ESSAYS ON SEVERAL Curious and Ufeful Susyecrs, In Specunative and Mrx’p MATHEMATICKS. Iluftrated by a V ariety of EXAMPLES. By THOMAS SIMPSON. Printed by H. Wooprant, jun. for J. “wees at the Lamb without Temple-Bar. M.DCC.XL, } : 0 rn - rors ee ITER ny er Or FRANCIS BLAKE, Orr Twifel, in the County of Dzrham, Efq; \e S our private Correfpondence ® occafioned my Drawing up fe- “e veral of the following Papers, I thence claim a Sort of a Right to addrefs them to You: But I well know the ( 1) common Style of a DEDICATION would to You be highly offenfive ; therefore all the Ufe I dare make of this Opportunity, is, to.declare Myfelf to be, STR, Your moft Obliged, Humble Servant, Tuo. Simpson, pau IE Reader, I prefume, will excufe me, ify Lie KEN inftead of acquainting bim, in the ufual Way, Y ayith the many weighty Reafons that induced 2 me to publifh the following Sheets, I foall 2 rake up no more of his Time than to give 4, concife Account of the Nature and Uje- eral Papers that compofe thts Mifcellany, zv The firft, then, ts concerned in determining the Apparent Place of the Stars arifing from the progrefive Motion of Light, and of the Earth in its Orbit ; which, though it be a Matter of great Importance in Aftronomy, and allowed one of the fineft Difcoveries, yet had it not been fully and demonfira- tively treated of by any ‘Author, or indeed thrown into any Method of Prattice. Now, however, I muft not omit to ac- knowledge, that in the lajt Volume of the Memoirs of the Royal ACADEMY of SCLENCES, for the Year 1737. ee a lately v1 Pity wee Ae Cr Bi lately publifhed at Paris, and brought hither a few Weeks fince, there 1s a Paper on this Subject? by Monfieur Clairaut, a very eminent Mathematician of that Academy ; to which he fubjoins a Set of Praétical Rules for the Aberration in Right-Afcenfion and Declination only ; wherein moft of bis Analogies are exactly the fame as thofe inferted in this Book, with which Dr. Bevis favoured me: For which Reafon I think it proper to affure my Readers, that my Paper, toge- ther with the Doctor's Rules, were quite printed off, and in the Hands of feveral Friends, who defired them, before Chrift- mas 1739. when the Severity of the Seafon interrupted for a confiderable Time the Impreffion of this Treati/e. The fecond Paper, treats of the Motion of Bodies affected by Projectile and Centripetal Forces ; wherein the Invention of Orbits and the Motion of Apfides, with many others of the moft confiderable Matters in the Firft Book of Sir Vaac Newton’s Principia, are fully and clearly inveftigated. The Third, fhews how, from the Mean Anomaly of a Pla- net given, to find its true Place in tts Orbit, by three fevera} Methods ; but what may beft recommend this Paper, is the Praétical Rule in the annexed Scholium, which will, 2 hope, be found of Service. The Fourth, includes the Mation and Paths of Projectiles in refifting Mediums, in which not only the Equation of the Curve defcribed according to any Law of Denjfity, Refifance, &c. hut all the moft important Matters, upon this Head, in the Second Book of the alove-named tiluftrious Author, are determined in a new, eafy, and comprebenfive Manner. The Fifth, confiders the Refiftances, Velocities, and Times of Vibration, of pendulous Bodies in Mediums, Lhe ‘d J tx 9) > 2 4 es Wit The Sixth, contains a new Method far the solution of Kinds of Algebraical Equations in N rb | more general than any bith derable Ufe, though tt perbaps may be objected, that the Me thod of Fluxions, qwhereon it 4: f junded, betng a-nvore exalted Branch of the Matbematicks, cannot be Jo f y appli what belongs to common iigevra. The Seventh, relates to the Method of Incremenis ; is illuftrated by fome familiar and ufcjul Examples, The Eighth, is a foort Invefigation of a Theorem for jind- ing the Sum of a Series of Quantities by Means of their Differences. The Ninth, exbibits an eafy and general Way of Invefiiga- ting the Sum of a recurring Series. Thefe three laft Papers relate chiefly to the Inventions of Others : As they are all of Importance, and are required in other Parts of the Book, I could not well leave them entirely untouch’'d; and if I fhall be thought to have thrown any new Light upon them, that may benefit young Proficients, 1 have my End. The Tenth, comprebends a new and general Method for find- ing the Sum of any Series of Powers whofe Roots are in Arithmetical Progrefion, which may be applied with equal Advantage to Series of other Kinds. The Eleventh, is concerned about Angular Sections and fome remarkable Properties of the Circle. ily te ae vill P R°E F AC &, The Twelfth, includes an eafy and expeditious Method of Reducing a Compound Frattion to Simple Ones; the firft Hints whereof I freely acknowledge to have received from Mr. Muller's ingenious I reatife on Conic Seétions and Fluxions- fra 2/0. ler Bho xl. Gaare. Lebreily 3. Cet: 6 fan 48. The Thirteenth and laft, containing a general Quadrature of Hyperbolical Curves, is a Problem remarkable enough, as well on account of its Difficulty, as its having exercifed the Skill of feveral great Mathematicians ; but as none of the Solutions bi~ therto publifhed, tho’ fome of them are very elegant ones, extend farther than to particular Cafes, except that given in Phil. Tranf, Ne 417. without Demonftration, I flatter myfelf that this which I have now offered, may claim an Acceptance, fince it is clearly inveftigated by two different Methods, without ree ferring to what hath been done by Others, and the general Conftruétion rendered abundantly more fimple and fit for Prac- tice than it there is. | ‘ SiS ot 8 On feveral Curious and Ufeful Subjects in Speculative and Mixt Mathe- maticks. Of the Apparent Places of the Frxep Sr ars, arifing from the Motion of Light, and the Motion of the Earth in its Orbit. PROPOSITION I. If the Velocity of the Earth in its Orbit bears any fenfible Pro- portion to the Velocity of Light, every Star in the Heavens muft appear diftant from its true Place; and that by fo much the more, as the Ratio of thofe Velocities approaches nearer to that of Equality. meeaesA~ OR, ifwhiletheLine fF E D C | Leeae ) CG is defcribed by res Nate a Particle of Light © Z| \ep/Ze@ coming from a Star ~ s 4a in that Direction, the Eye of an Obferver at T be carry’d, by the Earth’s Motion, thro’ TG; H and CT be a Tube made ufe of in T obferving; and a Particle of Light, from the faid Star, a | phys | B juf G 7% juft entering at C the End of its Axis; then when the Eye is- arrived at v, the Tube will have acquired the Pofition v D: parallel to TC, and the faid Particle will be at the Point m, where the Line CG interfeGts the Axis. of the Tube; becaufe GT:GC::Tv:Cm. Let now the Tube, by the Earth’s Motion, be brought into the Pofition Ew; then becaufe GT:GC::Tw:Cnx, the Particle will be at ~, and there- fore is ftill in the Axis of the Tube: Therefore when it en- ters the Eye at G, as it has all the Time been in the Axis of the Tube, it mu confequently appear to have come in the Dire@tion thereof, or to make an Angle with T H,. the Line that the Earth moves in, equal to CTH, which is different. from what it really does, by the Angle GCT :. Whence it is evident that, unlefS the Earth always moves.in a Right Line. dire@tly to or from: a given Star (which is abfurd to fuppofe) that Star muft appear diftant from its. true Place ; and the more fo, as the Velocity of the Earth (in refpect of that of Light), is increafed. And the fame muff neceflarily be the Cafe when the Obfervation is made by the naked Eye;. for the Suppofition and Ufe of a Tube neither alters the real nor apparent Place of the Star, but only helps to a more eafy De- monftration. PR Oz C3} PROPOSITION Il. To: find the Path which a Star, thre the aforefaid Caufe, in: one entire Annual Revolution of the Earth, appears to.defcribe. ET ATBA be Earth; S the Sun in one Focus; F the other Focus; T the Earth moving in its: Orbit from A. towards B ;: DTzaTangentat T; and: SD, FE Perpendiculars- thereto: Let Qu KRQ_ be Part of an indefinite Plane parallel to that of the Ecliptick,. paffing thro’ R the Centre of the given. Star; and take Tz to TR, as the Velocity of the Earth. in its Orbit at T, to that of a Particle of Light com- ing from the faid Star: Let "Tm be parallel to zR; PV perpendicular to AB 5. and QRK.: parallel to PaV: Then from the foregoing Propofition it is- manifeft, that a Ray of Light coming from R to the Earth: at 'T, will appear as if it proceeded from m, where the Line T m, produced, interfects the faid parallel Plane; and there- fore, becaufe Tm is parallel to Rz,. and any Parallelogram,. interfecting two parallel Planes, cuts them alike in every refpet, it is evident that Rm muft be equal to T 2, and QR m to VnD; wherefore fince D and P are equal to IB eaten two- (4) two Right Angles, DSP and DzP mutt be equal, alfo, to two Right Angles, and confequently QR m (= VaD) = DSP=AFE.. But Tx or Rm, exprefiing the Celerity of the Earth at T, is known tobe inverfely asSD; or, becaufe §D x FE 4s every where the fame, dire&tly as FE ; wherefore the Angles AF-E, QR m being every where equal, and Rm in a conftant Proportion to FE, the Curve QmK _ defcribed by wm, the apparent Place of the Star in the faid parallel Plane, will, it is manifeft, be fimilar in all ’Refpeéts to AEB de- fcribed by the Point E.: But this Curve is known to be a Cir- cle; therefore Qm K mutt likewife be a Circle, whofe Dia- ameter QR Kuis divided by R, the true Place of the Star, in the fame Proportion as the Tranfverfe Axis of the Earth’s Orbit is -divided by either.of its Foci. Wherefore, forafmuch as a {mall Part of the circumjacent Heavens may, in this Cafe, be con- fidered as a Plane pafling perpendicular to a Line joining the Eye and Star, it follows from the Principles of Orthographic Projection, that the Star will be feen in the Heavens as de- fcribing an Ellipfis, whofe Center (as the Excentricity of the Orbit is but fmall) nearly coincides with the true Place of the Star, except the faid Place be in the Pole or Plane of the Ecliptick ; in the former of which Cafes the Star will appear to defcribe a Circle, and in the latter an Arch of a great Circle of the Sphere, which by Reafon of its Small- nefs may be confidered as a Right Line. But thefe Conclu- fions will perhaps appear more plain from the next Propofition, where for the Sake of Eafe and Brevity, the Earth is con- fidered as moving in am Orbit perfectly circular, from which Aer real Orbit does not greatly differ, PRO- G 35) PROPOSITION UI. Having given, Srom Experiment, the Ratio of the Velocity of Light to that of the Earth in its Orbit, and the true Places of the Sun and a Star; to find the apparent Place of the Star from thence arifing. be the Earth’s pee tie Orbit, confidered as a Circle ; S the Sun in the Center there- of; r the Earth mo- ving about the fame Q from A towards Q3 re a Line, which being produced, fhall pafs thro’ the Eclip- tick Place of the given Star; AS parailel, and gr perpen- dicular, thereto: Let ef be perpendicular to the Plane of the Ecliptick, fo that rf being equal to Sv or Radius, re may be the Cofine of the Latitude of the given Star: This be- ing premifed, it is manifeft that the true Place of the Star, from the Earth, will be in the Direction 7 f, and with Ref- pect to the Ecliptick, in the Line re; therefore the Angle Sre (=QSr) being the Difference of Longitudes of the Sun and Star, is given by the Queftion, Let rg, the Sine of the Supplement of this Angle, be denoted by 4; its Cofine Sg, by c; the Sine of the given Latitude, or fe, by 5; and the Re- dius Sr, or fr, by Unity ; and while a Particle of Light 1s moving along fr, let the Earth be fuppofed to be carry’d in Gj its its Orbit from 7 to p, over a Diftance fignified by 7; and, pe, pf being drawn, make 7 2 and 2m perpendicular there- to: Then becaufe of the exceeding Smallnefs of pr it may be confidered as a Right-Line; and we fhall have 1 (Sr):d (72)2: 7 (r):rel—pm)s and tarsi c¢:7c(= rn) by the Similarity of the Triangles fru, Srg); whence as 1 (ff,) to s (fe) fo is rb, to rbs =(nm) the Sine of the Angle nfm: But fince the Sine or Tangent of a very {mall Arch differs in- fenfibly from the Arch itfelf, thefe Values rc and rds may be taken as the Meafures of the Angles ~ fn, and fm: Hence we have, as the Semi-Periphery Ar Q. (=3.14159, &c.) to 648000 re 3.14159, Se. (the Number of Seconds in the Angle rf”;) and as 3.14159 648000 (the Seconds in 180 Degtees,) fo is re to 648000 rsh &c.: 648000 :: rsd: aig oe, «= Af m: Therefore, as the Earth moves from r to p while a Particle of Light is de- fcribing fr, it is manifeft from the aforefaid Propofition, that the Star will appear removed from the great Circle pafling through its true Place, and the Pole of the Eclip- 648000 rc 3.14159,0¢. creafed by oe meron. D bid. 3.14159, Oe. tick by Seconds; and to have its Latitude in- COROL ENCE if C be the true Place of the Star, SCF its. Parallel of Latitude ; and about C, as a Centre, the El-- lipfis F PS T F, and Circle FH S$ OF be defcribed {0, that. FC may be a and TC, the Semi-Conjugate Axis, in proportion thereto, as s to 1; and if the Angle SCH be taken equal to the Difference of Longitudes: of the Sun and Star; then in the Point P, where the elliptical Periphery 1S interfected by the Right Line H Q falling perpendicularly on FS, the Star will appear to be pofited. For as 1. (Radius) : 9 (Sine of QC H)::CH:4xCH= E1.Q; but by the Rela- tion of the two Curves, CH: CT :: 6 x CH (=H Q): PQ: ; : ue 648000 r i 648000 r 56 that is, by Conftruction, 13 5.:: Ringe *3014159,0¢- C2 ih Le (8 ) = PQ; again as 1 (Radius): (the Cofine of QCH):: baie (aii) G48o007e =—C Q; which Expreffions 3.14159,0c¢- 3.14159,0¢. are the very fame as thofe above determined. © OR OF h, WCHEREFOR E it follows, that while the Sun appears to purfue his Courfe thro’ the Ecliptick, the Star will be feen as moving from F towards L and S$, and fo on, “till it hath defcribed the whole elliptical Periphery FLSTF; that its Latitude will be the the leaft at T; and its apparent Longitude the greateft poffible, when the Angle SCH, fhewing the Diftance of the Sun and Star in the Ecliptick, is equal to two right ones. It alfo follows, that the greater Axis of the Ellipfes, which all Stars appear to defcribe, are equal, and found by Obfervation to amount to 40 : Se- conds of a great Circle, very nearly ; the Term 20, 25 which frequently occurs in the practical Rules hereto annext, being put for the half thereof. It follows moreover, that the greateft Aberrations, or Maxima, in Longitude, will be as the Cofines of the Latitudes inverfely ; and the Maxima in Latitude, -as the Sines of the fame Lativudes directly. COROL. Ii. ENCE may alfo be found the. Stars apparent Right Afcenfion and Declination; for let ECP be the Pa- rallel of the Stars Declination, P the apparent Place of the Star when in that Parallel; make C A perpendicular to CH, ABD to SF, and BE to PC; and let HK, or the Angle HCK be any Diftance gone over by the Earth in the Eclip- tick, while the Star by its apparent Motion moves thro’ the correfponding Diftance PL: Let KmnG be parallel to HC, and (9) and Lrv to PC: Then, forafmuch as KL is parallel to HP, the Triangles GK L, CHP muft be equiangular, and there- foe GL:CP::KL:HP; but KL isto HP, as LI to QP, by the Property of the Curve; whence it will be GL: CP::L1I:QP: Wherefore, theSidesGL, IL, CP, QP about the equal Angles G LI, C P Q_being proportional, the Triangles GLI, CP Q_muft be fimilar, and therefore the Angle GI Laright one, and confequently the