluſion there brought out will an-
ſwer equally here; and conſequently that the ſum of all the
chances, whereby the exceſs of the poſitive errors, above the
negative ones, can amount to the given number m, preciſely,
will here, alſo, be truly defined by
+mtan2 (9) * om
n.n+1.n+(9—w) x nr
I
2
+
n nume I
ym
I 2
I 2
n
nI n
2
ym
I
2
3
I
2
3
nn+in+2
+
(q-2w) x
1.
3
n
3
&c.
&c.
But this general expreſſion, as ſeveral of the factors in the nu-
merators and denominators mutually deſtroy each other, may
be transformed to another more commodious.
Thus the quantity
n.n+1.642 (q), in the firſt line, by
breaking the numerator and denominator, each into two parts,
will become
nti.nt2.9+3......9.9+19+2.9+ 3......9+nI
n.nti.nt2.1+3...
which, by equal diviſion, is reduced to
9 +1.9+ 2.9 + 3
•n-
q+nI
3
9 to 1.9+n-2.9+1-3....q+r
I
2
3
2
3
I, 2
3
4
9
I
2
new I
I
2
3
fuppoſing p=9+n=tv m + n.
In the very fame manner, by making g=-w, and
p=+n(=p-w) it appears that "..n +1.n+2
(2-w)
I
2 3
home
of a Number of Obſervations in practical Aſtronomy 69
I
2
3
I
2
3
2
3
I
-3 (n— 1) *
n n-1.1-2 gues
I
. rx
2
I
2
3
1
I
2
3
I
2
3
P-1.2-2.5-3 (n-1) 8c. Conſequently our whole
given expreſſion (making p" =p-20, $" =p - 3w, &c.)
will be transformed to
+
p-1.2-2.8-3 (n − 1) * gente
"
_Þ-1.8-2.0-3 (n − 1) * nome
+8"-1.0"
"_ID"—2 D-3
p"— IP" 2 '-
-
&c.
&c.
Which expreſſion is to be continued till ſome of the factors be-
come nothing, or negative ; and which, when r=1, is the
very fame with that exhibiting the number of chances for
P points, preciſely, on n dice, having each w faces.
And, in this caſe, where the chances for the errors in ex-
ceſs and in defect are the ſame, the ſolution is the moſt ſimple
it can be ; fince, from the chances above determined, anſwer-
ing to the number p preciſely, the ſum of the chances for all
the inferior numbers to p, may be readily obtained, being given
(from the method of increments) equal to 1.5-2.93 (n)
p=1 p_2.p—3(n) x n +6"
(n) x n +$"1.0"52.93 (n) ***
p"— 1.0
p"-2.p""-3
+&c. The dif-
ference between which and half (60*) the ſum of all the
chances (which difference I ſhall denote by D) will conſe-
quently be the true number of the chances whereby the errors
in exceſs (or in defect) can fall within the given limit (m): fo
that will be the true meaſure of the required probability,
that the error, by taking the mean of t obſervations, exceeds
not the quantity * propoſed.
5
3
n n-I
I
2
3
3
2
(n) *.*7
n In
2
I
2
3
3
D
w w
But
70
Of the Advantage ariſing by Taking the Mean
But now, to illuſtrate this by an example, from whence the
utility of the method in practice may clearly appear, it will be
neceſſary, in the firſt place, to aſſign ſome number for v, ex-
preſſing the limits of the errors to which any obſervation is
ſubject. Theſe limits indeed (as has been before obſerved)
depend on the goodneſs of the inſtrument, and the ſkill of the
obſerver : but I ſhall here ſuppoſe, that every obſervation may
be relied on, to five ſeconds; and that the chances for the ſea
veral errors — 5", - 4", - 3", -2", -1", ", +1", + 2",
+ 3", +4", + 5", included within the limits thus affigned,
are reſpectively proportional to the terms of the ſeries 1, 2, 3,
4, 5, 6, 5, 4, 3, 2, 1. Which ſeries is much better adapted,
than if all the terms were to be equal ; ſince it is highly rea-.
ſonable to ſuppoſe, that the chances for the reſpective errors de-
creaſe, in proportion as the errors themſelves increaſe.
Theſe particulars being premiſed, let it be now required to
find what the probability, or chance for an error of 1, 2, 3, 4,
or 5 ſeconds will be, when, inſtead of relying on one, the
mean of ſix obſervations is taken.
Here, v being = 5, and t = 6, we ſhall have n(at)
=12, w(=UTI)=6, and pl= tu+n+m)
42 + m :
but the value of mi, if we firſt ſeek the chances whereby the
error exceeds not one ſecond, will be had from the equation
+1; where either ſign may be uſed (the chances being
the ſame) but the negative one is the moſt commodious: from
whence we have m(=-t)
-t) = 6; and therefore p=36,
Þ=30, p= 24, &c.
Which' values being ſubſtituted in the general expreſſion
above determined, it will become 35. 34. 33 (12) — 29 28 27
(12) × 12 + 13.22.210
(12) x 66
2 3
299576368: ánd this ſubtracted from 1088391168 (=x614),
leaves 78881480, for the value of D correſponding: therefore
the required probability that the error, by taking the mean of
fix obſervations, exceeds not a ſingle ſecond, will be truly mea-
ſured by the fraction
788814800
1988391168 ; and conſequently the odds
will
2 3
3
17.16.15 (12) x 220
I
I
2
3
of a Number of Obſervations in practical Afronomy. 71
8
to l.
IO
2; fo
Ihall m
will be as 788814800 to 299576368, or nearly as 2 to 1.
But the odds, or proportion, when one ſingle obſervation is
taken, is only as 16 to 20, or as
To determine, now, the probability that the reſult comes
within two ſeconds of the truth, let me be made =~2;
- 2t) =-12: therefore p=30, Þ
30, p=242
p" =18, &c. and our general expreſſion will here come out
36079407 ; whence D = 1052311761. Conſequently
1052311761
will be the true meaſure of the probability ſought:
1088391168
and the odds, or proportion of the chances, will therefore be
that of 1052311761 to 36079407, or as 29 to 1, nearly. But
the proportion, or odds, when a ſingle obſervation is taken, is
only as 2 to 1: ſo that the chance for an error exceeding two
ſeconds, is not -- th part ſo great, from the mean of lix, as
from one ſingle obſervation. And it will be found in the ſame
manner, that the chance for an error exceeding three ſeconds
is not here do part ſo great as it will be from one obſervation
only. Upon the whole of which it appears, that, the taking
of the mean of a number of obſervations, greatly diminiſhes
the chances for all the ſmaller errors, and cuts off almoſt all
poſſibility of any large ones: which laſt conſideration alone is
ſufficient to recommend the ufe of the method, not only to
Aſtronomers, but to all Others concerned in making experiments,
or obſervations of any kind, which will allow of being re-.
peated under the fame circumſtances.
In the preceding calculations, the different errors to which
any obſervation is ſuppoſed ſubject, are reſtrained to whole
quantities, or a certain, preciſe, number of feconds; it being
impoſſible, from the moſt exact inſtruments, to take off the
quantity of an angle to a geometrical exactnefs. But I ſhall tow
ſhew how the chances may be computed, when the error ad-
mits of any value whatever, whole or broken, within the pro-
poſed limits, or when the reſult of each obfervation is fup-
poſed to be accurately known.
Let
?
**
7 2 Of the Advantage ariſing by Taking the Mean
,
Fig. 20. Let, then, the line AB repreſent the whole extent of the
given interval, within which all the obſervations are ſuppoſed
to fall; and conceive the ſame to be divided into an exceeding
great number of very ſmall, equal particles, by perpendiculars
terminating in the ſides AD, BD of an iſoſceles triangle ABD,
formed upon the baſe AB: and let the probability or chance
whereby the reſult of any obſervation tends to fall within any
of theſe very ſmall intervals Nn, be proportional to the corre-
ſponding area NMmn, or to the perpendicular NM; then, ſince
theſe chances (or areas) reckoning from the extremes A and B,
increaſe according to the terms of the arithmetical progreſſion
1, 2, 3, 4, &c. it is evident that the caſe is here the ſame
with that in the latter part of Prop. II.; only, as the number
v (expreſſing the particles in AC or BC) is indefinitely great,
all (finite) quantities joined to v, or its multiples, with the
ſigns of addition or ſubtraction, will here vaniſh, as being no-
thing in compariſon of v. By which means the general ex-
prefion (F2
1:272.933 (n) — 1.
P2.83 (n) x n +
po 21.842,8"–3 (n)xn.". , &c.) there determined, will
bere become L. 2. § () - BB.B &
1.2.3.4(n)
xp- npa + n.
-
p"'", $c. (wherein
p = tv 7 m, p=p-v, p=p2v, p"=p-30, &c.)
and therefore, the value of D in the preſent caſe, being sun
x p" - n.pl" + n.
1
--.- 201", &c. it is
1.2.3()
evident that the probability
probability () of the error's not exceed-
ing the quantity (in taking the mean of t obſervations) will
be truly defined by
x 1
x &c.
1.2.3(n)
which may
be repreſented by the curvilineal area CNFE, cor-
reſponding
I
3
I
I 2
3
I 2
3
na Illy
n In 2
n.
2 3
2
I
N
2
1
of a Number of Obſervations in practical Aſtronomy.
7-3
t
n NI
there not
2
I
11
n I 7 m 2
I
2
reſponding to the given value or abſciſla CN (=). Now,
though the numbers v, P, and m are, all of them, here ſup-
poſed to be indefinitely great, yet they may be exterminated,
and the value of the expreſſion determined, from their known
relation to each other. For if the given ratio of to v, or of
CN to CA, be expreſſed by that of x to 1, or, which is the
fame, if the error in queſtion be ſuppoſed the x part of the
greateſt error ; then, m being tux, p (=tv I m) will be
tv I tux, and therefore
P
=txitx; which let be de-
noted by y: then, by ſubſtitution, our laſt general expreſſion
will become
Syn noy - 1+
Y у
21"
1.2.3(n)
2.7-3
y - 31", &c.
which ſeries is to be continued till the quantities y, y - 1,
y — 2, &c. become negative.
As an example of what is above delivered, let it be now
required to find the probability, or odds that the error, by
taking the mean of ſix obſervations, exceeds not a ſingle ſe-
cond; fuppofing (as in the former example) that the greateſt
error, that any obſervation can admit of, is limited to five
ſeconds.
Here t being = 6, n (= 2t) = 12, and x = 5
y (=txi-x) = 4,8; and therefore the meaſure of the
probability fought will be equal to 1pm
1.2.3 (12)
12x3,81 +66 2,8) 220 x 1,812 + 495*0,81"*
0,7668, nearly: ſo that the odds, that the error exceeds
not a ſingle ſecond, will be as 0,7668 to 0,2332; which is
more than three to one. But the proportion, when one ſingle
obſervation is relied on, is only as 36 to 64, or as 9 to 16.
In the ſame manner, taking x =
§ it will be found, that
L
the
we have
>
2
X
I
I 2
4,872
5
74
Of the Advantage ariſing from Taking the Mean
5',
the odds, of the error's not exceeding two ſeconds, when the
mean of ſix obſervations is taken, will be as 0,985 to 0,015,
nearly, or as 65; to 1; whereas the odds on one ſingle ob-
ſervation, is only as 64 to 36, or as it to 1: ſo that the
chance for an error of two ſeconds is not th part ſo great,
from the mean of fix, as from one ſingle obſervation. And it
will farther appear, by making x = 3, that the probability of
an error of three ſeconds, here, is not doth part ſo great as
from one ſingle obſervation : ſo that in this, as well as in the
former hypotheſis, almoſt all poſſibility of any large error is
cut off. And the caſe will be found the fame, whatever hy-
potheſis is aſſumed to expreſs the chances for the errors to
which any ſingle obſervation is ſubject.
From the ſame general expreſſion by which the foregoing
proportions are derived, it will be eaſy to determine the odds,
that the mean of a given number of obſervations is nearer
to the truth than one ſingle obſervation, taken indifferently.
For, if x be put (=1-x) = , and s= , then, y being
= tz, the quantity
1.2.3(7) Xy_n.y—11°+n."". y-21", &c. (expreffing
the probability that the reſult falls within the diſtance z of the
greateſt limit) will here, by ſubſtitution, become
nx% - $*+n. xz— 231", &c. which,
1.2.3 (n)
in caſe of one ſingle obſervation (when t=1, and n=2) is
barely z“, and its fluxion 2zż: therefore, if we now multiply
by 2zż, the product
Xuntz-1.2 Sl".zż tn.
-.-25)". zż,&c.
1.2.3 (n)
will give the fluxion of the probability that the reſult of t ob-
fervations is farther from the truth, or nearer to the limits,
than one ſingle obſervation taken indifferently. And con-
ſequently the fluent thereof, which is
into
1.2.3(n) n+2
2
24"
1
X2
2
{
n
41"
n I
2
ntz
Z
of a Number of Obſervations in practical Aſtronomy. 75
11
N
I
+
2-25Trafos
25.7-232*** +
X
I
2
5.737271
spate
25.% - 25
x"
nt I nt a
nt i
n + 2
&c. will, when z=1, be the true meaſure of the proba-
bility itſelf. Which, in the caſe above propoſed, where t=6,
and n = 12, will be found = 0,245, and, conſequently, the
odds that the mean of fix, is nearer to the truth than one
ſingle obſervation, as 755 to 245, or as 151 to 49.
&
Å
L2
A DE
Et
Deer $30etagetAgorta Sport Doetsetan
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SKY12265 22 Doe V 26 Vastastygaofskygas 42
A
DET ERMINATION
OF
Certain Fluents, and the RESOLUTION of ſome
very uſeful EQUATIONS in the higher Orders of
Fluxions ; by means of the Meaſures of Angles
.
and Ratios, and the Right-lines and Verſed-
fines of circular Arcs.
I
N order to treat the matter here propoſed with due
perſpicuity, it will be neceſſary, previous thereto, to
give a demonſtration of the two ſubſequent Lemmas.
LEMMA I.
The double of the rectangle contained under the co-fines of any
two arcs, fuppoſing the radius to be unity, is equal to the ſum of
the co-fines of the ſum, and difference of thoſe arcs.
Fig. 21. For, let AB and BD (= BE) be the two arcs, and CG and
Cn their reſpective co-ſines; likewiſe let CH be the co-fine of
their ſum AE, and CF that of their difference AD; making
nm parallel to BG. Then, Dn being En, it follows that
Fm =Hm; and conſequently that 2Cm= CH +CF: but,
by ſimilar triangles, CB : CG :: Cn: Cm; whence CG X 2Cn
CB x 2Cm CB X CH + CF. 2. E. D.
L EMMA II.
If A be any arch of a circle whoſe radius is unity, and n any
whole poſitive number ; then will
"
. .
co-fin.
14.A+n
co-fin. 1 -6.A+&c.continued
2
n I n-
2
2
3
to
The Reſolution of certain fluxionary Equations, &c.
77
to "t1, or in + 1 terms, according as n is an odd, or even num-
ber; in the latter of which caſes the half, only, of the laſt term is
to be taken.
For, by the preceding Lemma, 2 cof. Al = coſ. A + A
+ coſ. A — A = coſ. 2A +1;
A = cof. 2A +1; which equation multiplied
by 2 cof. A, gives 2” x coſ. Als cof. 2A X2 coſ. A + 2 cof.
A = coſ. 3A + 2 cot
. A (by Lem. I.) =coſ. 3A+3 cof. A.
Start
4
Multiply, again, by 2 cof. A ; fo ſhall 23 x coſ. A
coſ. 3A X 2 cof. A + 3 coſ. A x 2 coſ. A =
= cof. 4A
cof. 2A
+ + 3 cof. o (by Lemma I.) = coſ. 4A + 4 cor.
cof. 2A
2A + 3. In the ſame manner we have,
24 x coſ. A 5=cof. 5A + 5 coſ. 3A + 10 cof. A,
25 x cof. A' =cof.6A +6 col.4A +15 cof. 2A +10,
26 x coſ. Al' =cof.7A +7 cof. 5A +21 coſ. 3A +35 coſ. A,
&c.
&c.
where the law of continuation is manifeſt; the numeral co-
efficients being the ſame, and generated in the very fame man-
ner, with thoſe of a binomial raiſed to the 2d, 3d, 4th, 5th, &c.
powers, ſuceefſively ; except in the laſt term, when the ex-
ponent n is even, in which caſe one half only of the corre-
fpondent (or middle) term of the involved binomial is con-
cerned. Hence the propoſition is manifeft.
. ;
NI -
The ſame otherwiſe.
If the co-fine of A be denoted by x, it is well known that
À - Multiply the whole equation by V-1, ſo
-V
ſhall ÅVI
VI — XXXVI
whence, by taking the fluent, we have AV I hyp. log.
*+XX — I.
1. Let N be the number whoſe hyperbolical lo-
garithm
I
XI
ร
.
I
NI+xx
78
The Reſolution of certain fluxionary Equations,
garithm is 1; then, ſince hyp. log. NAVIG=AV-I) = hyp.
log. x + xx — ī, it is evident that NAVGI=x+vxx -1,
or M4 = x +Vxx – I (by making M=NV-). From
MA+M-A
which equation * (the co-fine of A) is found =
(and from thence (V 1- xx) the fine of A=V-IX
; but this laſt by the bye).
2
MA
- M-A
2
AR
2
Now ſeeing that 2 coſ. A = MA + M-A, we there-
fore have 2 cof. A = MA + M-A MA+M-+
nx M»~n
A+M-72.4 +1.1-1 x MM-4.A +M+4.A +&c.
by expanding MA +MA" and uniting, in pairs, the corre-
ſpondent terms (viz. the firſt and laſt, the ſecond and laſt but
one, and ſo on). But MrA + M-A, the firſt of theſe pairs, is
the double of the co-fine of 11A; for the very fame reaſon that
MA+M-A was found to expreſs the double of the co-line of A.
And thus, A-2.A + A=42. A will appear to expreſs the dou-
ble of the co-fine of n — 2.A, &c. And our equation will
therefore be reduced to 2* x col. A 2 cof. nA +
2n coſ. n - 2. A + 2n cof. n 4.A + &c. or
to cof. A)" པ་
of. nA + n cof. 1—2. A + n cof. n - 4. A
X
tn.
cof. 1-6.A +8C.
where, when the exponent n is even (the number of terms in
MA +M-A", expanded, being odd) there will be a middle
term (no-ways effected by M or A) which being an abſolute
number, muſt be taken ſingly, and conſequently, only the
half thereof when the whole ſeries is divided by 2, as is the
caſe in the concluſion, & E. D.
NI
2
no I
2
N I N
2
2
3
COROL-
by Means of the Meaſures of Angles and Ratios.
79
COROLLARY.
I
I
2
If Q be taken to repreſent an arch of 90 degrees, and the
complement (Q - A) of the arch A be put =B; then, by
ſubſtituting ſin. B for cof. A, and QB for A in our ge-
neral equation, we ſhall have ſin. B11 into cof. nQ-1B
+ n coſ. n --- 2.Q—n—2.B+w."... coſ. n-4.Q-1-4.B
+&c. being a general expreſſion for any power of the fine
of an arch (as the former was of the co-fine). But this ex-
preſſion may be reduced to a form ſomewhat more commodi-
ous, regard being had to the different interpretations of n,
with reſpect to even, and odd numbers. Thus, if n be ex-
pounded by any term of the ſeries 4, 8, 12, 16, &c. it is evi-
dent that nQ (in the firſt term) will be an even multiple of the
ſemi-periphery; and that n-2.Q (in the ſecond term) will be
an odd one, and ſo on, alternately. But it is well known, that
ſubtracting, or caſting off any multiple of the ſemi-periphery
no-ways affects the value of the ſine, or co-fine ; except, that
ſuch value, when the multiple is an odd one, will be changed
from poſitive to negative (and vice verſa). Hence our laſt equa-
tion will be reduced to fin. B
into cof. nB
2
N
I
n cof.
12.
. B + 7
&c.
2
4.B
cof. 14.B
into cof. nB — n coloñ
ñ— 2.B.+n." coſ. n—4.
&c.
And, for the ſame reaſons, the equation, when n is in-
terpreted by any term of the ſeries 2, 6, 10, 14, &c. will ap-
pear to be fin. B
cof. nB + n cof. n - 2.B
1
to be fin. Blº= into
N
-I cofon
in
2
4.B + &c.
But, when n is expounded by any of the odd numbers 1, 5,
9, 13, &c. we ſhall then (by rejecting the multiples of the
ſemi-
80 The Reſolution of certain fluxionary Equations,
11 I
2
1
I
اللهم
2
n-4.B
ſemi-periphery, &c.) have ſin. Bl"= 1 into cof. Q - nB
n cof. 'Q-n — 2.B + n. col. Q-n4.
&C.
into fin.nB - n fin.n-2.B + n. fin.
4.B-&c.
Laſtly, if n be expounded by any term of the ſeries 3, 7,
II, 15, &c. the reſult, or ſeries, will be the ſame as in the
preceding caſe, only the ſigns of all the terms muſt be changed
to their contrary.
But all theſe different caſes may be otherwiſe, more directly,
inveſtigated, by means of the two following Theorems; where-
of the Demonſtration is obvious, from that of Lemma I*
1°. The double of the rectangle contained under the fines of any
two arcs, ſuppoſing the radius to be unity, is equal to the difference
of the co-fines of the fum, and difference of thoſe arcs.
2º. And the double of the rectangle under the co-fone of the one
and the fine of the other, is equal to the difference of the fines of the
ſum, and difference of the ſaid arcs.
Hence it follows, that ſin. B x 2 ſin. B
cof. B+B
+ cof. B-B (by Theor. I.) = - cof. 2B + 1: whence,
multiplying the whole equation by 2 ſin.B, we have 2'xſin. B
fin.B
cof. 2B x 2 ſin.B + 2 ſin. B :
ſin. 3B + zfin.B (by
Theor. II.) = fin. 3B + 3 fin. B. Whence, again, by equal
multiplication, 23 x ſin. Bl4 fin. 3B x 2 ſin. B + 3 fin. B
cof. 2B
x 2 ſin. B = + coſ. 4B
3 cof. 2B + 3 coſ.o (by Theo-
rem I.) = cof. 4B
cof. 43 - 4 cof. 2B + 3.
In like manner, 24 x ſin. BS = fin.5B- 5 fin.3B+10ſin.B;
and 25 x ſin. BIE - coſ.6B+6 col.4B-15 cof. 2B+10,&c.
BA
3
Fig. 21.
By ſim. A's, BC(1): BG :: DE (2Dn) : DP(CF-CH)=2Dnx BG:
And BC (1):CĞ :: DE (2Dn) : Ep(ÉH-DF) = 2CG x Dn.
Whence,
3 ३
by Means of the Meaſures of Angles and Ratios.
8L
-I
Whence, univerſally, ſin. B1*
Х
2
+ ſin. nB # n fin. 1 -2.B + n.
fin.n
4. BF &C.
2
I
when n is an odd number ; and ſin. B"
X
2
2
2
n I
n In
2
n
n.
2
2
3
I
+ cof. nB 7 ncoſ.n - 2.B + n. cof. n— 4.B I &C.
when n is an even number: where the number of terms, in
the former caſe, will be us, and in the latter in tI; in
which caſe the half, only, of the laſt term is to be taken ;
and is always poſitive, as well as the laſt term in the former
caſe: whereby the ſigns of all the other terms (as they
change alternately) will be known.
If a, b, y, 8
d.. M be aſſumed to expreſs the terms (1, n,
&c.) of 1 + 1 raiſed to the nth power
(M being the middle, or greateſt term) it is evident that the fe-
cond caſe of our general equation (wherein n is even) will ſtand
thus, Gn. B)* 7 into + a cof. nA F B cof. 1—2. A
+ y cof. 1-4. A I 8 cof. ñ -- 6.A..... + M.
By the ſame method of proceeding, an expreſſion exhibiting
the continual product of the co-lines, or fines of any number
of unequal arches, may be derived.
For · (by Lemma I.) coſ. A x 2 cof. B = cof. A + B + coſ.
AB; whence cof. A x 2 cof.B x 2 cof. C
= coſ. A +B
X 2 col.C + coſ. A - B x 2 cof. C (by equal multiplication)
cof. A +B-C
= coſ. A +B+C + cof. A-B+C (by the Lemma);
cof. — A+B+C
whence, again (by. equal multiplication and the Lemma) we
M
have
2
1
82
The Reſolution of certain fluxionary Equations,
have cof. A x 2 cof. B x 2 cof. C x 2 cof. D cof.
cof. A+B+C-D
cof. A+B-C-D
cof. A+B-C+D
A+B+C+D+
cof. A—B+C+D+coſ. A-B+C-D
cof. —A+B+C+D
cof. —A+B+C-D
from which the law of continuation is manifeft.
I
nel
I
2
NI
2
M.
I
X
VI
PROBLEM I.
***
To determine the fluent of n being any odd affirmative
Vi
number.
If A be aſſumed to denote the arch of a circle, whoſe ſine
is x and radius 1, it is well known that Å VIS si and,
by the Corol. to Lem. II, it alſo appears that ** = to
x fin. nA - n fin. n - 2.A tn. fin.n—4.A
4.A().
Hence we have (= x*A)=+
Axf. nA -- Axl.n— 2.A+n." SÅ x 1.n— 4.A (+-).
But the fluxion of any arch, multiplied by the fine (the radius
being unity) is equal to the fluxion of the verſed-ſine :
therefore the fluxions of the verſed-fines of the arches nA,
n — 2.A, n - 4.A, &c. will be nÅ xſin. nA, ñ - 2. Ax
fin. ñ — 2. A, n—4. Å x fin. n — 4.A, &c. reſpectively ;
and conſequently the ſaid verſed-fines, the true fluents of theſe
fluxions : whence it is manifeft that the true fluent of our
whole expreſſion will be + T into Ix verf. fin.na
x verſed-fin. 1 - 2.
x verſed-ſin.n- 4.A
x verſed-fin. 1 — 6.A(*1). Wherein, of
the
I
n
2
2
I
2.A +
n4
2
N
nn 2
6 2 3
B
hy Means of the Meaſures of Angles and Ratios.
83
the ſigns to and before, the former, or latter ob-
tains, according as "+", expreſſing the number of terms, is
.2
odd or even.
PROBLEM II.
S
*"
I
4
NI
I
x
2
I
Х
I
To find the fluent of ; n being any even affirmative
number.
By the preceding Problem
Å; and, by the Corol.
to Lem. II, *=+
a cof. nA – Rx cofin-2.A ty x cof. n 4.A..... #M.
Therefore (= x^^)=+
+7
aA x coſ.nA BAXcoſ.n–2.A tyA * cof. 1-4. A...+MA.
But the fluxion of any arch, multiplied by the co-fine, is
equal to the fluxion of the fine, drawn into the radius: whence
it follows that nÅ x cof. na, n-2. Å x coſ. ñ 1 - 2.A,
n— 4. Å x coſ. n— 4.A, &c. are the fluxions of the fines
of the arches nA, 7 — 2.A, n— 4.A, &c. reſpectively; and,
conſequently that
x fin. nA x ſin.1-2.At ;
n-4.A x fin. n-6.A..+ MA
will be the true fluent fought : where a = 1, B=n, y
Bx"7', =*"??, &c. and wherein the ſign + or --
before , obtains according as in +1, expreſſing the
number of terms, is odd, or even.
+
a
B
y
X fin.
NI
I
N
n man 2
n
4
It
2
N
6
M 2
COROL-
84
The Reſolution of certain fluxionary Equations,
м х
m
m-
x
1
,
m
2
MX
х
I m
x
m
m
m
mempen I
Х
X
х
COROLLARY I.
Since the value (M) of the middle term of the binomial
it 17", expanded in a ſeries, is known to be "x"="x">?
in #1, by the law of the ſeries, it is evident that the
들기
​term next adjacent to it (on either ſide) will be expreſſed by
11/22
or by Mx ; making m= n. And, in the
inti
m+.
ſame manner, it will appear that the next term to this laſt will
be expreſſed by Mx
and the next to that, by
m+1 m +2
; and ſo on. Therefore, by ſubſtitut-
mti m + 2m +3
ing theſe values above, and inverting the order of the terms,
the general fluent, there given, will here be transformed to
SA
x ſin. 2A +
M
x-fin.4A
m of I
m+I m+-2
m+I
x1.6A+
m=-3x11.8A
m + 2 m+3
m+im+2 m +3m +4
&c. where the ſeries is to be continued till it terminates; and
M
where the value of the general multiplicator will be truly (and
I
moſt commodiouſly) expreſſed by
3 5
Х Х x*-1. For
2
4
6
2.1-3.
·
I. 2 3 4
들기
​hereof be multiplied by in.in-1.in. — 2.0 - 3...3.
2.1, and the denominator, at the ſame time, by its equal 1.2
•3.4. 11 / 들기 ​2.n-I
1.n, we ſhall then have M
n.nIn ņ3......3.2.1 1.2.3.4.5.6.7...
1.2.3.4....inx1.2.3.4....in 1.2.3.4...nx1.2.3.4...in
M 1.2.3.4.5.6.7.8......n
which divided by 2* gives
2.4., 6.8....nx 2.4.6.8...t
I.3.5.7..
2.4.6.8.......
2"
m
I
m2
Х
m-1 m2
Х Х Х
2
N
n.72
I.
M being
in+I. if the numerator
ll
1
2.11
22
2
NI
n
COROL-
1
by Means of the Meaſures of Angles and Ratios.
85
COROLLARY II.
n
;
1 - I
Х
2
3
ex
2
M
in
M
I
X
ty
NI
n'I
X
Х
4
2
Hence
may
the fluent of way 1 - xx be likewiſe deduced;
x"* XIXX
for this expreſſion may be changed to
or to
✓ I
**+27
whereof the fluent of the firſt term is
ŅI
VI
already found
X
4
A
x ſin. 2A +
xzfin.4A, &c. And, by
m+I
mt im + 2
making n=n+2, and mi n' (=m+1), the fluent of
.nt2
the ſecond term
in the ſame
vi
3
Ix
5
manner is given equal to
6
n'
m'
A
3
nto
x ſin. 2A, &C.
xung
m ti
4
n+ 2
A
m+I
x ſin. 2A +
m +I
х x; fin.4A, &c. Whence,
m + 2
m + 2
m + 3
by adding the fluents of both terms together, we have, after
1. 3.5.7....
proper
reduction
2.4.6.8...
non t2
m. m-7
1. M-17
A—M --Ixf.2AH
x 1.4A
m + 2
m + 2
m+ 3
m+2.m +3m +4
X
2.m31
x ſin. 6A +
x fin. 8A, &C.
m +2.m + 3.m + 4.m + 5
where the law of continuation is manifeſt, the differences (6,
10, 14, &c.) of the numbers 1, 7, 17, 31, &c. being in arith-
metical progreſſion.
I
---
1
X
X
2
m
n I
m. m-
m. M-
m-I. m - 2
I
ti
COROLLARY III.
Moreover, from hence the fluent of XetfxTgx++bx6
WI
3c. may be eaſily deduced : for, putting = xx
N
I
2
n
86
The Reſolution of certain fluxionary Equations,
***
=9, the fluent of the firſt term
ŅI
xe, is given, by Co-
m
fx")=**1*9f*
NI
m
M
eXA
m - I
X
in
X
fxA
х
m+2x ſ.2A +
gxA-
m
rol.I, = qexA x ſin. 2A+ Х xį fin.4A &C.
mti
mtim + 2
and that of the ſecond term
X
n+2
А
mti
m ti
x ſin. 2A +
X xşſin.4A&c. There-
m + 2
m + 2 m + 3
fore the fluent of the whole expreſſion will be found 9
x ſin. 2A +
x} fin.4A &C.
m+I
m to 1 m + 2
nt
in + I
x ſin. 2A +
m + 1
x fin.4A&C
nt
m + 2
m+2 m+ 3
nti
n +3
m + 2
x
X
m+1x1f.4A&C.
n + 2
n+4
m+3
m+3 m + 4
&c.
&c.
rt I
n
which, by making r= Xq, s xr, t
Sa
n+2
n+4
n+6
&c. will be reduced to
+ en + fr + gs + &c. * A
to I m + 2
.fr + gs + &c. x ſin. 2A
m ti m + 2 m+3
m ti
m+2 m toi
.fr+
m to I'm +2 m+2'm +3 m +3m +4
m-I m 2 m+I
F.fr +&c.x;1.6A
m+im+2 m+3 m+2'm +3 m +4
&c.
nt 5 xs,
s=1+3
M
m
1
egt
M
m-I
M
+
•egt
m
mn I
.eq+
&c.
SCHOLIU M.
V
a
From the fluents determined in the preceding Propoſition,
mp+-In
zmp + p-
thoſe of
abze x xmp+1p-iż, and
• ba?
bz?
xe + f + gz2P + b23P &c. (where m denotes any
whole
po-
fitive number, and p any poſitive number whatever, whole or
broken) may be eaſily deduced, by means of a proper transfor-
mation :
an
by Means of the Meaſures of Angles and Ratios.
87
99-
$
b
a
b
......
a
=
b
a
2a"
pbm the
ll
2a
n - I
x
n
pbm the
m
m
1
mation : for, va — bzł being
ad 1 _ bz?
let there
ba?
mta
be made =x*, orz?=x*; then zap+! = xx2m+1;
and conſequently, by taking the fluxion on both ſides,
*+1
mp + p.xmp+ip-=
x 2m + Tixama, orxmp+ {p-IŽ
+
mp + -
* xz.x2mě. Therefore our firſt expreſſion,
- bz”
stehen
will be transformed to
X (ſuppoſing n=
2m)
VI-XX
whoſe fluent (by Problem II. Corollary I.) will be given
I.3.5.7
X.
2.4.6.8.
A
f.2A+.
mm - 2
x-1.4A
m-I
m+I'm +2
mti'm +2 m+ 3
bz?
+&c. where A repreſents the arch whoſe line is
the
radius being unity.
In the ſame manner the fluent of our ſecond expreſſion,
va
2- bz® x xmp+Ip-'ż, will be given (by Corol. II.) equal to
ti
I:3.5.7...
X
2.4.6.8...
n+2
m.m7
I.M-17
+
x_ſin.4A
m+2
m+2.m+3
m+2.m+3.m +4
x_fin. 6A +
31
xu fin. 8A
ESC.
m+2.m+3.mt4.m+5
mp+ 1p-1
Laſtly,
xetfz + gz2P + b&IP &c. will be trans-
• ba?
formed to
Х Xe+++ ga
ga? x4 + &c. and the
VI I — **
fuent thereof (by Corol. III.) will therefore be given equal to
+eq + frx+gs x + &c. * A
十三
​egx
+gsxmxm2&c.xf.ZA
mti
mt2
6²
m+ 3
.. n I
20" to
pbm + 2
m-I fin. 2A +
m. 11 -
A
Х
m.
m1.m2.m
a
12
211
a SC
pom +1
20
pbm + 7
m+1+gs*
a²
x
88
The Reſolution of certain fluxionary Equations,
2a
M
x eqx mti
Bbm the
. n I
N
When /
а
z mp + že
z
mtot
X
+ fr x 2
응
​X
Х
&c. *}f.4A
m tim +2
b m + 2
m + 3
&c.
&c.
I.3.5.7
nti
n + 3
wherein 9
r = 9X
Se qe X
2.4.6.8.....
n+2
nt 4
esx n+ 5
t SX
&c.
n +6
bz
becomes equal to the radius, or unity, A will
be an arch of 90 degrees ; and therefore, the fines of all the
arches 2A, 4A, 6A, &c. being then equal to nothing, the
fluent of
will, in that circumſtance, be barely
✓ -ban
I.3.5.7.
X X A. Moreover, the fluent of
2.4.6.8...
n-2
a - bx® x zimp + ip-'z 'will then become
1:3:5.7.
2.4.6.8...
n +2
XA; and that of
xe +f2b+cº++ bx3P
a -
bz?
c
Iti fa n+In+3 ga?
+*+1.n+3.n+5 ba
&c.
nt2b
n+2'n+4' b n+2'n + 4n+6° 73
3.5......n-I
XA; where
q
and where, if
2.4.6...
m= 0, q muſt be taken = 1.
..n
20%
pom+]
..n I
va
m+I
2a
z mp + ip-
X
4
P6m +
etiam et
29a"
X
pbm 2
&
PROBLEM III.
To determine the fluent of 2 x coſ. mz X cof. nz X col.pz &c, in
which mz, nz, pz, &c. are any given multiples of the arch 2;
the radius of the circle being unity.
Make A mz, B = nz, C
= nz, C = pz, &c. and find (by the
method on p. 81) a ſeries of co-ſines of the multiples of z, to
expreſs the continual product (coſ. A x cof.B x coſ.C, &c.) of
the co-fines propounded; which ſeries, let be denoted by
a x cof.az + cof. Bz+coſ.yx+coſ.dz &c. (a, a, b, &c. being
conftant quantities) : then will our given expreſſion become
až
by Means of the Meaſures of Angles and Ratios.
89
aż x coſ.az + col. Bz+coſ. yx cof.dz &c. and its fluent will
therefore (by proceeding as in Prob. II.) appear to be = a into
+ +
fin. az
fin. BZ
ſin.yZ
+
fin.dz
o
&c.
a
B
?
I
I
2
2
Thus, for example, let the fluent of ź x coſ.mz x coſ.nz
be required; then will coſ. A x cof.B (= coſ. mm x coſ.nz) =
x cof. A+B + cof. A-B
— x cof.m tn.z+m-1.2:
+ -
therefore, in this caſe, a= 17 들 ​a=m+n, and B=m-1;
ſin
and conſequently the fluent fought
+
mti
2
1
into fin.m+n.2
2
fin.m
n.
m
n
I
4
In like manner, if the fluent of < x cof. mz x cof. nz X col.pz
were to be required ; then would cof. mz x coſ. nz x cor. pz
(= coſ. A x cof. B x cof. C) I into coſ. m +n+po+
cof.m + np.z + cof.mn+pix + coſ. --m+n+p.x;
fin. m ton + p. z
and therefore the fluent fought into
+
mtnte
ſin.m +1piz
fin. m m n + poz
fin. ntmtp.%
+
+
mtn
nt to
ntmtp
I
4
m
2
By the very fame method the fluent may be determined
when ſome, or all of the factors are ſines (inſtead of co-fines).
Thus, if there be given z x coſ. mz x ſin.nz, it may be wrote
Ż x cof. mz x coſ. 90°
nz; which is
=
into coſ. 90° +
n.z + cof.mt-n. 90° = into fin. 1 — m.
+ fin. n + m.z; and ſo the fluent (by proceeding as in
verled-fin. na
Problem 1.) will come out
into
+
verled-fin. n tm.z
n + m
m
ន
m. 2
I
m.z
22
12 m
N
PRO-
k
go
The Reſolution of certain fluxionary Equations,
24
!
PROBLEM IV.
From the equation ay + b3 + + + + &c. =0
(wherein a, b, c, d, &c. denote conſtant quantities); it is pro-
poſed to find the value of y in terms of z.
Afſume y = aMmm + BM"* + 3Mp% + 8Mqz &c. in which
M denotes the number whoſe hyperbolical logarithm is unity :
then will j mżaMmz f nzßM" + pżyMp% 30.
☺ = mºž’aMm2 to nºz*BMm2 + przʻyMp2 &c.
j = m3ž3&Mmz to n3z33M™2 + p3ž39Mp4 &c.
&c.
&c.
Which values being ſubſtituted in the given equation, we have
aqMmx + aßM+ ayMp4 &c.
bmaMmx + bn@Mn2 + bpyMpx & C.
cm” Mm2 + cn RM12 to cp Mp2 &c.
dm3cMmx + dn32M*2 +-dp3gMp3 &c.
&c.
&c.
From whence, by equating the homologous terms, we have
a + bm + cm + dm3 &c. = 0, a + bn + en + dn3 &c.
=0, a + bp + cp + dp3 &c. = 0, &c. that is, the re-
quired values of m, n, p, &c. will always be the roots of an
equation, a + bx + cãoz + dx3 @c.= 0, wherein the given
quantities are the ſame, in every term, with thoſe in the flu-
xional equation propounded. Therefore, when theſe roots are
known, the value of y will alſo be known: in which the coef-
ficients dis B, , d, &c. may denote
any conſtant quantities at
pleaſure; as is evident from the proceſs.
When ſome of the roots of the equation a + bx + cx? +
dx3 + ex4 &c. = 0, happen to be impoſſible, the values of
the correſponding terms of the ſeries am2 + BMW% + gMp3
+8M* &c. will then be expreſſed by means of the lines and
co-fines of circular arcs. Thus, for example, let the fuxio-
nary equation propounded be y =0; then we ſhall have
I — x=0; whereof the four roots are 1, -1, +VI,
and -VI; and, theſe being ſubſtituted for m, n, p, and
q
al
4
by Means of the Meaſures of Angles and Ratios.
gr
21
VI
we have
9, reſpectively, y will here become = «M* + BM- +
qM+v1 + $M-25. Now, to take
Now, to take away the imaginary
terms yMxv + $M_V=T, we may write k +1=y, and
k -1=d; whereby the ſum of the ſaid terms will be =
k x MxVG + M-- + 1x Mxv
+1xMV5 – M-V=
2k x cof.
zt x fin. (vid. p. 78): whence (putting h = / 는
​y = = AM + BM- + 2k x cof. % + 2h x ſin, z;
where a, b, b, and k denote any conſtant quantities, at
pleaſure.
dj
In like manner, ſuppoſing the equation given to be y +
= 0, we ſhall have 1 + dx3 =0; whereof the three roots
-d-, d-x+ V and di ✓
which, if s be put=d-1, and t = d-*x 1a, will be more
commodiouſly expreſſed by ---ss is tt-1, and samt
s-T:
And theſe values being ſubſtituted in the room of m, n, and po
we ſhall have
g
=&M-* + BM2+izv=;
«M-* + M* x BMIZVS+ 3M-2V= ; which, by reaſon-
ing as in the preceding caſe, is reduced to y =~M~ + M**
x 25 x ſin.tx + 2k x coſ, tz.
23
3
I
are
+
Alw
4
2
+9M}~zvoj
3
*
PROBLEM V.
dj
From the equation ay + + + &c. - AMP% +
BM9% + CM"* &c. to determine the value of y; ſuppoſing M to
denote the number whoſe hyperbolical logarithm is unity, and a, b,
c, &c. A, B, C, &c. any conſtant quantities.
Aſſuming y = PM1px + OM47 + RM!% &c. we have
$ = p2PM2% + q2QM + rzRM= 3c.
j = p*z*BMp2 + q*ż’QM9+ rʻz RMX &c.
&c.
&c
which
N2
7
92
The Reſolution of certain fuxionary Equations,
A
B
с
which values being ſubſtituted in the given equation, it becomes
APM1px + aQMax +
AQMq2 + aRM &C.
bpPMP2 + bqQM+ brRM” &c. = AMP% + BM9% +
cp PMp% + cq Max + crʻRM* &c.
CMr &c.
&c.
EC.
From whence, 'by comparing the homologous terms, we have
P.
R
at bp + cf* + dp&c. = atbą +593 + do? Ecc.
&c. whereby one value of y is known.
at brot cr² to dr3 &c.'
But the value or fluent thus found, in order to render it
general, muſt be corrected by the value of y found in the
preceding Problem, that is, by the quantity «Mm'z +- BM«z
+Mpz &c. wherein m', n, p, &c. denote the roots of the
equation a t- bx + cx* + dx3 &0.30, and a, b, y, &c. any
conſtant quantities. For, ſince all the terms ariſing from this
laſt part of the value of y, by ſubſtituting in the given equa-
tion, do mutually deſtroy one another, the other terms affected
with P, Q, R, &c, will be no-ways influenced thereby, but
remain exactly the ſame as above determined.
A
B
Q=
BM9%
+
COROLL AR Y.
If the equation given be mºy + = AMp2 + BM92 +
CM* + DM* &c. then (a being m”, c=1, and b, d, e,
&c. each = 0) we have P
&c.
mm + PP
mm to 99
and x
+ md=1; and conſequently y = «Mmzv=i+
AMPz.
BM-mV+
&c. = 2h x ſin.mz + 2k
mm + PP mm + 99
AMP
x cof. mz +
&c. (fee Ex. I. to Prob. V.)
mm + PP mm + 99
Hence it follows, that, if the equation given be mạy +
(= AMuzv=I + AM-mv-T + A'Mpv = i + A'M-PZW=&c.)
= 2A X coſ. 7x + 2A'x cof.ez &c. the value of y (by ſubſti-
tuting
- * = P,
q", -g=r gi s”,
A=B, A'=C, A=D, &c.) will come out = 2h x ſin.mz
+
BM92
+
:
2
>
by Means of the Meaſures of Angles and Ratios.
93
+ 2k x coſ.mz +
AMπαν-,
AM-rxv=
+
&C. 25 x ſin, inz
mm
TTT
mm -- NIT
2A
2A'
mint
77
mm
ያያ
+ 2k x coſ, mz +
x coſ. 7% +
x cof. pz &c. .
Which equation (wherein h and k may
denote
any
conſtant
quantities) is of ſingular uſe in determining the figure of the
lunar orbit.
A
B- UbP
In like manner, when the general equation propounded is of
this form, ay + + 3 + 8c. = AX +B2- +Cco-3
+ Dxu-3 &c. the value of y may be determined, by affum-
ing Pão + Qzo- + Rzu-2&c. = y; from whence, by ſub-
ſtituting in the given equation, and comparing the homologous
terms, there will be had P
Q=
R
C-V-1.6Q-V.V-I.CP D-U-2.bR-V-1.0-2.cQ-0.0-1.0-2.dP
SS
&c. where the ſeries will always terminate, provided v is any
poſitive integer ; and where, if to the value of y thus deter-
mined, the correction or ſeries (aMm2 + BM™+ MP4 &c.)
found by Prob. IV. be added, the general value of y will be
obtained
>
>
a
a
24
܀
PROBLEM VI.
To determine the value of y in any fluxionary equation of this
ay
bj
. form, ++ ++ dy A ; ſuppoſing A to repre-
ſent any quantity expreſſed in terms of z and known coefficients.
1°. Make y = Mpx x flu. 2PM-pz (wherein P denotes a
variable quantity, and p a conſtant one, to be determined): ſo
ſhall y x M-p% = Alu. ¿PM–px, or (by taking the fuxions)
jx M-px - y x pżM-px = 2PM-px; whence, dividing the
whole by żM->, we have - py=P.
2°. Make P = M12 x flu. żOM-qz; then our laſt equa-
tion will be transformed to - py x M-z flu. ¿QM–;
whence,
و
94
The Reſolution of certain fuxionary Equations,
{
...
Ý
23
whence, by taking the fluxions, - pý ~ M-*+ * -py
x— qżM–4* = żOM-,, or 3-p+.+ pqy = 2
by dividing the whole by żM7%.
3º. Make, now, R= M* x flu. ŹRM-%; then will
-p+9. + pgy x M-
fluent of 2RM-, or
-p+2...+paj x M-" + 3-p+q..+pgy x-
rżM-2 = żRM-r, or, laſtly, -p+9+rost
pq + pr+qr. - pory=R.
4°. Make, again, R= M* x flu. ¿SM-sz, and proceed
in the fame manner; fo ſhall-p+9+rts.
pq+pr+ps+ar+astrs.a- par + pas +prs+grs.... +
pqrsy = S: from whence the law of continuation is manifeſt.
Let, now, the ſeveral terms of the equation p+etrts
&c.=S be compared with the correſponding terms of the
given one, + &c. =A: ſo ſhall p+a+r &c. =
- a, pa + pr t ps + gr &c. = - b, pqr + pqs &c.
, &c. &c. Whence, from the geneſis of equations, it is
evident, that p, q, r, &c. are the roots of an equation 44
ax3 + bx2 + cx+d=0(or, xe taxams + bxn--2&c. = o)
wherein the given quantities are the very fame with thoſe in
the equation propounded. Therefore, when the values of theſe
roots are found (by any of the known methods) the values of
R, Q, P, and y may alſo be found, one from another,
ſucceſſively. Q: É. 1.
y
23
24
2:!:
24
23
The
by Means of the Meaſures of Angles and Ratios.
95
.
$
$
Mule:: Rile::Re: 2.14.
+&c.
3
Ai
&c.
ż
ż
222
I,
The ſame otherwiſe.
Let (if poſſible) y = AMp2 x flu. žAM-pz +- BM24 x flu.
ŻAM–9% + CM" x flu. ŻAM-* &c. (A, B, C, &c. p, q, r,
&c. being conſtant quantities to be determined): then, by
taking the fluxions, we ſhall have
= PAM**xflu. ŻAM-+AA+gBMr* x flu. ŽAM–9+&c.
Ep*AMPxx A.ŽAM-x+pAa+40+q*BM?zxA.ŽAM-97&c.
AÄ
P3AM1px x Alu.ŻAM-p% +pAa+ +
=p+AMPzx flu. ŽAM–2x+p+AA+PAA+PAG+
Which values being ſubſtituted in the given equation, and the
homologous terms being compared, we ſhall thereby get p4 +ap3
+ bp2 + cp + d=0,q+ aq + bg* + cqtd
= 0, &c.
also \p3+ap: + bp+cXA +-q3+ aq? + bq-tcxB
173 -+-ar* +br tax C +53 + asi + bs toxD
pa tap+bxA+a+aq+bxB+ya +ar+bxc+stastbxD=0
p+ax A +9+a x B tortax C otstaxD
A + B + C +D
Now, from the former of theſe equations, pt + ap3 + bp?
+ cp + d
= 0,94 + aq3 + ba+ cq +d=0,&c. it ap-
pears evident, that p, q, r, &c. are the roots of the equation
24 t- ax3 + bx2 + cx + d= 0 (or, more generally, of
2cm + axn- + bxn-2&c. o, n denoting the order to which
the Auxions aſcend in the given equation); which roots being
therefore found (by any of the known methods) the values of
po q, r, &c. will be obtained. But to find from thence and
the remaining equations, the values of A, B, C, &c. let the
laſt of theſe equations multiplied by a, be ſubtracted from the
preceding one, ſo ſhall pa t qB + rC+. sD=0: moreover,
let this new equation multiplied by a, be ſubtracted from the
laſt but two, and from the remainder let b + bB +6C +
bĐ=o be again ſubtracted, whence will be had pA + q*B
+
0.
96
The Reſolution of certain fluxionary Equations,
O,
Þ.SD
0,
r х
tqPC + 5D=0: and, in the ſame manner, from the firſt
equation, will be had p3A + q3B -f- oC + s3D= I (be-
cauſe 1, and not o, forms the latter part of that equation).
Now, from each of the equations (A + B +C+D=0,
PA + qBtr + sD=0, p’A + 9B + C + s-D = 0,
p3A + q3B +r3C + s3D=1) thus derived, let the preced-
ing one multiplied by p, be ſubtracted : fo ſhall
q- p.B +r-p.cts-p.D
9- p.qB + por tos
9 -p.qB +r-p.ricts-p.sD=I.
Moreover, from each of theſe laſt equations, let the preced-
ing one multiplied by q, be in like manner ſubtracted; whence
will be had
r-por-9.0 to sp.s-9.D= 0,
por-q.rCts-pis 9.sD=1.
Again, from the laſt of theſe, let the preceding one multi-
plied by r, be ſubtracted ; then will s
then will s-post
p.s-9.5-1.D
=1, and conſequently D=
-: whence it is
manifeſt by inſpection (becauſe p is the ſame with reſpect to A,
as s is to D, &c.) that A=
B:
P-9.-r.&c.
9-pigt.&c.
C-
&c. From whence the value of y(
I-por
AM*xflu.ŻAM-p7+BMx*x flu.ŻAM–9* &c.) will be known,
let the orders of fluxions in the equation aſcend to what height
they will. --Thus, for example, let the equation propounded
be + mºy = M*: in which caſe, a being = 0, b=m*,
c=0,&c. our general equation, *"* + ax"-ı + bx*–2 + 2x3
&c. = 0, will therefore become x + m = 0; whereof the
two roots (p and 9) are mv
and - mv
-I; from
whence A =
B
P-9
2mv - I
alſo, becauſe A is here = Muz, we have y = AMP*
2mv
I
S
pis
q.sg
I
I
I
9. &c.
I, and
I
I
I
I
I
X
$
by Means of the Meaſures of Angles and Ratios.
97
BMT2
TT
9
M"
AM2
x flu. ¿Mm-px + BM92 x flu. ¿M-z-q* = +
•
(by ſubſtituting the values of p, q, A, and B). But
T²tp²
in order to render the ſolution general, 'the value of y thus
found muſt, always, be corrected, or augmented by the quan-
tity Mp3 + BM9% +9Mr &c. (given by Prob. IV.) where a
B, y, d, &c. may denote any conſtant quantities whatever,
poſitive, or negative.--Other inſtances of the uſe of exponential
quantities, and of the meaſures of angles and ratios, in the
reſolution of fluxionary equations of different kinds, might be
given; but I ſhall conclude here, with obſerving, that a, in
this laſt ſolution, may denote any quantity wherein both
y
and z enter, as well as one in which is alone concerned in-
dependent of y.
국
​}
A
O
Α Ν
okortgetectores store cotonetietouto
Recente reaction to start to start stockisto
Focal
An INVESTIGATION of a GENERAL RUL e for
the Reſolution of Iſoperimetrical Problems of all
Orders.
L E M M A.
UP POSING a, b, y, d, e, &c. to be a ſeries of in-
determinate quantities,
S
and given
in
la
i &c.
R, S, T
R', S, T
B
R", S", T"
and that
are any quanti- y
R", S", T" ties compoſed of quantities;
2", R"', s'", Tun
&c.
It is propoſed to find an equation for the relation of a, b, y, d,
&c. ſo that the quantity 2+2+2"+2" + qilm &c. ſhall be
a maximum or minimum, at the ſame time that the other
quan-
tities R+R+R" + R" + R" &c. S+S' + S + S" + s'"
+ s' &c. and T+T' + I" + T"" + T'" &c. are all of
them given, or ſuppoſed to remain invariable.