Right Line SF the Locus of the Point G. Therefore, as the Angles ”, m, 7, v are all given, or continue invariable, let the Angle SCK, or the ecliptick Diftance of the Sun and Star be what it will, the Ratio of Cm to CG will always be given; but the Ratio of CG to Cr is given; therefore the Ratio of Cm to Cr is likewife given: Hence, becaufe rv is parallel to CE, the Ratio of Cm to Ev will be given. But Ev is the Difference of the true and apparent Declinations ; and C m, as the Sine of the Angle H CK: Whence it is manifeft, that the Aberration of Declination, at any Time, is as the Sine of the Sun’s Elongation from either of the two Points wherein he is, when the true and apparent Declinations are the fame; and therefore C 7 will be to Ev, or AC to EB, the greateft. Aberration, as QH to F 4, that is, as the Sine of HCF to the Sine of PCQ: But PCQ, being equal to the Angle of Pofition, is given, whofe Tangent, it is obvious, is to the Tangent of HCF, as QP to QH, oras CT toCO, or laftly, (by ConftruGtion) as the Sine of the Star’s Latitude to Radius: Hence the Angle HCF is given, from which, by Help of the foregoing Theorem or Proportion, the required Aberration of Declination at any Time, and in any Cafe, may be readily obtained. In like manner other Proportions may be derived for find- ing the Aberration of Right Afcenfion ; it being eafy to prove that it will be as the Sine of the Sun’s Elongation from where he aN ( 10 ) he is, when the true and apparent right Afcenfions are the fame; but the Method of Demonftration being the fame as above, it will be needlefs to repeat it. I thall therefore now proceed to illuftrate the foregoing Doc- trine by the praétical Solutions of the feveral Problems depend- ing thereon, as they were drawn up and communicated by Dr. Fobn Bevis, with fuitable Examples of feveral Stars, which, among many others, He has carefully obferved with proper Inftruments, and thereby, the firft of any one that I know ‘of, experimentally provd, that the Phcenomena are univer- fally as conformable to the Hypothefis in Right Afcenfions, as the Rev. Mr. Bradley, to whom we owe this great Difcovery, had before found them to be in Declinations.. PRAC- PRACTICAL RULES For Finding the ABERRATIONS FIXT STARS The Motion of Light, and of the Earth: in its Orbit, I N Longitude, Latitude, Declination, and Right Afcenfion. \ remain Senn enna ET A, the Aberration at any given Time. M, the greateft Aberration, or Maximum. ©, the Sun’s Place in the Ecliptick when the Star's Apparent Longitude, Latitude, . Declination, or Right Afcenfion, being the fame as the True, tends to Excefs. P,-the Star’s Angle of Pofition. Z, the Sun’s Elongation from. its neareft Syzygy with the Star, at the Time of O- For (me )) For the: Abberration in LLoNGITUDE. _©: is always 3 Signs after the Star’s true Place in the Ecliptick. PROB. I. %o find M. RU Le: Cof. Star’s Latit. : Rad. :: 20”, 25 .: M. Examepve in y U+fe minoris. OBE RAT LOIN. Log. Cof. Ar. Com. Star's Latit. 75° 13’———-_—__—~ + Log. 20”, 25 ee = Log. M 79”, 36 C593" a 1.5064 GES ee Re 1.8996 PROB Ty, To find A. RDG 3k. .Rad. : Sia. Sun's Elongat.from © ::M: A. ExaMPLe¢ in the fame Star. OPE RA TL ON. ‘Log. Sin. Sun's Elongat. from © 60° 00’ -++ Log. M 79”,36 — Log. Sin. Rad. =Log. A 68,72 — Merc. te oes} (ee EO ces ———— 1.8996 SS) X1.8371 Otherwife, without M. BLAS 35) 12. .Cof. Star's Latit. : Siz. Sun's Elongat. from © :: 20%,25 : A. ‘Same E x-a mM Pp.t£.as before. OPERANT 'F OW, ‘Log. Cof. Ar. Com. Star’s Latit. 75° 13’ ++ Log. Siz. Sun’s Elongat. ‘from © 60° 00’ + Log. 20”,25 pectin abe a 0952 ar or eet reer! O07 75 nce ree 10S om Log. Sin. Red, = Log. A687 ere enenpnrrnee 18371 For (143)) For the Aberration zz LATITUDE. © is always at the Sun’s Oppofition to the Star. PROB. I. Io find M. ROG: os Rad. : Sin. Star’s Latit. :: 20”,25 : M. Exampe te yin Usfe minris. ORERATION. Log. Siz. Star’s Latit. 75 ° 13’ sr er ag 99854 - + Log. 20/',25 — 1.3064 — Log. Sin, Rad. = Log. M. 19”,58 ——— eT 1.2918 PROB. I o fad A. R tho be8. ad. : Sin, Sun’s Elongat. from © :: M: A. ExaMPLeE in the fame Star. OPER A FOTN: Log. Siz. Sun’s Elongat. from © 60° 00’ +. Log. M 19,58 ———-———— SS og eee Ca 1.2918 —$_— —11,, 2293 — Log. Siz. Rad. = Log. A 16”,96 — Otherwife, without M. R Us fk. Radt +: Sin. Star’s Latit. X Siz. Sun’s Elongat. from @ :: 207,25 : A. Same Ex ampue as before. OPE RATION. Log. Sin. Star’s Latit. 75° 13! —————_——— 9-985 4 -}. Log, Siz. Sun’s Elongat. from © 60° 009 ————~———————-—_ 99-9375 +- Log. 20",25 1.3064 — 2 Log. Sin. Rad. = Log. A 16”,96 peer cessaee ee 212295 E Otherwile, (14) Otherwife, Rees: Cefec. Star’s Latit. : Siz, Sun’s Elongat. from © :: 20%,25 : A. Same Exampue as before. ork R At oO WN, Log. Cofec. Ar. Cam. Star’s Latit. 75°13’ -+ Log. Siz. Sun’s Elongat. from © 66° 00’ +E Log. 20",25 Ss 80.58 045 Saag saue ~.. OOS7> : ————_——___————. 1.3064 = hog. A 1 Geo ———— 1.2203 For the Aberration zz DECLINATION. PRO Bh eo Yimd-6; RU. LE. Sin. Star’s Latit..: Rad. :: Tang. P. + Tang. Z. Then, if the Star (in refpeét of that Pole of the Equator which is of the fame: cia as the Star’s Latitude) be in a Sign. 1. Afcending, and P be acute, Z taken: from the oppofite to its true Place,. gives ©; 2. Afcending, and P be obtufe, Z added to its true- Place, gives ©, 3. Defcending, and P be acute, Z added to the oppofite:to its true Place,. gives ©, 4. Defcending, and P be obtufe, Z taken from its true Place, gives ©,. provided, that its Deelination and Latitude be both North, er both South: But;. of one be North, and the other South, then for its true Plas read oppofite to its true Place, and vice verfa. ExaMPLe of Caf I. in the Pole Star. OTHER A TO NM Log. Tang. P 75% 21’ (acute) + Log. Sin. Rad. 20.5827 — Log. Siz. Star’s Latit. 66° o4’ North. 9.960g == Log. Tang. Z 76° 34,—— — —_—_— 10.6218 Therefore the Star’s Declin. and Latit. being both N. its Place aoe iet.. + {afcending) 6 Signs —_— oe a waa Bs) Log. Sin. Rad. — Log. Sin. Star’s Latit. 54° 25 ‘ (North) —————— 9.9102 = Log. Tang. Z 44° 29° 9.9920 Therefore, the Star’s Decl. and Latit. being both North, its Place 18 eects (defcending) +. 6 Signs Gee pee ee +Z =O Sse —— I 14 2g er oS ———_—— 1 97 43 pa ExamPpre of CafelV. iny Ur/e minoris. OP ER AA TO WN: Log. Tang. P 94° 48° (obtufe) ++ Log. Sin. Rad. — 21.075 — Log. Sia. Star’s Latit. 75° 13’ (North) —————-——- 9.9854 = greater than vv, the Projectile can never return. Now, therefore, putting p for the Area of a Circle whofe Diameter is Unity, and fuppofing greater than vv, the Area of the whole 3 eS F3 Curve, (being an Ellipfe,) will be iF ee ETE ce sidte (= pae); but as the Area ae is to== the equal Time OF its Defcription, fo is the pe of the whole Ellip- fis, to—= x ai or he — or the Time of one b/r b fr ~4rbb intire Revolution, Q, E. L COROL., I. ECAUSE 7, the Square of the Time of de- {cribing the Area RSP, is to —, the Square of that Area, as (1) the Square of a conftant Particle of @ Fea" Time to —j3- the Square of the Area defcribed in this laft Time, it is evident that the Square laft named will be ( 24) be to the Latus Rectum a as +b*: m* 3 which Proportion being conftant in all Cafes relating to the fame Center, it follows, that the principal Latera Recta of the Orbits of different Bodies, about a common Centre of Force, are direétly as the Squares of the Areas defcribed by the re- {pective Bodies, in the fame Time. COR OL. II. fe a 2 2 OREOVER, fince BS is =sd, and R=>—>—, we 5 v zx, : have 22 = er and «-.: 22 is tov, in the conftant BS Vr SB Ratio of 1, top FR: Hence it appears, that the Velocities are, univerfally, in the fubduplicate Ratio of the Parameters dire@tly, and the Perpendiculars falling from the Center of Force on Tangents to the Places of the Bodies, inverfely, and therefore, in the fame Orbit, the Velocity will be, barely, in the inverfe Ratio of the Perpendiculars fo falling. COROL. Il. INCE Pis=~r re x at, or, in a conftant Propor- tion, to a3, let v, s, and d, be what they will; it: follows, that the Periodic Times, about the fame Center of Force, whether in Circles or Ellipfes, will be in the fefquiplicate Ratio of the principal Axes. Cc OnRw.L. ( 25 ) ¢ 0. RB .O.. Bey. ECAUSE neither the Values of a nor P are affected by s, it follows, that the principal Axis, and the Periodic: Time will be the fame, ifthe Velocity at P be the fame, let. the Direction of the Projectile at that Point be what.it will. C O: ROE. 7 i. \ NV HEN “dew ene v (=a) is = 2d, or, which is the fame,, La r arbb when v = Je ; then d being the mean Diftance or Semi- Tranf{verfe, tie Point P’ will fall in one Extreme of the Con+ jugate Axis, and 4 /22 the Velocity there, will. be juft fuf- ficient to retain a Body in a Circular Orbit at that Diftance (d) from the Center of Force; and this Velocity, in refpect of different Orbits, will, it is obvious, be inverfely as the Square Roots of the mean Diftances : Wherefore the Velocities: Bodies in Circular Orbits about a.common Centre, are reci-. procally in the fubduplicate Ratio of the Radi. COROL. VI: rr v be= bjt or the Square of the Velocity be juft twice: as great as that whereby the Projectile might defcribe a circu- lar Orbit at its own Diftance from the Center of Force (Cor.V.): then a, the Tranfverfe, becoming infinite, the Ellipfe degene- rates into a Parabola, whofe principal Larus Rectum is 4.4553. whence it appears,. that the Velocity of a Body moving in a Pa- rabola is inverfely as the Square Root of its Diftance from the Centre of Force, and that it will be, every where, to the Ve- H locity 7 ih ars Hi Ta | jt rinl 1 K La | i ‘ 1 S| | Hilly PTE I Ha Lun +) yp Mt) } i 1 CP Hith { i | (Ni PTH Watt HANH) } | j i | | t 1 Bh | | ra tai | | i I wih Hh hill | rir aa } 1 1 HE ih a Ht) HH WN tH | 4 NG A} aA | 1 ate PT He 1 HRT | I ma q in iH | Hl| Ht WL LT iM it | Hi WHA 1} | bi 1} i Hi WA HH a 1 mt ya \ | | | ( 26 ) ‘locity that might carry the Projectile in-a cirenlar‘Orbit, at its own Diftance from the Centre, as the Square Root.of two, £0 one. C0 K-04... VIL B° T if v be greater than 4 ee , the TrajeGtory will d ‘be an Hyperbola, whofe principal Axisis Zoo _, (=—a) as 4rbb ZL has been before intimated, and therefore e ee a) will af v 3 rsa* be ——22i22—. Hence, from the Nature of the Hyper- Jdv?—4rbb bola, if R be affumed for Radius, a Zar (=5) will be the Tangent of the Angle which the A- fymptote makes with the Axis, or the Supplement to 180° of the utmoft Elongation the Projectile can poflibly have from the loweft Point of its Orbit. PROP. (27 ) REESE a ne PRE GQ BGS EeF ayo. N. al. A. Body is let go from P, at a given Diftance PC from C the Centre of Force, ina given Direction Pb, with a gi- ven Celerity ; To find its Trajectory? the centripetal Force being as any Power (n) of the Diftance, and the abjolute Force at P given. ET R bea Point _4 in the required Trajectory, and 7 ano- ther indefinitely near it; and with the Centre C, let the Circular Arches Pep, RD, our. defcribed, and having drawn CRe, Crf, &e. let C P= a,,CR (2 @ U} sa Pen aa (=U v) =x, rn=y,Rr =z, and s=the Sine of the Angle C Pé@ to the Radius 1; and let Pd (m) be the Space that might be defcribed in (1) a given Particle of Time with the given Celerity ; and r the Diftance a Body would freely defcend in that Time, by an uniform Force equal to that affecting the Projectile at P: Then the Space which would be uniformly defcribed in that fame Time, with the Celerity acquired by defcending thro’ the faid Diftance7, it is well known, will be equal toe rv. But, from hence to ( 28 ) to find the Celerity at R, with the fame Velocity that the. given Projedtile is let go from P, towards 4, let another pro-~ ceed from the fame Point, in a Right-Line paffing directly thro’ the Center of Force ; and let the Celerity at U, or the Space that would be uniformly defcribed therewith in 1, the abovefaid Particle of Time, be denoted by v: Then, as a”, the Centripetal Force at P, is tox”, that at U, fo is 27, the Velocity that might be generated by the former in the Z2rx # , that which would be gene- given Particle of Time, to nerated by the latter inthe fame Time: Wherefore, as 1, 4 that Time, to *-: {© is 7, the Time of defcribing U v, a U a to v, the Velocity acquired in this Time: Whence, by mul- ° e ° = H e tiplying Means and Extremes, &c. we get vv=———» and at Ly ats therefore vy = —2 - + fome conftant Quantity d; which 2-1 xa” to determine, let U coincide with P, x be = a, and pti bee v=m, and the Equation becomes = se Sa xa +d; i I hence d= = + —— ; which Value being fubftituted above, “a-tl U2 2 bie a+ we fhall have — = pow Cf. 2t4eteo.. anditherefore U 3 - aot a-+1 Xa” =m? +24 eh *s But this is likewife the Celerity of a-+-1 nb iXa z. the firft Projectile at R: For fince both Bodies have the fame Velocity at P, their Velocities, at all equal Diftances from the Centre muft be equal ; and therefore V mp — or its ({ 29) its Equal V xx+)9 will confequently be the Time of the faid given Projectiles moving thro Rr, or of defcribing the the Area - =RCr, by Radii drawn to the Centre of Force: Wherefore, fince — is the Area of the little Triangle PCé that might be uniformly defcribed in, 1, the given Particle of Time, with the Velocity atP; and, becaufe the Areas are as the -Times, it will be, as — to 1 (the faid Time), fo is “”, to 2 v="-b": Hence we get y = = or, Vv V atu? —ms*a® e SMaAX 4 rx"t3 by fubftituting inftead of : rax” VA m* x paar — m*s*ar a I a+ixa” v, its known Value, as above found. But as Cr, isto rz, wre ae oe Cf (=#@) ° sf Gath TEER gta ge =A. (= ef) nd 2-1 xa” «whofe Fluent Pe is the Meafure-of the angular Motion; from which, when found, the Orbit may readily be conftructed ; ‘becaufe, when ‘Pe, or the Angle PCR, is given, as well as CR, the Pofition of the Point R 1s alfo given: But this Value of A is indeed too much compounded to admit of a Fluent in -general ‘Terms, or even by the Quadrature of the Conic Sec- tions, except in certain particular Cafes, as where z is equal to 41, —2, —3, or —-5, or the Law of centripetal Ah as the firft Power of the Diftance dire&tl y, oF We 72 4i9", “OL ‘sth Powers thereof inverfely ; therefore, in other Cafes, can only be had by infinite Series, &F¢, or Curves of a fuperior Order. Q. BE. LL J CoROL = ( 30 ) C.4) Roa) 1. | F, inftead of the abfolute Celerity of the Projectile at r; the Ratio thereof to that which it fhould have to de- {cribe the Circle Pe, be given, as pto 1, and not only the fame Thing, but the Ratio between the Celerity at-any o- ther Diftance CR, and that which a Body muft have to de- ferib= a circular Orbit at that Diftance, be required: It will be, as a”, the centripetal Force at P, tox”, that at R (or UY, fo is r, the Diftance a Body would freely defcend by the former of thefe Forces in 1, the given Particle of ~ 2 ° e o~ Time, to —, that which it would defcend by the latter in a the fame Time: Therefore, if Us be taken equal to on, at and sf be made perpendicular to AC, it is manifeft, that Uz, being indefinitely fmall, will be the Diftance which a Body muft move over in the aforefaid Particle of Time, to defcribe the Circle UR: But Uz, by the Proe. Ls = ° ° ° Z rch I perty of the Circle, is in that Circumftance = ./——__ ;, je oh pl : St -, tov, or its Equal, wherefore we have, as 4/ eel ee Ht — I a a oe (above found) , fo is the Velocity a Bo- dy muft have to defcribe that Circle, to that with. which. the given Projectile arrives at R: Therefore, when x is =a,, * . ° F e > » n= and R coincides with P, the Proportion o en) to até 2 GHEH 4 Ly tH vie ] , Wee etree. WMaehuthere 15, 25 47 ora, tO. 7, 19- a-ri ye 4 nia given as I, to p, by Suppofition; whence, multiplying Ex- tremes ( 31 ) tremes and Means, we get m=~./2ra; which being fab: {tituted inftead of m, in the Value of v, it will become I i pe BH ater] ? V2prrapie 2°" op pe pit 2 ye AU oi al nb xa® Mp1 apr oa-p1 tt easier piG 7 ~ ° ew 2 oa ° 2 at} j therefore this divided Dy v 2) "oF ino ane ; n ui x1 ni) a for the Ratio that was to be found. And, in like man- ner, by fubftituting for m in the Value of A, we get spaax fo laaiteaN 2 2 +3 ° lita, 2 Xx? — 7? 