Let 2, R, S, T denote any correſponding terms of the ſe-
ries's 2+2+2"+2"+2" &c. R+R + R" + R"
+ R"" &c. S + S + S” + $"' + '" &c. T+T' + I"
+T" +T" &c. reſpectively, expreſſed in terms of u, any
one of the propoſed quantities a, b, y, d, e &c. moreover let the
fuxion of Q.la being variable) be denoted by qà; that of R by
ră; that of Q' by GB; that R' by vß, &c. &c.
It is evident that the quantity 2+%+2"+2" +2."
&c. cannot be a maximum or minimam, when RR+R" +
R" + R" &c. S+S + S" + $"" + s'" &c. and T + 1
+ I" +T" + T'' &c. are given quantities, unleſs the part
2+2+2+ " is a maximum or minimum, when the
parts
:
;
An Inveſtigation of a general Rule, &c.
99
q
O
-O
O
parts R +RER + R", S+S+s' + s", and T + T
+ I + I" are given quantities ; becauſe the terms in theſe
parts may be alone made variable, while the other terms
are ſuppoſed to remain the ſame, whereby the whole ſums,
R+R+R" + R" + R" &c. S + $' +S" + " t shi
&c. will remain the ſame, and the quantity 2+2+2
+2" + " &c. will be a maximum or minimum, when the
part + 2+ % + 2" is ſo. But when 2+2+
+ 2" is a maximum or minimum, and R+R+R + R",
5 + 5 + 5 + s", and T+T + T'+I" are given (or
conſtant) quantities, their fluxions will be, all of them, equal
to nothing; whence we have theſe equations,
que t që tas to's
ri tra trêt ru
sit to see +58 - s'n
tir ttà + t'B + t"
In order now to exterminate the fluxions i, è, , j, let
theſe equations be reſpectively multiplied by 1, e, f, g, (yet
unknown) and let all the products thence ariſing be added
together; whence will be had gt er tfs + gtxut
ater+fs+gtxä ta' ter' tfö' +8t'x8+"ter" +-f5"+gt" xje
Make, now, 9. ter +fs + gt = 0,
ster' + fs' + gt'
9"+er" tfs" to gt"
From whence, there being as many equations as quantities,
e, f, g, to be determined, the values of thoſe quantities will
always be given from thence, in terms of the quantities q, r, s, t,
6, r, s, t, q", 1", s", ť" (excluſive of q, r, s, t,). Now, ſee-
ing all the terms of the equation 9 tert.fs togexit
ater+fs+gtxå ta' ter'+fś+sť' xß++er" +f;" +gt™xģ
o, after the firſt (@ter tfst gtxu) do thus actually
vaniſh (by their coefficients being taken equal to nothing), it is
evident, therefore, that a ter+f5+ gťmuſt alſo be =0
(or flux. 2te flux. R + fx flux. S+gx flux. T=0); where
0.
0,
O,
02
By
***.""** *********YWITTEET
HS,"
B
IOO
An Inveſtigation of a general Rule
e, f, g being quantities depending intirely upon q, r, s, t, &c.
(excluſive of q, r, s, t), they muſt neceſſarily be invaria-
ble, or continue of the ſame value, let q, r, s, t, ſtand for
which terms you will of the correſponding ſeries's
, "ta"&c.
p!' &c. becauſe the quantities q, r, s, t; 5,5, 5, 7, 9"
go", s", t", (on which e, f, g, depend) have themſelves a deter-
minate value each, in the required circumſtance, when
2+2+29&c. is a maximum, or minimum.
gi
رک
PROPOSITION.
Sufpoſing y and u to be two flowing quantities, and that Q
R, S, T &c. are quantities expreſſed in terms of y, u, and
given coefficients ; 'tis propoſed to find an equation, expreſſing the
relation of y and u (or of , R, S, T, &c.) ſo that the flu-
ent of Qj, correſponding to a given value of y, Mall be a
maximum or minimum, and the fluents of Ry, Sý, Tj, 30.
all of them, at the ſame time, equal to given quantities.
Let Q, Q', 2", 2", &c. be the different values of Q, that
will ariſe, when y is, ſucceſſively, expounded by the terms of a
given arithmetical progreſſion whoſe common difference is the
indefinitely ſmall quantity ý (a, b, y, 8; &c. denoting the re-
fpective values of u), and let R, R', R", R", &c. be
the correſponding values of Ř, &c. &c. Then it is well
known that the ſum of all the quantities Quý + Q'y to Q'y'
+"y"+""y'&c. will be=fluent of ; and the fum of all
the quantities ky' + Ryt-R"y"+R"y' + R'''y' &c.= fluent of
Rý, &c. But, by the Lemma, it appears, that .Q+&+?.
+2+ &c. or Qy' + Qý + 2"1" + " +- &c. (becauſe
j' is conſtant) will be a maximum or minimum, and the
quan-
tities Ry' + Ř'y' +- R"y' + R"'y', &c. Sy' + S'y' + S'y' + 'S"ly
&c. at the ſame time equal to given ones, when the relation
of y and u (or of Z, R, S, T, &c.) is expreſſed by the
equa-
tion, flux. &texflux. R+fx flux.S+gxflux.T=o: where e,
f, g, &c. denote (unknown) conſtant quantities; and where,
in taking the fluxions of Q, R, S, T, &c. the quantity u is
alone,
for the Reſolution of Iſoperimetrical Problems. IOI.
*
alone, to be conſidered as variable; becauſe the ſucceſſive va-
lues of y, entering reſpectively into Q, V, 2", 2", &c. are
conſtant quantities, being (by hypotheſis) ſuch as ſucceſſively
ariſe from the terms of a given arithmetical progreſſion. Hence
we have the following
GENERAL RULE
for the Reſolution of Iſoperimetrical Problems of all orders.
Take the fluxions of all the propoſed expreſions (as well that re-
Specting the maximum or minimum, as of the others whoſe fluents
are to be given quantities, making that quantity, and likewiſe its
'fluxion, invariable, whereof the fluxion (as well as the quantity
itſelf) enters into the ſaid expreſſions; and, having divided every-
where by the fluxion of the other quantity made variable, let the
quantities hence ariſing, joined to general coefficients 1, e, f, g, &c.
be united into one Jum, and the whole be made equal to nothing :
from which equation (wherein the values of e, f, g, &c. may be ei-
ther poſitive, or negative, or nothing, as the caſe requires), the re-
quired relation of the two variable quantities will be truly exhibited.
To illuſtrate the uſe of the rule here laid down by an example, Fig. 22.
let x and
y be ſuppoſed to repreſent the ordinate (PQ) and ab-
ſciffa (AP) of a curve ADQE; and ſuppoſe AFRG to be an-
other curve, having the fame abſciſſa, whoſe ordinate PR is,
every-where, axmyon ; 'tis required to find the relation of x
and y, ſo that the area BFGC, anſwering to a given value of BC,
ſhall be a maximum or minimum, at the ſame time that the cor-
· reſponding area BDEC is equal to a given quantity. Here, by
hypotheſis, the fluent of ax"y"ý is to be a maximum or minimum,
and that of xy equal to a given quantity: taking, therefore, the
Auxions of both expreſſions, &c. (inaking x alone variable,
according to the rule), we thence get maxm-'yj-ej =0:
ha; ax"
and conſequently axmyn (=PR)
Therefore, ſeeing PR is in a conſtant ratio to P& it is
evident, that both the curves will be of the ſame kind; and
that they will be both byperbolas, or both parabolas, according
as the values of the exponents m-1, and n (in the general equa-
tion
whence xm-004
11
102
An Inveſtigation of a general Rule
ma
yx3
yy
yy
tion to them guna =) are like, or unlike, with regard to poſitive
and negative. If m-- I be poſitive, the equation gives a mini-
mum; if negative, a maximum; but when m-Io, or when
m = 0, the equation fails; in which caſes there will be neither
a maximum, nor a minimum.
For another example, let the fluxions given be and X; the
fluent of the former (anſwering to a given value of y) being to
be a minimum, and that of the latter, at the ſame time, equal
to a given quantity. Here (3 being concerned independently,
either, of its fluent x or fluxion ä) let the fluxions of both ex-
preſſions be taken, making * alone variable; whence, after
3yäx
Зухх
dividing by š, we have and 1: therefore, in this caſe
te=o: whence å=al y- j (ſuppoſing a= e); and con-
ſequently x=2ałył ; being an equation anſwering to the com-
mon parabola. The ſame concluſion may be otherwiſe derived
(without ſecond-fluxions) by aſſuming =V; whereby our
two given expreſſions will be transformed to yjo' and jv: from
whence, by the rule, we get 3vịyj + ej=0; and therefore v
al yet; whence a ały-ij, and conſequently x =
za{y, the ſame as before.
Fig. 23. If the abſciſa (AP) of a curve AQC be denoted by x, and
the ordinate PQ. by y, and p be taken to expreſs the meaſure
of the circumference of a circle whoſe diameter is unity; it is
well known that the ſeveral fluxions, of the abſciſla AP, curve-
line AQ, area APQ, ſuperficies of the generated ſolid (by a ro-
tation about the axis AP), and of the ſolid itſelf, will be, re-
ſpectively, repreſented by x, Väx+jy, yx, 2py Vää+jj, and
py's: if therefore, the fluxions of theſe different expreſſions be
taken, as before (making : alone variable) we ſhall get 1 +
+fy + + hy’ = 0; being a general equa-
√xxtiy
xx+jj
tion for determining the relation of x and y, when any one
of thoſe five quantities (viz. the abſciſſa, curve-line, area,
ſuperficies; or "folid ) is a maximum or minimum, and all,
*
j
( )= y;
ež
or
į
for the Reſolution of Ifoperimetrical Problems.
IO3
f
ex
Vaa - y
V **+ jö
or any number of the others, at the ſame time, equal to given
quantities ; wherein the coefficients e, f, g, and b, may be
po-
fitive, negative, or nothing, as the caſe propoſed may require.
Thus, for inſtance, if the length of the curve only be given,
and the area correſponding is required to be a maximum, our
equation will then become
+fy = SO, or a mi?
Väst jä
yj
y* x *x + jj (by making ş); whence ==
and conſequently x=a-V aa-yy, or 2ax - x=y'; an-
fwering to a circle ; which figure is, therefore, more capacious
than
any other under equal bounds.
If, together with the ordinate (which, here, is always ſup-
poſed given) the abſciſſa, at the end of the fluent, be given
likewiſe, and the ſuperficies generated by the rotation of the
curve about its axis be a minimum; then, from the ſame
equa-
gyx
tion, we ſhall have it = 0, whence (making a =
ау
is found =
and from thence x = ax hyp.
vyy-aa
log.
i which equation, by being impoſſible when
is leſs than a, ſhews that the curve (which is here the cate-
naria) cannot poſſibly meet the axis about which the ſolid
is generated; and conſequently, that the caſe will not admit of
any minimum, unleſs the firſt, or leaſt given value of y, exceeds
a certain affignable magnitude.
When any, or all of the above-ſpecified quantities are
given, and the contemporary fluent of ſome other expreſſion, as
2x +jj)" xymi luz, is required to be a maximum or minimum, ,
our equation (by taking the fluxion of this laſt expreſſion, and
joining it to the former) will then be x3 + jj)*-? x 2nxiy"js-
+ fy + + by = 0: which, when
Vice + jj
m=1, and n=-1; will be that defining the ſolid of the
leaft reſiſtance; and this, when the length of the axis, only, is
fuppoſed to be given (without farther reſtrictions) will be ex-
preſſed
I
ytv yy-maa
a
-28
eë
gyi
+d+vää
tij
104
An Inveſtigation of a general Rule
O; but
O.
I
2
you
ex
we have
+ fyt Tixt
V
xx + y
V Xx + yg
tyy
Växtuj
preſſed by **+-jy!" x 2x+yj' + d=0, or 2yj?k=dx
iš + jjl ; being the caſe firſt conſidered by Sir Isaac New-
TON.-If both the length and the ſolid content be given, the
equation will be - 2xyj' x *x + jj!-- + d + hy
)= 0;
if, beſides theſe, the ſuperficies is given likewiſe, it will then
be- 2xyypx ** +jj-+++ Time to s
gyä
thy
Thus, in like manner, by aſſuming m=--
5, and n=
+d+
+ hy* = 0;
being the general equation of the curve of the ſwifteſt deſcent ;
which, when e, f, g, and h are all of them taken equal to
juli
nothing, will become +d; which is the caſe confi-
dered by many Others, anſwering to the cycloid. When the
length of the arch deſcribed in the whole deſcent (along with
the values of x and y) is given, the equation will then be
tega
= 0, or e ty-Hi*xx'=dºxxx +jj.
√xx tij
✓ xxt jy
And thus may the relation of x and y be determined in any
other caſe, and under any number of reſtrictions; provided
that one of theſe quantities, only, enters into the ſeveral
expreſſions given.-When both x and y are concerned, as well
as their fluxions, the conſideration becomes more compli-
cated; nor does it ſeem practicable to arrive at a General
Rule, to anſwer equally in ſuch caſes. Nevertheleſs, if the
ultimate values of x and y are ſuppoſed given, or the required
curve is to paſs through two given points, without being con-
fined to farther limitations, except that of the maximum or
minimum (which caſe is the chief, and the moſt uſeful that
can occur); then the method of ſolution may be as follows:
Take the fluxion of the given expreſſion (whoſe fluent is to be a
maximum or minimum) making s alone variable; and, having di-
vided by ä, let the quotient be denoted by u.
Take, again, the fluxion of the ſame expreſſion, making x, alone,
variable; which divide by *: then will this laſt quotient
From
} у
es
+d++
u.
for the Reſolution of Iloperimetrical Problems.
10:5
From which equation the value of u, and the relation of x
and y will be determined.
.: ", " hii , *: 1'7','
yx?
;
j
.
>
yy
yy
= U, and gyis
it
;
u
f+gx
4 U
3yä?
yy
Thus, for example, if the expreſſion propounded (whoſe
fluent, correſponding to any given values of x and y, is to be a
minimum) were to be f + gxx
then the fluxion thereof,
when x alone is made variable, being f+gxx
3yxkä
and, when
x alone is made variable, equal to syt* we here havef+gxx.3842
=i; the latter of which, divided by the former,
gives
whence hyp. log.u=hyp. log. F +381 +
hyp. log. d (d being any conſtant quantity). Conſequently
dxf+gx14; which value being ſubſtituted in the equati-
on f + gx x U, we thence have f + g******
dy-j", or f +8x1* x* = cy-bj (making c = and
conſequently by taking the fluent again, we have 3xf+8x13–35*
2cy}; expreſſing the general relation of x and y, ſuppoſing
them both to begin to be generated together. If f= 1 and
g=0, the fluxion propounded will become yx" (the ſame as
in the firſt of the former examples);- and here, á being
cy-lj, x will be 2cy, anſwering (as before) to the com-
mon parabola.-But if f =o and g = 1, then our given
fluxion will become xy:, and the reſulting equation will be
of
); which alſo
anſwers to a parabola, but of an higher order.-The very
fame concluſions will, in like manner, be brought out by
making j and y, ſucceſſively, variable (inſtead of s' and x).
For, here, the two fluxions reſulting (after having divided by
P
and
48
1
yy
j
!
106
An Inveſtigation of a general Rule, &c.
13
;
u
y
and j) appear to be f +8* *— 2443
= 1, and (+3^**? =u:
whence, dividing the latter by the former, we have =
and therefore hyp. log. ył + hyp. log. a = hyp. log. u
(a being any conſtant quantity). Conſequently, ay-t=ů=
ft gx x - 2y3:3
and f+ gx* * E cy-j (c' being put
-a). Hence, by taking the fluent again, we have
3xf+gali - 3fi
=
2cy}, the very ſame as before.
>
48
+
!
Of
At
Of the Reduction of Algebraic Equations, by the
Method of Surd Divisors; containing an Ex-
planation of the Grounds of that Method, as it
is laid down by Sir Isaac Newton in his Uni-
verſal Arithmetick.
T
3
*HE reduction of equations by furd diviſors, which is
looked upon, by many, as a very intricate kind of
ſpeculation, is founded on the ſame principles with
the method of extracting the roots of common qua-
dratic equations, by compleating of the ſquare, with this diffe-
rence only, that the ſquares on both ſides of the equation are,
bere, affected by the unknown quantity; whereas, in the com-
mon method, the ſquare on the right-hand ſide is a quantity
intirely known. What we, therefore, have to do, is, to
Separate, and ſo order the terms of the equation given, that both
fides thereof may (if poſſible) be complete ſquares.
Case I. If the given equation be a biquadratic one, let it be
** + px' + 9x2 + rx +'s=o, and let there be aſſumed
xx+1px+2) - Ax+Bi =x* +px' +qxº +rx+s(=0);
that is, let the values of the quantities 2, A and B be ſuppoſed
ſuch, that the coefficients of the powers of x, when xx+ px+2.
and Ax + B are ſquared, ſhall agree, or be the ſame, in every
term, with thoſe of the equation given. Then, the ſaid quan-
tities being actually ſquared, our equation will become
** + px' + 22**
tip*x* + pQx +23 =x*+px?+qx2 +rxts.
- A Ax? - 2 ABx - B
From whence, by equating the coefficients of the homologous
powers, and putting a =9
App, we have,
1. 22+ ip - A = 9, or, 22= A t«;
2. pQ-
or, p2= 2 AB +r;
3. - B* = S, or, 2= B' +'s.
Now,
**
2
ಸು
2 AB
1,
ری
P 2
108
Of the Redu&tion of Algebraic Equations,
ap, and
2
Now, if the value of Q, as given by the firſt equation, be
ſubſtituted in the other two, we ſhall get PA 2 AB = B,
and A + A B’ = %, ſuppoſing B=r-
S
='s-. In which equations the unknown quantities
appertaining to the latter of the two aſſumed ſquares are only
concerned, and from which their values might be found. But
as the reſulting equation, when one of the quantities is exter-
minated, riſes to the ſixth dimenſion, and would, perhaps, re-
quire more trouble to reduce it than, even, the original one
propounded, little advantage would be reaped therefrom. In-
ſtead, therefore, of proceeding farther in a direct manner, it
may be of uſe to try, whether ſome property, or relation of
theſe quantities cannot from hence be diſcovered, whereby we
may be enabled to gueſs at their values; which may be after-
wards tried by means of the equations here exhibited.
Firſt, then, it is evident, if both A and B are either integers
or rational quantities, that the equation ** +1px + 0
Ax+B1 (= ** + px' +9x2 +rx) =0 will, even after it is
reduced to x* + px + = Ax + B, be intirely free from
radical quantities. In which caſe, the method of rational di-
viſors taking place, a reduction by means of ſurd quantities, or
diviſors, as they do not naturally ariſe in the conſideration,
cannot be of uſe. But the relation of the given quantities
p, q, r, s (which we ſhall always, hereafter, conſider as inte-
gers) may be ſuch, that the values of A and B ſhall be radical
quantities, commenſurate to each other ; in which caſe, where
the method of rational diviſors fails, we may aſſume vn for
the common radical diviſor, and expreſs the quantities them-
ſelves by k/ n, and I'm; that is, we may make A=kv'n,
and B=Wn; by which means our two equations, derived
above, will be changed to 'pk’n — 2kln = B, and k*n +
ak*n — In = {, or to pk’ — akl and k*n tak
21=, reſpectively.
Now, ſince n is ſuppoſed to be an integer, it is plain from
hence (conſidering k and I alſo as integers, or the halves of
fuch
B
2
n
by the Method of Surd Diviſors.
Iog
n
n
fuch) that and 25 muſt be integers likewiſe, or, at leaſt, the
halves of integers ; and conſequently that n (whoſe value we
are here ſeeking) ought to be ſome common integral diviſor
of B and 21
Moreover, with regard to k and l, it is evident from the
firſt of thoſe equations (pk — 2l = that the former (k)
21 =
=)
B
ought to be ſome diviſor of
e
n
; and that, if the quotient be
-1)
2
at nk²
>
2
N
taken from 7pk, the remainder (pk
will be the dou-
ble of 1.
It farther appears, from the equations Q=4*+a, and Q=
B' + s, by ſubſtituting for A and B’ their equals nka and nl”,
that 2 will be =
and l?
! =
QQ-
From the former
of which 2. will be known, when n and k are known; by
means whereof and the other equation, I may be, a ſecond
time, found ; and the agreement, or coincidence of this value
with that before determined for l, will prove the ſolution in
all reſpects ; becauſe then the conditions of three original
equations (2Q = A' + a, p= 2AB +r, O=B’ 7 s)
will be all compleatly fulfilled. It is true indeed, that no
immediate regard, in the concluſion, is had to the ſecond of
thoſe equations; but then it ought to be obſerved, that the
equation 1pk
k
whereby 1 is, the firſt time, found, is a
conſequence thereof, being derived from that, and the firſt
equation, conjunctly: and it is known, that, whatever values
are diſcovered for unknown quantities, by means of equati-
ons derived from others, ſuch values do equally anſwer the
conditions of the original equations propounded.
Seeing the method of ſolution, above traced out, depends
upon the affuming proper diviſors of B, 25, and, for the
values
B
II
Of the Reduction of Algebraic Equations,
at nk²
22
}
2
values of n and k, it may therefore be expedient, firſt of all,
in order to bring the work into leſs compaſs, to reject ſuch
diviſors of thoſe quantities (if we can by any means diſcover
them) which we know are not for our purpoſe. And this
inay,
in ſome meaſure, be effected, from the confideration of
the properties of even and odd numbers.
In order to which Q= being previouſly trans-
formed to nk* (= 20-a) = 2Q-9+p=pl-f
(by putting a 2Q=f), it is evident, from thence, that if
p be an odd number, pa — 4f, and conſequently its equal 4nk”,
will likewiſe be an odd number ; becauſe an even number (48)
ſubtracted from the ſquare of an odd one, always, leaves cdd.
Therefore, ſeeing 4k+ x n is here an odd number, both n and
4k' muſt be odd (for the product of two even numbers, or of
an odd one and an even one, is even, and not odd). Whence
it follows, becauſe 2k) is odd, that 2k muſt be odd too; and
conſequently k the half of an odd number.
Now, ſeeing P, n, and 2k are all of them odd numbers
(when p is ſuch) they may, therefore, be expreſſed by 2a +1,
26+1, and 2c + 1, reſpectively; a, b, and c being integers :
in conſequence of which affumption the equation 4nk” = p?
4f, will, by ſubſtitution, be changed to 8bc* + 8bc + 2b
+ 40 + 46+1= 4a+ 4a + 1 - 4f, or 2bc2 + 2bc +
b + c +c= ata-f. From whence it is mani-
feſt, as all the terms, but b, are known to be integers, that.
b muſt be an integer likewiſe: and ſo, b being an even num-
ber, it follows that 11, or 2b -+ 1, muſt be the double of an
even number (or a multiple of 4) increaſed by unity. There-
fore all the diviſors of 8 and 23 that have not this property
may be ſafely rejected, as not for the purpoſe.
In like manner, if p be even, the ſame limitations will take
place, provided that r is odd; which will be the caſe when Q
is the half of an odd number (For, when Q_is an integer,
A=pl - f) and B'(=Q
- s) being integers, their
product A'B' will be an integer, and conſequently the ſquare
Foot thereof AB (being rational) will likewiſe be an integer ;
and
Sorter
w
by the Method of Surd Diviſors.
III
and fo, pQ_and 2AB being both even numbers, their diffe-
rence r, as given by the equation p= 2AB + r, would be
even, and not odd). Therefore, ſeeing Bº, or its equal nl”,
is here equal to the ſquare of half of an odd number (Q)
joined to an integer (-s), in the ſame manner as nk" was in
the preceding cale; it is evident, from the reaſoning there laid
down, that the value of n is ſubject to the very fame reſtricti-
ons here, as there.-Other limitations might be pointed out,
from the properties of even and odd numbers, were the thing
worth purſuing farther. What is already delivered on this
head is ſufficient for the purpoſe, and for the underſtanding
of Sir Isaac Newton: I ſhall therefore, from the ſeveral
concluſions above derived, now lay down the ſubſequent
R U L E
for the reduction of an equation (x* + px' +9x2 +rx+s=o)
of four dimenſions.
Make a=9- p", B=r ap, and {=s so-
put for n ſome common integral diviſor of B and 28, that is neither
a ſquare, nor diviſible by a ſquare, and which being divided by 4,
ſhall leave unity, if either p or r be odd. Put alſo for k fomé di-
viſor of, if p be even, or half of the odd diviſor if p be odd :
take the quotient from pk, and call half the remainder I. Make
Q=
try if n divides 22- s, and the root of the
quotient be equal to l; if it ſo happen, then the propoſed equa-
tion, by means of the values thus determined, will be reduced to
xx + px +2=+V nx kx +1.
That the diviſor n ought not here to be a ſquare, is evident
from what has been already remarked, ſince both A and B
would then be rational quantities ; and that it ought not
to be diviſible by a ſquare, will alſo appear, if it be conſidered
that k and I in the equations ky n = A, and I n=B, are
to be taken the greateſt, and n the leaſt, that the caſe will ad-
mit of.
1
ao; then
4
8
n
at nkk
and
2
No
7
II 2
Of the Reſolution of Algebraic Equations,
1
αα
No regard in this Rule is had to that circumſtance, in which
B happens to be nothing. Sir Isaac Newton here directs,
to take k alſo equal to nothing. The reaſon of which depends
on the equation pk akl = , which in this caſe becomes
pk --- 2kl = 0; where one root, or value of k muſt, necef-
farily, be nothing. Therefore Q being a, we have
Wn(=Ve- s) Va — S; ſo that, by a direct
pro-
ceſs, our given equation is here reduced to x2 + px ta
s, wherein a is given =9— *pp.
The celebrated mathematician MACLAURIN, who, in his
Treatiſe of Algebra, has commented largely on the diſcoveries
of our Author, ſeems to repreſent this part of the General Rule,
as not well grounded; laying down, at the ſame time, two
new Rules, in order to ſupply the defect. Which Rules, I
muſt confeſs, to me appear unneceſſary; ſince it is certain, that
the method of ſolution, as laid down by Sir Isaac NEWTON,
is more direct and eligible in this particular caſe than in any
other.
It muſt be allowed, indeed, that the manner of applying the
Rule, in this caſe, is left fomewhat obſcure ; but as to his di-
recting, to take k=0, when B=0, it cannot, I am fully
perſuaded, admit of any well-grounded objection. For, though
“ it does not neceſſarily follow that k muſt be = 0, when
B=0," yet the taking of k thus o, involves no abſurdity;
ſeeing one value of k (at leaſt) will be nothing. The truth is,
there are three different values that k may admit of (as ap-
pears by the ſubſequent note *); all of which will, equally,
fulfil the ſeveral conditions required, and bring out the very
ſame concluſion. Thus the value of k, in the equation
* If the ſquare of half the ſecond of the original equations, 2Q-a= AA,
PQ-r= =2AB, QQ-s=BB, be ſubtracted from the product of the other
two, there will be obtained the equation Q - R + pr-sxQ+
2 2 X
xas - Ir=0; wherein the unknown quantity Q is alone concerned ;
which equation being of three dimenſions, the root Q, and conſequently k
Q-
-
appears, that Q muſt always be a diviſor of the quantity as - rr; which is
a circumſtance taken notice of by our Author.