7? gt — ——— for the other Quantit; wal HT Se tik ahi Xa? tT! Ca y required, ©) OF Re OP LSE ENCE, if the Angle CPS be. fuppofed. to be di: o He eo) 6 . e e 5 ae -b1 2 minifhed 2” znfimtum, and p? +—~— san " xX ( i aiti x2 ral *, the faid Value of v, be’ taken=o0, we fhall have: =I ppxnti “E r] 7b Sy (5 Ae Height to which. the Body would afcend, if projected directly upwards ; there-- J 3 f oe ks t fore, 2p px ay neh! Yee A pat joie Diftance, it miuft freely defcend to acquire the given Velocity 3. which Diftance, therefore, with an uniform Centripeta] Force,, where 2=9,. will be = 424 ; and with a Force Inverfely, as» the Square of the Diftance, == 27%. But when 9 is- 2— 9? =1, or the Velocity of the Projectile at P is jut fufficient’ tov ( 32) “to retain a Body in the circular Orbit P@, A P then becomes I aa x@ — a; which in the faid two Cafes, willbe: a, Z and .2 refpectively ; but infinite when x is =—3. COROL. UI. ‘ aie m-t-1 is a pofitive Number, the Velocity VV 2raxXp?+— — atet ty: ,vat the Centre :C, a--l al x atti ‘where x becomes=o, will, it appears, be barely -equal to f2rax p* + > but, when 2-1 is negative, or the ‘Law of Centripetal Force more than the firft Power of the Diftance inverfely, it will be infinite ; becaufe then, the In- dex being negative, x*+1 (or its Equal 07+) will come in- to the.Denominator. COQ RK, O-L., IV: KR F OREOVER, when x-+1 is negative, and x N infinite, the faid Velocity will alfo ‘become 4 2 : Le Oe st = becaufe then, for the Reafon above f{pecified, xt: will be =9: And therefore, when the Cen- ‘tripetal Force is more than the firft Power of the Diftance nverfely, a Projectile moving from P with the given Velo city pf 2ar (=m) along the Right-Line PA, will af ecend even to an infinite Height, and have a Velocity there TS eRe SS ° e e fignified by ra x p* + Sri or in Proportion to the gi- Ve TTR EM , 2 2 ‘ven Velocity, as \/ p* iMate rae to p, provided p* + =e be ( 33 ) be pofitive; for otherwile the Thing is impofiib Root of that Quantity being manifettly fe, ble, the Square CLOMR, Ovbaiwy: y ENCE, if the leaft Velocity that can carry the E Body A Hf to an infinite Height, or that which it would acquire by freely defcending fos the fame Height, be required: By making p? eS =o, we fhall have p= diss 3; which, 4 sack , ‘ genres rl fubftituted in p / 247 gives »/ —— xX ay Bar 2252 ¢/ era for the Value fought ; and this, it is manifeft, is to J 24ar, the Velocity a Body muft have to defcribe the Circle Pe, fay ‘ : as / > to Unity: Therefore, when z is lefs than — 3, or the Law of Centripetal Force more than the Cube of the Diftance inverfely, a lefs Velocity will carry a Projectile to an infinite Height in a Right-Line, than can retain it in a circular Orbit, was it turned into a proper Direétion. CrOsRN GO eva: HEREFORE, if it were required, how far a Body muft defcend by an uniform Force equal to that affecting the Projectile at the Point P, to acquire the fame Celerity that another Body, by freely falling from an infinite Height (as above) has at its Arrival to that Point; then, by fubftituting the Value J fis 2 , as found in the laf Article, inftead of its Equal, in4 f = we Cor. II.) there comes out rae for the Value fought: And hence it appears, that ae Velocity with which a Body, falling freely from an in- K finite Ose } finite Height, would impinge on the Earth, is no greater than that which another Body may acquire by an uniform Gra- vity, equal to that at its Surface, in falling freely thro’ a Space equal to its Semi-diameter, SCHOLIUM. R OM < the Ratio found in Corollary I. between the Ve- locity with which the Body arrives at any Diftance (x) from the Centre of Force, and that which it ought to have to defcribe a Circle at the fame Diftance, it will not be difficult to determine in what Cafes the Body will be compelled to fall to the Centre, and in what other Cafes it will fly ad znf- nitum therefrom. For, firft, if the Body in moving from P, begins to defcend; or the Angle C Pd be acute, I fay, it will continue to do fo ’till it atually falls into the Centre of Force, if the Quantity ( Ree eet “sit in its Accefé thereto, be not fomewhere greater than Unity ; or, which is the fame in effect, unlefs the Body has fomewhere a Velocity more than fufficient to retain it in a circular Orbit at its own Diftance from the Center of Force: For, if it ever begins to afcend, it muft be at a Point, as D, where a Right-Line, drawn from the Centre, cuts the Orbit perpendicularly,. and there, it is manifeft, the Celerity muft be as above fpe- cified, otherwife the Body will ftill continue to defcend, or elfe move in the Circle DL about the Center C, which is equally abfurd. On the contrary, if the faid Quantity, in approaching the Centre, increafes fo as to become greater than Unity, or be every where fo; then, the Velocity at all infe- rior Diftances, being greater than the Velocity that is fuffi- cient to retain a Body in a circular Orbit at any fuch Dif tance, the Projeétile cannot, it is evident, be forced to the Centre. But ( 32) But, on the other hand, the Angle CP 4, being fuppofed obtufe, it will evidently appear from a like Reafoning, that, if the faid Quantity be always greater than Unity, or the Body in its Recefs from the Center, has, in every Place thro’ which it pafleth, a Velocity greater than is fufficient to retain it in @ circular Orbit at the Diftance of that Place from the Cen- ter of Force, it muft, of confequence, continue to afcend ad infimtum. Now, therefore, to find in what Laws of Centripetal Force thefe different Cafes obtain, let the Angle C P 4 be firft fappofed acute, or the Body moving towards the Centre, and ‘ : a+r x in the abovefaid Quantity ./ p* + a — to be x 7% infinitely fmall; then it is evident, that that Quantity will become either // or infinite, according as m-+-1 is a. negative Number, or otherwife ; wherefore, in the latter of thefe two Cafes, the Body can never be forced into the Cen- tre; neither can it in the former, when 2 has any Value be- twixt —1r and — 3, as is manifeft from above, becaufe an is greater than Unity (Rectilinear Motion being here excepted: ) Nor will either of thefe Conclufions hold lefs true, when the Angle CPS is obtufe; for it is obvious, hat if the ProjeCtile cannot be forced to the Centre, when dire@ed towards it with the leaft Obliquity, it never can, when the Obliquity is increafed: But on the contrary, if m-+.1 be either equal to or lefs than —2, and p be vaa9 lefs than 1; then the faid Value zie not being greater than Unity, the Projectile muft inevitably be drawn in- : to the Centre ; for, the afore-mentioned general Expreffion not. exceeding ( 36 ) exceeding Unity, neither at the given Diftance a, nor at the leaft aflignable Diftance, cannot at an intermediate Diftance ; becaufe, in the Defcent of the Body, the Exprefiion muft ci- ther: increafe or decreafe continually, there being only one Dimenfion of the variable Quantity («) pitecaicd But, when # is greater than Unity, other Things continuing the fame, I fay, the Body, if it efcapes the Centre, and once begins to afcend, it will continue to fly from the fame ad infinitum, For, fince the Part D L, &c. of the Trajectory, which it will begin to defcribe.on its leaving the loweft Point D, is in every re{peét equal and fimilar to DR, &c. if another Body projected upwards from P, in the oppofite Direction, with the fame Ve- . locity, continues to afcend ad infinitum, our firft Projectile, after it has pafled the loweft Point, muft do fo too, and vice verfa ; therefore p” gras being there affirmative, and the Se Sadakb: wie. Angle CPé obtufe, the Quantity // p? ++ aida — sei ous when x is infinite, will alfo be infinite; whence from the above Reafoning, the Pofition is manifeft. Hence we conclude, Jirft, that when 7 is greater than —- 3, or the Law of Cen- tripetal Force, as any Power of the Diftance dire@ly, or Jefs than the Cube thereof inverfely, the Body cannot pofli- bly fall into the Centre, except ina Right-Line, And, /econdly, that, when the Force is, as the Cube, or more than the Cube of the Diftance inverfely, it muft either be forced to the Centre, or fly an infinite Diftance therefrom, unlefs it moves in a Circle. Furthermore, becaufet he a bovefaid Quantity, when w is in- finite, in all Cafes where z+-1 is negative, and PP greater than , appears to be greater than Unity, it follows, that in all thofe = (239-)) thofe Cafes, the Body may afcend, even to an infinite Height, and a@ually will do fo, when has any Value betwixt —1 and —3; becaufe then, tho’ the Body fhould at firft approach towards the Centre, its Afcent cannot be anticipated by being drawn into it, as it may, when the Value of z is {maller, as has been above fhewn. Note, The fame Things may be otherwife determined by Help of the laft general Value of A; for if ppt a SiN x 4-1 ae XXX, the Square of its Divifor be n+ika” made equal to nothing, the affirmative Roots of that Equation, or Values of x, will give the greateft and leaft Diftances of: the Projectile from the Centre of Force, and therefore in thofe: Cafes, where it is found not to admit of two fuch Roots, the. Body muft. either fall into the Centre, or fly it ad infimitums —— p* S* Q* — L PR OP.. ( 38) PROP OST TI1-0.N.. up Lo find the Motion, or Angular Diftance of the Apfides, in Orbits nearly circular ; the centripetal Force being as any Power of the Diftance. ET ArPa bethe propofed Orbit, A and P two Places of the higher and lower Apfides, Ae EaA, and nPbn, Circles defcribed with the Radii AC, CP about C, the Cen- tre of Force ; let 7 be a Point in the Trajec- tory taken at Pleafure ; and let the Velocity of the Body at the higher Ap be to that which it ought to have to retain itfelf in the Circle Ae E, as,/ 1 —e, to 1; calling AC, 13 re, y; Cr, r—y; andAe, A: Then, by fubftituting, 1 fora, 1 for s, (/I——e for py 1-9. for spaax x, and y for x, in 1 EOE PRET ar 2-1 atixaea’? Tl the general Value of A, as found by the laft Prod. the fame it i—e xy ; is manifeft, will become —_ (“~~ it : oe ek eee for the Value of A in this particular Cafe; which by redu- cing ( 39 ) cing 1— yl? and Ty" t3 into fimple Terms, is y Jt —el ry Ls oe K1—2 yf yy—1-e— aX 1—n7+3X y-a+3X eee) a, yf tae Sere 3 SAG up ee ied — > , 9 V 2 ey—n$3 XV) or — y C2ey becomes = 0, or A=A Fluent of re when viz —y , eH, Vny3 >? is equal to a Semi- circle whofe Radius is Unity, or to 180 I 180 ; Degrees ; therefore Fra x 180° = 7 7 Degrees , is the Meafure of the Angle ACP. Qe. 1. cOROL I. ‘ A THEN 2 is equal to, or lefs than —3, then the Value VY <8 +) of the Angle ACP, becoming either infi- it follows, that if the Law of Centripetal / 24-3 nite or impoffible, han the Cube of the Diftance inverfely, Force be, as the Cube, ot more t ( 40 ) inverfely, the Trajectory cannot have more than one Apfae : And, therefore, the Projectile in all fuch Cafes muft inevita- bly either fall into the Centre of Force, or fly from it ad in- jinitum, unlefs it moves in a Circle; which is agreeable to the Scho/ium aforegoing. But, if 2 be equal to 1,0, —1, or —2; then will the Angular Diftance of the two Apfides De, Oyen ete 55.) I2g° 17, Or T8a° eo", TE fpectively ; the firft and laft of which we are affur’d of from. other Principles, iro bh OL." i. iG the Diftance (D). of the Apfdes be given, and the Law of Centripetal Force from thence be re- quired: Then, by making ar equal to D, we fhall ar3 have hea. =n, for the Value fought: Hence, if D be 360°, or the Body takes up one intire Revolution in going from one 44/e to the other ; then, muft the Law of Centripetal Force be reciprocally as that Power of the Dif- tance, whofe Exponent is 23; but, if either 4%/, from the Time of the Body leaving it, to its Return again, has mov'd forward only a very {mall Diftance, E, or D be = 180° 4 = the Force will then be inverfely as the 2p Power 3 of the Diftance, very nearly. SCH OLIU M. F x be any Diftance of the Projectile from the Centre of Force, and the Law, by which it tends towards the fame Centre, be every where, as cx*-+-dx™-exi+fx7, &c. c,d, &c, n,m, &c. being determinate Quantities ; pod i ( 41 ) if a be the Diftance of one of the Apfdes from that Centre, the ees Diftance a thofe Apfides will be tag ab +fal, 32x: ced FEexie bE From th MEAN ANOMALY of a Planet given; to find its PLACE im its ORBIT, ET AOB be the given Orbit, S the Sun in.one of the Foci, AC the Semi-Tranfverfe AN Axis; CO the Semi-Con- y,/ ; i jugate, AEHBA a Cir- iP cle circumf{cribing the El- lipfis, and let # ‘be the Place of the Planet at a- ny given Time after or. before its pafling, A, the Aphelion ;_ thro’ which draw ExP perpendicular to AB, and having joined the Points E §, E‘C,S, and made SD perpendicular to ECD, take the Arch EH equal to SD, and the Arch Aa equal to SC. Then, the Sector ECH being equal to the Triangle ECS, ACHA will be equal to ASEA;; inafmuch as the former of thofe Areas is com- pounded of the Sector AC E and ECH, and the latter of the fame Sector and the Triangle ECS: Where- fore, fince the Area ASEA, is ‘to AEBCA, half the Circle, as the Elliptical Area AmSA, to the Semi-Ellipfis AzxBCA, by a known Relation of the two Curves; if, M inftead A a hk : oy & ( 42 ) inftead of ASE A, its Equal be fubftituted, we fhall have, as ACHA: AEBA:?:: AnSA: AnBA; but ACHA is to AEBA, as the Arch AH toA EB, the Semi-Cir- cumference ; and therefore it wall be, as AnBA : AnSA -- AEB : AH: Wherefore fince the Areas AmBA, AnSA, defcribed by Radii drawn to S, the Center of Force, are as the Times of their Defcription, it. will be, .as the Time of defcribing An BA, or that of Half. one Revolution, is to the given Time of deferibing AnSA, fo is AEB to AH; which, therefore, is the given Mean Anomaly in this Pofition, or the Arch proportional to the Time of the Planet’s moving thro’ A x. Fé now A C =a ne S=e, Addoiee SHE equal FE, ‘ts Sine EP equalx, and its Co-fine CP=y. Then, from the Similarity of the Triangles CEP, CS D, we fhall have, EC: EP:: Aa (SC): EH (SD) and, con- fequently, AE + —~ X Aa=AH, o E+xx Aa, —D; which Equation, it is manifeft, will hold equally» whether the Arches AE, Aa, and AH, be taken in De- grees or in Parts of the Radius: But.now, in order to folve the fame, let the required Arch, or Value of E, be eftimated.