X4
(22
72
by the Method of Surd Diviſors.
113
}
7
2
22
** +2x' - 37x*— 38x+1=o (propoſed by this gentleman)
may be o, 3 or 4; or, which comes to the fame, the equation itſelf
may be reduced to ** ** - 19=+6V 10, *2 + x +
= IV 5* 3x +}, or to x* +*—3=+V 2x4x + 2.
All which are, in effect, but one and the ſame equation, as
will appear by ſquaring both ſides of each, and properly tranf-
poſing; from whence the given equation ** + 2x - 37**-
38x + i=0, will in every caſe emerge. The ſecond of
theſe equations is that brought out by Mr. MACLAURIN; but
the firſt, which is that found by our Author's Rule, is not
only more commodious, but eaſier to be determined, being
derived by a direct, and very ſhort proceſs. — And ſo much for
equations of four dimenſions.
CASE. II. If the equation to be reduced is of ſix dimenſions, let
it be 46 + px' + ax* + rx' + 3x2 + tx+v=0; and let
there be aſſumed x + xpx+0x + R) Ax? + Bx +C"
46 +px' +9x* trx tsx* + tx tol=0); which, by in-
volution and tranſpoſition, will give
+2Qx+ + 2Rx
+- p*x* +pQx' +pRx"
A*x* + 2ABx' +
to'x' +2QRx+R'
2 ACx2 +B’x2 +
-qx+ -rx?
2BCx+c.
From whence, by equating the coefficients of the homologous
powers, and writing a=9- PP, we have,
1. 2Q- a= A';
2. 2R + PQ-1= 2AB ;
3. pR + Q - ś= 2AC + B’;
t = 2BC;
5. R? —V= C.
If now the value of Q(=A+) as given by the firſt
of theſe equations, be ſubſtituted for Q, in the ſecond, we
ſhall get 2R +pA' + pa-r=
r=2AB; and conſequently
R=AB - PA +iB (by putting
po) : which
value, together with that of Q, being ſubſtituted in the three
remaining equations, we ſhall have,
Q
3
Q+
sx²
tx
. V
4. 2QR
r
I.
I14
Of the Reduction of Algebraic Equations,
1.PAB -*p*A++B+4A*+&A’+-s=2AC+B”,
2. A'B - ipAt +BA+AB paa+al-t=2BC,
3. A'B'- PA'B+BAB TopA - PBA+iB-v=C':
which, by putting y=s - PB, ?=y, n=t-1«ß,
and @ ABB, will be reduced to
PAB — *pA+ A+ + «A?
+ A + QA’ - S=2AC + B’,
A'B - PA* +¿BA’+ aAB paA-9=2BC, and
A'B'- PA'B + BAB + +p*A*- PRAP-0=C, reſpec-
tively.
Now, if the values of A, B and C are ſuch, as to admit of
ſome common ſurd-diviſor, let that diviſor (as in the preceding
Caſe) be denoted by vn, and the quantities themſelves by
kV ñ, W n, and mv n, reſpectively: then, ſubſtitution being
made and every equation divided by n, we ſhall have,
I. pkl - p*ktink*-+ ak" } = 2km + %,
2. nk'l — pnk* tilk-+ akl - pak — = 2lm,
3. nkºl— pnk'l+ Bkl + rapºnk* – plk— .
From whence it appears (ſince k, l, and m are here conſidered
as integers, or as the halves of ſuch) that 2, , and 0 ought to be
all of them diviſible by n; or, which is the ſame, that n ought
to be ſome common integral diviſor of the quantities šo no
and A*
Furthermore, with reſpect to the limitations to which k is
ſubject, let the ſeveral terms in the former part of the firſt of
our three equations, in which k is found in order to abbrevi-
ate the work) be denoted by Fk; then will the equation itſelf
be changed to Fk 2km + 1. And, in the very
fame
0
m.
n
§
* Sir Isaac Newton directs to take n, fome common diviſor of 25, 4,
and 20 (inſtead of S, M, and 6); but this makes no difference, becauſe all
diviſors of & and are alſo diviſors of 28 and 20; nor are there any divifors of
28 and 24, but what will likewiſe be diviſors of & and 0,. if we (as Sir ISAAC
has done) admit into the confideration fuch fractions as have the powers 2 for
their denominator ; which ariſe from the value of p, in the aſſumed equation,
being a fraction of this kind, when p is an odd number.
manner,
A
115
by the Method of Surd Diviſors.
n
Fko
Hks
80
Gkn
2km ;
n
n
2n
nn
2nn
a
2nn
manner, our other two equations will be changed to Gk
= 2lm, and Hk - = m*, reſpectively.
Let now the ſquare of half the ſecond of theſe equations be
ſubtracted from the product of the firſt and third, then will
FHk?
+
2G*k* +
***
which, by dividing the whole by 2k, and putting a=3-**,
will, at length, become
FHk — G*k – 1Hx+1Gx-Fx+=m":
where ?, n, and being all diviſible by their common diviſor
n (as is ſhewn above) it is manifeſt that in order that m
may be an integer, or the half of an integer) ought alſo to be
diviſible by its diviſor k, that is, k ought to be ſome diviſor of
the quantity
Again, with regard to I, let the value of R (= AB - - PQ
+ r), as given by the ſecond of the five original equations,
be ſubſtituted in the fourth; by which means we have 2QAB
potrQ_t=2BC, or ¿Qnkl — potrQ-t=2nlm
(becauſe A = k/n, B=Wn, C=m/n), and conſe-
rQ-pQ² -t
quently
= 2m --- 2kQ; where 2m -- 2kQ_being an
integer, it is evident that e-pill- muſt be an integer alſo,
rQ-PQ-t
and, conſequently, I fome diviſor of the quantity
From whence l being found in numbers, the value of R(=AB
iPQ+r=nk] PQ+r) will be had likewiſe ; and
then, by means of the three laſt of the five original equations,
the value of m may be alſo found, three ſeveral ways, and
the truth of the ſolution thereby confirmed : for theſe equati-
ons, by ſubſtituting for A, B; and C, their equals kyn, n,
and my n, do become R’ V == nm", 2QR
”-t
2nlm,
and pR +0-5= 2nkm + nll; from the firſt of which
Q_2
nd
nl
n
M
*
116
Of the Reduktion of Algebraic Equations,
RPU
& 전
​m
;
n
nl
2nk
✓
from the fecond, m = QR- *'; and from
the third, m=
QQ to PR - nll
— nll - $; which values, therefore,
when Q, R, &c. are rightly aſſumed, will be all found equal
among themſelves; and our given equation, x6 + px' + q**
+ rx' + sx* + tx+0=0, or x + 1px* + Qx. + RI*
Ax? + Bx+01 = 0, will then be reduced to x + px' +
Ox+R=+ Ax+ Bx +C) =+vnxkx* +- lx + m.
As to the limitations in the diviſors to be tried, with reſpect
to even and odd numbers, the reaſoning thereon is the very
ſame as in the preceding Caſe; which, therefore, it will be
unneceſſary to repeat.-One circumſtance there is, indeed, that
merits a particular regard, and that is, when a=0; in which
caſe k (or one value of k at leaſt) will alſo be = 0, and the
reduction will be performed by a direct proceſs. For, k being
nothing, the three equations wherein k, l, and m are firſt in-
troduced, will become — =l, -=2lm, and — =mº;
whence W n=v, mon=V-6, and conſequently a (=
Cô - x) =
19*) = 0, as it ought to be. Therefore, by ſubſtitut-
ing theſe values, and writing alſo inſtead of Q and R their
equals a and B, the equation given, is here reduced to
x' + px* + ax +iB=+*V—5V= 0.
1
0:
1
CASE III. If the equation is of eight dimenfions, let it be
48 + px? + qx6 + rx + 5** + tx' t vx' + wx + %
Then, by aſſuming ** + px' + 0x2 + Rx + S
Ax + Bx2 + 2x +D = 48 + px? + 9x6 + rx' + sait
+ tx' + vx' + wx + 2(= o) and proceeding as in the
two former Caſes, we ſhall have here,
]. 2Q-a.
A',
2. 2R + por
3. 2S + PR + QR-s=2AC + BB,
4. ps + 2QR - 7
2AD + 2BC,
5. 2QS + RR-V = 2BD + CC,
6.
2AB,
t.
V.
by the Method of Surd Diviſors.
117
7. SS
6. 2RS
2CD,
DD.
Put now (as before) A = kvn, B=ln, C = myn,
D n'ſ n; put alſo (to ſhorten the work) R= O'n tja,
R= R'n + B, S = S'n tir; that is, let the quotients of
Q, R, and 'S, when divided by the common diviſor n, be Qi
R', and S, and the remainders-, , and -->, reſpectively: then,
to determine theſe laſt, which muſt be firſt known, before n
can be known, let ſubſtitution be made in the ſecond and third
equations, every-where diſregarding ſuch terms wherein n and
its powers are involved. Thus, ſubſtitution being made in the
ſecond equation, we have 2kn +B+POʻn + por=
r=2kln;
where the homologous terms, in which n enters not, are B,
ipa, and — r: the others, therefore, being here diſregarded,
we have B tipo r = 0, or B
po. In the very
fame manner, from the third equation, yt pltiacS=0,
and conſequently y=s.- PB
- PB - 1a.
Let ſubſtitution be now made in the fourth, fifth, fixth, and
ſeventh equations (ſtill diſregarding all ſuch terms as would
involve the powers of n), and there will come out,
I. ipy + al
2. ar + ABB
r T
t
3. By
2
2,
4. 477
Now, as all the other terms, that would ariſe in theſe equati-
ons (beſides thoſe put down) are affected with n, and are there
fore diviſible thereby, it is manifeſt that the four quantities
ipy taß-t, ary + ABB-V, By - W, and cry
here brought out, muſt likewiſe be, all of them, diviſible by
the ſame common diviſor n, when the equation given is capa-
ble of being reduced. If, therefore, no ſuch common diviſor
(under the reſtrictions ſpecified in the preceding Caſes, depend-
ing on p, r, t, or w being an odd number) can be diſcovered
(which will moſt commonly happen) the work will then be at
an end.
}
From the ſame method of operation, which may be looked
upon as a ſort of examination, whether the equation be redu-
cible:
118
Of the Reſolution of Algebraic Equations,
&
cible or not, we may find all the quantities to which n ought
to be a common diviſor, when the equation given is of 10, 12,
or a greater number of dimenſions.
Thus, let there be given x2 + pxzemi + 9x21–2 +- rx2-3 +
sx2–4 + tx2cm5&c. = 0, and let there be aſſumed
*+1px-torta x xemz+R'n+1 Bxx-3+Sintiyxxenec
-nxkxes +- lxe+ mx-3&c.) = 42€ + pxzemi +qxae- &c.
then, by ſquaring me to premat O'nt a X x-&c. and
tranſpoſing x2 + pxse-1 + 9x2e? &c. it will appear, that the
terms of this equation, in which n enters not, will be
B
д &
**24-2+ipa ***20-3
1
+i
pa
Xx24-47
-9
al
tay xx.2-6
ABR
a
E
PBT
1/4 pod
>X 322-5
r
a²
¿
t
U
S
&c. From the former half of which terms, all the quantities a,
B, %, &c. will be determined, by aſſuming the coefficients
equal to nothing: thus we have a =9- PP, B=r-
- pa,
g
PB Lac, d=t -paß, &c. And then,
theſe quantities being known, the coefficients of the remaining
terms will likewiſe be known; which ought, all of them, to be
diviſible by n, in order that the reduction may ſucceed ; that
is, they ought to be ſuch, as to admit of a common diviſor (n)
under the reſtrictions before ſpecified.
For example, if the equation given were to be of twelve di-
menſions, as x2 +px" +qx1° +7x9 +s48 +tx? + vxo + axs
+ bx*+cx' + dx* + éx+f=o, we ſhould have a=9- PP,
B pes, yes-PB
-PB-Q, =t-py-B, and
v- po
po ay - iBB; and the coefficients of the other
ſix terms (whereof n ought to be a common diviſor) would be
ipe tad tiBy-a, ae tiba tiryb, Be tyd 6,
Age + 188 d, de é, and we — f.
Theſe operations, for finding of n, as this fort of reduction
is ſeldom poſſible in high equations, will moſt commonly end
the work. If ſuch a value, however, ſhould be found for ny
as to anſwer all the conditions above ſpecified, it is not by pur-
ſuing the ſame method of diviſors, laid down in the reſolution
of
r r
EV
by the Method of Surd Diviſors.
119
of the preceding Caſes, that the values of k, l, &c. can from
thence be determined, without a prodigious deal of trouble.
There are indeed various other means of trying theſe quantities,
by aſſuming ſome of them, and finding the others from thence;
and ſo proceeding on, changing the values continually, till all
the conditions of the ſeveral equations, ariſing from the com-
pariſon of the homologous terms, are fulfilled. But as this is
exceedingly laborious, and ſeeing after all, the uſe of ſo great
reductions (as the ſagacious Author himſelf obſerves) is very
little, there not being, perhaps, one caſe in a thouſand in
which they can ſucceed; I ſhall, therefore, defiſt here.
1
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THE
R Ε S Ο L U Τ Ι Ο Ν
OF
Some GENERAL PROBLEMS in Mechanics, and
PHYSICAL ASTRONOMY.
Fig. 24.
I
desk ck
AP
BP
A
AP2
X
PROBLEM I.
Suppoſe a ſyſtem of bodies A, B, C (connected together) to re-
volve about a center, or axis (P), with a given angular celerity;
it is propoſed to find the momentum (k) which, acting at a given
diſtance QP from the center, ſhall be juſt ſufficient to ſtop, or take
away
the whole motion of the ſyſtem.
F the given angular celerity of the ſyſtem, at any di-
ſtance PG from the axis, be denoted by v, the cele-
rities of the ſeveral bodies A, B, and C will be truly
CP
expreſſed by PGxv, PGxv, and
PG
xv, reſpectively. Hence
(by the property of the Lever) it will be, as PQ is to AP, fo
is (AC xº x A) the momentum of the body A, to
QPX PG
VX A, the momentum, which acting at Q, is a juſt counter-
poiſe to the action of A. And, in the
very
ſame manner, the
momentum, acting at Q, ſufficient to take away the motion of
B, appears to be
Whence it is
manifeſt, that the ſum of all theſe, muſt be the true momen-
tum required; or that k=uX
A XAP? + B x BP + C x CP2
PG X QP
2. E. I.
COROLLARY I.
If the motion of the ſyſtem is that which might be pro-
duced by any given momentums a, b, c (or forces capable of
producing thoſe momentums) acting on the bodies A, B, C, in
directions
BP2
QP xPG XUXB; and ſo on.
The Reſolution of fome General Problems.
I 21
AP
AP
BP
CP
directions perpendicular to AP, BP, and CP; then (by the
property of the lever) the force a x @P acting at Q, having
the ſame effect to turn the ſyſtem about its axis, as the force
a, acting at the diſtance AP, &c. it follows that the force,
which, by acting at Q, is ſufficient to deſtroy the whole moti-
on of the ſyſtem, will here be a x
QP
which being ſubſtituted in the room of k, our general equation,
in the laſt article, will become v=PGx axAP+ bxBP +exCP
AXAP? +BxBP:+CxCpzi
ſhewing the angular celerity at the diſtance PG, produced in
the ſyſtem by the action of the given forces; which celerity is,
therefore, in proportion to the celerity ) that the given force
(or momentum) a is capable of producing in the ſingle body
A, as
ax AP + bx BP + cx CP
to unity.
AxAP? + Bx BP2 +- Cx CP2
op +by
tcx op
響
​3
AxPG
х
COROLLARY II.
If the momentum k be given equal to that of the whole
ſyſtem (A, B, C) in a direction perpendicular to the line PGQ
paſſing through the common center of gravity G; then the
length of the lever (PQ) by which k acts, may be determin-
ed from hence. For, the celerity of the point G being repre-
ſented by v, the momentum laſt named will, by the property
of the center of gravity, be rightly defined by vxA+B+C;
which being ſubſtituted in the room of k, we thence get
AXAP? + B X BP + CXCP
; exhibiting the diſtance of
A + B +CxGP
the center of percuſſion (Q) at which an immovable obſtacle
receives the whole force of the ſtroke.
QP
COROLL ARY III.
If a ſingle body S, equal to the ſum of all the bodies A,
B, C, be ſuppoſed to revolve (independent of the others)
about the ſame center, with the common angular celerity of
R
the
I 22
The Reſolution of ſome General Problems
SP
SP
the ſyſtem, its momentum páxvxS, or pe xvxA+B+C,
will be in proportion to the momentum k (given by the Pro-
AXAP? + B x BP2 + Cx CP2
poſition) as QP x SP to
By mak-
A + B + c
ing theſe two quantities equal to each other, we have SP =
AX APP + B X BP2 +Cx CP2
for the diſtance of the body S from
A + B + CxQP
the axis of motion, when its momentum is equal to the mo-
mentum k, or when equal forces, applied to the ſingle body at
S, and to the ſyſtem at Q, can take away, or produce equal
angular celerities in both, about the common axis of motion P.
COROLLARY IV:
Hence, if the point Q be ſuppoſed to coincide with S, our
AXAPto B x BP? + C x CP2
laſt equation will become SP
A +B+C
Thewing the diſtance of the center of gyration, or the place
of the body S, where the ſame force can take away, or pro-
duce the whole motion of the ſyſtem A, B, C, as can take
away, or produce the motion of the ſingle body S, equal to the
ſum of all the former, and revolving with the ſame angular
celerity.
COROLLARY V.
But if the point Q be taken in the common center of gravity
G of the ſyſtem, and Gg and Ss be drawn perpendicular to
the horizontal line TP; then, the force of gravity by which
the whole ſyſtem is urged in the direction Gg perpendicular to
the horizon, being the ſum of all the weights (A +B+C)
it is plain that the part of it acting in a direction perpendicular
to PS, whereby the motion about the center is accelerated, will
be A+B+Cx Ps. But the force whereby the weight S, in
PG
a direction perpendicular to the fame PS, is accelerated, is
Pg
equal to S x
- A+B+C * (becauſe S = A +B
PG
Ps
+ C, and
PC). Therefore, ſecing the forces acting at S
and
Ps
PS
PS
in Mechanics and Phyſical Aftronomy.
123
1
and Q are here equal, it is evident, from Corol. III. that the
diſtance SP, ſo that the fame angular celerity may be produced
in the ſingle body as in the ſyſtem, will be truly exhibited by
AXAP2 + B x BP+CxCP>
the general equation SP
there
A to B +CxQP
derived; the point S thus determined being the center of oſcil-
lation, and the ſame with the center of percuſſion, found in
Cor. II, having its diſtance from the axis, equal to a third pro-
portional to the diſtance of the center of gravity and that of
gyration, determined in Corol. IV.
Pg
Ps
and S x
>
PG
1
only the
GS
COROLLARY VI.
Hence, alſo, the preſſure on the axis of ſuſpenſion P may
be deduced : for, ſince the angular celerities, produced in
the ſyſtem, and in the ſingle body S, by the equal forces
A+B+CX
are the ſame, it is manifeſt
PG
PS
that the abſolute celerity produced in G, during any given
time, will be but the of that produced in S; ſo that
part
PS
PG
Ps part of the gravity of the ſyſtem is employed in
accelerating its motion, the other part is being loſt on the
axis of ſuſpenſion; which axis will therefore, in a direction
perpendicular to PG, ſuſtain a force expreſſed by A+B+C
Pg
But this is not the only force by which the axis
is affected; ſince, beſides the other part of the force of gravity
(A+B+C x Core in the direction GP, the centrifugal force,
acting in the ſame direction, is to be taken into the conſidera-
tion; whereof the quantity will be the ſame, as if the whole
maſs of the ſyſtem was to be placed in its common center of
gravity G. For, if upon PS the perpendiculars Aa, Bb, Cc be Fig. 25.
let fall; then, the centrifugal forces of the ſeveral bodies A,
B, C being as the maſſes drawn into the reſpective diſtances
from the center P, the effect of thoſe forces in the direction PG,
will
GS
X Х
PG PS
R 2
I 24
The Reſolution of ſome General Problems
UX
PG?
;
will therefore be expreſſed by Ax Pa + BxPb + C x Pc,
which (by the property of the center of gravity) is known to be
equal to A +B+C XPG.
But the preſſure on the axis may be otherwiſe deduced, in-
dependent of the center of oſcillation : for the angular celerity
generated in the ſyſtem about its center of gravity G (which is
the ſame with the angular celerity about the point of ſuſpenſi-
on P) is intirely the effect of the action on the point of ſuf-
penſion ; and the momentum, or force, ſufficient to produce
that celerity, is found (by the Propoſition) to be
A x AG* + BxBG’+CxCG”, which is to the abſolute mo-
mentum vx A + B + C, generated in the ſyſtem, as
A XAG’ + Bx BG? + Cx CG2
to PG. Therefore the force act-
A+B+CxPG
ing on the axis of ſuſpenſion, in a direction perpendicular to
PG, muſt be to the force employed in accelerating the motion
of the ſyſtem (in the like direction), in the ſame proportion
above ſpecified ; ſo that, to have the true meaſure of each, the
force of gravity muſt be divided in that ratio : whence (taking
GS
A XAG? + B x BG? + Cx CG2
it will be, as GS +PG
AB+CxPG
(PS) is to GS, ſo is the force of gravity, in a direction perpen-
dicular to PG, to the force acting on the axis of ſuſpenſion,
in the like direction.
That the proportion here determined is the ſame with that
found above, and the point S, the center of oſcillation, is thus
CXC6")
made to appear.
2
Since AP? GP + GA 2GP x Ga,
BP = GP' + GB’ + 2GP x Gb,
CP = GP: + GC 2GP x GC,
it is evident that A X AP + B x BP + CxCP' (as given
above) will be
A+B+CxGP' +AXGA' + B x GB*
+ C X GC
2GP X A X Ga B x Gb + C x GC
A+B+CxGP +AXGA+B x GB’+CXGC”, barely;
becauſe (from the property of the center of gravity) all the quan-
tities AxGa-BxGb + CxGc deſtroy one another. Hence,
by
1
in Mechanics and Phyſical Aſtronomy.
1 25
:
GP +
;
by ſubſtituting the quantity here found, inſtead of its equal (inCor.
A + B + CxGP+ AⓇGA? + Bx GB+ Cx GCS
V.) we get SP=
A + B + C x GP
A X GA + Bx GBP + CX GC2
; and conſequently
A + B + C X GP
(SG) the diſtance of the center of oſcillation, or percuſſion from
the center of gravity
A X GA + B x GBP + C x GC?
the
A+B+CX GP
very ſame as above. Hence it alſo appears, that, if the plane
of the motion remains unchanged, the rectangle under SG
and GP will be a conſtant quantity; and that, if S be made
the point of ſuſpenſion, then P will become the center of oſcil-
lation ; and, laſtly, that the oſcillations will be performed in
the ſhorteſt time poſſible, when SG and GP are equal to one
another, and equal, each, to JAxGA’+BxGB?++ x GC?
it
A +B+C
being well known, that the ſum of two lines, whoſe rectangle
is given, will be a minimum when the lines themſelves are
equal to each other.
The ſame method laid down above, for finding the preſſure
upon the axis of ſuſpenſion at reſt, anſwers equally when that
axis is ſuppoſed to have a motion, or when the ſyſtem, or
body, has a progreſſive motion, as well as an angular one (as
is the caſe of a cylinder, which, in its deſcent, is made to re-
volve about its axis, by means of a rope wrapped about it,
whereof one end is made faſt at the place from whence the
·motion commences): the momentum of the rotation about the
center of gravity, generated in a given particle of time, being
always as the force producing it, drawn into the diſtance of
the point where the force acts, from the center of gra-
vity, as well when that point is in motion, as when it is at
reſt.
Another thing it may be proper to take notice of, which is,
that in the foregoing conſiderations the bodies A, B, C are ſup-
poſed to be very ſmall; ſo as to have all their parts, nearly, at
the ſame diſtance from the axis of motion. But, to have the con-
cluſion accurately true, every particle of matter in the ſyſtem
ought
I 26
The Reſolution of fome General Problems
ought to be conſidered, and treated, as a diſtinct body: from
whence, by means of the method of fluxions, the ſum of all the
momenta will be truly found : but this relating merely to mat-
ters of calculation, I have no deſign to touch upon it here. I
thall only add, that the center of oſcillation may
be otherwiſe,
very readily, computed, from Corol. I. even in caſes where the
forces acting on the bodies A, B, C have any given relation to
each other. For, if a, b, c be taken to repreſent the, reſpec-
tive, meaſures of the ſaid forces (or the momenta they would
produce in a given time) it is evident, from thence, that the an-
gular celerity that would be generated in the ſyſtem (at the dif-
tance !, from the center, during the ſame time) will be truly ex-
аҲАР -- bx BP + cx CP
preſſed by
:; which, in caſe of a
A X AP + B x BP + Cx CP
ſingle '
body S, acted on by the force s, becomes (or
SxSP"
SSP). Therefore, by putting this laſt value equal to the
former, we have SP
A x AP+. B x BP + C x CP
š
a X AP + b X BP + cx CP
ſhewing at what diſtance from the point of ſuſpenſion the
ſingle body S muſt be placed, to acquire, by means of the
foree s, the ſame angular celerity as the ſyſtem itſelf acquires,
from the action of all the other forces given.
SX SP
Х
;
wan
L E M M A.
If a given angle AOB be divided into two parts AOC, BOC,
the product (or ſolid) contained under the ſquare of the fine (CD)
of the one part AOC, and the fine (CE) of the other BOC, will
be a maximum, when the tangent FC of the former part is double
the tangent GC of the latter, or when the fine of the difference of
the parts, is one-third of the fine of the whole given angle
.
Fig. 26.
For, if the fine (CD) of the former part be denoted by x,
and that (CE) of the latter by y, it is well known that (m.) the
celerity of x's increaſe (ſuppoſing C to move from A to B) will
be in proportion to the celerity (– j) of y's decreaſe, as the
co-fine
in Mechanics and Phyſical Aſtronomy.
127
O, and
co-fine of FCD to the co-fine of GCE; that is, as pe to cc
But, when x*y is a maximum, we have 2xxy + x^j
conſequently * : - j :: : 2y. Hence, by equality, tc: GC
:: x: 2y; and therefore FC=2GC. Let, now, OH be drawn
to biſect FC in H, and let HM and GN be perpendicular to
FO; then, ſince it is proved that FC = 2GC, it follows that
FH FG, and that HM, by ſimilar triangles, muſt likewiſe
be = {GN: but HM and GN are fines of the angles HOM
and AGO, to the equal radii OH and OG; whence the latter
part of the Lemma is alſo manifeft.
+
PROBLEM II.
Suppoſe that a plane ABC, moving with a velocity and
direction repreſented by bB, is acted on by a medium, or fluid,
whoſe particles move with a velocity repreſented by DB, and in
directions parallel thereto; to determine the efféet of the fluid
on the plane, in the direction of its motion BH, and alſo what
the angle of inclination ABD muſt be, that the effect may be
the greateſt poſſible.
Becauſe a particle, impinging on the plane at B, moves thro’ Fig. 27.
the ſpace DB in the time that the plane itſelf, from abc, arrives
at the poſition ABC, it is evident that the diſtance (De) of the
faid particle from the plane (produced), at the beginning of
that time, will be the meaſure of the relative celerity where-
with the particles of the fluid approach the plane in a direction
perpendicular thereto ; and, conſequently, that the force of the
ſtream in that direction, will be as Del (it being well known
that the force of a ſtream upon any plane-ſurface, is always as
the ſquare of the relative celerity with which the particles ap-
proach it; in a perpendicular direction). Hence, by the reſolu-
tion of forces, it will be, as the radius, is to the fine of the an-
gle ABH (or abH), fo is the force Del', to its required efficacy
in the propoſed direction BH.
1
More
I 28
The Reſolution of ſome General Problems
3
Moreover, with regard to the latter part of the Problem,
the angle bBD, which the directions of the two motions make
with each other, being given, as well as the ſides Bb, BD con-
taining it, the remaining angle BbD will from thence be
known, as likewiſe Db: and ſo De being the fine of the angle
Dbe, to the given radius Db, the effect (Del' x ſin. abH) will
therefore be a maximum, when fin. Dbel x ſin. abH is a maxi-
mum ; that is (by the Lemma), when the fine of the difference
of the angles Dbe, abH, is equal to part of the fine of the
whole given angle BbD: from whence the difference being
given, the angles themſelves will be known.—The geometri-
cal conſtruction from hence, is extremely eaſy; for, having
from the center b, with any radius, deſcribed the arch mr, on
rb produced (if neceſſary) let fall the perpendicular mp; take
PI=of mp, and draw qs parallel to pr, cutting the circle
in s; then biſect the arch ms by the line bae, and the thing is
done : for the ſine su of Sr (or of the difference of the angles
Dbe, abH) is by conſtruction (=pq) of mp the fine of
the whole given angle BbD; as it ought to be, by the Lemma.
But, if
you
had rather have a general Theorem expreſſed
in algebraic terms, then let the velocity (6B) of the plane be
put = a, and that (DB) of the fluid=b; and let the fine, and
co-line of the given angle DBb (to the radius 1) be denoted by
m, and n, reſpectively; alſo, having drawn BFL perpendicular
to bFe, put bF = x, and BF =y; then, fince FB and FL are
tangents of the angles FbB and FbL, to the common radius
bF, it appears, by the Lemma, that FL is (= 2BF) = 2y;
whence (ſuppoſing LR and DQ to be perpendicular to BbQ,
we have (by ſimilar triangles) as Bb (a) : BF (y) :: BL (34)
: BR
3yy
, and therefore bR (BL — Bb) =
Al-
fo, Bb (a) : 6F (*) :: BL (3y) : LR = 3*y. But the value
of DQ being mb, and that of bQ=nb a, we have again, by
ſim. triang.) mb : nb-a :: 3xy : 39y --- aa
and conſequently
3yy
X 3xy, or 3yyyy
! x 3x7
(be-
Зуу
a
nb
nb
aan
mb
mb
}
in Mechanics and Phyſical Aſtronomy.
129
nb
d
3*
X
2,
(becauſe yy + xx = aa); whence
+
and from thence, by completing the ſquare and extracting
9
2+
3 x nb -- a, equal to
the tangent of the angle bBF, the complement of the re-
quired angle abH, or ABH. 2. E. I.
the root,
nb
x
у
4
mb
2
mb
COROLLARY I.
gaa
466
1
If the given angle DBb be a right-one (which is the caſe
when regard is had to the wind ſtriking againſt the fails of a
windmill); then, m being =1, and n=0, our expreſſion
for the tangent of bBF (which here is equal to the angle of
inclination ABD) will become 12+ +3; and this, if
a be taken = 0, or the plane be ſuppoſed at reſt, will be
V2, barely; anſwering to an angle of 54° 44'. But if the
velocity of the plane be ſuppoſed , , or of the velocity of
the medium or ſtream, then the angle of inclination ABD
will be found from hence equal to 58° 14', 61° 27', or 66° 58',
reſpectively; ſo that, the greater the velocity of the plane is,
the greater alſo will be the angle of inclination. Hence it ap-
pears that the fails of a windmill, that the effect may be the
greateſt, ought to be more turned towards the wind in the ex-
tream parts where the motion is ſwifteſt, than in the parts
nearer to the axis of motion ; in ſuch fort, that the tan-
gent of the angle formed by the direction of the wind and
the fail, may be, every-where, equal to 12 + gaa
+
466
the velocity a being proportional to the diſtance from the
axis of motion.
za
223
COROL
S
COROL-
130
The Reſolution of ſome General Problems
COROLLARY II.
If, inſtead of the angle DBH (or DBb), the angle DBA,
which the direction of the ſtream makes with the plane, be
given; then it will appear, that the effect will, in this caſe, be
a maximum, when the fine of the angle ABH, made by the
plane and the direction of its motion, is to the fine of the ſaid
given angle DBA, in the given proportion of BD to Bb. For,
the force in the perpendicular direction FB being expreſſed by
Del', its effect in the direction BH will, therefore, be defined
or its equal
(ſuppoſing BA produced
to meet DeE in E). Now DB and the angle DBE (as well as
Bb) being ſuppoſed given, DE is given from thence. But it is
well known, that the ſquare of one part of a given line, drawn
into the other part, will be a maximum, when the former part
is the double of the latter. Conſequently De muſt here be the
double of Ee; which laſt, or its equal BF, will therefore be
BF
De* x
Dalix Ee
by De
BOS
Bb
{DE.
>
But, fin. B6F : radius :: BF (DE) : Bb;
And, radius : fin. DBA :: BD : DE;
by compounding of which, we have the proportion above laid
down. "But that proportion, it may be obſerved, can only take
place when Bb is equal to, or greater than of DE: for,
when Bb is leſs than of DE, Ee (which is always leſs than
Bb) cannot be equal to of DE ; but will approach the
neareſt to it, when BF coincides with Bb, that is, when the
angle FbH, or ABH is a right-one; and in this caſe, the effect
will be a maximum, when the direction of the inotion is
per-
pendicular to the plane.--If the given angle DBA be a right-
one (which poſition appears from hence to be the moſt ad-
vantageous, becauſe DE then becomes = DB) it follows that
the ſine of the angle ABH, which the required direction
makes with the plane, will be to the radius, as part of the
velocity of the ſtream is to the velocity of the plane (or
fail).—Hence, if the force of the wind be capable of pro-
ducing a degree of celerity in a ſhip, greater than part of
its own celerity, it is evident that the ſhip may run ſwifter up-
on
in Mechanics and Phyſical Aſtronomy.
131
on an oblique courſe, than when ſhe fails directly before the
wind *.
PROBLEM III. .
Suppoſe that a thread ACnCA, having two equal weights A,
A, Jufpended at the ends thereof, is hung over two tacks C, C, inz
the ſame horizontal line ; and that to the middle point of the thread
(n) equally diſtant from the tacks, another given weight B is fixed,
which is permitted to deſcend by its own gravity, ſo as to cauſe
the other two weights, at the ſame time, to aſcend : it is propoſed
to find the law of the velocity by which the ſaid weights aſcend and
deſcend; abſtračting from the reſiſtance of the air, the weight of
the thread, and the friction on the tacks.
Let v denote the velocity of B (meaſured by the diſtance Fig. 28.
that might be uniformly gone over in one ſecond of time),
and let b (=.32 feet) the meaſure of the velocity
which gravity can generate in a falling body, in one ſecond;
putting CE =a, En = x, Cnry, and the tenſion of the
thread :
=w: then being the time in which B would, uni-
formly, deſcribe the diſtance x, we ſhall have, as I (ſecond) is
to , fo is b (the velocity generated by gravity in one ſecond)
the velocity generated (or deſtroyed) by gravity in the
bi
to,
v'
*
time
0
N
уу — аа
Y у
:):
yjnyy
aa
уух
Moreover it will be, as BC (y) : En (yy-aa) :: 0 :
the velocity with which the weights A, A aſcend;
whoſe fluxion yy — aa x yu + aav;
(
Jy --- aa x yu + aavy is there-
yy
fore the increaſe of that velocity, in the time but were
* In the above confiderations the velocity of the plane (or fail) is, all along,
treated as a given quantity ; becauſe the ſame direction that gives the effective
force the greateſt, when the velocity is given, muſt neceſſarily give the
velocity the greateſt poſſible, when the force, alone, is given.
not
܀
S 2
132
The Reſolution of ſome General Problems
bx
V
bi
bi
to
уух
уу - aN X YUU + aavvy
:
2WX
y
2wx
y
2W*
not the ſtring to act on the ſaid weights, their velocity (inſtead
of being increaſed) would be diminiſhed, and that by the quan-
tity o* (as is found above). Therefore the whole alteration of
motion ariſing from the tenſion of the ſtring is to that ariſing
from the action of gravity, in the proportion of
yy - aa x yü + aavi
+
; and, conſequently, the tenſion of
the ſtring (w) will be to the weight of the body A, in the ſame
proportion: whence we have w=Axit
byyxx
Again, it will be (hy the reſolution of forces) as Cn (y) is to
En (), ſo is 2w (the double of the tenſion of the thread) to
the effect of that tenſion to retard the deſcent of the
weight B; which being ſubtracted from the gravity (B), the
remainder B - will be the force by which B's motion is
accelerated. Hence we have, as B is to B fo is a lot.
)
the velocity that would be generated by the gravity in the
time to that (0) generated in the ſame time, by the force
o'
B-
From whence, by multiplying extreams and means,
=bx
we get vi=bi— 25xiw
-
2Ax
aa x yvui + apoy
By
By
Вуз
(by ſubſtituting the value of w) = bi
2b Ay
2 Avv
+
208AX
you
vaj
(becauſe yý = xx) = bx — 2543
2 Avu
+
2y?ve -
20°yy
: and, conſequently, by taking the fluent,
bx
2bAy
+ 4A * + d (d being the necef-
ſary correction); which equation, if be put =m, will be
2by + 2dm
-
; Thewing the true ve-
mt 1. yy - ABS
locity
y
time
2wX
у
=
2bAxi
a
rendere
2014,
B
edhe
yš
B
a?A
Х
B
g²
y4
|
Αυ?
B
2
B
VU
yy
B.
2A
2bmx
=
reduced to v=YT
in Mechanics and Phyſical Aſtronomy.
133
.
locity of the body B ; whence that of the body'A (
2by + 2dm
will alſo be known. X. E. I.
m+ 1. yy - aa
2bmx
✓
lue of v will be y/2bmx = 2by – 2bmf + 2bg
mt1.yy
2
COROLLARY I.
If the firſt values of w and y, when the motion commences,
be expreſſed by f and g, reſpectively; then, v being
O,
when x=f, and y=g, we ſhall have o=2bmf - abg + zdm,
and conſequently adm = abmf + 2bg ; ſo that the general va-
-
2hm x x-f-2b xy-g
y
mt 1. yy
From whence the greateſt diſtance through which the weight
B can deſcend, before its whole motion is deſtroyed by the
other weights A, A, may be eaſily determined: fór, ſince the
velocity, at the loweſt point of the deſcent, vaniſhes, or be-
comes equal to nothing, we ſhall, in that circumſtance, have
2bm x 2 -- f --- 2b xyg=0, or mx
o, or m x'x-f+8(=y=
✓ xx taa ✓ xx+88 - ff; which, ſquared, gives
m* x*— F1° + 2mg xx-f= xx — ff: whence, dividing
by x — f, we have m* x x - f + 2mg =x
*+f; and con-
ſequently x =
1 + mm. f; exhibiting the diſtance of the
point n below the horizontal line CC, when the whole motion
is deſtroyed, and all the weights begin to move the contrary
way. But it muſt be obſerved, that this can only happen
when m is leſs than unity, or when the weight B is leſs than
the ſum of the other two : for, if m be equal to unity, x will be
infinite; and, if m be greater than unity, the value of x will come
out negative; which thews the thing to be impoſſible, or that
the weight B muſt continually deſcend; except when m is leſs
than unity, or B leſs than 2A: in which laſt caſe, it
appears
that the bodies will oſcillate; backwards and forwards, continu-
ally; in ſuch fort, that the two extream diſtances from the
horizontal line CC will be expreſſed by f and 2mg — 17 mm.f
whereof the latter, when f= 0, will become
2mg
I
mm
I
mm
2ma
1
mim
134
일
​The Reſolution of ſome General Problems
She ha
712
I
m112
ing f and 2ing — 1+mm. f
I
mm
B
2A
xCC; ſhewing the loweſt deſcent of n, from the line
CC, when the motion commences from that line.—By mak-
1+ mm.f equal to each other, we get f=mg
xg, or f:g :: B : 2A. From which it appears, that,
if the firſt poſition of the weight B be ſuch, that En is to Cn
in the given proportion of B to 2A, no motion at all will en-
fue, but the weights remain in equilibrio. Whence it is evi-
dent, that, if the motion commences from any point below
that here determined, the weight B will firſt of all aſcend,
till the diſtance from CC is 2mg - 1+mm
It mm.f; after which it will
again deſcend, to its firſt diſtance f; and ſo on, backwards and
forwards, continually.
I
mm
COROLLARY II.
2bmx
(v=y
maa
aa
If Cnc, in the firſt poſition of B, be ſuppoſed to coincide
with the horizontal-line CEC, and the body B be impelled from
thence with any given celerity c (meaſured, as above, by the
ſpace that would be uniformly gone over in one ſecond of
time); then, v being C, when x = 0 and y = a, we
ſhall, by ſubſtituting theſe values in the general equation
2by + 2dm
=yJ2
2ba + 2dm
obtain c = a
ad
m + I. yy
✓ 2dm — 2ba
and conſequently 2dm = mc + zba; ſo that v
2bmx
is here W
2by + 2ba to mo?
ya
which, when b=0, or,
m + I. yy
when the bodies are not acted on by gravity, will become v=
And in this caſe, the time of moving thro’En
m+Iyy
m +1.gy-- aa m+1.*x + ma?
(whereof the fluxion is
may be readily found, by means of an hyperbola, whoſe tranſ-
verſe
aa
1
mªcy
аа
I
V
incy
micv xx + ac
1
in Mechanics and Phyſical Aſtronomy.
135
2a
ز
za
3
2
3
를
​;
verſe and conjugate axes are
and 2a; it being in propor-
✓m
tion to the time of moving uniformly over the fame diſtance (x)
with the given celerity at E, as the arch of the byperbola is to
its ordinate x.
PROBLEM IV.
Suppoſing a ſpherical body of ice, or any other matter, revolving
about its axis, to be reduced to a fate of fluidity; to determine
the change of figure thence ariſing.
It is demonſtrable, that the figure of an homogeneous fluid, Fig. 29.
revolving about an axis (PS), having all its particles qui-
eſcent with regard to each other, muſt be that of an oblate
ſpheroid OAPES (ſee Art. 395 of my Doctrine of Fluxions); and
that the particular ſpecies of ſuch ſpheroid, anſwering to any gi-
ven tiine of revolution p, will be truly defined by the equation
p=9/
where 1 : 1+tt :: PS’ : AE”; PS be-
3 + tt A-37
ing the axis; and AE the equatoreal diameter; alſo A the
circular arch, whoſe radius iš unity, and tangent t; and 9
the time wherein a ſolid fphere, of the fame magnitude and
denſity with the ſpheroid, muſt revolve, ſo that the centrifugal
force at the equator thereof, may be exactly equal to the at-
traction, or gravity. Now it is evident, that, whatſoever
figure a fluid, revolving about an axis, at any time hath, the
momentum of rotation about the axis will be no-ways changed,
with the figure, by the action of the particles on each other :
ſo that the momentum of our propoſed fluid, ariſing from the
ſphere of ice, will, at all times, be the very fame with that of
the ſphere itſelf. From whence it may be eaſily proved, that
the time wherein one intire revolution of the fluid, con-
ſidered as a ſpheroid, might be uniformly performed, muſt
be always as AE”: therefore, if we make e = AE, and put
d the diameter of the ſphere (or of the fluid, when
AE PS) it follows that the ſaid time * will be truly expreſſed
* At Art. 399. of my Fluxions, this time is, by miſtake, put down
-Źx
xs (inſtead of a xs); whereby the remaining part of that Article is
by
,
}
rendered erroneous.
4
::::..........!
136
The Reſolution of ſome General Problems
WX+
3 ttt X A
372
dt
X
13
35²
AE
vittt
I + it
€3
Vitott
d4
I
ex
73
by axs (ſuppoſing s to denote the given time of revolu-
tion of the body, when under the form of a ſphere) :
113
which being put (= 9
ad
the time where-
in the revolution is performed, when the particles are in
292
equilibrio, we ſhall thence have 3+ tt x A – 30
But, becauſe PS (= :) the maſs of the ſphe-
roid will therefore be as (= AE* x PS); and that of
the ſphere, as d': which two quantities being made equal to
each other, we have
And, this value being
it tt
3 + tt X A - 3t
wrote in the room of its equal, we have 3+ * XA
293
3 ttt xA - 3t
292
xit tti From the
reſolution of which equation the value of t, and the ſpheroid
itſelf, will be known. The ſpheroid thus determined, is that
under which the fluid might remain in equilibrio, were the par-
ticles to be, once, quieſcent with reſpect to each other: but the
particles, in their receſs from the axis, do, through the cen-
trifugal force, acquire a motion from the axis, which is not
immediately deſtroyed, on the fluid's aſſuming the figure, or
| degree of oblateneſs above determined; the equatoreal parts
ſtill continuing to recede from the axis, till the gravitati-
on, by degrees, prevails, and in the end quite overcomes the
ſaid motion. After which the equatoreal parts will begin to
ſubſide, and again approach the axis, in the very fame manner
they before receded therefrom: and ſo will continue oſcillating,
backwards and forwards, ad infinitum. But if the fluid is fup-
poſed to have ſome degree of tenacity, the oſcillations will be,
every time, contracted, and the parts of the fluid will then
converge to an equilibrium, under the form above deter-
mined.
L E M
or
I
Х
3
I + tt
23
352,
13
353
in Mechanics and Phyſical Afronomy. 137
C
CX
cami
3
2
ca²*
:
C
3
c²a²
2
L EMMA.
Suppoſing a body to move with an uniform celerity, in a right-
line AD; to determine the rate of increaſe of the relative celerity
by which it recedes from a given point C, out of that line.
Make CA (perpendicular to AD) = a, and AB = x; and Fig 30.
let the meaſure of the body's celerity, or the ſpace gone over in
a given time g, be denoted by c: then will &* expreſs the
time of deſcribing : (or Bb); and it is well known, that
(=cx AB) will be the true meaſure of the cele-
✓
xx taa
rity with which CB increaſes; whoſe fluxion,
is
** + aal
therefore the (uniform) increaſe of that celerity, in the time
gx : hence it will be, as
gi
:g (the time given) ::
xx+aa 2
*Х АС
(=
the required increaſe, that would uni-
СВІ
formly ariſe in the given time g: which increaſe, ſince
repreſents the paracentric velocity of the body (in a direction
perpendicular to CB) will be, always, expreſſed by the
ſquare of the meaſure of the body's paracentric velocity, ap-
plied to the diſtance (BC) from the given point, or center.
& E. I.
COROLLA RY.
It is evident from hence, that if a force, which in the given
time
8
is ſufficient to generate the ſaid increaſe of velocity,
be ſuppoſed to urge the body towards the center C, and there-
by deflect it from its rectilineal motion, the celerity with which
CB increaſes will then be uniform ; becauſe the force applied,
each moment of time, is juſt fufficient to deſtroy the increaſe
that would ariſe, in the ſame moment, from the body's being
ſuffered to continue its motion uniformly in a right-line. If
the direction (Bb) of the motion is perpendicular to CB, the
body, thus acted on (as no celerity is generated in the direction
CB), will move in the circumference of a circle. Conſequently
T
thc
2010)
3
2
**+ aal
CXAC
CB
138 The Reſolution of ſome General Probleriss
C
2
Eb
ED*
>
or
2CB
2
2CE
VCB 1 BỘ
the force above determined is the fame with the centrifugal
force in a circle, when the diſtance from the center, and the
angular celerity are the fame.
But all this may be made to appear in a different manner,
by ſuppoſing Bb exceeding ſmall: for, if BE be made
perpen-
dicular to ČB (produced), BE will then expreſs the length
whereby CB would be uniformly augmented, in the time (2 x Bb )
of deſcribing Bb; and therefore eb, the exceſs of Cb above
CE, will be the ſpace through which the force muſt cauſe the
body to deſcend, in order that the increaſe of the diſtance from
the center C may be the ſame as would uniformly ariſe with
the firſt celerity, at B. But it is evident that this excefs eb
(which, by the property of the circle, is
?
alſo expreſſes the effect of the force, neceſſary to cauſe a body
to revolve in the circumference of a circle Eef, with the ſame
angular celerity. To determine, from hence, the velocity
which this force would generate in the given time g, we have,
(the ſquare of the time of deſcribing Bb, or Be) is to
E61
*ХЕВ
g", ſo is
the ſpace through which the
ball might fall, by means of the ſaid force, in the given
c? XED
XĀCI
time g; the double of which, (or its equal
CB X Bb
CB X CB
is, therefore, the true meaſure of the velocity ſought; becauſe
the diſtance gone over by a falling body is but the half of that
which might be deſcribed in the ſame time, with the velocity ac-
quired at the end of the deſcent. The quantity here determin-
ed (as has been before obſerved) is the meaſure of the force by
which the body is made to recede from C with an uniform ce-
terity: if a force, leſs or greater than this, be ſuppoſed to act,
the difference will cauſe an increaſe or decreaſe of celerity in
the line CB, proportional to the ſaid difference.
gºx Báº
as
CC
2
to
2CB
A
#
}
PRO-
3
in Mechanics and Phyſical Apronomy.
139
a,
X,
2
ť,
V,
Share
all,
PROBLEM V.
Suppoſe that a body, let go from a given place A, in a given
direction, with a given celerity, is continually ſolicited towards a
given point C, by a given centripetal force ; to determine the path
ABP in which the body will move.
From the center C, through A, let the circumference of a Fig. 35.
circle ADK be deſcribed ; and, ſuppoſing B to repreſent the
place of the body,
(the radius CD of the circle,
the radius vector CB ..
the arch AD, meaſuring the angle ACB
put the time of deſcribing the angle ACB
the meaſ.of the celerity with which the line CB incr.
the meaf.of the celer, with which the area ACB incr.
the meaſure of the centripetal force
23
where, by the meaſure of a celerity, I mean the ſpace that
would be uniformly deſcribed with that celerity in a given
time g; and by the meaſure of a force, I underſtand the
meaſure of the celerity that might be uniformly generated by
the force, in the ſame given time.
Since the celerity with which the area ACB increaſes is ex-
preſſed by au, it is evident that the paracentric velocity of the
radius vector CB, at the middle point (b), will be expreſſed by
and that of the body itſelf by the ſquare of which
laſt, divided by x, will give for the true meaſure of the
centrifugal force (by the Lemma); whence it appears that
(tamine -Q) the exceſs thereof above the centripetal force 2,
is that force whereby the celerity v is accelerated : therefore
we have g:t::
Q (the meaſure of the celerity gene-
rated in the given time g): 0. But, becauſe the paracentric
velocity of the body is that of the point D (deſcribing the
T 2
circular
1
를
​au
2au
4a´u?
ges
40*u
x3
2 au
2
140
The Reſolution of ſome General Problems
2au
Х
2a24
circular arch AD) will be
or ; and ſo we have
(the diſtance deſcribed in the time g) : ž (the di-
ſtance deſcribed in the time i): whence, by equality, Q
29?u
8
8:t::
4a?u
x3
2au
4a´u
x3
x?ż
2a4
: 0 :: : 2; and conſequently v = QX
Again, the ſpaces ż and (deſcribed in the ſame time) be-
ing in the ſame proportion as the celerities and v, with
2a2u
2aaux
xxż
To ex-
aň
2auri
>
zauw to 201
2uż
2użXI-W
: 0 -4au
܀
x3
2ału
2a2u
2U XI
w
I
w
W
W
I
uw
2,
ZZ
aa
aa
uzz
4au’xi-w
which they are deſcribed, we alſo have v=
terminate w and ☺ out of this, and the preceding equation,
make W, (or w at1); then v
and
2auw
*
ох
Qx?z
z
Qż
(by writing for its equal x); which
equation may be reduced to
e
+
expreſſing the general relation of w and z, or of x and z, ac-
cording to any value of u. But in the caſe propounded, where-
in no force is ſuppoſed to act, beſides that tending to the center
C, the celerity au with which the area ACB increaſes, will
be a conſtant quantity ; and therefore, i being here = 0, our
equation becomes
aQ
: from whence,
44% XI-W
when Q is given in terms of x (or w), the relation of w and
z may be determined.
COROLLARY I.
Hence, if the centripetal force, by which a body deſcribes
a given orbit at reſt, be known, the increaſe of that force,
when the orbit itſelf is ſuppoſed to have a motion round the
center of force, may be eaſily deduced : for, let the angular
motion of the orbit, be to that of the body in the orbit, in the
conſtant ratio of m to l; then, the whole angular celerity of
the
a²w
:
I-W
in Mechanics and Phyſical Apronomy.
141
x3
x3
ز
the body, here, being in proportion to the angular celerity
when the orbit is quieſcent, as m + 1 to 1, the centrifugal
force here, will therefore be to that (40*) in the quieſcent or-
bit, in the duplicate ratio of m + 1 to I (by the Lemma), and
ſo will be truly expreſſed by m tilx
4a´u
From whence it
42?u
appears, that mm + 2m x is the increaſe of the centri-
fugal force ariſing from the motion of the orbit: which quan-
tity, therefore, muſt be that whereby the centripetal force
ought to be likewiſe increaſed, in the moveable orbit ; ſo that
the difference of the two forces, whereby the motion of the
body in the line CB is accelerated, may be the ſame here, as in
the quieſcent orbit; in which caſe the value of CB itſelf, in all
contemporay poſitions, muſt neceffarily be the ſame. Hence
it appears, that the increaſe of the centripetal force, in order
to the deſcription of a moveable orbit, will be always inverſely
as the cube of the diſtance ; and will, moreover, be to the
centrifugal force in the quieſcent orbit (in all contemporary po-
ſitions), in the conſtant ratio of mm + 2m to 1.
COROLLA RY II.
If the centripetal force (Q) be ſuppoſed, inverſely, as the ſquare
of the diſtance, and the given value thereof, at the lower apſe
A, be to the centrifugal force there, in any given ratio of 1 me
to 1; then, as the general value (4*) of the centrifugal
force, will, at A, become = the centripetal force there
will be expreſſed by XI-e; and conſequently that at B
I-eX 4u
I-eX 4x2
by xong
or its equal
Tel'. Which
value being ſubſtituted for Q, our equation here becomes
aw
=e-w: whence, multiplying by w and taking the
aw?
fuent, we get
(where, the angle CAB being
ſuppoſed
4u?
a
4u2
X 1-
a
܀܀
w?
w?