-pretty near the Truth, and let this aflumed Value be denoted -by As=E, its Difference (se) from the Truth, by E; and Das Ason ax Aa=D—E~—~«x x Aa, the Error of the the Equation, byR; make vr parallel to A B, and let sd bea Tangent to the Circle at the Points: Then, as sd, by reafon of its Smallnefs, may, inthis Cafe, be‘confidered as equal toe s, and becavfe of the Similarity of the Triangles CZs, s7¥, r we ( 43 ) we fhall have as 1 (Cs) Fy (Ck) 3: E:y x E — rd, or er, very nearly; whence E=E +E: Aiichs ip ee =x y x E, which Valuestherefore being fubftituted in the general Equa- tion E+ xxAa=D, there comes out ovo xen Sn ye DiikirwAg ot +yXE xAa=D very nearly ; wherefore eae ‘es Ae =) =a or, 7a nearly : Hence it appears, that, if the Error of the Equation be divided by 1-+¢ y, and the Quotient added to, of fubtraéted from the firft or affumed Value of E, there will arife a new Value of that Quantity much nearer the Truth than the former: And if with this new Value, and thofe of x and y correfpondiag thereto, we proceed toa new Error, or compute the Value of R, and that of the Divifor 1--ey, &ec. it is. likewife evident, for the very fame Reafons, that a third Value of E may be found, by the fame Theorem, ftill nearer the Truth than the preceding, and from thence another, and fo another, &c. ’till .we arrive to any Accuracy defired, each Operation, at leaft, doubling the Number of Places; fo that in the moft excentric of the planetary Orbits two Operations will be found fufficient to bring out the An- gle ACE to lefs than.a Second: And when thatis known, as EP and SP are then given, the Angle SP may be ea- fily -had; for, by the Property of Curve, it is AC: CO 5B: Pa= A, and §.P' :: eA EPadiis AC ( Radius) ; ao =the Tangent of AS” QE. JL. Otherwile, (44) Otherwife, Let Radius EC=, and the general Equation AE + — xAa=AH, oE=D—~ x Aa be again refumed; then, the Orbit not being very Excentric, E will, it is evident, be nearly equal to D; and, confequently, (x ) the Sine of E, nearly equal to the Sine of D: Therefore, if the Sine of D be fubftituted for », and the faid Sine be de- noted by x (fignifying the firft Value of x) it 1s obvious, that D—= XAa will be nearer to the true Value of E, f than D, and, confequently, that the Sine of D bi oe x! Ava (which I call x) nearer to x than (x) the Sine of D; wherefore D — x x Aa, mufit be, ftil, nearer the Truth, or the required Value of E, than D— = x Aa, and, confequently, its Sine (which I call x) ftill, nearer x, than (0) the Sine of D— = xAa@: In like manner, the “a Sine of D—— xAa (or x) will appear to be nearer x Ms than x, and D— =x Aa, nearer to the required Value “i than D— — x Aa, &c, &c. Whence the following Me- thod of Solution is manifeft. Let 1.758123, the Log. of (57.2958) the Number of De- grecs in an Arch equal in Length to Radius, be added to the Logarithm (45) Logarithm of the Excentricity, and from the Sum dedud& the Logarithm of Half the greater Axis; the Remainder will be a 4® Logarithm (L) ; which, being once computed, will {erve in all Cafes of that Orbit: To this Logarithm add the Logarithmical Sine of the given Mean Anomaly reckoned to or from the Aphelion; the Sum, rejecting Radius, will be the Logarithm of an Arch in Degrees ; which, being taken from the Mean Anomaly, and the Sine of the Remainder added to the faid Logarithm, the Sum, rejecting Radius, will be the Logarithm of a 24 Arch; which, in like manner, being taken’ from the Mean Anomaly, and the Sine of the Remainder added to the fame Logarithm, the Sum, rejecting Radius, will be the Logarithm of a 3¢ Arch; from whence, by repeating the Operation in the very fame manner, a 4% Arch will be found, and foa 5%, Ge. “tll we arrive to any affigned Exaétnefs; the Error in the Anomaly Excentri, of Angle ACE, which Angle is to be exprefied by the Difference of the Mean Anomaly and the laft of the faid Arches, being always much lefs than the Difference of the faid Arch and that which immediately precedes it, from which Angle the true Anomaly is had as in the above Cafe. Q,£. L. Otherwife, The foregoing Conftruction being retained, let Radius (AC) —1, the Sine of the given Anomaly ACH =a, its Co-fine =4, and let Em be the Sine of EH: Then willa Cu—éxEm — x, the Sine of the Difference of thofe Angles, by the Elements of Trigonometry ; but EH being =ex, Em (by the fame) ° 3 xy 3 5 4.5 2 will be esp ee So fe. and (Carat aos 2.3 2.3.4.5 2 t : ® e a ee 2 , Se. whence, by Subftitution, Ge. we get Tbe xx ~ + ( 46) Bi new 3 y 3 4 ue + e ® aenne SE ~~, Gc. =a, where, by inverting 1 ° a 2ase™ 34 ws, tbe the Series, « comes out = et ia ane. ae” Ol oo ae. ues 2bb—da aa—3bb mn, GO. =aX I—be+ ar A ee ng Ps c. from whence, if the Excentricity be not very large, the Angle ASn may be had as above to any Degree of ExaG- neis 20. Hou: Note, That, in thefe Solutions, when E is greater than a Right Angle, its Co-fine y is to be confidered as a negative Quantity. Dike ty aa 8 ts Wa. As the foregoing Methods of Solution may feem te- dious or perplexed for common Practice, a fhort Approx- imation, tho’ limited in Point of Exactnefs, may be of Service. In order to this, we have have given EP 2b6— 5 a@a—3bb (apo eee ee RE xe 3! Ce as above, from whence PC (=y=V/ 1— xx I—xx) is=b+ ate— 3 5a? e 644—2aa 65a*—6454 _ Re es: x ba*e+, Sc. But, by the Nature of the Curve 1+-ey is =Sx, and Pz =\/1—ee . ( _ OCXEP ew Lone) f AC Pa\... x Xx € oe) and therefore 1 -pey (= Aer) will ety 1-- ey (= “| its Co-fine: Hence, by the Elements of Tri- be the Sine of the Angle AS», or true Anomaly, and ty Ma lt See 18 aetay —bx4/1—ee gonometry 4x ——— — bx i-pey Iey will be the Sine of the Difference of the Angles HCA, AS», or of the Equation of the Orbit; wherein, by fubftituting, inftead ( 47 ) inftead of x and y, their refpective Values, and contracting the whole by Divifion, &c, there will come out 2¢e into 1— che - 366 44 245aq 27 6 ol ; ita ar iia ible ti x 60 = ;A8.6e Wet do x 3, Se, which, when e¢ is not very large, will appear to be equal to 2a ¢ 254 Ri pbd , he ee Se ey Bea | for this, converted to pt Oren : ° be ab? e3 8 Ss a Series, is 2¢e — 4 A — + Joaibes e 3p 4 . ; ery 7 sb uasch —, €c, from which, if the former Series be ta- aac 17b6 5S aa byte eed, ken, there will remain only = fc, Hence is deduced the ee PRACTICAL KU E ie For finding the Equation of the Centre from the Mean Anomaly given. 4s Radius, to the Co-fine of the given Anomaly, fo is - Parts of the Excentricity of the Orbit, to a fourth ee which Number add to half the greater Axis, if the Arwmaly be lefs than 99, or more than 270 Degrees, atherwife fi Jubtrae&t from the fame: Say, as the Sum or Remainder, is to Double the Excentrictty, fois the (Logarithmic) Sine of the given Anc- maly, to the Sine of a firft Arch; from three Times which Sine dedutt the double Radius, the Remainder will be the Sine of a fe cond Arch, whofe + Part, taken from the former, kaves the Equation fought. A n A. ( 48 ) And it muft be noted, that this Ru/e, in the Orbits of Saturn, ‘fupiter, and the Moon, anfwers to a Second, and in thofe of the Earth and Venus to lefs than 3, of a Second. And, in thefe two laft, the Arch firft found will, without farther Correction, be fufficiently exa@t to anfwer to the ni- ceft Obfervations, the Error never amounting to above 2 or 3 Seconds; which is more correct than either the noted Hy- pothefis of Ward or Bullialdus, as will appear from the fol- lowing Examination of thofe Hypothefes, which, as they have been much celebrated, and come near the Truth in ma- ny Cafes, may here alfo deferve a particular Confideration, And, to begin with the latter, which fuppofes the Angle AFn made at F the upper Focus by the Aphelion and {n) the Planet to be the Mean Anomaly, and therefore SnF the Equation. Becaufe a the Sine and 6 the Co-fine of the faid Angle are given, by the Nature of the Ellipfis, r+2eh-tee Das ceasey ronnie alfo given; whence, by Plain Trigono- metry, it will be, ag Sw ae (oe ys See OR TI +e ] h ° 2 ; a M rere. fe to the Sine of SzF, which, put in a Series, is oe x Ted + 200 — 1 X eed 3 — 4 db xbe}, Be. and this taken from 2@¢x I — gee + 32° —*x ev, &e. 4 eb 66 aa 2 fi leaves 2a¢ X rs +> 7 x ee, ce. for the Error of this Hypothefis, But now for the other, where, ExP being perpendicular to AB, AF E is fuppofed the Mean Ano- maly. Let SC and Cm be perpendicular to FE; then it will be as 1 (EC) : @ (the Sine of AFE, or CFE) 2: e (CF) : ae=mCm, the Sine of CEF; whence Em, its ( 49 ) its Co-fine, =f 1—2@7e7: Again, as 1 (the Sine of CmF) :e (CF) :: —é (the Sine of FCm) : —eb =F m, equal alfo to Cm, becaufe CS is =FC; for which Reafon S/ ig double to mC; wherefore it will be, as / “q — ate? eb El) Siy tis Radi t ees «the (E/) : 2ae (S#) r (Radius) ets Tangent of SEF; which, in a Series, will be 2aex wreath 1-—f0-— 2 +. 75%, &c. whence the correfponding Sine is eafily found = 2 ae x Tee eb bre? — 3", Be. and this RST de : taken from 2aex 1—~ 2 4 4 — axe, gives 2ae in~ b bb aa : bs to—— +t xe, €f¢, for the Error in this Cafe Hence it will appear, that the greateft Error of each of thefe Hypothefes, in the Or- bit of Mars, where e is up- wards of .og, will be about 5 or 6 Minutes, and in the other planetary Orbits, ac- cording to the Squares of their Excentricities (in Parts of their own Semi-Axis) nearly; it alfo appears, that towards the Aphelion the Circular Hypothefis will be the more correct, and near the Perihelion the other ; and, laftly, that both Hypothefes make the Equation too large in the higher, and too fmall in the lower, Part of the Orbit. O Having ge) Having fhewn how much thefe two noted Hypothefes, (by many fo much efteemed) differ from Truth, it may be proper to proceed now to give fome Examples of the pre- ceding Methods, whereby the Problem is more correctly folved. EXAMPLE I. ET the Excentricity of the propofed Orbit be ,3, of the Mean Diftance or Semi-Tran{verfe Axis, and the given Mean Afiomaly 72° 12’ 36” =72.21 Degrees; and let the Anomaly Excentri, by the firft Method, be required. It will be, as 1, the Semi-Tranfverfe, to .o5, the Excentri- city, fo is 57.2958, the Number of Deg. in an Arch, equal in Length to Radius, to 2.86479 =the Arch Aa; wherefore, the general Equation, in refpect to this Orbit, will be E +. 2.86479 x x =D, and by writing therein the given Anoma- ly, inftead of D, it will, in this particular Cafe, become E +- 2.86479 x x= 72.21. Now, becaufe 2.86479 x x muft be lefs than 2.86479, E, it is evident, can neither be much lefler, nor much greater, than 7o Degrees ; therefore, I eftimate the fame at 7o Degrees, and then fay, as Radius, to the Sine of that Angle, fo 1s 2.86479, to 2.692 ; whence E— D-+ 2.86479 x x equal 0.482, which is the Error, or firft Value of R: Again, for the Divifor 1--ey, as Radius, to (y) the Co-fine of 70°, fo is .05, (4) O171 =ey; therefore 25+ emj=,iokza, and hi eeiiet cake (Opa I--e;} 1.0171 69.536 for the next Value of E; wherefore, it will be, as Radius, to the Sine of 69.536, or 69° 32'.3%, fo is the faid Co-efficient 2.86479, to 2.68401 ; from whence the next = 0.464; which, being taken from 70°, gives (51) next Value of R is found equal o.o1001; and this, divided by the next Value of 1--ey, or even by 1.0171, the laft Value, and the Quotient taken from 69.536, leaves the true Value of E (= 69° 31” 34”) to lefs than a Second, Ex MP EB MIE ET the fame Things be propofed, as in the preceding Example, and the Anfwer according to the fecond Me- thod be required. The Value of L,. or the Log. of the Aré-Aa, as found by the laft Example, being 457093, I add thereto the Logarithmic Sine of 72° 12’ 36", or 72° .21, the Sum, re- jeGting Radius, is the Logarithm of 2.73, the firft Arch, which fubtracted from 72.21, the Remainder will be 69.48; to whofe Sine adding the faid Value of L, the Sum, dedut- ing Radius, will be the Logarithm of 2.683, the fecond Arch; with which, repeating the Operation, the third Arch will come out 2.6838, &c. and this taken from 72.21, leaves 69.5261, Se. or 69° 31° 34” for the Anomaly Excentri ; from whence the True Anomaly will come out 66° 52’ 50°. EXAMPLE III. HE fame Things being given; by the practical Rule, T it will be, as Radius to the Co-fine of 72° 12° 36°, fo is .0625 (=i of .05) to .o1g!; again, aS I-+.0191, to 0.1, the double Excentricity, fo is the Sine of the fame Angle, tothe Sineof 5° 21 41”; three times whofe Log. Sine, minus double the Radius, is the Sine of 2’ 48”; the 4 Part whereof being taken from 5° 21’ 41’; leaves 5° 20° 45” for the Equation of the Center, and :. this taken from 72 12° 36” will give 66° 52° 51”, equal to the Irue Anomaly very nearly. Of ~|-e xr Ip ; _ : P q- Mm . Of the Motion of Projectiles in refifting Mediums. PROPOSITION IL ‘Suppofing that a Body, let go from a given Point, with a _ given Velocity, directly to.or from a Centre, towards which it uniformly gravitates, is refed by a fimilar Medium, tn the Ratio of certain Powers of the V elocity, whofe In- dices are reprefented by the given Numbers, r, s, t, &c. And fuppofing the Part of the whole Refiftance, at the faid given Point, correfponding to each of thofe Powers, as well as the Force of Gravity, to be given; “tis required to find the Relation of the Times, the Velocities, and the Spaces gone over. WET P be the given Point, DPC the Right Line in which the Body moves, and D, e, any two Points therein indefinitely near to each other: Sup- pofe the Velocity at P to be fufficient to carry the Body, uniformly, over a given Diftance g, in a given Time 4; and let m be the Space, which would be de- {cribed in the fame Time with the Velocity, that would be generated in that Time 7” vacuo by a Force equal to the Body’s fpecifick Gravity in the given Me- dium; let the Part of the Refiftance, which is as the 7 Power of the Celerity, at the aforefaid Point, be fuch, that the Body in moving over a given Dif tance 4, with its Velocity uniformly continued, would from that Part alone, meet with a Refiftance fufficient to take away its whole Motion; or which is the fame, let 4 be i the (53) the Diftance that might be defcribed with the Velocity at P in the Time that the Body would, by the faid Part alone, have all its Motion deftroyed, was the Refiftance to continue the fame as at the firft Inftant; and let the like Diftances, with refpect to the other Parts of the Refiftance, that are as the Powers of the Celerity, whofe Indices are, s, 7, &c. be c, d, &e. refpectively 5 laftly, let P D=x, De=Pq4 =x, the Time of defcribing P D=T, and the Space the Body would move over in the given Time 4, with the Velocity at D, =v. Then it will be, asg:4::x (Pq): * the Time of de- gs ° e* the Velocity de- {cribing Pg, and as O:g:: x (Pg): 4 ftroyed by that Part of the Refiftance, which is as the r Power of the Celerity, tn that Time; therefore, the Velocity at D being to the Velocity at P; as v tog, that deftroyed, by the fame Part, in the fame Time, from the Body’s leaving D, ie x = , becaufe this Part i tee will confequently be of Refiftance is as the 7 Power of the Velocity: But the Time of defcribing De, is to the Time of defcribing Pg, as g to v; therefore the Refiftance arifing from the afore- s ° e . 7 ~ yey faid Part in defcribing De, muft be == x 2 =2 75 bg7—!} g bg?—2 from whence, it is manifeft, by Infpection, that the other Parts of the Refiftance, or Quantities of Motion deftroyed yi-Tlox wi : cues, €sc, And therefore the cl whole Velocity deftroyed by the Medium, in defcribing De, thereby, will be pgs . ie A 9 Ser Ee z F te is ———— = += a5 &c, But, the Time of defcribing bate eee 3 De, being to a that of defcribing Pg as gto v, will d Pp be ( 54 ) be reprefented by —-; and therefore it will be, ash: m :: Vv be, “=, the Part of Velocity generated or deftroyed in VU that Time, by the Force of Gravity, which added to, or taken from, the former Part, arifing from the Refiftance, according as the Body is in its Afcent or Defcent, the Sum ‘ pa yr—ls ys—ly or Difference = —~ 4- ———* 4 Ber str cme , Se. mutt, it is manifeft, be equal to (—v) the whole Decrement of Velocity : e m= 7) TU Hence we have x = = : 5 = seem 8 = > EF. ee Moreover, becaufe T, the Time of defcribing De, is found equal to i. we have fh = x; which being fubftituted in- itead thereof in the other Equation, &c. there will come out T= ee ig kB OT Et fe he —— » &. ChOOR Ob "4, ENCE, when the Refiftance is barely in the fim- ple Ratio of the Velocity, then c, d, &e, being infi- nite, our Equations become «= —=%"_, and T equal =t im 4. 2S b : — £—vVvxb ps. ~e: Whence x = ces > = 2°_ into the Hyp. Log, eo =t im = o* It gvatms bh. comm h of Eng » and Teo into Hyp, Log. ‘boserork COROL, (55) CO RO Level. UT, when the Refiftance is, as the Square of the Velocity, r being equal to 2, and ¢, d, &e. infi- nite (as before) the Equations will. be x equal to —— 5 =m + —hy — = x Eog. £ Pe co — baat if v be taken =0, we fhall and T equal to Hence x is found equal to have 2 ~ x Hypb. Log. 1-4 -£,, for the Height of the whole Ateint; but, if Nala . taken =0, we fhall have / m6 equal to the greateft Velocity the-Body can poffibly acquire by ee ; laftly, if g be taken =0,. there will Be— x Log. = a for the Diftance gone over when: the ae falls ia Reft ; therefore, in that Cafe the Log.. 5s, being => if nm be put for the abfolute Number mn 8 a —>’ wil Hyperbolic Log. is — == we thall get —"— = = i and confequently v= mb'* X= Moreover, with refpect to the Time, becaufe T in the Defcent of the Body is = —#42_ the Time it felf will, in. in bl2 +u this Cafe, beacaw/, ce into. the Hyp. Log. =; x zo =f, and therefore, when the Body defcends from Reft, — iL 7 mol? +2 I : a a mit? Le me is barely =— // = x Log = nee wherein, if the above. Lae Be found Value of v be fubftituted, it will be — / — x Log. ( 56 ) Log. “= ;:: But, in the other Cafe, T being = eehie 1 1— 112 Ll So mtbhtouv T will be equal to Ee drawn into the Difference of the two Circular Arcs, whofe Tangents are g and v, and whofe com- mon Radius is ./mb. And, in like manner, the Values of x and T may be exhibited by the Quadratures, &c. of the Conic Seétions, in any other Cafe, where the Refiftance is barely as a fimple Power of the Velocity, whofe Exponent is a rational Number, and alfo, in many Cafes, where the Refiftance is in the Ratio of two different Powers, by Help of the laft Problem of this Treati/e. C O'R O L,I. F m be taken = 0, or the Body be fuppofed to be aftect- ed by a Medium only, and the Refiftance be barely as a fimple Power (7) of the Velocity ; then x becoming equal ae a. eh and T= — bhg"" *u "4, x, in this 5 ca ba i te eee Cafe, will therefore be = Bee —, and T equal bhg' shel Trae Ii—_—7 : Where, if r be taken =0, 1, 2, . c b bu ec. fucceflively, x will be — — age? | Bs =, poe. =. — b+ -+, Be. and T equal to — x ie End Loe. aes eh g g x i, a s——, Ge. peeve ; from whence, by g'h bb exterminating v, we have x = Sox i—, T= ra Log. 3 g bxtins pied ods Hi latest T= bx “f= Se, expreffing the Relation of the Times and Spaces in the faid Cafes, refpectively. SCHO- C37 SHY O Louw: N Fluids void of Tenacity the Refiftance is in the Du- plicate Ratio of the Velocity; and it is found, that a Body in fuch Fluids, by moving over a Space, which is to ;of its Diameter, as the Denfity of the Body, to that of the Medium, with its Velocity uniformly continued, would meet with a Refiftance fufficient to take away its whole Mo- tion: Therefore, if this Space be taken to reprefent the Va- lue of 4, in Cor, II. by Help of the Theorems there given, the Velocity, Time, or Space gone over, will be readily ob- tained. For an Inftance hereof, let a Ball, whofe Dianieter is + of a Foot, and whofe Denfity is the fame with that of common Rain-Water, be fuppofed to be projected upwards in a Direction perpendicular to the Horizon, with a Velocity fufficient to carry it uniformly over a Space of 300 Feet in one Second of Time; and let the Heighth of the Afcent, the Times of Afcent and Defcent, with the Velocity generated in Falling, be required. Becaufe, the Denfity of Rain-Water, isto that of Air, as 860 to 1, 6 will, here, be eR ERS, 860) 764.4. Feet; and fince the Velocity, which a Body would acquire in one Second of Time by freely defcending zn vacuo, is fufficient to carry it uniformly over a Diftance of 32.2 Feet in that Time, it will be, as the abfolute Gravity, to the fpecifick Gravity ; or, as 860, to 859, fo is 32.2, the faid Feet, to (32.16=) m, b being equal to the Time above- mentioned: Wherefore, if for g, 5, b and m, their re{pective Values 300, 1, 764.4, and 32.16, be fubftituted in the afore- {aid Theorems, we thall have, firft, = x Hyp. Log, 1 + = (=x) = 630 Feet for the whole Heighth of the Afcent, 28, x Arch, whofe Tang. is g, and Rad 4/md, = 5.48 Se- oe conds, ( 98 } conds, the whole Time of Afcent; 3°, = .21455;3 4h y= 139, the Diftance that would be uniformly defcribed in one pete p. Second with the Velocity acquired by falling ; laftly, vee s t+/ I—fi1—z Ball be fappofed to move in Water with the fame given Velocity 5 then, the fpeeifick Gravity in that Fluid being nothing, the Body may be confidered as moving by its innate Force only; and, therefore, the Number of Fect gone over, in any Num- ber of Seconds, denoted by T, will (by Cor. III.) be 2, into the Hyp. Log. of 1 + 337.517. 72 x Log. = 6.85, the Time of Defcent. But if the fame Oe ees iis Si) el pe ee PROPOSITION Il. To find the Refiftance and Denfity of a Medium, whereby a Body, gravitating uniformly in the Direction of Parallef Lines, is made to defcribe a given Curve ; the Law of Re- fitance being given, partly as the n Power, partly as the 2n Power, partly as the 3% Power, &c. of the Celerity or as aC *4+bC 27-4 ¢6C37, &c. where C denotes the Celerity, and n, a, 6, c, &c. any determinate Quantities. ET ArC be the propofed » Curve, and AH the Axis thereof, or a Right-line in which the Body gravitates, to which let rn and em be parallel, and Hr and 4m perpendicular, r and m being any two Points in the Curve taken indefinitely near to one a- nother: Suppofe the Body arrived to 7, with a Velocity in the (59) the Direction re, reprefented by v; let AH=x, HA (rn) =x, Hr=y, nm (re)=y, rm=z, and let D be as the Tequired Denfity. Then, fince the Velocity in the Direction re is v, that in the Dire@tion rz will be <=; and therefore J 2x2" ail be the Flaxion of the fame, or the Increafe J of Velocity in that Dire@tion during the Time of defcribing rm; wherefore, if from this we take the Part arifing from the Refiftance of the Medium, which is ae (becaufe it is y to v, the Alteration of Velocity in the Dire€tion re, as x to y) there will remain 2% for the other Part arifing from the J Force of Gravity, in the fame Time and Direétion ; therefore, the Refiftance in the faid Direétion, is to the Force of Gra- vity, as —. —— to —=, or, as tee to 1; and, confe- J Z quently, the abfolute Refiftance, in the Direftion rm, to the Force of Gravity, as ——= to 1: But (22 the Part of UK i y Velocity, arifing from Gravity, being as the Time ~ of UV defcribing 7m, may be expreffed thereby ; whence we have “* =, or v?~ = yy; and therefore in Fluxions 2vux Vv d ¥ a Vv +%v*?x =0, Or — 7S ~~, which fubftituted in the fore- 2% to 1 for the Ra- going Proportion ——= : 1 gives Uv * s 2x tio of the Refiftance to the Gravity. Moreover, becaufe the ( 60 ) the abfolute Velocity is bs the Refiftance, by Suppofition, J ; wel eal? will be as D into a x 22 -- bx = &c,. or, becaufe ves x : z : an : == is=—:, as D into ax —+bx=—, Ge. which J ot xz x Quantity muft therefore be as =, andconfequently D as : x te QE.L. oth hes fre Seer teet heer 5 5g hen G.% Xn 72 bz ¢ Eee 3 wipe Se. Cit kh Ori ALR Y. ENCE, if the Law of Refiftance be only as a fin- gle Power (2) of the Velocity ; then, by taking b, c, d, Sc. each = @, anda = 1, we have ——~—. for the 2" x Za Denfity ‘in that Cafe; which, therefore, when 7 is = 2, or the Refiftance direétly as the Square of the Velocity, will be barely as ——. Pex AMP LL Ef ET it be required to find the Denfity of a Me- _) dium, wherein a Body moving, fhall defcribe the common Parabola. Here x being =" , we have x equal Zyy See p ( 6 ) 20) yw 2ye, and w=0 3 and therefore D=0; which p p fhews, that a Body, to defcribe this Curve, muft move ur Spaces entirely void of Refiftance. EXAMPLE HU. , © find the Denfity, @c. when the Curve isa Circle, ‘ and the Refiftance as the Square of the Celerity. Becaufe x, in this Cafe, isaxza—/ 44—J), there will IJ | ‘be *« = ——————, 2 = > em a2 a ang) asx /aa—yy /aa—yy aa—yy ay 32°)? 5. therefore the Denfity (=~) will here be, as aa—JJ x rae , or, as the Tangent of the Diftance from the higheft Point direétly; and the Refiftance will be to Gra- vity, as 3y to 24, or, as 3 Times the Sine of the fame Diftance to twice the Radius, SCHOLIUM. WF the Denfity of the Medium be given, the ‘Curve it BR felf may be determined by the Conftruction of the foregoing fluxional Equation: So, in cafe of an uniform Denfity, and a Refiftance, as the Square of the Velocity, where we have D= 4, of Dv V gy + XX — %; BN Dy 4 D\* y* ep : Se oo 9:0» ov + ae D is conftant, and the Refiftance barely as the Velocity, it R. | taal ° 2 q x will be found = = rae And when (62 } will be x= — 35, x Log. 1 2; ; — = ix: p; in ei= ther Cafe, being twice the Radius of Curvature, or the Para- meter at the Vertex; both which, and the true Value of D, may be eafily computed from the Velocity at A, and. the given Denfity of the Medium, PROPOSITION IU The Centripetal Force being given, and the Law of Refftance, as any Power (n) of the Velocity ; to find the Denfity of a: Medium.in each Part thereof, whereby a Body may defcribe a given Spiral about: the Centre of Foree.. ET RmH_ be the given Spiral, R and m two Points therein as near as. may be to each other, and’C the Centre of Force; and let RO be the Radius of Cur-- vature at R ;» making OD and mp perpendicular to RC, and. calling RD, s; RC, x; Rp, x; Rm, z; the Centripetal Force, C; and the Velocity; v. Forafmuch, as R.O.is to RD, as the abfolute Centripetal Force at R, to that, which tending to. the Centre O, would be fufficient to retain the Body in the Circle, whofe Radius 1s RO, and, becaufe the Centripetal ’ Forces. in Circles, the Velocities being the fame, are inverfely as the Radii; RD, itis manifeft, is the Radius of the Cir- ele which might be defcribed with the Velocity and Centri- petal Roree at R; Therefore, fince the Centripetal Force, in ( 63 ) ini Circles, is known to be fuch, as is fufficient to generate or deftroy allthe Velocity of the moving Body, in the Time it. is uniformly deferibing a Diftance equal to the Semi-diameter of its Orbit, we have, s (RD): z% (Rm)::v: (= ) the. Velocity which the Centripetal Force would generate in ano- ther Body freely defcending from Reft at R, in the Time: the former is defcribing Rm; wherefore, by the Refolution: of Forces, it will be = (Rm): x (Rp) =: <= :==, the Velocity generated. in. the fame Time,- by the Body defcri- bing Rm ;. which, therefore, added to vy the Excefs of the- Velocity at R-above that at m, the Sum => -- v will, it: is manifeft, exprefs the Velocity taken away by the Medium in that: fame Time: But the Velocities generated or de- ftroyed in equal Bodies, in equal Times, are as the Forces: by which they are generated or deftroyed ;. and, therefore, . it will be as = +4 v : 2% ,.0r as -+- =X to 1, fo: 5 Ss & is the Refiftance to.the-Centripetal Force. . But, the: Velo-- cities in Circles being in the fubduplicate Ratio of the Ra-- dii and Centripetal Forces conjunétly, v. wall be as y/sC,., and confequently — = 7. + aS. whence, . by. Subftitution, , > ° a : : ( ° ° Ec it will be as Stes ope reat or, aS 2% + S°X —- he Z& 2% Z&-- 2 :-C, fo is the Refiftance to the Centripetal Force; _ but - 2% © is the Centripetal Force, and therefore 2 Ke SEX shite. 3 2© is the required Refiftance ; .which-being divided by @C!=) the ( 64 ) the 2 Power of the Velocity, becaufe the Refiftance is in the Ratio thereof, and the Denfity of the Medium conju:ct- 9 —- ~ will, it is ma- ee TE VLE ee? 2Ciz es 2 nifeft, be as the Denfity of the Medium. 2, BE. ly, the 252 eC Ex AM? & E: ET the Refiftance be in the duplicate Ratio of the Ve- locity, the Centripetal .Force as fome Power, .m, of the ‘Diftance, and the Curve propofed the Logarithmic Spiral ; and, Radius being 7, let c be the Co-fine of the common Angle, which all the Ordinates make with the Spiral: Then s, by the Nature .of the Curve, being = x (= FR 2a, 2 x will be: 2% = 22, and therefore 2*t! 4 ,#¢ . 4° 2% r 22 + C é 2r : me £ mE. hence we have, as 723: 2%) fo is 2% F, - ; 2 c the Refiftance to the ‘Centripetal Force. And the Denfity.of ats; the Medium will be as —— that is, when ¢ and m are -given, reciprocally as the Hee rces fram the. Centre of Force: ‘But when m is —3, then ets becoming =0, it -ap- pears, that the Body in this. Cafe mui move in Spaces entire- ly void of Refiftance to defcribe the propofed Spiral: And, therefore, the Law of Centripetal Force being mare than Sh: Cube of the Diftance inverfely, the Defeription of this Curve will, it is manifeft, be impoffible from any refifting Force avhatfoever. OF (65 ) Of the Motion and Refiftance of Pendulous Bo- dies in a Medium. PRO FP AD lege ise Suppofing two equal Pendulums, whofe Bobs are in Form of the Segments of Spheres, to be moving with equal Velocities in a refiting Medium, and the Thicknefs of each Bob with the Diameter of the Sphere from which it is formed, to be given; To find the Ratio of their Refiftances. ET AKBA be one _y Side, or Half, of one of the propofed Bobs, and EAKBFCE Half the whole Sphere whereof it 1s a Seg- ment; and let the faid Seg- ment be conceived to be divi- ded into an indefinite Number of indefinitely fmall Lamina, by circular Planes perpendi- cular to the Axis KC, and equi-diftant from each other; and let AeBSA be one of thofe Lamine, included be- tween any two adjacent Planes, and 2 be the Thicknefs thereof, or the common Diftance of the faid Planes; calling AC, a; AD, c; any Ordinate RQ, Js MY; J and KD, x. Then, by the Property of the Circle, we have DR = Story ; which, Radius be- ing a, is the Sine of the Angle that the Surface at Q makes S with ( 66 ) with m Q, the DireCtion of its Motion, or the Incidence of the Particles it ftrikes againft: Therefore, fince the refifting Force of the Medium, on any Surface the Velocity being the fame, is as the Number of Particles falling thereon, and the Square of the Sine of their common Incidence conjunétly, the whole Refiftance of that Part of the propofed Surface, reprefented by Quy, will be as 2 xy 5 becaufe ny is evidently as the Number of Particles: Hence, by taking the Fluent, we have 72 xcc—z+y* for the Refiftance of A Q; the Dou- aa 3 ble whereof, =, when y becomes equal c, will_con- 34a fequently be the whole Refiftance of the faid Lamina: Which Refiftance, if the Axis KD (x) be, now, fuppofed to, flow, and x be put inftead of , will, it is manifeft, be the Fluxion of the Refiftance of the propofed Segmen; AKBD: But « being = a2—,/aa—cc by the Property Cae of the Circle x will :- be= 7—=—==, and confequently the aac e Acte above-faid Fluxion equal to 3; whofe Flu- 3aa W/ aa—ce AK XICK ADxDC 1A DZ +» 2 Ta Z x i a CK? 3 therefore the Double thereof as the whole Refiftance of the given Pendulum. 2, 2.1. ent will be and C.O. R: Ogi Y WY ENCE it appears, that the Refiftance of the whole i BB generating Sphere will be exprefsd by EK x KC, or the Area of the Semi-circle EKFE; and therefore is to that of its circumfcribing Cylinder, moving in the Direction wf its Axis, exactly, as i to 2, COROL. ( 67 ) | © O KO Bore, F the Refiftance, as above found, be divided by 3.141 59; &c x K D?