ете
2zz
2
142
The Reſolution of fome General Problems
ari
ez
or
✓ zew
Home WW
✓ zew
WWV
BC =ex
AF
AC
AF
AC
AC-CF
AC
I
CF
CD
I
CE
CB
Fig. 32. ſuppoſed a right-one, no correction is neceſſary); ſo that we
eriv
have ź
: but the laſt of
theſe quantities is known to expreſs the fluxion of a circular
arch (A), whoſe verſed-fine is w and radius e ; therefore i be-
ing=Ă, or a :e::%: A, it follows that the arches x and A
(which are in the ſame proportion with their radii a, e) muſt be
ſimilar, and conſequently their verſed-fines, AF and 'w, in the
ſame proportion above ſpecified, or as a tu e: whence we have
AC
w== x AF, that is, 1–
But
(ſuppoſing BE perpendicular to AC);
therefore I -
AC
and conſequently BC - AC
BC
=BD) = exCB -- CE; from which equal quantities take
away e x BD, ſo ſhall BD-ex BD=ex AE, and therefore
Ime:e:: AE : BD; which is a known property of the conic
ſections, with reſpect to lines drawn from the focii. Hence it
appears, that the trajectory will be an ellipſe, parabola, or hy-
perbola, according as the antecedent 1 -e is greater, equal to,
or leſs than the conſequent e; or according, as the centripetal
force at A, is greater, equal to, or leſs than half the force fuffi-
cient to retain the body in the circular orbit ADK.- As to the
particular ſpecies of the curve, correſponding to any given va-
lue of e, it is, from hence, very eaſily determined : for, if
AO be made to repreſent the ſemi-tranſverſe axis, then will
AO: 0C (:: AE : BD, p. conics) ::1-e:e; therefore, by
diviſion, AO: AC:11--e:1-21; whence AO is known.
ex I
CE
CB
COROLL ARY III.
If to the foregoing force, varying in the inverſe ratio of
the ſquare of the diſtance, another force, which is inverſely
as the cube of the diſtance, be joined (which, at A, is to the
former part in any given ratio of s to i-e), the place of a
body, thus acted on, may be found in the ſame conic ſection
Fig. 33. A'PRS, above determined, fuppofing it to have a motion
about
in Mechanics and Phyſical Aſtronomy.
143
about the focus C, which is to that of the body in the
ſection (referred to the fame point C) in the conſtant ratio
of m to 1; the value of m being =Vits
-
I, of fuch
that mm +2m:1::3 : 1 (by Corol. I.) – If the centripetal force
be barely as the cube of the diſtance inverſely, the curve A'B will
degenerate to a right-line ; in which the body will continue to
move with an uniform velocity, while the line itſelf BA' (al-
ways touching the circle in A') is ſo carried along by the mo-
tion of the radius CA', that the angle ACA ſhall be to the
angle A'CB, in the conſtant proportion above ſpecified; the ra-
tio of the centripetal, and centrifugal forces at A (and conſe-
quently in every other poſition) being expreſſed by that of
s to its.
COROLLARY IV.
If the centripetal force to be as any power of the dif-
tance, whoſe exponent is n, and the given value thereof, at
A, be in proportion to the centrifugal force
44
*
as r to I;
a
4ru?
4ru
I
X-
X
I
a
a
a
a²w
I
w
I
2%
a²w²
wa
w
2żż
2
W
we ſhall then have Q=
; and here
-wl
aQ
our equation,
will become
4u2X I w
wl
a-ü
: which being multiplied by w,
ol nota
and the fluent taken, we thence get
+
in te
nti and conſequently
aw
Ž
; from whence the
✓
+
nti
value of %, by infinite ſeries, or the quadrature of curves, may
be found.
But when r differs but little from unity, and the orbit is
nearly circular,
(becauſe of the ſmallneſs of w)
-71- I
2r. I
w
ar
2w
W²
nto I
7
-
n 42
1
W
will
144
The Reſolution of ſome General Problems
I
r
r r
%%
I
-
W.
will be nearly equal to rxi+n+2.w; and there-
a” w
fore
. W
n + 2 .rw = I
aาพ
n+3.w, nearly; and conſequently
7 + 3. 22
nt3
Fig. 33. Put f = tand A = Vn+3 x %, that is, let A re-
preſent an arch A'Dof the circle ADK, which is always
to the arch AD (or %), in the conſtant ratio of Vn+ 3 to 1:
then À À being = n+3 x żż, our equation, by ſubſtituting
a²w
theſe values, will become f-w; which differs in no-
A A
a w
Carti
ie
2.2
- n +3
thing from that
w) reſolved in Corol. II, excepting
only, that A and f are here uſed, inſtead of z and e: whence
it is manifeſt, that the value of w (there repreſented by ex verf-
ed-fine of z) will here be truly expreſſed by fx verſed-fine of
A: from whence and what is there demonſtrated, it alſo
ap-
pears, that the place of the body will be in the periphery of a
given ellipſe A'BR, revolving about its focus C, with an angular
celerity, which is to that of the body in the ellipſe, in the con-
ſtant ratio of the arch AA' to the arch A'D, or as I
to vn +3. And it is evident, that the motion of the apſides
will be to the motion of the body in the ellipſe, referred to the
focus C, in the ſame given ratio ; and that the angle deſcribed
by the body in moving from one apſide to the other (becauſe
AD is always
will be = 1800 x
n3
180°
All which concluſions, as well as thoſe derived in
the preceding Corollaries, exactly agree with what Sir ISAAC
Newton has demonſtrated, by a very different method, in the
third and ninth Se&tions of the firſt book of his Principia.-As
to the motion of the apſides of the lunar orbit, with the other
inequalities depending on the ſun's action, theſe require the
uſe of other principles, and the ſolution of the following
PRO-
I
I
A'DX
Int3
int3
V x + 3
in Mechanics and Phyſical Aſtronomy.
145
Rt
g
PROBLEM VI.
The fame being ſuppoſed as in the laſt Problem, and that, beſides
the force tending to the center C, another force, whoſe meaſure
is R, acts continually on the body, in a direction perpendicular
to the radius-vector BC; it is propoſed to determine the curve ABP
which the body, ſo acted on, will deſcribe.
Everything in the preceding Problem being retained, Fig. 31
we have nothing more to do here, than to get an equa-
tion for u, by means of the new force, whereon, the increaſe
or decreaſe of u intirely depends. In order to this, we
have, as g (the given time) is to t, ſo is R, the velocity gene-
rated in the time g, to the velocity generated in the time
t; in a direction perpendicular to BC: whence the cor-
reſponding increaſe of the celerity au, with which the area
Ri
Ri
ACB is generated, will be expreſſed by xş,
that is,
will be rai. But, as it has been proved that g: 1 ::
ż,
i
; and therefore ai =
Rx
3 (becauſe x=7
.
Hence, by taking the
Rz
fluent, we have u = c +flu. (c? being put for the
neceſſary correction, or the value of 'u' when z=o). From
uw
e
which equation, and that
4au-XIwl
derived by the preceding Problem, the relation of u, w, and %
may be determined, when the law of the forces Q and R is
aſſigned. 2. E. I.
8
Х
8 2
2aºu
xaż
Rx3z
we have
or 2uii
2aau
4aau
g
IR
2
a
a3
I
w
I W
I
W
w
I
someone
аа
aa
uzz
U
CO-
146
The Reſolution of ſome General Problems
을
​>
201 2
w
I
aa
w
X
2Σ
22
qa
aa
;
w
axi-W
2 II
a? Exi
А
w
w
X
2Σ
I
2%
2 II ż
COROLLARY
If the forces Q and R are ſuppoſed to be in proportion to
(4) the centrifugal force at A, as A to 1, and 11 to 1, re-
ſpectively (4 and 11 being any variable quantities whatever),
and if the celerity (au) with which the area ACB increaſes be
ſuppoſed in proportion to (ac) the firſt value thereof at A, as
4cc II
1 /
£' to 1; then, Q being = 464, R
=
our two equations, by ſubſtituting theſe values, will become
w
and " +
¿
I=It-flu.
or £=1+flu. and
+w=1-
ΣχI
making AC unity; which laſt equation will be rendered ſtill
more commodious, by writing for its equal 3; whence
Ilw
will be had +w=1
ΣΧΙ--μ
into A X
A X 1W
xi-W13: from which the
values of w and £ may be found, when thoſe of A and I are
aſſigned: by means whereof the time (t) of deſcribing the an-
gle z, will alſo be known, from the equation
(a-
bove derived): which by ſubſtituting c £? and
ū
for their
equals u and gives i =
&
To exemplify the uſe of the equations here derived, by the
reſolution of a caſe on which the determination of the lunar or-
bit depends, let the force A, whereby the body is folicited towards
the center, be conſidered, as compoſed of two parts; whereof
the principal (6 x 1—201) is in the inverſe duplicate-ratio of the
diſtance; the other part, which is ſuppoſed ſmall in compariſon
of the former, being as the diſtance directly, drawn in-
A)=5
22
EzX I MW
w
Пх
32
g
20au
I
I-W
2
Х
20
x1
XI-W
I
)
I-W
to
in Mechanics and Phyſical Aſtronomy.
147
I
1-
II:
W
3
IMU
to a ſeries (P'xcof.pz+Qxcof.qx + R'xcof.rz, &c.) of co-
fines of multiples of the arch >, joined to ſmall, given, coeffi-
cients P, Q, R', &c. and let the force i, acting in the per-
pendicular direction, be alſo ſuppoſed, as the diſtance directly,
drawn into a ſeries (Pxſin.pz + xſin.qz+Rxſin.rz * &c.)
of fines of multiples of the ſame arch, joined to ſmall, given,
coefficients P, Q, R, &c. According to theſe aſſumptions, by
ſubſtituting bx i 2013 * P'coſ.pz+coſ. qz &c. and
FxPfin. pz+Qfin.qz &c. for their equals A and II, our
two equations, E=I+ 2 fluent
and + w =
**£-Axi—21 - Axxi—1?, will here become
£=i+2 fluent Pż fin.p% +Qž fin. qz &c. x 1—2014, and
W
+w=
into 2-6-P'coſ.pz + cof.qz
£.
XP fin.pz+Q fin. qz &c. X I-wi
Now, the orbit being ſuppoſed nearly circular, we may,
in
order to a firſt approximation, neglect w in both the factors
I-201–3 and 14, as being very ſmall in reſpect of unity;
by which means £ will become = it2 flu. på ſin. pz+Qż
2P
fin.
9%
&c. itd-
* cof. qz &c. (ſee
p. 82:) whered, repreſenting the neceſſary correction to the flu-
ent, muſt be taken = + &c. fo that £ may be =1,
when z=0.
This value of £ being now ſubſtituted in the ſe-
cond equation +w=x2-6~P'coſ. px-Q_coſ. qz &c.
(where * P fin. px+Q fin. qz &c. on account of the ſmall .
-3
I
Σ
&c. XI-W
ZZ
W
-4
1
x cof. pz-
2Q
9
2P
2Q
P
9
w
* This aſſumption is not the lefs general by the multiples of z being taken the ſame
here as in the value of A; becauſe, if any multiple of 2, in the one value, enters
not into the other, it is but fuppoſing the correſponding coefficient in this laſt, to va-
niſh or become equal to nothing.
U 2
neſs
148
The Reſolution of fome General Problems
neſs of
ż
of )
there cometh out
บ
2P
2Q
22
9
P' cor. pz
Q cof.qz &c. where P"
2P
J-99
P
9
P
I
-99
4
I
W
S
+w=x+d-6-*+P'xcoſ.pz +Qxcof.qz&c.
P
where the general multiplicator may be alſo omitted, as differ-
ing very little from unity: this being done, and the fluent being
taken, according to the method on p. 92,we thence find w=itd
-6ta cof.z-
+P;
IPP
Q"=
2Q
+ O', &c. and where the term a cof.%, by which
,
the fluent is corrected, muſt have its coefficient ſo taken, that
w and x may have their origin together, that is, a muſt be made
=+
"
+ &c.
I-PP
Having thus found a value nearly equal to w, we may by
help thereof, proceed now to a ſecond approximation, by ſub-
ſtituting that value for w, in the factors 1-27-3 and
wherein it was before neglected; and, to facilitate the computation,
the terms in the value of 1-w (whoſe reciprocal is the diſtance
of the body from the center of force) may be expreſſed by the
general ſeries of cofines, ex i-Bcol.lz-Ccof.y—Dcof.oz &c.
(as it appears from above, that the value of w will conſiſt of
ſuch): by which means the ſame terms before determined
will be again brought out, together with a number of others,
ſerving as a farther correction. But, ſince the former
opera-
tion is made, more with a view to diſcover the form of the feri-
es, than to be regarded for its exactneſs, I ſhall have no far-
ther reference thereto, but proceed to determine the value of
the ſeveral quantities e, B, C &c. de novo, by a method fome-
thing different from that uſed above.
Firſt, then, from the equation
1-w=ex-Bcof. Bz-C cof. yz- &c. will be had
=+xB coſ.Bz+C cof. yz &c. + &c.
and
w
wt3
}
'in Mechanics and Phyſical Aſtronomy.
149
I
and i
4
€4
출
​Pż
2P
24
24
e4
P
Ccol.pty.Z &c.
I
x col. px -
X
+
*
pet
+ x B cof. Bz+C coſ. yz &c. + &c.
which laſt value being multiplied by Pź x fin. pz (accord-
ing to the purport of our firſt equation, £=i+2 fluent
Pž fin. pz+Qž fin.qz &c. x122-4), and a proper regard
being, at the ſame time, had to the Theorems on p. 80, the
product will ſtand thus,
xſin.pz+ X-Bfin.6—p.z+Bſin.B+p.z &c.
or thus,
Pż
2Pż
x ſin.pz + xB1.P-B.z+B1.p+B. z+Cl.p-y.z+ Cl.pty.z&c.
whereof the fluent will be
2P Bcol.p-B.Z Bcorpt.z Ccoſ.pag.z
e4
.
+
P-B
pats
ptr
In the very fame manner, the terms ariſing from the multi-
plication of Oz x ſin. qx will be exhibited; and we ſhall there-
fore have
itd-
x cof. pz +ş -x cof.qz +
x cof. rz &c.
4P Bcor. p-B.Z Bcol. 7-+-B.z
C cof.pty.
+
+
+
paß
AB
p-
pty
40 Bcof.qß.z
Bcor.9+B.Z
Ccor.q-2
C col.q+yz
+
+
+B
at?
L
In which the quantity d, aſſumed to denote the neceſſary cor-
rection, muſt be ſo taken, that may be
P
that is, d muſt be equal to X ++
9
B
B
С
C
* +
&c. +
pams p+B p-
pty
B
С
+ +
&c. * &c. Proceeding
q
9+
aty
now to ſubſtitute in our other equation (+ wx 2 – - Et
b + P' coſ.pz+Q_cof. qz8tc. x1 +2013+
2
P
R
х
24
Ccor. payz
na
&c.
e4
X
+
et
I-B
qon
&c.
&c.
I,
when z=0
2
R
&c. +
4P
X
+
+
40 x
B
С
+
€4
qruary
150
The Reſolution of ſome General Problems
w
x Plin.pz+Qfin. qz &c. x i7-4=0) we ſhall, in the firſt
place, by taking the fluxion of w=Icetex
Bcof.Bz+Ccof.yz&c.
have ex® B fin. BätyClin.yz &c. (vid.p. 82 and 83.)
whereof the fluxion being, again, taken, we get
w
=>x3B fin. Bxty Clin.ym &c. and conſequently
w
t-w=1metex I-BB x B cof. Bz+Imrx Ccof. yz &c.
22
Hence 7.7 w x£ - £=-et
B P
IBB. х
9
ex1 —PBxBcof. Bæti-myyxCcof.yz &c. X 14 dana x cof.pz+ 9x col.gz &c.
which (by an actual multiplication of terms of the two ſeries's
into each other, and a proper application of Lem. 1, on p. 76, neg-
lecting at the ſame time, all terms wherein two, or more di-
menſions of the quantities B, C, D &c. would ariſe) will be
reduced to
-£+i+d.IBB.eBcof.Bz+5+d. 1-yy.eCcoſ.yz+itd.1-1.6Dcof.dz &c.
xcof.pzz.z+cof.B. 48.2+ x cof.q=2.24.cof.qp.z &c.
-Imqayo 2x x col.fy.zt.cof.pty.z+
xcol.Ex.3tcof.pt.7.2+ xcof.
x col.q-yzf-cof.aty.z &c.
&c.
&c.
In like manner we have P'col.pz+Qcof.qx&c. XI-W
P'coſ.pz+Qcoſ.qz &c. x + x B cof. Bz+Ccof. zz &c.
ex P' coſ.pz+Q coſ. qz+R' cof.rz &c.
3P
xBcol. B.z+B col.p+B. 34Ccoſ.py.z+C cor.P+y.z &c.
С
P
I
+
+3
x Bcof.q—B.%+Bcol.q+B.z+Ccof.qmayoz+Ccof.q+y.z &c.
2e3
Ľ &c.
&c.
Laſtly becauſe
exlBſin.czty Clin. yx &c. it fol-
lows
in Mechanics and Phyfical Aſtronomy.
151
W
P
1
23
1 dlo
&c.
lows that x Pfin.pz + Q fin. qz &c. XI-W
ex 3Bſin.Bz+yCſin.yz &c. x Plin.pz+Qfin.qz &c.
x x x B col.fz+Ccof.yz&c. whence, by proceeding as a-
bove (neglecting the latter part, 4x Broſ. Bz+Ccoſ.yz &c.
of the laſt factor, as producing terms involving two dimenſions
of the quantities B, C, D, &c.) will be had
X BBx cof.DB.z-BB col.ptB.z+yCcol.pyzamy Ccoſ.ptv.z &c.
X BB coſ.q--ß.z-BB coſ. 9+B.ztyCcof.q-y.z~yCcof.q+y.z&c.
=. z
&c.
Now let the three values, thus determined, be collected to-
gether, taking inſtead of E, its equal, as given by the firſt equa-
tion, putting f=itd, and dividing the whole by itd.e; by
which means our equation is, at length, reduced to
1-BB. B coſ.Bz+I-oy. Ccof.yz+1-08. Dcof. dz &c. า
I+ + +Pixcol.pz+22+Q'x coſ.gz&c.
ef eff
4P
3P
BP
2.
PB
e*f*
4P 3P,
BP
+
I-8.
col. Ptk.z
PAB
etf
4P 3P'yP
+
etf
3P
yP
Р
IYY.
col.pty.
4f
&c. &c.
But as the coefficients of the terms in this equation are much
compounded, it will be proper to make a ſubſtitution for them:
P
Thus, let P=+ @=
e4f
e*f
PBT=4
paß
2P
4
3P
X
Pts 2P
c4f?
PC
b
I
2P
X
9
B.
+
t
P
X
p
cor. -.Z
2
2
B
+
x
ll
2
2
С
+
Iyy.
Х
cof. powy.
?
2
2
4P
с
+
+
%
Part
2
X
P
2
pi
2
Х
+ Extreme
&c.
р
9
3P
B
I-BB
PB
X
2
P
+
+
eff"
PB2=
PB
+
1
IBB
P
2
152
The Reſolution of ſome General Problems
4
y
2
3P
y
2P
2
etf
*
6 b
O
3P
I-YY
PC
PC,
+
X
2P
P
Af?
4
I BB PC
PC2
+ + +
X &c.
ptg
ſuppoſing theſe ſubſtitutions to be continued on, to take in the
terms affected by the other given quantities Q, Q, R, r, &c.
(which are had from thoſe above, by barely writing Q and q &c.
in the place of P and p&c.) This being done, our equation
will ſtand thus
1-BB. Bcof.ßz-+-1-yy.C coſ. yu+1-88. D cof.dz
&c.
-it +Pxcoſ. pz HỌx cot, qẽ đc.
ef
+PBi x cor.p-B.: +PB2x cor.p-t-B.
+ Bixcuſ. q-B.2 + QB2 x cof.q+-B.z &c.
+PC I x cof.p-y.% + PC2 x coſ.pty.z
+QC1xcof.qy.c + QC2 xcof.qty.z &c.
&c.
&c.
From whence, by comparing the multiples of the arch 2, in
the firſt and ſecond lines, we have y=p, d=1, and ſo
on, to as many values (n) as there are quantities p, q, r, &c.
And, by equating the correſponding coefficients of thoſe
ē
equi-multiples, we alſo have C=
D--
I-pp
E-
and ſo on, ton terms. Then, the value of the firſt
n terms of the ſeries B cof. Bx + C coſ. yz + D cof. d % &c.
(excluſive of B coſ. Bz, of which more hereafter) being thus
known, the terms in the 3d, 4th, 5th, &c. lines of our equa-
tion, being compounded of them and the given quantities P, P,
Q q, &c. will alſo become known; from whence, by
continuing the compariſon with the terms of the upper line,
after thoſe already taken, a new ſet of terms, involving two
dimenſions of the quantities B, C, D, &c. P, P, Q, Ï &c.
will be determined: by means whereof, ſtill continuing the
operation in the ſame manner, a third ſet of terms may be ob-
tained; and ſo on, at pleaſure.
As
Р
of
.
I-99
R
Irr
pi
in Mechanics and Phyſical Aſtronomy.
153
b
As to the firſt term 1-BB. B cof. Bz, whereof no uſe has
been yet made, it is reſerved to take off, or deſtroy any other
term, or terms, of the ſame ſpecies, that may ariſe in the
ge-
neral equation. If no ſuch term ſhould occur, it is but making
the coefficient 1-BB. B=0, and every thing will be right:
But that ſuch terms do actually ariſe, will appear in the fol-
lowing illuſtration of the general method of proceeding, applied
to a particular caſe, whereon the determination of the lunar or-
bit depends.
Let P=P, Q=P, and Q, R &c. all equal to nothing *;
then our equation will become
1-BB.B col.ßz +1-47. Ccoſ.yz+1-08.D cof. dz+1-et. E cof.ez &c.
-1+ +Px col.pz+@+PB ixcol.p–B.z+PB2xcol.ptB.z
+QB1 +QB2x cof.Bz+PCIxcol.p. z+PC2xcol.pty.z
+ Qui+QC2 x coſ. yz + PDI Xcof. D-.% + PD2 xcor.p+8
.
+ QDi+QD2x cof. dz+PET xcor.
pe. Z + PE2x cor.pt-%
+ QET QE2 x cof. εzt &c.
Make now, y=p, d=p-B, and e=p+B; then the equation
will ſtand thus
1-BB. B coſ. Bztiryy. C coſ. y + 1-0. D cof. :z* &c.
It +Pxcol.pz + + PB1xcof. DB.Z+PB2xcol.p+B.z
ef
+ B1 +QB2 x cof.B2+PC1 +PC2 xcof.2pz+QC++ QC2 x col.pz
+PDix coſ.B2+ PD2xcol. 2p-B.z+QDI+QD2 xcol.p-B.Z, &c.
Put B'=QB1 +QB2 + PDI + &c.
C'=P+QCT+QC2 + &c.
D' =PB1 + QD1 + QD2 + &c.
E = PB2+ &c.
F = PC2 + &c.
G'=PD2 + &c.
Then, expunging the terms which have no multiple-coſine
in them (as alſo deſtroying each other) we, at length have
* That theſe aſſumptions are ſo made, as to expreſs (nearly) the forces
whereby the ſun diſturbs the moon's motion about the earth, will be ſhewn
b
hereafter,
X
I
*
The Reſolution of fome General Problems
154
C
D'
D
I
DP
E?
23
14PP
2
сех
F cof. 2p%
+ &c.
T-BB. Bcol.But I myy.Ccoſ.yztiadd. D cof.dz + 1ES. E coſ.cz &c.
+ B'cof. B2 to C'coſ.pzt. D'cof.D-B.2+ E'col.p+B.z =0
+F'coſ. 2px + G'cof.2p-B.zt H'coſ.2p+B.zt &c.
whence 1-BBxB=B', C=
I---el
F
G
E
F
H
I-EB1
1-2p-pl
&c. and conſequently
I-20-4-B)
1-w=ext
=exi-B cof. B2-C cof. y z &c.)
C'xcol.pz
D'coſ.B.Z E' coſ.pt Boz
1-B cof. But +
+
1-PP
I-31
Imptoß
Gʻcof. 2paß.z H'cor.2p+B.z
+
+
+
I-4PP
I--2-maß I-2p+81
In deriving the equation here brought out, all terms involving
two, or more dimenſions of the quantities B, C, D, &c. are
neglected; it will, therefore, be neceſſary to ſhew now, how
the effect of thoſe terms may be computed, and the approxi-
mation carried on, to any farther degree of exactneſs deſired.
Previous to which, it will be proper to obſerve, that, in the
preceding calculations, the quantity 1-21-3 was taken
+ x B cof. Bx +C coſ. yz &c. barely, and 1-2
+ * B cof. B+C cof. y % &c. whereas the true value
of wi
or ex I-B cof. B
C cof. zz&c.
is
6
e
+ 2 x Bcoſ.Bz+Ccof. yz&c. t • * Bcula Az+Ccof.yz&c. *+&
and that of 1-W
4 B cof. Bx + C coſ. yz &c.
+xBcof.
Bz+Ccoſ.yz&c.! Now Broſ.Bz+ Ccof.;z&c.*
(before neglected) is evidently equal to B' x coſ. B21+
2 B C x col. B x x C cof. y z &c. Bºx1 + coſ2x +
BCxco.z-B.z+col.7+2.2 + &c. all which cofines, to-
gether with their coefficients, will be known, ſeeing B, B, Y, C
&c.
1
I
4
-3
3
I-
I
4
I
I
in Mechanics and Phyſical Aſtronomy.
155
&c. are given (or nearly fo) by the preceding operation. And,
in the ſaine manner, a ſeries of cofines expreſſing the value of
B cot.B +-C cof. yz &c.' (or of the moſt conſiderable terins
thereof) may be found, ſhould it be neceſſary to continue the
approximation ſo far.
To find what alteration theſe quantities will produce in the
value of 1-W (before found) let any new term thus ariſing in
the value of 1-2-3, be denoted by x cof. æz, and the term
M
e3
4
-101
N
N
xcoſ, az=
3
II
N
e3
+
€3
anſwering to it in the value of 1-w by - xcof.az: then the
correſponding increaſe in the value of — e£(= €-2ex flu.
P żfin. pz+Q z fin.qz. &c. * 124,) will be
21 x Au
P zſin.pz + Q ż ſin. 92.
&c. X
into flu.
Pixſin.pma.x+fin.pta.z+Qxſin.q-az+fin.q+-a.z&c.
Pcof. Panima. Z
Pcoſ.pta.z Qcof.quaz, Qcof. 9+a.z
into
+
pa
to
pta
qator
ata
&c.
In like manner, the increaſe of the ſecond member
(P'coſ.pz + O'coſ. qz &c. * I-W
&c. x 1 W1-3) of our general equati-
on (ſee p.147.) appears to be P' coſ.px+Q'col.qz&c.xx cof.az
M
xP'col.pa.z+P'col.pta.% +Q'cof.qa.ZtQ'cor.q ta.% &c.