~ bases WS ana the folid Content, or Quan- 2 tity of Matter in the Pendulum, the Quotient will, it is mani- feft, be as the Retardation of its Velocity arifing from that Re- fiftance ; and this, if b be put for the Axis or Thicknefs of 36bd bb+aai” the Bob, and d its greateft Diameter, will be equal to very nearly. ' C"O-KR Of Fy oF HEREFORE, if 4 be taken=d, we thall have = for the Retardation of the Globe, | whofe Dia- 4 meter is.d; and therefore the Retardation of the Pendulum 356d to that of its circumfcribing Sphere will be as meri 3 ——— 2 to-2, oras2bd : bb+-dd/ nearly. 44 Note, If the Bobs of Pendulums be in other Forms than thofe of Segments of Spheres, the Refiftance will be readily had as above; fince it is evident, that AC (a) being ta- ken for the Normal of the generating Curve K A, the Re- fiftance of the Lamina AeBSA will be i as is there found, let that Curve be what it will. be Isp ee MA: The Refiftance of a Body ina Medium, 1s to the Force of Gra- vity, as twice the Space thro which the Body muft freely fall by that Gravity to acquire the given Velocity, to the Space Space over which it might move with that Velocity in the Time wherein the faid Refiftance, uniformly continued, would take away the Body's whole Motion. For the Velocity acquired by freely defcending from Reft, thro’ any Space, is known to be generated in the Time that the Body, with the Velocity fo acquired, would move uni- formly over double that Space: But the Forces, by which the fame Motion would be uniformly generated or deftroyed, are {inverfely as the Times in which it might be generated or deftroyed ; and therefore inverfely, as the Diftances de- {cribed with the fame Velocity in thofe Times. Sr ee a ee ae PROP O'S LT1.ON oll Suppofing that a heavy pendulous Body, ofcillating in a Cyclotd, is refifted by an uniform Force, and at the fame time by a rare and fimilar Medium, in the duplicate Ratio of the Ve- locity ; To find the Excefs of the Arc, defcribed in the whole Defeent above the Are, deferibed in the fubfequent Ajcent, and the Time of one entire Ofcillation. ET ABD be the whole Cycloid, BC its Axis, EB the Arc defcribed in the Defcent, and BI’ that de- fcribed in the fubfequent Afcent; draw GHE, Ff, &e. parallel to AD, let S be any Place of the Body, and JR, the | ( 69 ) the Diftance thro’ which it muft freely defcend in vacuo, by a Force equal to its fpecifick Gravity in the given Me- dium, to acquire the fame Velocity as it has in that Place: Suppofe that Part of the Refiftance, which is uniform, to be to the Force of Gravity, as m to 13; and let d be the Space over which the Body muft move with its Velocity uniformly continued, to meet with a Refiftance from the other Part, fufficient to take away its whole Motion ; or, which is.to the fame Effe@, let d be to the indefinitely fmall Arc $2, as the whole Motion of the Body at S, to that deftroyed by the Medium in moving thro’ S23 cal- ling BD, 5; BE, a; Rb, x; ES, #5 Sz, x. Now, the Force of Gravity being reprefented by 3, that Part of it whereby the Body is accelerated, at the Point S, is —— (= a And, by the preceding Lemma, we have, as @: 22:2: 1; == for the latter Part of the Refiftance, or that in the dupli- cate Ratio of the Velocity ; which being added to m, the a—x former Part, and the whole taken from ——, gives a an m — == for the whole Force whereby the Body is accele- rated at the faid Point: ‘Therefore the Velocity, there, being known to be, as \f 2%, that generated in == the Time 2% of defcribing Sz will be defined by ——=— 1% — — x = ; which muft therefore be equal to, —= , the Fluxion 4 z NV 2% or Increafe of, / 22, the aforefaid Velocity: Hence, we ze 22 : : Lae ’ have Jae ee Mmm oe 5 which, Ey ymin if inftead Za ac da ) w T e 4 of a— mb, Be. becomes — — 4. 4. == 4. = equal 0; we La em whence, by folving the Eaton z is found = <* =» b x x7 — aa a cl Pads . ES¢, Or, if p be on aoe (0. 367878) the N the Number, whofe hyperbolical Loga- dd+d dx rithm is —1,= I—p~ — eee But when the Body arrives at F, the Hight of its Afeent,.z, or its Equal re 5 — | c. become l ; str x ™ ar de Sas S equal O; which Equation folved, gives x= 2¢ — ee zh ae , 8. 3 — EBF: therefore FG ism 2a—2e4 40 16%" &e, ‘ 34 odd? e2 ee: =2mb+t. ——, &c. (by refuming 2 m4 inftead of its Equal ek oui which, becaufe m and — are fuppo- fed very fmall, will be 2m6+4. =" — very nearly. Q. 2.2. Moreover, fince the Time of sfeshantie the leaft pofii- ble Diftance Sm, is as es, by fubftituting therein the Value of z, as above found, we fhall have : Ha S. 2ex d+ 2 a ie a Pre 7 b bd ine sae 1+3h ae * er eee Cg for the Fluxion of the requi- Spel Gal eo) ans ee een red Time: But, becaufe—>; — x + —, &c. the Square te of the Divifor of the latter Part or Factor thereof, when «x becomes Cor) 16¢3 | gd z 7 be equal to Nothing, if 7 be put to denote the Value of 2e¢— 4.e¢ 1623 3d gdd? becomes = 2¢ = 4/7. &c, appears, from above, to &c. or the Root of the Equation, it is manifeft 25 ze that —- —r + 3a I —_— d 2r2 a 5 eee : confequently » — a gute ee Ge. =——; which hoe being fubftituted inftead thereof, our faid Fluxion will be- bz a come 2e|2 a eae 4r3 ei ae: &y jz X*2 3d de aes Poe ie eB Gangs ae 8 ; : = 3 xX #3 f = I= se x rtex rrr txx . XZ mee n — 2 ey e r—sx int xX —~ +4x uae ee, Lite Ta ARES Wo Sat ridin Gece a. Rene I oe, ie r+x rer xe x x eae 2 x into: I—2 Xp hax age ee. peel xr=al and laftly, by converting 1—2 x, 0. ee into a ra- Et b2 x 2 rr rx tional Series, equal to ——>;———— into 1 4 Sto z $ a I += xr— x |? Fugy &e. Now, the Fluent of eg ee. ( = ) when « r— x12 Vrx—xx is = 7, is known to be equal to the Periphery of the Circle whofe Diameter is Unity; wherefore, if that Periphery be put equal f: the required Fluent of our given Expreffion Lee rox 3 pe Se ‘tet au ad “++, Ge. will then appear to —— eT » Savi * zx eps -~-+ 77, &e. from p. 118.. of to b= be my Book ee Fluxions ; which, by reftoring the known z Value of 7, will become p 4* x 1 -- | Soy =, + 2 3e * me é e e3 Se, =pb*xI— oe add adie Ole XI Gs aat oat De = zr es Ge. = pb* xi + eg — Gav» Se. and this is, it is mani- feft, as the Time of one entire Ofcillation, Q. EF. L. Coen Orin tT HEN gd is infinite, then FG becoming barely equal amb, it will be as 20: FG::1:m.s Hence appears, that the Excefs of the Arc defcribed in the whole Defcent above that defcribed in the fubfequent Afcent, when the Refiftance is uniform, is to twice the Length of the Pen- dulum, or DBA, as the refifting Force, to the Force of Gravity. COMB, OC CLS 10 UT, when m is=o, or the Refiftance barely in the duplicate Ratio of the Velocity; the faid Excefs will be in the duplicate Ratio of the Velocity, or Arc defcribed nearly. COROL. IU. F m be confidered as negative, or the Pendulum, inftead of being refifted by an uniform Force, be accelerated thereby, fo as to continue its Vibrations in the fame given mo A eS Se Se Are ; (73 ) Arc; then, fince 2md +- 7 (= FG) becomes = 0, —m will be=- 37 : And, avibetsbe it is manifeft, that the Force, which acting uniformly on the given Body, is fufficient to counter-ballance a Refiftance in the duplicate Ratio of the Velocity, or to keep it vibrating in the fame given Arc, muft be to the Weight of the Pendulum, as 2aa@ : 34dnearly: And, therefore, the Arcs, which a given Pen- dulum fo aétuated, will continue to defcribe, by different Forces, will: be nearly as the Square Roots of thofe Forces. COROL. IV. HEN both mand + are equal to nothing, that is, when the Ofcillations are performed without Re- fiftance, the Time of Vibration will be barely p 2 *; which is to the Time, wherein a Body freely defcending from Reft, would fall thro’ CB, Half the Length of the Pendulum, as the Circumference of a Circle, to its Diameter. C.O;R O1L4:.¥ M OREOVER, when only — is = 0, or the Refit tance uniform, the Vibrations will, alfo, be Ifochro- nal, and performed in the very fame ei ‘as if the Pen- dulum was not at all refifted: 2 €. OF O.k.). VE. UT if m be equal to nothing, or the Pendulum be refitted, only, in the duplicate Ratio of the Velo- city ; the Time of Ofcillation will then be pd? x U I+ ( 74 ) a 3 , l--+ oy ot €c, Therefore, the Excefs of the Tims of one whole Vibration, in a Medium refifting in the duplicate Ratio of the Velocity, above the Time of Vibration in the leaft Arc poffible, is to the Time of Vibration in this Arc, aa zig 3 : : ee aS eT g “odda? &c. to Unity ; -or, becaufe — is very. fmall, as qq» to.1 very nearly: Hence it fhould follow, that the faid Excefs, is in the duplicate Ratio of the Arcs very neatly, I fay fhould follow, becaufe I know very well, that Sir I/aac Newton, Princip. Prop. 27. B. Il. makes it to be, nearly, in the’ fimple Ratio of ‘the Arcs: This I confefs had made me more than a little fufpeét, that I might have here fallen into an Error; and yet upon re-examining the Procefs with more than ordinary Attention, I have not been able to difcover any Miftake therein committed; but, if any fuch fhould occur to my Readers, I fhall readily ac- knowledge my {elf obliged for the Difcovery. or WT OL FU MM. F inftead of a Cycloid, the Ofcillations be performed in I a Circle, the above Conclufions will ftill hold, provided the Arc defcribed be but {mall; excepting thofe that relate to the Time of Vibration, which is fhortned or prolonged, independent of the Refiftance, from the particular Nature of the Curve, acgy ding as a fmaller or greater Arc is defcribed. But, if to the Time pb? x 1+ a EFF » &c. found as above, be added the Excefs of the Time of Vibration in the Arch a, of a Circle whofe Radius is 4, above the Time, in the leaft Arc poffible, which, by p. 140. of my Book of | Fluxions, (75) 1 —— x Fluxions, is pb* xi &c. we fhall then have p4* » Zz loi <6 7h a &c. for the Time of Oscillation in the Arc a of that Circle nearly. Hence it will not be difficult to determine, how much the Times of Vibration, in {mall Arcs of Circles, are increafed or decreafed from the different Weights of the Atmofphere. For, if the Force by which the Pendulum is kept in Motion, be always the fame, the Arc defcribed, by Cor. II]. will be as / bd, that is, in the {ubduplicate Ratio, inverfely, of the Denfity of the Medium, or Height of the Barometer: ‘Therefore, if / be put for the Height of the Barometer, at the Time when a given Arc ¢ is defcribed, the Length of the Vibration correfpond- pie z ing to (y) any ether Height thereof, will be “2” ; this there- yr fore being {ubftituted inftead of a, and — inftead of d, I 2 2 2 in the above Expr i z NIN SEL preflion, gives p57 x 1 epee tore Ge for the Time of Vibration correfponding to this laft Height; which, when y=, is pe x1 + the Difference of the Times of Vibration an{wering to the two Heights of the Barometer 4 and y, if abe put equal » €&c. therefore to the Difference of thofe Heights, will be 2: vee c2 ye 2 : . aris 3 EEE Gag» Ge: nearly, excepting by fo much as it is varied thro’ the different fpecific Gravities of the Pendulum, &. in a rarer or denfer Atmofphere; which is eafy to be com- puted. However, after all, it-is not to be fuppofed, that the Alteration in the Time of Vibration, above {pecified, will happen ( 76 ) happen immediately upon the Rife or Fall of the Mercury ;. becaufe the Pendulum, thro’ its v/s zmertiz, will be fome Time before it can be brought to perform its Vibrations, either in a greater or {maller Arc = And, indeed, the Alte- rations, both in the Time and Arc defcribed, from the above: Caufe, are fo fmall, when compared with thofe arifing from. Fri@tion and Expanfion,. as fcarcely to come under the niceft Obfervation.. PROPOSITION Ul. Suppofing that a heavy Body, ofcillating in a Cycloid, is refifted. by a rare and fimilar Medium in the Ratio of a given Power of (n) of the Velocity; to find the Excefs of the Arc defcribed in the whole Defcent above that defcribed in the fublequent Afcent, and the Number of Ofcillations that will be performed before any other given Arc is defcribed, or the Pendulum bas loft. a given Part of its Motion. ET ABD be the whole Cycloid, BC its Axis, EB _4 the given Arc deftribed in the firft Defcent, BF that defcribed in the fubfequent Afcent, and FG the required Difference of thofe Arcs; and, fuppofing the Body to be arrived at any Point S, let its Velocity there be the fame as it (97) it would acquire in freely defcending from Reft thro’ the Arc eS by an uniform Gravity equal to its {pecifick Gra- vity in the given Medium ; and let d be to the Length of the Arc BD, as the faid Gravity to the Refiftance, which the Body would fuffer with the Velocity that it might acquire, from that Gravity, by freely falling thro CB: Draw EHG, SR, Gc. parallel to AD, and let b, a, A®, and x, ftand for BD, BE, Be, and BS refpedtively, ‘Therefore, the Velocity acquired zz vacuo, being in the fub- duplicate Ratio of the Diftance perpendicularly defcended, RAE, or its Equal “A—zz (from the Property of the Curve) will be as the Velocity at S$; and therefore =iAtes a the Fluxion thereof, as the Increafe of that Velocity: Bur, this Increafe depends upon two Caufes; the one, the Force of Gravity, and the other, the Refiftance of the Medium. If the Medium did not refift, A would be conftant, and VA—szx? wherefore, the other Part, arifing from the Refiftance, mutt therefore the Increafe of the Velocity barely as / ae Medium, to that Part of the Gravity by which the Body is , and, confequently, the refifting Force of the be as accelerated, as —— : ae or, ‘as Ato2zz%: But this Part of the Gravity being to the whole, as z to d, by the Property of the Curve; the Refiftance will, it is mani- felt, be to the fpecific Gravity, as A to 2bz. Moreover, be- caufe the Velocity, at the fame Point S, is to. the Velocity which would be acquired by freely defcending (‘as above) along BC, as “/ A—2z:6; the Refiftance, with the for- mer of thofe Velocities, will be to the Refiftance with the “ x latter, un Aa eek 2 B2—t # SMT 4 | latter, as A—22l2 tod , or, as to 6; and, con- fequently, the Refiftance at $, to the Force of Gravity, ET ae ie ; (ean ee ad: Wherefore, it will be as ee 1 dic A : 262; whence “ = wee ee , from which Equa- 2 db*%—2 tion A may be determined by the known Methods of in- finite Series, @c.. be the Medium. what it will: But, in a very rare one, fuch as is fuppofed in the Propofition, the Thing may be, otherwife, much more eafily effected. For,. then Ee, being, at its greateft; exceeding fmall, ca — 22%, may, without fenfible Error, be fubftituted in our- Equation for A—223; which done, we have A. equal 2 zy aa—zel* - And then the Fluent. of the latter Part dbt—2 thereof, when z is equal a, and man even Number,. will n+1 a ; 2 4-., 6 8 me out ——— into—— x—..x —x—>;. be found to co ange : aa >? ce, to. — Factors: 5. and. when 2 equal a, and.” an odd: Se n—+-1 = I 2 granted, s oats — WWD. 00 8 aon Su“ _—_ Tr—t Quantity whofe Increment is muit confequently a ‘ -_— —s_ be = : ¢ Whence to find the Value of a FraGtion rnaX Aue a p™, =I of this. kind, from its Increment given, there arifes this Rule. ‘Strike out the leaft Factor in the Denominator of the given Increment, and inftead thereof put the ReGtangle of the common. Difference of the Factors into the Number of remaining Factors; the Refult will be the Va- ue that was to be found. hx A MP LE. i it be required to find the Sum of the infinite Bad Series Dit gta, or one finele Frac- For oe + a Se, ne fingle tion ( 93 ) tion (if poffible) that fhall exprefs the Value of, er. = 45 + aE +. nies Here, if the required FraGtion be confidered as made up, or generated by a continual and regular Ad- I I dition of the Terms, &c. —-» — >» —- of the ‘propofed Se- 45 3-4 2.3 ries, then a being the next fucceeding Term of the Pro- greffion, or the Increment of the faid Fraction, the Fraction I it felf, by the foregoing Rule, will be = —~ IX2—1X2 oe In like manner will be found 12) 23) 34 4S xi ea pene Ge, a ee 3-45-60? ~ aaa 7 Ge. = ay notte ort + Gag ee N.B. That in finding the Value of any Quantity by the Methods foregoing, it ought to be well confidered, from the Nature of the Queftion, whether that Quantity confifts barely of fuch Fraction, Produ@, or Products, as are {pe- cified in the Propofitions, or thofe joined to fome invaria~ ble Quantity (as is done in Fluxions } and Allowances is to be made accordingly. Bb An An Inveftigation of Sir Waac Newton’s Theorem for finding the Sum of a Series of Numbers by means of their Differences. ET a, 4, ¢, d,e, f,g,6, Gc. be the given Series of . Numbers: Then, by taking each of them from the next fucceeding, there will be —2+, —d+c, —e Ja, —dte, —f+g, for the firft Differences: Again, taking each of thefe Differences from its fucceeding one, we have a—2b+c, b—2c+d, c—2dte, d—2e+-f, Se. for the fe- cond Differences, In like manner the thirdjDifferences will be found —¢+346—3c+d, —b+3¢—3¢d+e, —c+ 3d—3e+/f, &e. and the fourth, a—4b+4 6¢c—4d+e, b—4c+6d—4e+/f, &e. Sc, Let the firft Difference of 4? the firft Order be called D, the firft of the fecond Order D, the firtt of the third Order D, Gc. then we fhall have a=a, b=a-+-D, C=—a--25 1D, d=a—3b+ 3c-4D, &e. and from thence by Subftitution, a=a b=a-+ D c=a+t2D+ D d=a+3D+3D+4D ema+4D+6D+4D+D, where the Law of Continuation is manifeft, the Unciz of the Values of ¢, d, e, &c. being thofe of a Binomial, raifed to the fecond, third, fourth Powers, &¢. Therefore, if be put for the Number of Terms in the propofed Series 2 +-6 +-¢-+d, &c. whofe Sum we are about to find, the Value of (95) of the next Term in the Progreffion after the laft in that Series, or that whofe Place is defined by m-+-1, will, it is plain, be equal to a@--xD+nx ——D+2x far x —— D +2 =x x — D, &c, And fo much will a faid Sum be incited by augmenting 2, the given Number of Terms by Unity ; which Sum, therefore, a the firft of the two foregoing Propofitions, isna+n <2 Dpaxtcixt x's B, Ge QE L EXAMPLE, UPPOSE a, 4, ¢, d, &. to be a Series of Squares, as g, 16, 25, 36, &c. whofe Roots are in Arithme- tical Progreffion ; then will the firft Differences be 7, 9, 11, 13, &c. the fecond, 2, 2, 2, &c, the third, 0, 0, Sc. Ge, Therefore zg = 9, D= a: D=2, D=o0, D=0, &c. and confequently 2@--n x= D, &e. equal 9% +- nx-> x7 + nx 2x2 x2= 9416 25+ 36 +49, Se. con- tinued to z Terms. In the fame manner the Sum of a Series of Cubes, Biquae drates, Sc, may be found, An eafy and general Manner of inveftigating the Sum of any recurring Series. PROPOSITION. Suppofing py %, 7, 8, Sc. to be any Quantities, either pofi- tive or negative, and A4+B+C+D- +E, &c. a recurring Series, or one whofe Terms A, B, C, &c. are fo related, that any one of them, being multiplied by p, the next follow- ing, by q, the next in Order, by r, &e. the Sum of all the Produéts, thus arifing, fhall be equal too: To find (x) the Sum of fuch a Series. ECAUSE, by Suppofition, pA+ ¢B+4rC, &. is equal o, pB +9C-+1rD, &. equalo, pC + 4D + rE, &c. equal o, &c. &e. it is evident, that the Sum of (pA +tqB+4+7C+45D, &e. } pB+¢qC+t+ rD+sE, Ge. | 3 Cc D E F, &ec. | muft confe- all thefe, J E* +9D+7r ; re ¢ \ auienily Be “e pD+gE4 7rF +5G, &e. | equal o. pE+qF +7rG+sH, &e. L Se Ge. Ge Se. Se. Where, becaufe A+B +C+D, ec. is = x, B+C+D, Ge = xa A, CH DEE, Se, = x—A—B, &ec. &c. the Value of the firft Column towards the Left-hand will be px; ¢hat of the fecond, gx x—A; that of the third, rxx—A—B; (an emt i eet that of the fourth, sxx—A—B—C, &c. &c, and therefore px (97) petgxx—A+rxx—A—B+sxx—A—B—C &. or Oe Ty pane +sx—sxA+B+4C, «203 hence w= 2AtrX APB APE CH MC APEECED Sh es ae acoder tres, oe Cec. Quay XA M Poe § ET the Series ay + y? $2- , Be. (= x) be propofed, where p equal — y, g=a, r=0, s=0, Ge. A=ay, a3 > _ gqA+rxKA+B, Ge. ay B=yy, &c, then will x = eee become as in this Cafe. BX AM? Lie I ro ae 2 +5,— 2 uy 2s &c, = x, where p= 1, 9=223, r=2°, s=o, t=0, &. Then A being =1, B= — >, &e, tAFPXATR ES (54) will 6: jor 8 here become ae Mit oe ce PROP. i Ses a ( 98 ) A Method for finding the Sum of a Series of Powers, &c. PROPOSITION To find the Sum of any Series of Powers whofe Roots are in Arithmetical Progrefion, as m+-d\" +--+ m+: 2M) Wt. se Rah wate, Bol x", m, d, and n, being any Num- bers whatfoever. i ET Fe DF fh Bx. Te py OF =p he > 4 Fx” 7, &e, — K, if poffible, be al- ways ki to md" i es ie x”, and A, B, C, &c. determinate Quantities: Then, if any oleate. ber in the Progreffion m+ d, m+-2d, MEA 9.0 ons xd, x+-2d, x+3d, Ge. as w-+d, be fubftituted inftead of x, the Equality will ftill continue; and we fhall have —_—7z+1 ——____. ——_—_———_, — Se Ax x +d! +Bxw+d)4+C xxadi ‘Dyed ce, —K equal ma)" 4 madi” Ae x4d''; from which taking the former a there remains Axx-+d" au uae + Bxx+dl" a Wear ly Tg Be = x+-dl", fhewing how much each Side is increafed by aug- menting in Number of Terms in the given Series by Uni- ty; where, by tranfpofing x+di", and throwing the feveral Powers of x + ¢ into Series, we have a | | Oo = (99) IQ) “GP Gt ee ing) y > . * é 2G) “HP gig ¥ 2 XI TEP gemy*X OXI % * 783 “9? 6 u Eo X = ag TP zany * a eo Ke +P yy Xe + & we ; = & z 2, Beets 763 ‘sP ¢—u ®VE X TEX = =X Ihe FP ee XV TZ Xx —XIfe bey eV Xibe te yevite g z [82 8? fae FX TK P tem * X o @e +X capes sie ee + waa 8 Fe. abr et ee eee ec, each Side of whichr bemg 2 3.4 2.3.4,550 3 incteafed by Unity, and the whole multiplied by d” gives: x % I a rae "43d!" 44a!" .... Pivorlin ee Ae Sanat ee aX n—it XK n—2 x" 8. —a- +--+ 2.44/56 ni 2a A ae X 1—2 &r 304, 2-450" Be xOAUM Pb BR. ET it be required to find the Sum of a Series, con- fifting of 100 Cube Numbers, whofe Roots are, : 3 5 ees 2 I, 25 +) 3> Se. Here ( 103 ) Here d, the common Difference of Roots, being equal —, %= 3, and x = 0, let thefe Values be fubftituted in the E- quation in Cor, II. and it will become ( im in, = seat ++ set a ) 3187812.5 , the Number that was to be found. EX A.M P L-E! if. - 2? oo => d=— . Then the Equation in the laft Corollary will become aa fal = a} Lt. i) sin eb eee On ae very nearly’. fo that, taking x equal 4, it will be ey + = Bh 3, 4 + 1= 3.0731; which differs from the true Value by lef than ;<2.< ; and if more Terms had been ufed, the Anfwer would {till have been more.exact; but never can come accue rately true, when # is negative or a Fraction, becaufe then both Series run on ad tnfinitum. SC. BH Onhily UM. HE ‘Theorems, above found, are not: only ufeful in finding the Sum of a Series of Powers, but may be of fervice alfo in the Quadrature. of Curves, &c. efpe- cially as the €onclufions will be accurately. true, and the Reafoning thereupon. {cientifie, rey This: Bere ee an eee ae eee ( 104 ) "This I fhatl endeavour to thew by the following Inftance ; wherein AC, being fuppofed a Curve, whofe Equation is =z (AB being equal z, and CB equal y) the Areg ABC is required. Let AB be divided into any Number, x, of equal Parts, as Ad, bc, cd, &e. and from the Points of Di- vifion let Perpendiculars be taifed, cutting the Curve in the Points, 1,2, 3, &e. and having made pi, 972, 73; s4, &c. parallel to A B,-Iet the Bafe Ad, 4c, cd, &e. - of each of the ReGangles p 4, gc, rd, Gc. be reprefented by d: Then d1, c2, 23, Gc. the Heights of thofe Rect- angles, being Ordinates to the Curve, will be 2", 2d”, 2 al Sc. re(pectively, -each of which -. being multiplied by d, the common Bate, and the Sum of all the Products taken, will give d into pee aid” at (=Apig2r, Gc. CBA) for the Area of the whole cir- cumf{cribing Polygon ; and this Series, according to the above- ° A 4 : fH +1 x 2 faid Theorem ( Cor. III.) is equal to d rt in, -& Fo Gein ti axel” | sd &e.* =~ a €c. or, becaufe dx =z, it will é 1 3 5 rN dnt be = eX a ce aca &e, Now, if from this the Diffe> rence of the Infcribed and circumfcribed Polygons, or the Grom - é 4 Re ~|. i Rectangle BD = dz” be taken, there will remain ae oe , for the Area of the infcribed Polygon. Hence, it — is manifeft, that, let d be what it will, the infcribed Poly- gon can never be fo great, nor the circumfcribed fo fmall, as n+ 1 ABXBC ae (= na —\: And therefore this Expreffion muft be accurately an to the required Curvilinear Area ACB. Of Angular Sections, and fome remarkable Pro-- perties of the Circle. PROP OS T7706 Bae The Radius AC, and the Chord, Sine, or Co-/ine of an Are, as Ar, being given; to find the Chord, Sine, or Co-fine of ARa=mxAr, a Multiple of that Arc, ET RH betaken = AR, and the whole Arc AH be divided into as many equal Parts, Ar, rf, &c. as there be Units in 2m; andthe Chords Bv, Bf, &c. are dra wn, as alfo the Radii Cr, CR, CH, and the Perpendiculars rf, RE; calling AC, 1; Batya Cp, wee Ce, Be 7p, 2; RE, U;A4, 23-and ABS Z s... Phen; :becaute any one of thofe Chords, as Bf, is to Br + BR, the Sum of the 2 next it, as BC to Br, by a known Property of the Circle, we fhall have yx Bf. Br + BR, or yx Bf —Br=BR;; and for the very fame Reafon, yx BR— Bf = Bg, and yxBg —BR=BA, Gc, &c. Hence, it ap- pears, that the Values of the Chords Bf, BR, &c. (which Re | to eS Se ess sss SS ( 106 ) to a Radius equal A B, will be Co-fines of the Angles A Bf, ABR, &c.) may be readily had one after another, by ta- king continually the Produ of the laft by y, minus the laft but one, for the next following: And thus are had, ym2=By, yi—37= BR, yt—4)* + 2=Beg, 5 — 573+ 57 = Bd, yom 64+ 99? —2 =BH, Se. &Se, And generally, fuppofing A y"~’ —By” > 4+ Cy" 5, &. to denote any one Chord of the foregoing Order, and A y zi ( 107 ) — By So + Cy "~* @c, the next to it; then the Chord next following thefe will be Ay”t* — By”! 4 Cy" u—1 Ge, — Ay +By" 3—Cy” >, &e. =Ay™™ i 9 3 Fe { Sc, at h ; n—t ae: ; "3, &e. From which x ty oar Seer ie -(by the Method of Increments foregoing) A will come out pot a nu—« = 1 Baw Ca —, D= — ae E = 2-2 2 X as x xe x1, &c. and confequently Ay ”— By = 2 if 2 be taken equal to the given Number m, it will become m — 3 pew iy &c. equal BR; but n— 4. u—=—2, u— — OY %, Bey” ny 2x73 y" aa LP treads state —8 % ASS y” fs ee a : Ly” ©, &c. wherein yo — my" "mx if n be equal 2m, then it will be y°— my” 74 2=3 2 ce &c, equal BH; where the Series are to be con- tinued till the Exponents become negative. Hence, be- caufe Bf is equal 2, and the ArccAH=mxAf, it fol- lows, that the Chord HB will be = 2X! mm MK DK Lis 9a & — xaxt” *, &e. and therefore, X (= CE) the required Co-fine being equal ? H B, we have K = 2%) zal m 2 a —— "—2 a m—3 m— 4. - i, — =X 2K + x73 xawl "2 y Uk, an wee un ~~ m—6 m8 og AD 5 oe ae ae ee &c, fhewing the Relation of the Co-fines; from whence U i ii BO Ni ig a Bit ek th a Ii a Baki ha ibe? eRe EC re ee ( 108 ) hl | em ene, i —= % U ( = “1—X*!) comes out =V {—xxl, in, 2x| 7 —— 2 ha EH ase Phas ae — 7 x 26! ie a ay | aoe 4. fe LesiG, ) toes ae x 4 x na x20! , &c. Furthermore, becaufe ae 2 y ae ny Yo +n = y BEARS, | , AE will yA . the® Numbers sderres {ponding to thofe Logarithms muft be equal: Hence, by adding together the two Equations, we have 2X = te of wena a! ee ge by taking their Difference, 2\/X*—1 = 6 o/ og eT x— fxx—il”; from whence, by expanding the latter Part of each of the Equations into Series, and dividing the whole by 2, there will come out X=x” + mx “= I m2 x Bl EATER ee RE Sa ts ES¢, and VY X?—1= SV xx—I iN, mx™—* 1 myx nee Vi —=Z X [— K™~3 xox —1, Ge. the former of which being 2x1” m semen (lh & reduced into fimple Terms, givesX = ea 2 ys —, 1 — + 2x23 youl” *) Be. the very fame as above found. And the latter, by multiplying by / — I, to Gg take ( rip} take away the imaginary Quantities, and fubftituting U and w inftead of their Equals MP 4/7 Ieee a1 becomes TD Usain mxt—uid > 4mx2 x 7 Ku? oii a el — m—t1 m—2 mM — 2 m— 4. K [uy 4 Te Ae a ee ee + 2 3 4 5 m—5 1—uui 2 x44, Bc. which, in like manner, being redu- 2 ced into fimple Terms, will be U=mu—m»~ ~—! x 2.3 oll Bos ae “3 mx = x 2S x us — mx A x aes “5, &:3 Ss 2 23s — is hen yen de STOR OL. 1 ECAUSE the laft Equation, as appears from the Procefs, will hold as well when w# is a Fra@tion as when a whole Number; let m, or the Multiple Arch AR (= mx Ar) be fuppofed indefinitely ait then will mz I m * — mx 7 xui+ mx" x ee xXus, &e. the Sine of that Arch, or the Arch it felf (which in this Cafe may be confidered as equal to it) become mz oe = 1 9 ale gD ass gi &c. and theref = (meena ( A ore E 2.54% 2414.5-6.9.° the Arch A» AR ° ° e ° ° e gl whofe Sine is #, wiil, it is manifeft, be = w—. 3-345 3.3-5-5 47 3-3-5.5.7.7 9 at 2-3+4.5 = 2.3.4.5.0.7 2.3.4-5.0-7.8.9 ? Se. COROL., Cre Civ oO Be F Ar be fuppofed indefinitely {mall, and m indefinitely great, fo that the Multiple Arch mx Ar (=A) may be a given Quantity ; then fince z may be confidered as equal to Ar, mu will be equal toA, andmu—mx — 2 a 5 ° mix mirus xu3, Ge, the Sine of A, equal to mu — a A3 As A? or A— — —— — —— ., &e. becaufe 2.3 2.3.4.5 2.3.4.5.0.7 7 e 1 9 25, Gc. in the Factors 7? —~1, 2* ——Q, ec. may here be rejected as indefinitely {mall in comparifon of mm’, SCTOL fU.M. PECAUSE wm.) moteur egy ae te ‘ ——} 7 is found above to be univerfally = 2x! _. » %: awit ie W2—~ 2 Cpe al mn — 4. har lS 2M ae i ee ee Xx ———- ¥ He oe —— m—6 : : 2 ; 2 x | , Ge. it is evident, by InfpeGion, that Lo ee aE UAE newer eee med ——, m—— 2 aE —; m7— 4 2x!” 4omx2x| + mx" x25! » Se. and ., a SSI, J oa | J we mee Der VT eee TAS ae ee Pe or — 2 ee . ° my FP abs mx 73 y™ ta, &e. (by fubftituting -7. in the room of x, and rr in that of Unity) let » and 2 y be what they will: Therefore, ify" 4. my”~? pe 4 Us pleat ge eee ease RE oe eres . ( 82°) mx 3 y™— 474 om x A e+ oe: yo ro tomy 3 S. Rioicere i m—6 30 eek tee Pace NORE a Ce. be orc equal to fome given Quantity c, there will be given 2_ zy 7 = jig ue ; eso ee FORT |” 7 mio = ¢; and therefore 4 4 / eager ss rar?” —- iis i ges aaa ‘ Pia 4 2 ae =¢¢; wherefore, the double Rectangle of J. rs S22 rr : ii into 2. and I eS c¢ 2m c Ge 2m |m | arr art A Swi tee and there- c ance 2 WH hay iG te fom 9 — yee ar a +-—-Y= ee e Which ( 113 ) Which may be ufeful and ferve as a Theorem for the So- lution of certain Kind of adfected Equations, comprehended in this Form, viz. y"-+- my”? r? + mx 2 y"~4 r4, ec. =c: For an Inftance hereof, let the cubic Equa- tion x3 + 4x =h be propofed ; then, by comparing this with y” — my” *r?, Se. we have m=3, you, mr? ; s i At DP Onn = 1% c=b* and confequently x= 24. ge? ate, By hb ie ee +o fey pine telat PROP OTS Et Ona If on the Diameter AB, from any Point C, in the Circle ACB, whofe Centre is O, the Perpendicular Ck be let fall, and the Arc AC be divided into any Number, m, of equal Parts, as Aa, am, &c. and if the whole Pe- riphery be alfo divided into the fame Number of equal Parts, beginning at the Point a, as ab, bc, cd, &c. and from any Point P, in the Diameter AB, or AB produced, Lines be drawn to the Points a, b, c, &c. I fay, Pa? x PS? x Pc? x Pd’, &c. the continual Product of the Squares of all thofe Lines will be equal to AO?” = AO” X% 202 XOP” + FO. UT AO.