And the increaſe of the laſt term, or member,
P fin.pz + Qlin. qz &c. *x flux. 2201, is had =
Maz
P fin. px + Q fin.qz &c. X Х
x fin.az
3z
Ma
P ſin. px + Q fin.qz &c. *•
x ſin. az
aM
643
xPcoſ. Daz-P cof. ptazt Q.col.q-az-Qcof.qta.z &c.
Let theſe three quantities be now collected together; from
whence the new terms entering into the general equation,
when the whole is divided by ef, will ſtand thus
P
2e3
I
e3
3e3
X2
156
The Reſolution of ſome General Problems
N
MP aM
N
MP ?
+
+
aM
6
M
N
e4
MP
P
X
+
6
P
N
MP
X
aM
x
6
MO?
aM
X
6
1
N
X
quæ
N
X
q-a
x
I-atal
Р
P
X
ZP -- 6 xcol.p-a.ztcafita
+
etf p-o a
xcol. Ptaz
2P
+ &c.
So that, from the general method of operation before laid
down, it appears that the proper correction for the value of
1-W, ariſing from the terms, xcof.azand x cof.az, be-
fore neglected in the reſpective values of ī and 1-w-4,
N
aM cof. p-d. %
will be expreſſed by e into
+
X
enfp-a
2P
1-pai
cof. pta.z
+ +
十
​eff pa
2P
1- pta
cof. 1-2.
er
+
etf
2Q
ΜQ'.
aM
cof. q-ta.
+ +
+ &c.
e4f
ata 2Q 6
In reſpect to which it may be obſerved, that no regard has
been had to ſuch quantities as would ariſe from the multiplica-
tion of the new terms in E, by the ſeries 1BB. B cof
. Bz
+1-uy. C cof. yx +&c.(as ought, in ſtrictneſs, to have been
done); becauſe theſe terms being very ſmall, they will, after
multiplication into the ſmall quantities 1-BBxB, 1–27xC &c.
be ſo far reduced, as to become quite inconſiderable. And it
may
be obſerved farther, that even the whole product ariſing
from the multiplication of the value of £. into the ſaid ſeries,
except the terms-e8+1-BB.efBcof.Bz+1-78.ef Ccof.yx
&c. might be alſo rejected, without producing an error of
more than a few ſeconds, in finding the place of the moon in
her orbit.
Having ſhewn what alteration will be cauſed in the value of
1-w (the reciprocal of the diſtance of the body from the cen-
ter of force) from the taking in of any ſmall, aſſigned terms
in the values of 1–2013 and 1–201-4, it may be a pro-
per place here to ſhew (as it is of great importance to be
known) what change will ariſe in the ſaid value of 1-2
from the addition of any ſmall, new terms,
and
A fin.az,
1
in Mechanics and Phyſical Affronomy.
157
1
en-
A'
A 2
X
4
B
I-BB
AB
X
More
78
2
4
+3
I-BB
100
AB
X
2A
2
T
A
cor.az
A 2
х
T
IT
4
3A'
B
1-BB
X
AB
X
9-3
2A
2
and A' coſ. 7 %, to the reſpective forces P ſin. px + Q fin.
qz &c. and P' coſ. pz + O'coſ. qz &c. whereby the motion of
the body is diſturbed. What this alteration, or correction
ought to be, is eaſily diſcovered from the general equation on
p. 151; from whence, by ſubſtituting 7, Ā, and A'inſtead of
P, P, and P', reſpectively, the new terms affected by ,
tering into the ſaid equation, will appear to be
+Ā *
x coſ. 7 %
etf
3A'
+ +
x cof.-ß.x
2A
enf
3A
+ + +
x cof. + +B.z. &c.
eff
and conſequently the increaſe in the value of 1—w ariſing
therefrom=e into + Äx
enf
+
coſ.T-B.Z
+
enf --- 1
3A
cof. +B.%
+
a +8
eff
to &c.
I+B)
where, in many caſes, the firſt term alone, will be ſufficient.
In making the different corrections above pointed out, it
will be found neceſſary to have a particular regard to ſuch
multiples of the arch 2, as are, either, very ſmall, or nearly
equal to unity. (of which two kinds, thoſe whoſe exponents
are p-26, and p-B, will be found the moſt conſiderable.)
For, though the coefficients of ſuch terms ſhould appear, at
firſt, to be ſmall, they ought not, therefore, to be immediately
rejected; becauſe the diviſors which they afterwards receive
(the former in obtaining the value of 14w, and the latter, in
finding the anomaly from thence) are ſuch as may render the
effect, or quotient, too conſiderable to be intirely diſregarded.
And it may be eaſily conceived, without the help of cal-
culation, that a term, or force of the former kind, expreſfed
by the fine or co-fine of a very ſmall multiple of the longitude
%, muſt neceſſarily have a much greater effect, than another
(having the ſame coefficient) which is proportional to the fine
or co-fine of a large multiple of the fame angle : becauſe, when
the
++
4
B
1-23 AB
x
х
7T
1
2A
2
ļ
1
158
The Reſolution of fome General Problems
܀
20
the index, or multiple is a very ſmall one, the term itſelf, while
z increaſes, will continue, for a conſiderable time, nearly of
the ſame value; and confequently, will have its whole effect
exerted in the ſame direction; but when the multiple is a large
one, the changes from poſitive to negative, and from thence to
poſitive again, are ſo quick, that ſufficient time is not allowed
for producing any conſiderable inequality in the body's motion,
before that effect is again deſtroyed by the ſame force, acting
equally in the oppoſite direction.
The value of iw (the reciprocal of the body's diſtance from
the center of force) being, by the formulæ laid down above,
approximated to a ſufficient degree of exactneſs, we may from
thence, and the equation i * El xir* (given at page
146.) proceed to compute the time (t) of deſcribing the angle
z; whereby the difference between the true and mean anoma-
lies will alſo be known; which, in the lunar theory, is the great
point in queſtion, and is beſides, abſolutely neceſſary in or-
der to introduce the proper quantities of the forces whereby the
moon's motion is diſturbed.
Let, therefore, the value of £ (as given by the firſt equa-
tion, at p. 149,) be here repreſented by
fxi-Bcoſ.ßz-Ccof. y - Dcof.dz&c. fo ſhall
xi-Bcol.fz-Ecof.yz &c.) Xemx 7 - B cof. BZ &c.
But 1 - Bcof.B2-Ccoſ.yz&c.
=1+x B coſ. Bz &c. +
**Bcof. Bz+Ccof.yz &c.] =1+xBcol.Bz+Ccoſ.y &c. +
{ x BB x1 +cof. 2Bz+BĆ x cof.gob.z + cof. + B.z&c.
Andı-Bcor.Bz-Ccof.yz&c. -=+2 x B cof. Bx &c. +
3*, BBxit.cof.28z + BC x coſ.y-B.z+coftBiz &c.
Which values being multiplied together, we thence get i
t =
2 x Bcof. Bz+Ccof.yz &c.
it
XB cof. Bz+C cof. jy z &c.
X
20
x
1
that
gz
2cceft into 1
+{
+
in Mechanics and Phyſical Afronomy.
159
Х
2cef}
3x, BBxit.col.2ßx+BCxcof.yR.z+coſ.t-B. &c.
+XBBxi +cof. 2B%+BČxcol.z-Biz+coſy-tB.z &c
xBBxit-cof. 2ßz+B+-Bēxcol.zBzt-c.7+1.2 &c.
Put h=i+xBB+CCT-DD &c. + XBB+CC+DD &c.
+xBB+CC+DD &c. (B)=2B+B, (BB)={BB+ in
BB+BB, (BC) =3BC+ BC+BC+1BC, &c. conti-
nuing on theſe ſubſtitutions, ſo as to take in the terms affected
by the other quantities D, E, F, &c. (which are had from
thoſe above, by barely writing one letter for another);, by means
whereof our equation is reduced to
5+(B) xcof.Bz+(C)xcoſ.yz+(D)xcol.dz &c.
gż
i
(BB) x coſ. 232+(BC)xcoſ.g-B.z+coſ.y+B.z
(CC) x cof. 2yz+(CD)xcoſy-dzcol.ytd.z
&c.
&c.
From whence, by taking the fluent &c. we have
(B) xſın.Bz (C)xſin.y% (D) x fin.dz
z+
he
+
+
hy
gh (BB)xſin.23%, (BC): ſin.y-B.z
Х
+
+
2ceºft
258
b
—В
rts
(CC) xſin.zyz (CD) fin.gd.Z
fin.y
to.z
+
* &c.
+
zby
h
y+
In reſpect to which it will be needful to obſerve, that the firſt
. term, x, will, when z = the whole circumference,
2ceºf?
be the true meaſure of the mean periodic time; becauſe all the
other terms being compoſed of the lines of multiples of the arch %,
they will, while z keeps increaſing, change from poſitive to nega-
tive, and from thence to poſitive again, and ſo on continually;
and therefore can have nothing to do in the mean motion; being
themſelves no other than the proper equations whereby the
mean and true motions differ from each other; ſo that, the true
motion being defined by 2, the mean motion will be expreſſed
(B) fin.bz (C)fin.y%
by % +
&c. as above determined.
he
From
bos
+ &c.
fin. y+B.Z
Х
X
+ &c.
X
gh
+
by
160
The Reſolution of ſome General Problems
,
Fig. 31.
gh
From the expreſſion xz, here found, the proportion
2ceft
between the mean periodic time of the body in its orbit ABP
&c. and the periodic time in the circular orbit ADK, that might
be deſcribed, independent of the perturbating forces, by means
of a centripetal force ſufficient to cauſe the body to move there-
in, will be known: for the quantities e, f, and h being here,
each, equal to unity, the ſaid expreſſion will, in this caſe, be-
xz: whence it is evident, that the periodic time
in the circle, will be in proportion to the periodic time in the
orbit ABP &c. as unity is to
eft:
conie
8
20
b
APPLICATION to the Lunar Orbit.
Fig. 34
In order to apply the concluſions derived in the preceding
pages, to the determination of the lunar orbit and the different
inequalities of the motion therein, it will be neceſſary, firſt of
all, to inveſtigate the ſun's force to diſturb the motion of the
moon about the center of the earth; from whence all thoſe ine-
qualities, except that ariſing from the excentricity, are pro-
duced.
Let C, S, and B repreſent any three cotemporary places of
the earth, ſun, and moon, reſpectively; and, upon the dia-
gonal BS, let the párallelogram BCSH be conſtituted; mak-
ing BF perpendicular to CS.
If k be aſſumed to denote accelerative force of the earth to
the ſun, the accelerative force of the moon to the ſun will
be truly repreſented by k x which force may be reſolved
into two others, the one in the direction BC, expreſſed by
kx
and the other in the direction BH, expreſſed
SB) SC
SC
by k x
X
from which laſt, let the force k in the pa-
SB)
rallel direction CS be ſubtracted; fo ſhall the remainder
k x
SC2
SB)? ;
SC? BC
x
.
2
SC?
SB;
in Mechanics and Phyſical Aſtronomy.
161
3
SC, acting in the
CB
SC
x
2
CF
SC-SB
kx
be that part of the force, acting in the direction
SB3
BH, whereby the motion of the moon about the earth is dif-
SC-SB
turbed: But this quantity, kx is evidently equal to
SB
SC? +SCxSB+SB”
kx SC-SB X
; which, as SB (by reaſon of the
SBY
great diſtance of the ſun) is nearly equal to SF, will be alſo
SC+SCxSB+SB
CF
equal to k x CF x
or to 3kx sc, very near ;
SB3
becauſe SC being nearly equal to SB, it may be here ſubſti-
CF
tuted inſtead thereof. Now this force 3k x
direction BH, parallel to SC, may be, again, reſolved into two
others; the one in a direction perpendicular to BC, expreſſed
CF
by 3k x x ſin. SCB=3kx x ſin. SCB x cof. SCB=
SC
3k
CS
x ſin. 2SCB; and the other in the direction of CB, ex-
CB
preſſed by 3 K x SC X cof. SCB=3k x 5c x cof.SCB+=
3k CB
XI cof. 2SCB: from which laſt, let the force
SC BC
kx x (orkx BC ) above found, acting in the oppoſite
SBT
direction BC, be ſubtracted, ſo ſhall the remainder
k x sex. + 1 cof. 2SCB be the diminution of the centri-
petal force to the earth, ariſing from the action of the ſun. .
To exterminate k and CS from theſe expreſſions, let the
given quantity 0,0748, expreſſing the mean periodic time of
the moon, in parts of an year (as found by obſervation) be de-
noted by m; then (by the proportion on p. 160,) as 277 is to 1,
ſo is m to
meafi
-) the periodic time in the circular orbit ADK,
that might be deſcribed, independent of the perturbating forces,
by means of a centripetal force ſufficient to cauſe the moon to
revolve therein. But in circles the centripetal forces are
Y
known
2
X
SC
SB
CB
3
b
b
:
162
The Reſolution of ſome General Problems
CA
CS
pro-
bh
CB
CS
ed by 3m'ctf x CB
2hh
X fin.
X
CA
2bb
known to be as the radii directly, and the ſquares of the perio-
dic times inverſely; whence we have, as 1 (the force in the
circle ADK) is to k, the mean force whereby the earth is retained
in its orbit about the ſun, fo is mefni to : from which
k
portion
merf*:And the forces kx C9 * +1 coſ.2SCB,
CS hhxCA
kxCB
and x} fin. 2 SCB, whereby the motion of the moon
about the earth is diſturbed, will therefore be truly defin-
3m²e4f CB
CA X coſ.2SCB+, and
2 SCB.
If the fun and moon be ſuppoſed to move from the line CA
at the fame time, ſo that the angle ACS may be the ſun's ap-
parent motion about the earth, whilſt the moon in her orbit
moves from A to B, it will be, as I:m::2(=the angle ACB):
mz = the angle ACS, nearly; which would be ſtrictly true,
were the true motions of the ſun and moon to be exactly in the
fame proportion with the mean motions. Hence SCB (=ACB
-ACS) will be had =%-mz, nearly; and conſequently
2SCB=pz (by making p=1-m X 2): and ſo the forces
found above, by ſubſtituting this value, and writing in the
room of its equal will, by means thereof, be reduced to
СА?
3m%e4f
m*e*f
and
3m*ety
ſin. pz: which
quantities, with contrary ſigns (becauſe they diminiſh the centri-
petal force to the earth, and the area deſcribed, inſtead of in-
creaſing them) being compared with the two general ex-
preſſions
P' coſ. pate cof. qz+R' cof. rz &c. and
P fin.pz+Q fin.qxR fin rx &c. as given on p. 147,
* In deriving this concluſion, regard is had to the moon's relative motion about
the earth's center; but if we conſider the motion, as performed about the common
center of gravity of the earth and moon, the reſult will be exa&tly the ſame.
CB
I
I
x col.pzt has
X
x
Х
2bb
2bb
I
X
I
we
in Mechanics and Phyſical Apronomy.
163
X
bb
X
2
2
bh'
bh
we ſhall, in this caſe, have q=0, r=0, &c. P
3m² ,etf
etf
m2e4f
etf
-0,0084 x
=–0,0028 x
Q
(=P'), P= 37* X (=P), Q=0, R=0 &c. But
here, inſtead of h, or its equal, it1 BB+CC+DD &c.
(given at p. 159,) it will be ſufficient to make uſe of the firſt
term of the ſeries only (till a more exact value, by means of the
ſubſequent calculations, can be known); becauſe alf the quan-
tities B, C, D, &c. being ſmafi in compariſon of unity, the
ſquares of them, which are here neglected, will be ſtill ſmaller,
and of leſs conſequence.
If, now, the caſe under conſideration be compared with that
laid down, and reſolved, at p. 153, they will appear to be
the fame; ſo that we have nothing more to do here, than to
compute, in numbers, the different values of the algebraic
quantities there brought out. But one thing previous there-
to muſt be taken notice of, reſpecting the principal (B cor. B2)
of thoſe values, on which the great elliptical equation, ariſing
from the eccentricity, depends, which cannot be known but
from the obſervations of Aſtronomers; ſince it is owing to the
projectile-velocity which the moon, firſt, received, more than to
the perturbating force of the ſun, whoſe effect we are about
to calculate. But, though the term B x cof. Bz, cannot be deter-
mined by theory alone, yet the value of its exponent B, on
which the motion of the apogee depends, may from hence be
deduced. This value, in a former operation (ſee page 148,)
was found to be an unit (as the circumſtance of the problem,
when the apogee is at reſt, abſolutely requires); the true value
cannot, therefore, differ much from an unit, which may be
uſed inſtead of B, till a more exact value, by means of a ſecond
operation, is known. Making, therefore, B=I;
0,0084; P=P; Q=0; Q=P=P; m= -0,0748,
(=1-mx2) = 1,8504 y=p, =pB, Esp+B, &c. (as
before determined) and ſubſtituting in the equations on p. 151,
152, 153, we have
ż
P
P
eff
Y 2
164
The Reſolution of fome General Problems
PE
eP
petf
P
e4f
+
2x—0,0084 -0,0843–0,001748 ;
2Q
0,0084
ēc
0,0028;
PB1(= +
0,048B;
e*f
tit
0,0285B;
a ଧା
t
getf
euf
3
Q
4
I-BB
P
PB
Х
1
Donß
2
4
I-BB
PB
Х
PB2(=p7B
+
+
7
1
ololololololol
1
I-do
1
E
eff
4 + 2 +
2
7
4
PC2=
Iy
PC
十​三十
​X
Ptr
0,0404C;
e4f
PD
PDI(++
0,0414D;
P
4
-do
PD
PD2(=
x
0,0273 D;
pti
ctf
4Q
BO
+39
OBI (=
PB
BBx ex
q-ß
220f
0,0042 B (becauſe Q=0, and Q=P); and in the
very
ſame
manner, QB2=-0,0042B; QCI=-0,0042C; QC2=-
0,0042 C; QDi= 0,0042 D, QD2
QD2 = 0,0042 D, &c.
&c. *
Whence.
* )=
2
2
9
* In theſe calculations, all terms whoſe diviſors are found equal to no-
thing, will themſelves be nothing, and not infinite, as might at firit be ima-
coſpj.Z
gined: thus the term
by having its diviſor p =0, intirely va-
nilhes. The reaſon whereof will appear evident, if it be conſidered, that this
term,
cof p—4.2, ariſes by taking the Auent of —¿X fin.py..,, orixo:
which is, manifeſtly, equal to nothing. But an objection may, perhaps, be
brought from hence, againſt the truth and univerſality of the fluxionary
calculus; ſeeing thereby an expreſſion is here derived, which, though actually
equal to nothing, appears nevertheleſs under the form of an infinite quantity.
cof.pammy. Z
To clear up this point, it muſt be obferved, that is not the complete
fluent of -zx fin. py, but the variable part of it only; the corrected Au-
cof.py.Z
ent being
X-Itcof.py.z. But, whatever
panny
value pay is ſuppoſed to have, the cofine of my.Z will, it is known, be ex-
I
I
+
Or
- 4
preſſed
in Mechanics and Phyſical Aſtronomy.
165
Whence it is evident, that
B'(=QBI+QB2+PDI &c.) =-0,0084B-0,0414D,
C'(=P+QC1 + QC2 &c.) ==-0,01748—0,0084C,
D'(=PB1 +QDit QD2 &e.) =-0,048B-0,0084D,
E(= PB2 &c.) = -0,0285B,
F'(=PC2 &c.) =-0,0404C,
G'(=PD2 &c.) = -0,0273D, &c.
But B'being=-1BR.B, C=-my.c, D'=-1SD.D,
&c. the ſecond and third of theſe equations, will by ſubſti-
tution, be changed to i-y.C=0,01748+0,0084C, and
1-8. D = 0,048 B +0,0084D; whence C is given
0,01748
0,01748
-0,007186; and D=
lyga0,0084 I-1,85047-0,0084
= 0,178 B: which values being ſub-
1-0,85041-0,0084
ſtituted in the other equations, they will become B=-0,0158B,
E=-0,0285 B, FP = 0,0003, Gʻ= 0,005 B; whence
E(=-
- 0,004B, F(=
-p+B!
0,000023, G= 122) 0,0008B; and conſe-
1-2-310
quently 1-w= fex i-Bcoſ.Bz~Ccof.yz-D cof.dz &c.)
= e into 1-B cof. Bz+0,007186 cof.pz-0,178 B cof.Biz
+0,004Bcof. +3.z—0,000023coſ.2pz+-0,0008Bcoſ.2p-Biz
&c. expreſſing the reciprocal of the moon's diſtance from the
earth.
As to the quantity B, which enters into the greater part of
the terms of the ſeries here found, it cannot, as has been al-
0,048 B
E
F
2
1-4PP
G'
preſledby1_277*** +3.)*** &c. So that, the Auent in queſtion will be
I. 2
1. 2. 3. 4
I
pomy. 7+z4
1.2
truly repreſented by X
+
&c. or its equal --
+
px?z* &c. which entirely vaniſhes, as it ought to do, wher
1. 2. 3.4
-7.22
1. 2
1. 2. 3. 4
ready
166
The Reſolution of ſome General Problems
ready intimated, be otherwiſe determined than from the ob-
fervations of Aſtronomers: nor will the equation above expreſſing
the relation of B' and B, afford us the leaſt help therein: For,
by ſubſtituting -0,0158 B inſtead of its equal B', that equation
will become -0,0158 B -1-BB.B, or 1-BB=0,0158;
where, Bintirely vaniſhing, nothing in relation to it can,
therefore, be determined. We have here, indeed, an equa-
tion for finding the valuc of B; which from thence is given
✓ 1-0,015850,99206; by means whereof the motion of the
apogee will be known: for it will appear, by cor. i and 3 to prob.VI.)
that i-w=exI--B cof.Bz is the equation correſponding to a
moveable ellipſe, turning about the focus, or center of force,
with an angular celerity which is to thatof the body in the ellipſe,
every-where in the conſtant proportion of 1-B to B : whence
it follows, that the mean motion of the apogee, ought to be in
proportion to the mean motion of the moon, as 0,00794 to
unity: which differs from the real proportion (of 0,008455 to
1) as given from obfervation, by about is part of the whole
value: nor ought this to ſeem ſtrange, as a number of (ſmall)
terms yet remain to be introduced into the value of 1-2,
the corrections pointed out on p.156 and 157: beſides which,
the difference ariſing in the co-efficients of the terms already
found, by ſubſtituting this, new, value for B, will amount to
fomething conſiderable, from whence, alone, near half the
error would be taken away. But, to avoid the trouble of
repeating the ſame operation, again and again, with the new
values of B, thus found, I shall here, at once, take ß equal
to the true value (0,991545) as given from obfervation; but
if the eccentricity B, be ſuppoſed to vaniſh, i-w will then become = ex
1+0,007186 cof.pz0,000023C08: 2pz &c. which, when pz=0, or the moon is
the fyzigy, will be rexi +0,007186~0,000023=1,007163 Xe; but when
2pz=90 degrees, or the moon is in the quadrature, it will then be = ex
1-0,007786–0,000023=0,992791xe. Therefore, when the orbit is with-
out excentricity, the diſtance of the moon from the earth, in the fyzigy is in pro-
portion to the diſtance in the quadrature, as ----
or as
1,067163 xe
0:992791 00 -1,207163, that is, as 69 to 70, vary near; the ſame as is found bye
Sir Ifaac Newton in his Princip. B. III. prop. 28.
fhall,
1
I
to
2
O992791 x
in Mechanics and Phyſical Aſtronomy.
167
ſhall, at the ſame time, put down the ſeveral terms ariſing in
the equation for the value of B; by means whereof it will ap-
pear, in caſe both ſides are found to be equal, that the va-
lue of the root B has been rightly aſſumed; and that the mo-
tion of the moon's apogee (which has been the ſubject of ſo
much ſpeculation and controverſy) is intirely conſiſtent with
the general laws of gravitation.
Now, of the quantities above determined, PB1 and PDI are
thoſe that are the moſt affected by altering the value of B:
theſe being, therefore, computed a-new (making B equal to
0,991545, inſtead of unity) the former will here come out
0,04747 B, and the latter =
0,04168 D: from
whence, by proceeding as before, we have D=0,1869 B,
and 1-BB(=-1) =0,01619. As to the values of E, F,
G, &c. they are ſo ſmall, in themſelves, and fu little affected
by B, that to compute them a-new, would be quite unnecef-
ſary; the difference not producing an error of a ſingle ſecond,
in the place of the moon.
To apply, now, the obſervations laid down at page 156,
in order to obtain from thence a farther correction of the value
(ex
Si-Bcoſ.Bz+0,007186c0f.px0,1869Bcoſ.p-B.2
x{
+0,004Bcor.p+3.2-0,000023 cof.2pz +0,0008 Bcof.2p-B.z)
above found, the ſeveral powers of the ſeries B cof. Bz
0,007186 col.pz+0,1869 B coſ. pmb.z &c. (before omitted)
muſt be now taken, or ſuch terms thereof, at leaſt, as are of
conſequence enough to merit regard. Thus, in the ſquare, or
ſecond power,
the terms which appear conſiderable enough to
merit an examination, at leaſt, will be thoſe ariſing from the
ſquares and the double rectangles of the three firſt terms of the
root, which are vaſtly larger than the others. Theſe will be
B + Bcof. 2ßz - ,007186 B cof. P-Box-,007186
cof.p+B.z+0,1869 B* col. 2B.z +0,1869 B* col. pz +
,000025 +,000025 cofine 2px - ,00135 B coſine ße
,00135 B cof. 2p-B.2+,0175 B'+,0175 B* cof. 2p-2ß.z.
But, in the value of 1-W13, theſe terms are affected with the
common multiplicator (vide p. 154); and, in the value of
܀
6
There
168
The Reſolution of ſome General Problems
IO:
IMWI
6
10
N
84
P
N
cof. p.
X
x
2P
6
P
N
X
+
+
x
, by the common multiplicator ſo that, in order to
find the effect of the firſt of them (Bº) from the formula at
M
p. 156, we muſt compare - Bºx: and 1 Bºx, with
23
X
cof. az, and x coſ. az, reſpectively; whence M=3B',
N=5B’, and a=o: and conſequently,
MP aM
eff
+
+
pa
Impomat
MP aM
col.pta.z
eff
2P
pta
6
&c.=,029 B’ coſ. pz.
Iptal
In the ſame manner, with reſpect to the ſecond term, i
B'
cof. 2Bz, we have M=3B", N=5B’ (as before), and a=2B;
from whence the correction, anſwering thereto (excluſive of the
general factor e) will come out, 0,3178 B' cor.p2B.: +
,0023 B'cof.p+28.2 +0,0028 cof. 2ßz.
Again, in relation to –,007186 B coſ. p-B.2, we have
M=~,431B, N=4,0718 B, and a=p-B; from which
will be found ,000735 B x
,000073 B coſ. 2p-B.z
+,000464Bcoſ.p-B.z; where the firſt term is of that ſpecies
on which the motion of the apogee depends, and where the ſe-
cond is too ſmall to be farther regarded.
In like fort, from 4,007186 B cof.p+Biz, we ſhall get
50ообо Вх
beſides two other terms; which, on ac-
1-BBY
count of their extreme ſmallneſs, may be intirely neglected.
By proceeding on, in this manner, two, or three ſmall
terms, more, (producing equations of a few ſeconds, each)
will be found; which being joined to thoſe above, we ſhall from
thence have, +,03 B’ cof. pz + ,00055 B col. p-B.z +
0,3028B'col.p—2B.z+,0072B'coſ.2Bz+,0070B'coſ.zp—2B.Z
+,ooo16 Bx ; which, drawn into the general multiplica-
1-BB
tor e, will give the whole increaſe of I-w, by taking in the
ſquare of the ſeries aforeſaid. As to the increaſe ariſing from
the
cof. BZ
IPB
cor.bz
i
cof. 8%
in Mechanics and Phyſical Aſtronomy.
났
​169
coſ.cz
I-318
the cube thereof, few of the terms will be found conſiderable
enough to deſerve notice, the only ones of any conſequence to
be collected from thence, being-,00004B+0,7B'x col.p. 1.2
-0, 11B'x coſine p-3.8.2 —,08 B' x cof. 2p-3B.z
--,055B' x : But, from the biquadrate, we ſhall only have
one term (1,1B*coſ.p-2B.z) that can produce an equation of
more than a ſecond, or two; though the effect of this one term
will, alone, amount to near a minute; which is owing to the
ſmallneſs of the exponent p-2B, conſonant to what has been
before inculcated (at page 157). And it may be worth while
to take farther notice here, that all the terms hitherto deter-
mined, except two or three, belong to one, or the other of
the two kinds there ſpecified.
If, now, the ſeveral quantities above brought out, be collected
into one fum, we ſhall have 0,03B” cof.pz+,00051B-0,7B'
x cofine p-B.z +0,3028B' 7 1,1B* x coſine P-2B.x
+0,0072B'coſ.zßz+,0070B'coſ.25—23.2-0,11B'col.p-38.2
-,08B'cof. 2p-3B.z+,000163—,055B'x of all which
terms, 0,3028B* +1,1B* x cof.D—2B.z is by far the greateſt,
and ſo conſiderable as to produce other terms, or correcti-
ons, of conſequence enough to be regarded here. The new
terms ariſing from the entering of this term into the firſt
power of the ſeries, for the value of 1-w, will appear to be
--,0032B*cof.222--0047B’coſ.mp—23.3+,0026B'coſ.p—2B.%
(which are found as above, by making M=3* -0,3028B”,
N=4x=0,3028B", a=p-23, and neglecting 1,1 B*) :
And by the entering thereof into the ſquare of that fe-
ries, from the double rectangle under it and the firſt term,
there will be had +,088 B'coſ. 2p-3B.z
,019B'coſ.p—3B.z+,03B' x which, together with the
quantities before found, being joined to the former value of
I-W, we have at length
cor.Bz
I-B
cof. Bz
I -BB
N
170
The Reſolution of ſome General Problems
>
I - Wex
1s
138
1-38
1
كردم
1
I--Bcof.sz+.007186+,03B"xcol.pz-0,1864B+0,7B'xcol.priz
1,004Bcoſ.p+Bezt0,3054B²+1,1B4xcoſ.p-2B.Z-,000023coſ.2p%
t.coo8B cof. 2pmb.z+,004B? cof. 262 +0,0023B” cof. 2p - 27.2
[+0,13B:cof 1-3B.z
In which equation the terms of the ſpecies, cof.Ez, are not put
down; becauſe the original term-B cof. (ez, which is equivalent
to them, is ſtill retained.
Thoſe terms, however, though of no uſe in this equation,
are not to be intirely diſregarded; ſince on them the motion of
the apogee depends. By collecting them, and the former value
of —B, into one ſum, we get the equation - B=_2016193
,00016B-,055B3
,03B3
201603B+,025B3
+
+
: from whence,
when B is aſſigned, the value of B will be found, but fill ſome-
thing ſhort of the true value, as given from obſervation.
But it ought to be now remembered, that all the equations,
hitherto brought out, are derived on the hypotheſis, that the
true motion of the ſun is to that of the moon, in a conſtant pro-
portion: which is near the truth, only; fince the diſtance of
the ſun from the moon, whereon the perturbating forces de-
pend, is ſometimes greater, or leſs by two degrees, than ac-
cording to that hypotheſis: which difference muſt, of conſe-
quence, render other corrections neceffary. How theſe may
be
introduced I ſhall here inſtance, by firſt conſidering the noti-
on of the earth about the ſun, as uniformly performed in a cir-
cular orbit; leaving, to be determined by a future operation,
the other equations ariſing from the eccentricity and parallax,
which will be obtained in the ſame manner.
It is found at page 159, that the moon's mean motion, an-
ſwering to the true motion %, is rightly repreſented by % +
(B) xlin.Az (C)xſin.yz (D) xſin.dz
+
+
&c; from whence the mean,
hy
or true motion of the fun (for, in a circular orbit, they are the
(B) xſin.Ez (C)xſin goz
fame) will be had = mx zt
&c.
by
(B) x fin.cz
=mz+mz, ſuppoſing z'.
---
the
13
ho
+
be
(C)Xſn.7Z &c. Hence
ba
br
in Mechanics and Phyſical Aflronomy.
171
3
tot
ha
&c. =
the true diſtance of the moon from the ſun, will be
mz', and the double thereof equal to 1-mx2z_2mz', or to
pz_2mz': of which the ſine is =fin.px x coſ. 2mg-col.pz x
fin. 2mz';. but the arch 2mz' being exceeding ſmall, the fine
thereof will be nearly equal to the arch itſelf, and the co-
ſine very nearly equal to the radius, or unity; and conſe
quently the fine of the double diſtance of the moon from thę
201(B) x fin.bz
ſun ſin. pz.col.px x 2mz'=fin.pz-cor.pxx
m(B)
ſin.px + x fin. p-R.2
fin.pt-B.z + " Tony
(C)
hß
x ſin. --7.2-ſin.ptr. z &c.
In like manner, the co-line thereof will be had (=
col.pz x coſ: 2mz'+fin. pz x ſin. 2mz'=cof.pzt-fin.pz x 2mZ
m(B)
nearly) = col.pz + x cof. P-2.2 col. pt.3.2 +
be
m (C)
X col.p-.2 -- col.ptv.z8oc.
by
Which two quantities being multiplied by P, it will ap-
pear, that the new terms ariſing in the two expreſſions of the
forces (beſides thoſe, P ſin.pz and P cof.pz. in the former
MP (B)
mP (C)
hypotheſis) are x ſin. 0-B.z-ſin.pt-b.xt
mP (B)
x ſin.p-y.z-pty.z&c. and x cof.p-3.2col.p+2.7
be
mP(C)
+ x coſ.p.ptyiz &c.
Now it will appear, by a compariſon with the formula on p.
157, that the effect of the two firſt, correſponding terms here
mP (B)
exhibited, making a=p-B, will be
+ 1x
b Betf
mPB (B)
+ to
hzetf
7
mPB (B)
cofint +Z
g +++ -
x
&c. which, be.
ats
bisexf
I
cauſe m=,0748,
, =--,0084, and h=1, will be equal to
14f
Z 2
h3
hy
by
cof. #7
2
X
ITT
i
B
4
IM3B
cof.
2x
X
x
+
7T
2
77
I am
4
I-33
X
P
172
The Reſolution of some General Problems
sa
col. az Z
4
B
IPB
B
IT
TB
2
Bcor. 7
4
X
1-33
X
titi
Ccof.y.Z
2
I
4
7
1
Iyy
Х
T
2
I-
B
y
>
B
,00063(B)
into + 1x +
+1-
4
B cof. tom.z
+
+
TAB
1-4+3,1
ti
+ &c. =-,0081 (B)
x coſine p-Box +,0187B (B) x coſ.p-2B.z +,0011 B(B)
x cof.pz&c. where all the terms, after the two firſt, may, on
account of their extreme ſmallneſs, be intirely neglected.
In the very fame manner, expounding # by p+3, pg
(=o), pty (=2p), p-(=B), and p+8=20—3,) ſúcceſ-
,00063(B) ,00063(C) ,00063(C)
ſively, and ſubſtituting +
(
,00063(D), and +300063(D), reſpectively, in the room of
,00063(B)
(B); there will, in the firſt caſe, be found the term
-,00015 (B) x cof.p+B.z; being the only one producing an
equation of more than a ſingle ſecond; in the ſecond and third
caſes, no term at all
, worth notice, will be found: but, in the
fourth (where =) two pretty conſiderable ones ,00222 (D) *
cor, 8%
and 0,0233 D(D) x coſ. p2B.z, do ariſe ; whereof
the former is of that ſpecies on which the motion of the apogee
depends, and gives, an increaſe of that motion, of about part
.
By purſuing the ſame method, the effect of the ſecond order
(BB) x fin. 282 (BC)xſin. 7-B.z &c. may
&c. may alſo be comput-
but here it will be ſufficient to make uſe of the firſt term
of the general formula alone: from whence are obtained the
quantities +,0045 (BB) x cof. p-2B.z, and — ,0059 (DD)
x cof. P~2B.z; which are both of that kind, ſpecified at p.
157, rendered confiderable by the ſmallneſs of their expo-
nents, and are the only ones bere, that merit regard. Theſe being,
therefore, joined to thoſe found
above, the whole correction fought
will be, -0,0081(B)x col.paß.2-0,00015(B)xcoſ.P+B.2
+
1-BB
of terms,
h.28
5.B
ed;
I
1
in Mechanics and Phyſical Aſtronomy.
173
I-Wex
I-BB
I-BB
I-BB
+0,0187B(B) -0,233D (D) +0,0045 (BB) –0,0059 (DD)
X coſ.p—2B.z. But (B)being =2B, (D)=2D (BB)={B”,
and (DD)=_D’, nearly, vid. p. 159 (B, C, D, &c. being
neglected, as too inconſiderable to be regarded here) our expreſſi-
on may therefore be changed to 0,0162 B col.p-B.z
-0,0003 Bcoſ.pt B.z+0,0441B' -0,0554Dºxcol.p-28.2
which being added to the value of 1-W, before found, we
thence have
--Bcof.Bz+0,007186 +0,03B* xcol.pz-0,20263 +0,7Bsxcol.p- Boz
+0,0037Bcof.p+boz+0,03495B2-0,0554D +1,1Bxcor.p-2B.%
--,000023 cof. 2pz + ,0008 B cof. 2--B.z +,004B2 coſ. 2B z +
20023Bạcof. 2p28.340,13B3 col.p-38.2
0,00222(D)xcor.bz
And, by writing the coefficient of the term
in the equation for the motion of the apogee, it will here be-
0,01603B+0,025B3 0,00222(D)
come - B
0,01693B+0,028B3
becauſe (D)=2D=2x0,2026B+0,7B':
1-BB
whence 1-BB = 0,01693+00028Bº=0,017015 (ſuppoſing
B=0,05505); and conſequently 1-B=0,00854; which
value is, now, about tó part greater than the true value, giv-
en from obſervation, and is near enough to ſhew, that the
force of the ſun is ſufficient to produce all the motion of the
moon's apogee, without ſuppoſing a change in the general law
of gravitation, from the inverſe ratio of the ſquares of the di-
ſtances. Several
Several very ſmall terms belonging to this equation,
- have been omitted, beſides thoſe ariſing from the ſun's excentri-
city, and the inclination of the lunar orbit; which together,
may very well be ſuppoſed, fufficient to cauſe a difference equal
to that abovementioned.
If, now, the value of, B expreſſing the mean excentricity,
in Sir Iſaac Newton's lunar theory, be expounded by 0,05505,
according to that author, the general equation of the orbit,
found above, will here become
3
7
I -
174
The Reſolution of fome General Problems
1 -Uex
1-0,05505coſ.B2-49,007276col.pz0,01127 col. D--B.z
1-wex} +0,000204 col.p+B.z+0,0010623c0f.p—23.2-0,000023 col. 2px
40,000044cof. 2p-B.z+0,000012cof. 282 +0,000007cor. 2p-22.7
40,00002) coſ. p3k.z
But this value muſt now be corrected by the difference ariſing
from our having, in all the preceding calculations, taken the di-
viſor bh=1, inſtead of the true value it xBB+CCFDD&c.?
(vid. p. 159.) which value, becauſe B==0,05505, C=-
0,007276 &c. is given =1,0097. From whence and the equa-
tions, on p. 153 and 163, it appears, that all the terms above
exhibited, in whoſe exponents the quantity p is,
fingly, concerned,
ought to be diminiſhed in the ratio of 1 to 1,0097 ; and that
ll thoſe, where 2p is in like manner concerned, ought to be di-
ininiſhed, in the duplicate of that ratio: by which means our
equation is, at length, reduced to
1-0,05505 cof.Bz-+0,007206coſ. p2--0,01116 col.paß.z
0.co0202c01.p+3.7 +0,001092 col. pn-2B.2-0,000022 cor.2px
40,000043coſ.2p-B.z+0,00001 2cuſ.232 +0,000007cof. 22-28.7
+0,00002 i col.38.2.
From whence all the great equations of the moon's motion,
and all the ſmaller ones, except thoſe depending on the ſun's
excentricity &c. are obtained, within leſs than half a minute
of the truth ; fuppoſing the mean excentricity (B) to be here
truly aſſigned. If it ſhould be found neceffary to augment, or
diminiſh the value thereof, then the term -0,01116col.pß.2
(producing the equation, called the evection) muſt be alſo
aug-
mented or diininiſhed, in the ſame ratio; and the term
+0,001052c0f.pm-22.2(which is the next conſiderable of thoſe
wherein B enters) muſt be augmented or diminiſhed in the
duplicate of that ratio. As to the reſt of the terms, they are
ſo ſmall, that a little alteration in the value of B will produce
no difference in them worth notice.
In the fame manner, the inequalities cauſed in the moon's
motion by the ſun's excentricity, may be computed. For the
mean motion of the moon being given, very nearly, by the pre-
ceding calculations, the mean motion of the ſun, being in pro-
portion thereto as m to 1, will be alſo known; from whence,
and the excentricity, the ſun's true anomaly, and diſtance from
the
1
1
in Mechanics and Phyſical Aſtronomy.
175
izen
X
20
the earth will be had, in a ſeries of fines and co-fines of the
multiples of the arch 2; whoſe exponents and coefficients are all
intirely known: by, means of which, the ſun's true diſtance from
the moon being (nearly) obtained, the new terms ariſing from
the eccentricity, in the general expreſſions for the perturbating
forces, will be had, in conſequence thereof; and, laſtly, the effects
themſelves, produced thereby in the general equation of the orbit.
When the equation of the orbit, or the value of 1 --w (the
reciprocal of the moon's diſtance from the earth) is thus deter-
mined to a ſufficient degree of exactneſs (by repeating the ope-
ration if neceſſary), the difference between the true, and mean
motions of the moon, muſt, from thence, be found, in terms
of the latter: this may be done by firſt finding the mean mo-
tion in terms of the true (as is fhewn at p. 159) and then
reverting the ſeries; or, otherwiſe, without finding the mean
motion at all, by the reſolution of the fuxionary equation
ETXE
; according to the method made uſe of
of in a former paper, in this collection. The proceſs (which
is more laborious than difficult) I ſhall not inſert here, as my
deſign, in this miſcellaneous work, is not to exhibit every opera-
tion neceſſary to the forming of a complete theory of the moon's
motion; which, for its importance, and the great variety and
intricacy of calculations ariſing therein, may very well merit
to be the ſubject of a volume, by itſelf.
It
may
fuffice, in this place, to have pointed out (by a me-
thod not very perplexed) how the different inequalities of that
motion may be determined, and the moon's true place, acccrd-
ing to gravity, aſcertained. At another time, if health permits,
I may, not only give, at large, the application of the equations
and precepts here delivered, but alſo a new ſet of lunar tables,
deduced therefrom; which (though a work of much labour)
I ſhall the more chearfully undertake, as Dr. Bradley has
very obligingly offered to affiſt me with any obſervations,
that may be wanting in order to the compleating of the
deſign, and eſtabliſhing the theory on a proper baſis. And
I have ſome reaſon to hope, that, when that part of the data,
which can be only known from obſervations, is truly ſettled,
the
2
176
The Reſolution of ſome General Problems
the place of the moon may be always obtained, within about a
minute of the truth, even, without the uſe of a multitude of ta-
bles, by means of proper contrivances, reſpecting the ſmaller
equations, or the leſs conſiderable terms of the general ſeries.
I ſhall conclude what I have to ſay on this ſubject, at preſent,
with obviating a difficulty in the application of the equations
abovementioned, when the effect in the moon's motion, de-
pending on the ſun's eccentricity and parallax *, are computed
thereby. In this caſe, a new ſpecies of (very ſmall) terms, af-
fected with the co-fine of the arch 2, will be found to enter
into the general equation (on page 151) whereof the effect can-
not be determined in the fame manner with that of the other
terms, affected with the co-lines of the multiples of that arch:
For if, according to the method of proceeding there laid down,
we aſſume a term (as I-77. II cof. nm) in the upper line,
1-B2.B cof. Bztimyy. Ccoſ. 7z+1-dd. Dcof. dz &c. in or-
der to compare it with a term, g cof., of the aforeſaid ſpecies,
we
* To determine the perturbating forces, ſo as to take in the effect of the
ſun's eccentricity and parallax ; let a=CS, x=CB, and y=CF; then BS be-
CS2 x CB
ing=V aa-2ay+*x, the force k x
in the direction CB (vid. p.
BSI
kx
2y
kx
39
CS3
160) will be = aximă
+
Xit
&c. and that (kx SBI
aa
a
2y
-k) in the direction BH, equal to k XI-
+
k=kX
aa
3y
15 yy
3**
+
2aa
&c. which, becauſe y=*X cor. BCS, will be reduced to
a
2aa
a
AL
kx 3.x
Xit x cof.BCS &c. and kx 3 x cof. BCS + x9+15coſ. 2BCS&c.
4aa
reſpectively. But the laſt of theſe forces may, again, be reſolved into two o-
thers; the one in the direction BC, expreſſed by k cof. BCS X
3*
3kx
x cof. BCS + x9+ 15 col. 2BCS, (= x1 + cof. 2BCS+
4aa
3kxx
8aa
X II coſ. BCS+ 5 coſ. 3BCS); and the other, in a direction perpendicular
3x
thereto, expreſſed by kſin. BCS X Xcoſ, BCS +
4aa
X9+ 15 col. 2BCS
(=
a
2a
in Mechanics and Phyſical Aſtronomy.
177
we ſhall then have a=1, and conſequently 1-1=0; and
fo, the whole (1-77. col. Tz) intirely vaniſhing, we have
nothing left to compare with the term propoſed (8 col.z) where-
by its effect can be known. Nevertheleſs
, it is by means of a
term (II cof 7%) of the ſame ſpecies, entering into the ſeries for
the value of 1-W, that the effect in queſtion muſt be exhibit-
ed: for, though indeed the term II cof. (or I cof. z), by re-
ceiving the co-efficient 1–7 (=o) intirely vaniſhes, in the
firſt line of the general equation, yet, that will not be the caſe
in the other parts of the equation; wherein it will be found,
affected in the ſame manner with the term B cof. B2 (whereof the
determination hath been given, in computing the motion of the
apogee); which muſt neceffarily appear to be the caſe, if it be
conſidered, that the terms 1 cof. 72 and B cof. Bz are alike
concerned in the original ſeries 1 - Il coſ.nz-B cof. Bz-C cof.
72 &c. from whence the equation itſelf is derived. Hence we
not only have a proper term to compare with the given one
(8 cof.z), but alſo an eaſy way to diſcover what the reſult will
34*
2a
ܨܐ
2a
3**
Xfin. 2BCS + Saa x fin. BCS + 5 fin. 3BCS). From the former
Rix
3*
of which, let the force XIt X cof. BCS, in the oppoſite direction be
3txx
fubtracted, and the remainder, xi+3cof.2BCS +
х
8aa
3 cof. BCS+5 col. 3BCS, will be the force whereby the gravity of the moon
to the earth, is diminiſhed. But the quantity k (which muſt be exterminated)
is, at the mean diſtance (d) of the ſun and earth, found to be equal to d x
m²e4f
mºe4fd3 metf
(vid. p. 162 ); whence, a?:d2::dx ThxTA: aa
X
the
bhxCA
hhxCA
general value of k; which being ſubſtituted inſtead thereof, the force in the di-
rection of the radius vector will become
m2e4f. d3 CB
CB
15
*23*CA X1+3coſ.2BCS +
CS
*coſ. BCS + cur. 3BCS; and
4
that in the perpendicular direction,
m²e4fd3 CB
CB
15
26" ** CA * 3 fin. 2BCS + T5 x fin. BCS +
fin. 3 BCS:
4
Where m-=0,005595, and hh=1,0097, nearly (vid. p. 174); and where
no regard need be had to the value of c4f, as this quantity, when ſubſtitution is
made in the general formula (on p. 157.) intirely vaniſhes.
2564
Аа
bes
178
The Reſolution of ſome General Problems
to be
I
g
I--B3
be; for ſeeing that II cof. and B cof. Bz have there the ſame
coefficients, and that the ſum of thoſe belonging to Bcof. Bz
has been already found, in computing the motion of the apogee,
=-1-BB, it follows, that -1-BB. 17 cof. z (or
BB.11 cof.z) will expreſs the value cf all the terms of this
ſpecies, ariſing in the equation from the introduction of II cof.
into the original, or aſſumed ſeries :. whence, by compar-
ing this quantity with the given one, g coſ.z (that is, by
making-1-B2.II coſ. % + g coſ. z =o) we get 11 =
Ses; and conſequently Il coſ. z= x cof. z; whereby the
required effect of the quantity g cof.z, is known.
Tis true, indeed, that the new term II cof. z, thus deter-
mined, will introduce an infinity of others; but, of theſe, none
will be conſiderable enough to merit the leaſt degree of atten-
tion, except (perhaps) ſuch as are expreſſed by the co-fine of
1- -B.z; which, becauſe of the ſmallneſs of 1-B, may, for
reaſons before ſpecified, produce an effect not to be neglected
without a proper examination. To give here the quantity of
this effect, it will be neceſſary to obſerve, that, of all the terms
in the values of 1 --W1–3 and 1—27-4, that which ariſes by the
multiplication of 4 x coſ.z into 2DxCof. dz, and comes from
the part a xrīcoſ. 2+Bcoſ.Bz+C coſ. yz+D cof. dz &c.
will ſo much exceed, in its effect, the others wherein II is con-
cerned, as to render them, in a manner, inconſiderable. But this
cof. zx2Dcof.dz (or cof. 2 X 2 col.-B.x) is
IOTID
reſolved into x cof.p-B+1.2,and xcof.D-B-1.2;
whereof the former part only, in which the difference of B and
1 is concerned, is for our purpoſe: from whence (as appears
by the obſervations at p. 156) a new term expreſſed by
10IID cor.B1.%
c*f* 1-
x col.B-1.2, nearly) will
ariſe in the required value of 1-.
Of the two terms here
found, the former gives a very ſmall equation depend-
ing
IONI
IOII
1οΠD
term
IONID
eP
ер
IND
ett
X
Х
BI
577(or px
IS
in Mechanics and Phyſical Apronomy.
179
ing on the moon's diſtance from the ſun's apogee; and
the latter, an equation ſomething more conſiderable, whoſe ara
gument is the diſtance of the moon's apogee from that of the ſun.
By having ſhewn above, that the effect of ſuch terms, or
forces, as are proportional to the co-fine of the arch 2, is expli-
cable by means of the co-fines of that arch, and of its multiples,
(no leſs than the effects of the other terms that are proportio-
nal to the co-fines of the multiples thereof) a very important point
is determined: For, ſince it appears thereby, that no terms enter
into the equation of the orbit but what by a regular increaſe and
decreaſe, do after a certain time return again to their former
values, it is evident from thence, that the mean motion, and
the greateſt quantities of the ſeveral equations, undergo no
change from gravity.
THE END.
c3z
r.
2a
3
A
E RR A T A.
Page 5 line 18, for R read R'; 1. 21, for R' r. R; p. 30. 1.9, for ha tbh r.
ha+b; 1. 13, for tr. h; p. 39. 1. 14, put the comma before ros; p. 58. 1.
11, for PSO r. PŚC; p. 59. l. 15, for 1. 22, for Fr. N; p. 114,
1. 32. r. powers of; p. 127, 1. 9, for AGO r. AOG ; p. 128. 1. 30, for BL.
BR; p. 131. 1. 23, for BC r. nC; p. 138. I. 15, for Eef r. Ee; p. 143, 1.
15 dele to; p. 152, l. 2, for BB, s. yy; p. 160. l. 27, for
SC S
In fig. 18, at the interſection of the circumference of the circle and the right-
line Fm, place an e.
BC
BC
1.
So
!
I.
f
%
* Printed for J. Nourse the following Books, all written by Mr.
THOMAS SIMPSON, F. R. S.
SSAYS ON SEVERAL CURIOUS AND USEFUL SUBJects, in
E
ſpeculative and mixed Mathematicks; in which are explain-
ed the moſt difficult Problems of the firſt and ſecond Books
of Sir Iſaac Newton's Principia ; being an uſeful Introduction to
Learners for the Underſtanding that illuſtrious Author ; Quarto,
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III. The DocTRINE OF ANNUITIES AND REVERSIONS, deduced
from general and evident Principles, with uſeful Tables, ſhewing
the Values of ſingle and joint Lives, &c. at different Rates of
Intereſt. To which is added, a Method of inveſtigating the Va-
lues of Annuities by Approximation, without the Help of the
Tables. The whole explained in a plain and ſimple manner, and
illuſtrated by a great Variety of Examples; Oétavo, 1742.
IV. AN APPENDIX, containing ſome Remarks on a late Book on
the ſame Subject, with Anſwers to ſome perſonal and malignant
Miſrepreſentations in the Preface thereof; Octavo, 1742.
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ſtrated and applied in many uſeful and intereſting Inquiries, and
in the Solution of a great Variety of Problems of different Kinds.
To which is added,
The Geometrical Conſtruction of a great Number of linear and plane
Problems, with the Method of reſolving the ſame numerically.
The Second Edition, with large Additions ; Octavo, 1755;
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Efſay on the Maxima and Minimna of Geometrical Quantities,
and a brief Treatiſe of regular Solids; alſo the Menſuration of
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tion and Application of Logarithms, Octavo, 1749.
IX. Select Exercises for young Proficients in the Mathematics,
Octavo.
Α.
A А
Fig.1
Fig.2
2.
Fig. 3.
P
D
D
R
ALA
R
\R
72
А.
G
0
G
F
F
O
a FH
F
a
e
E
P
GM
a
Fig.6.
Fig. 4.
R
Fig. 5.
GM
P
P
VEC
B
FK
H
E E
T4
FI
H
O
10
Vh
W
a
I
NG
NG
Fig.9
C
Fig. 8.
GI
G
Fig. 7.
P
Q<
K
D
C
HL
a
E
S
or
S
O
G
S
HN
C
Fig.12.
G
Fig.10
K
Α.
m
Fig.11. P
E
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