=to.1, PO; = to.x, AP? = to I wx? U, Ok =to 5, 2m =to n, and the Square of any one of the Chords Aa, Ad, Ac, Ad, &c. equal to z: Then, fince any one of the correfponding Arcs Aa, Abd, Abc, &e. reckoned forward a certain Number of Times, brings us to the fame Point C, or, is equal to AC, or AC plus a certain Num- Gg ber Saye ere age ee ict pape RSS ( 114. ) ber of Times the whole Periphery, it appears from the laft aaah 1 Propofition that +2” uz” "= uX—— & —2 =nx- 2—5 mM ak 3, &c. continued to m Terms, is= AC?, or becaufe AC* is=24+25 (ABxAZ) it will be 2” x Wi—=T eT 3A UN1——2Z — ae —3 = wan he + 2x2 — 2x — x — Uke = 2426 =0, let x ftand for the Square ns which of thofe Chords you will: Wherefore, the Roots of this Equa- tion being the Squares of the Chords Aa, Ad, Ac, &e. they muft be all pofitive, their Sum =”, the Sum of their ae a) Soke Oe common Algebra. Now, if se be made perpendicular to AB, we fhall have A P? Sere b Aicee Pies = AP* + Ac? = APx ao = AP? oop ox Bel, which n—3 in See ee et ee Pale eh oes Late ze? ee see SAD See eS eee [ 25.) in Species, is Pet=vtxx Ad’: And, for the very fame Reafons, Pd6*=vu+x¥xAb*, Pcotzu+txx P eas ye therefore the continual Produé& of vy +x x Aa* into vu+x*xAd* into v+xxAc*, &c. is equal to Pa? x P6*xPc?, Gc. But in the former of thefe Produéts, it is evident, that when the feveral Faétors are actually drawn into one another, the Co-efficient of the firft Term or high. eft Power of v, will be 1; of the next inferior Power, the Sum of all the abovefaid Roots A a7, Ab2, &c. into x, of the next following, the Sum of all their Produéts into x2, Sc. and, therefore, the Sum of thofe Roots being already 1—3 2 found =”, their Produéts = 2~ Mm » Se wellavet. | mm aac ti nxu” * +ax— xy” Tt — 2 Bere te u2—4. u—e +nx oe 5 —s Zo — 1 SO Wires ay i “OX TE RL Ke et OE oe eataiar ate I Ne ob Pa*x Pd*x Pc, Sc. Or, by fubftituting for v, its Equal ——— ° o mie thes Sa mS a= % Twxf it will be rax! +nuxxIinxl? ” + 2x = x*Xraxl?4....-2+26 xx” = Par y% Pb { Pc, &c, (becaufe 2m=n): This in fimple Terms is AE 1—nax+ "x MEM REED MES eae! | 2 3 We aX = xx? + aX KAS 4 3, &Se. * * AL 1x SF yx? —Nny» oe Xe, Se, pe 2 , See * * * -- 1x x 3, Se. &e, +2+6xx” | (976 1) Which contracted, by adding together the homologous Terms, becomes 1 * * *, &ec. . Hence it appears, that the -Co- efficients do every where deftroy one another, except in the firft, laft, and the middlemoft of the faid Terms; and that the middle Term would likewife vanith, if inftead of 2+2bxx™, the correfponding Term of the above Series 1nxl*+tnxx ianxl*-2, or that where the Expo- nent of x is m, was to be added ;_ wherefore this ° a! > I . m m1 Term being N X*—— X See ee Into (=2x")as is eafy to perceive from the Law of Continuation, we have Pa gvn” aa eeto ba x Pe* xP? Cc, or, AO?” +202xAO”— xPO7+ PO?” = Pa*xPb?, &e. And, when the Point & is taken on the other Side of O, Oz becoming _O%, AO?”—20kxAO”—'xPO”* + PO?*” will be equalto Paz xPb2x%Pcz, &. Q, E. D. CO OR» O Lins ci F C be taken at B; then will OL=AO, and Pa? xP 62x Pc*2, Se. =AO’*” +2A0” xPO” ++ PO*”; where, by taking the Square Root on each Side, we have PaxPdxPc, &. =AO”+4+PO”., COROL, I. UT if C comes into A; then A being —o, andO£% =AO, AO?” —20kxAO”"xPO7"=Pa?z x P62, Gc, will therefore become AO?”W2A0” x PO ( 117) PO“+PO2” =Pa2xPb:xPc2, Ge. And Pax Pd, &. =AO7nPO”, COROL UL ENCE it is manifeft, that if ahy Circle ABCD, grc. be divided into as many equal- Parts as there are Units in 2m (m being any whole Number what- foever) and if in the Radius OA, produced thro’ A, any one of the Points of Divifion, a Point as P be aflu- med any where, either within or without the Circle, PA x PCxPExPG, &c. will be == AO”nPO” PBxPDxPFxPH, &. =AO"+PO”, and PA xPBxPCxPDxPE, &. =AO’"nmPO*”. Hh PROP. 118 ) Of the Redudtion of Compound Fradtions into more fimple ones. Ay Be Z tf fer = A) ei PLivt. eo wh Wot. bf ie ROPOSITION. To divide a Compound Fraétion, Ge atbxtex*tdx3?.., ae > into as many fimple ones as there are Units in p; fuppofing m to be any whole poftive Number, not exceeding p, and the Denominator reducible into binomial Faéfors. ET r—«x be any one of the given Faéors into which the Denominator may be reduced, and let —— r—x =e At+BxetCxr?7+Dx3.... +. +x P—2 ey St gethxttixi....fPxP! _—-___#**—_________; then, by Reduttion, we atbxtcxttdx3....Qxh~! 4x? rA-brBxtrCx*rDxifrExt oo... rixP—24 * * — Ax— Bx?—C x3 —Dxt.. cae eeeee —txp—! have =O. sfsguesh x7 sits ps kxt ccc ce ceees +sPxP—! be affumed equal to becaufe r—-xx f+ gx +t hx, Se at bx-+cx*, Se, Hence, by comparing the homologous Terms, i get JF s _ fs gs a eo ES Cy ORM” oa ae we gs oe laftly ¢, or sP= — ec wae ee =e oe ——— : wherefore fs --grs+bsr*-isri+tksrt.... cs. vy rm frtgrtbbrsirthrs oo. Pr? But, sPr? aor” “ands= SRE pe MIMS SS Sry ee ae i P - P es = rT idee 4 : Biot ata re es ry E . : 5 A a ee ee ee a ae RS ee I OE = ee OE ee epee ea eee ace ee eS eee : Bs z i ( 119 ) But, becaufe r—xxf+gx+ hx + ix3....4Pxe—! ism atbw owt dxt,,... Qe?Th-L x? we rf prextrhatrixs rket..... rPxP—1 * have * —fx— gxr*—hxi—i xt Sa SILSh inh Laem ete —— Pa =0- — Aa —b yee 0K md xi ment ii... QxP—! Se ae te Ae b a b c and nel chee — $= z+—, b=[Gt+at- i= pres ¢ eH es es -+—, and P= aera Se. Whittice a,grz==atdbr, br3= aN tr¢s=atbrtcrz—tdr3, Ge and..Pr?=a+4 br + cr2+dr3—+er+,...Qr?—'; wherefore (by adding - ‘all thefe Equations together) there will be fr + gr? + br3 -+ Pr?=pa+t p—ix br + p—2xer? + p—3 Xdr4.....Qri—! ; which-laft Value being fubftituted in that of + gives f= 2 ee ee patp—i XbrpHz Xcr*-+p—3Xar3 .o. .Qree! for the Numerator of one of the required fimple. Frac- tions; whereof the Denominator is »—»; from whence, by Infpeétion, the Numerators anfwering to. the. other given Factors or Denominators are eafily obtained: For, JE OR ee My Bee ey oT we jes > be the” faid Factors Into “which . ¢ 4 6 x 6%? =e OMI virus ge Omer -+ x ?is reducible, or Rx x S—x x T—wx, &c. be =a+bx4- I 75 Oe te eee ———TE>E™ETO a He patp—i XbR+p—2X cR? +7—3 KAR3...QR?P! i b> e put =A, ——__——_—_- sae Linda eee b P > 9a 4 pi XES-E pees pax 8? ‘ erat : Ee, . vx m—t Pb tec x +dx3 ae ae ae, ee BL. Ll — x (xc, €$¢. \it 1s evident, that will ‘be AvR!# —— Bas ~ o—x equal to + EXAMPLE I. —I ET the Fraé€tion a = =e Then will a=2, 6=—3, c=1, pe O,e=0, &e, V=1,p=2,m=2, R=1, S=2, T=0, &. A=1, B= , be propofed. EXAMPLE IL. x AI —_—— ; then, by com- rt2bx® x? I F the given FraGtion be paring a+ bx +cx’ tdx3..... Oni 4.9? with 1+2bx” 4.%7", we have a= 1, p=2n, the Coefficient of the middle Term = = 2.4, and all the reft except the lait =0; wherefore A (=—————$$—$$ ———_______ } patp—i1 Xbr+p—2Xcer?... .Qri—! will in this Cafe become = > Bb 2 Fe, a 2 4 2n X 1st br 2“aXi1t55" ( 121 ) A General Quadrature of Hyperbolical Curves, PROP O 5 liPaeen There are two Curves AC, HDG, baving the fame com- mon Mbfciffa AB (x) whofe Ordinates BC and BD are repr n—r—l d —*___+ ; “Yo find the Area of —_______., and — rae dx ®t x2”? rat dx”™ + x2” each; fuppofing r and n to be any whole pofitive Numbers, and the Denominator 1 = dx” +. x7” not reducible inte two binomial Faétors. ET 2x be taken in ¢ as often as poffible, and the Remain- der be denoted by m; and let Ax*>—*-+4 B rma Ny r—3n 2a+em n-+sm ; ym pettm tan” ne ae 7 ie | ee ie med xe" het a—?7 Wie , be affumed = an ; c being any given Quantity : ; ~—— AX li Then ie fs Ny a fe t B iis Py i te 4 & Fe 3 es 4 E = a ; Wend = sae Se ee - ee ees Panes ae ao : Lat ee I ee ee EN Ee en En he ee ee Eee oe a ea 2 ee ee ee ee eee = ( 122 ) Then, by reducing this Equation in- nam eo to one Denomination, we fhall have ee D fe Bde iia ft icles Ni | +A +B * moms C x@=—omO; and Qu * * \+ — as| —dly x2ntmt 7g hee [+4 oe) ey thepiote Aa BadA, C= dB —~A, DidC.—B, Egan ea ey fas 7, vV=—S5. But, now in order to conftruct thefe feveral Co-efficients ; with the Radius 1, and Centre O, Fig.2. let the Circle AB be defcribed ; take OX = } d, C & perpendicular to A B, meeting the Circle in C, and the Arch CBU to the Arch AC, as r—m to nm; and let the fame be divided into as many equal , at the Points R, S, T, &c. Te eae Parts as there are Units in 2 and let ¢ be now fuppofed = C&; then will Ck, Rr, &c. the Perpendiculars falling from thofe Points on the Diameter A B, be equal to the faid required Co-efficients A, B, C, &c. re- {pectively : For, fince the Arcs AC, CR, Ge. are equal, by a well known Property of the Circle Rr (B) is = eo xCk(=adA), ~$5(C) = 20X2* _ CZ (= dB— A), AO —Tt(D)= 20x — Rr (=dC—B) &c, Hence, Wwe ( 124 ) u-Lp me ae sda =k RPS kl we have ———___—— x” 7 dehy m1 we Uv&x atm iy; an Be ia eer erry x I— ax 2% ettr—i sg hs a5 nnd@ietore a into Ce XR” HE Rr x t—dx4% ex #2 Cx Tr Be % x ri 2 ees toa OS Sea i xUox «T_T Kx Ce t—det@ty, 2% a — i —27 Fe 3% ee elem ef Ben a Se ee r—n Ke r—2nXe r—3"Xc SET ar ‘ Be wy Ge est Te ne te plus the Fluent of aces Se Ey atm m ‘ 4 ° uX x +T tx ‘ will give the Area 1n the firft Cafe. Raid we yeu 2m And this Method of Solution, it is manifeft, will hold equally, when the fecond Term of the Divifor is pofitive, or the given Denominator 1 ++ dx” + x°”, if &, inftead of being taken towards A, be taken at an equal Diftance on the contrary Side, the Center from O towards B. But ees that above-named will be obtained, take Aa to AC, as 1 to mand OP =x; and beginning at the Point a, let the whole Periphery be divided into as many equal Parts, ab, bc, cd, &c. as there Units in 2; letting fall the Per- pendicular am, and putting 6=id (= O2) and Om=f; then, becaufe Pa*xPb*xPctis=1—2be" + x%’”, (as is proved in the Propofition preceding the laft) and be- caufe 1—2fx+xx (=Oa?—20mxOP+ OP?) is Pa*, we fhall, by feigning 1—~ 2f/ «+4 xx equal to No- thing, now to find the Fluent of from which 3 (3125 ) thing, get f-/ ff—1—x, and f—V/ ff—1—x for two of the (2 2) imaginary binomial Faétors into which the faid Quantity 1—24x%” + x", or its Equal f+//f—1—* % f—A/ ot oe —x x Ge. is reducible : Wherefore, if f+ /ff—1i be put =f, and f—VSff—i=z, , mm qi then. will f_____— , and ——1____,, By the 2nX1—bp* Xp— x 2nXi1—bq"Xq—«x Jaft Propofition, be two of the fimple Fraétions into which eS Df may be divided; thefe being added toge~ ee SS t—2bx%" +. x2” ther (to take away the imaginary Quantities ) give: p™ g-fa™p—axp™ ta" ix pMa7tt 9m pF tb baxp My pamp” 2 into 1—bxp” +4” + 459% 9” into p—x X g—* which, becaufe pg Seah V IPH IAS + Viftie=r, will. be mar pares uae xg 49 ashi it 2a into 1+-b5—bX p* +9” into1—2fx+txx ; n i Ree i ! ‘ But, fince # +2 is the Co-fine of the Multiple Arch AC (=2xAa) and Jee that of mxAa, &e. as is. manifeft from Page 109. if AH be taken = mx Aa, and Af=m— 1xAa, and am be put = g, Hd the Sine of AH=G, and O/=F; our Expreffion will be thus exhibited, Cof. of AAS En” Cof..CH . Bot a AH —. Aais=Af, AC—AH=CH, Ge. we have by the Elements of Trigonometry the Co-fine of Af= Ff + Gg, Cof. of CH = 6F + cG, and the Cof. of Cf= bgG + bfF +cfG—cegF; and tierefore our Expreflion will ftand thus, K_k Die Sa a aaa a hist UR a ps a ck cee lac hy aC am 4 FF Ce IRE oie apc tae IN OC eee E ae FRE a seen: eS ————— es Pi 8 nS gt cI i i i ar ol in a ace cine age (126 ) Ff+Gg—- bt fF — 52 2G —befGpbegh be FHF FEC» | neeKi—2fxpxex P where, by fubftituting 1—cc inftead of bb, it is, at length, reduced to {Phe eFteeGrefGtex$G— <8 And this drawn | cnXi—2fx+tre into x; is one of the m rational Fraétions, (whofe Denomi- nators ate cnx Pa, cnx Pb*, cnxPc?*, Ge.) into which the Quantity oe whofe Fluent we are feeking, x ~ may be divided. Now, therefore, the Fluent of —F7— being oy (PaO) or, sa into the Arch meafuring the Angle on and that of - equal to the fame A : Arch into + , plus (AO: Pa) or the Hyperbolical Lo- garithm of a x? the Fluent of OPEL Pati Nae ae FE pbgF $egG—b7GX+4+bC—FXx* | that of thofe caXi—2fx bee ; FraGtions. whofe Denominator is cz X Pa*) will be = 2 — Ex (0a: Pa) + x ae Or, CH HAxXOk ObXOk eae Te ae into (Oa: Pa), + HCE hg H? into (PaO): From whence the Fluents of the le of the Fractions, which make up the required Value, whofe Denominators are ncxP4?, ncxPd*, Se. are deter- mined, by Infpection, fince the Manner of Conftruc- tion muft neceflarily be the fame in all of them. Next, from hence to find the Fluent of eee For the very fame Reafon that AH was taken =X A@ in find- ing See ES ee aa ag) ee ae Saad ing the Fluent of ——*—_# 1—2bx% 4 x 22? let A H be (now) ta- ken = m+uxAm, and let HA be perpendicular to AB; then er Ee into (Qa: Pa), + XE 4 Hb into (PaO) ee, &c. &e. will confequently be the Va- lue fought; but AH—AC being = AH, H4AxOf— Céx—O4 will be AOxH4A, and —~O4xOL+CA x hs =OxAO, and Pye the faid Value equal to xo (O¢:Pa) +~-=— a (:P2O) Ge. &e. &e. Now, from the two foregoing Fluents that of ic x UvXx*t"_ Ty MB. : ° t ° ieee = 1s readily determined ; and, if TQ_ be taken — mx Aa, or ACQ = — x AC, and Q zx per- ils: to AB, will come out = = (O4:Pa) —— (PaO) &e, &e. &c.. that is, if, for the fame Reafon. that TAQ is made = mx Az, TAQ be made mx Ab, Pet a TAQ=mxAd, or QQ— QQ _— QQ, Ge = mx ab, and the Perpendiculars Qu, Qn, Be. be let fall on the Diameter AB, it will be ee axck ( 129) Qn(O0a Q2z (04: iPr) + OmC P20) 4 ? —Qn (Oc Bois NO wxCk Qn(Od: —Q2 (Oe: :Pa)+On( PaO)? PS) Dg, FeO) Pd) + On (—PdO) Pe)—On (—PeO) 1 &e, ee. 5 : Ck xt" Ree 2 This therefore added to—="Z>- + —GoaxKe ? Se. continued ’till the Denominators become nothing or nega- tive, (as above found) will be the required Area in the farft Cafe. But for the Area in the other, where the Ordinate is te ee I tela pte “pe eF let « be put equal to ee or y=—3 then s yrfatt far! Peake, will Sk. , and x=—; and top dx? 4 2%~” roe dy® ty?” BP —r—i.» . therefore “| or the Fluxion of the propofed Area tae dx™ +x?” sung Meee y ABDHEA, equal to —— , and confequently pan age ee y Hea 30 ‘ ; which Expref{- P that of BF GD equal to I= dy zz +7 Zn fion being the very fame in Form with the Fluxion of the -Area ABC; /it is manifeft, that, if P, in the fore- going Conftru@tion, inftead of being taken at the Diftance x from the Centre, be put at the Diftance — (= 9) therefrom, as at P, and the Signs of all the Indices of x be changed, the Expreflion fhewing the faid Area ABC, will give that of BF GD, or the Value required in this Ll Cafe SR SO OIE ee ee OS EO OF EEE NEN Ee Tie eter Ls Sie ( 130 ). Cafe: Which therefore is £4 4 RYX*™ — Sixx?) rm r—22 r—3n ? yer &e, into ~——, plus ae into Qz (Oa: Pa) -. On (PaO) &c. Gc. But OP (x) being to Oa (1), as Oa toOP “es i the Triangles OP ¢, and Oa P, will be fimilar, and therefore the Angle P20 =O P a, and Oa: Pa, OP: Pa, &c. wherefore the faid Area will be Ck Rr¥x% Ss x 2% 7 go tz Fr rz a fhi-ee Gs r—3% » Se. sO Cc f Qu (OP: Pa) + Ox ( OPa)7 Qz (OP: P4)—On( OPS) —Q2(OP:Pc)+0z( OPc)} Qn (OP: Pd)+O7 (—OPd) —Qu (OP: Pe).07 (~OPe) . &Se, EF, J . tse pile yy sd plus = into < Oc O Low I ENCE the Area of a Curve, whofe Abfciffa is x and Ordinate ee n u— =e Td? 120 may be eafily obtained: For, let the Radius of the Circle A B, now be denoted by g, the reftas before; then, fince every Term in the required Area muft confift of the fame Number (7—z) of Dimenfions, by fubftituting the feveral Powers of & for thofe of AQ, or Unity in our former Area » It will become toe —_—_—_——., Ck ( 132 ) elt Ck RrXg? a7? Ss Mg2# 20 3 gen Ck into r—n ae r—~24n fe rm 3H . Ge. iaCk into Qnz (Oa:Pa) + On(Pa2O) + Ge. Ge. for the Area in this Cafe, ‘Cr Oris. L. 7/TI. [| ENCE, alfo, may the Area of a Curve, whofe Ab- ‘ bP aes {ciffa is 2; and Ordinate , where a Pon gh" pg h 4g 2h f denotes any Number at pleafure, and r and x as above, be ? boc eafily derived: For, putting z * =x,a"=g, andd=f x es dy hi Say e rests mi 1 Of a ek SP , we have zz? = - a — nlp —y 7 1 x . eS ee mony oe n P x b x — xx ; and 22 SR an ee — ST a7? gh! pe P +27? § stg dn” fe 2” wherefore, if in the foregoing Area, anfwering to the Ab- espe getty ae vata Ee os thefe Values of d, g, x, be refpectively fubftituted, and the whole be {cifla x, and Ordinate multiplied by a it will become the Area in the prefent + TP =P Cafe, which therefore will be**" _—s into S4 4 pPXCA r—?n RrXa?lz? semen 4 4 Pie T 2F ‘ ue a = , &c. continued ’till the Denominators become nothing or negative, ce Toe ere ni re Sib nie : et Tee : eiriacan dela ies eT ee sy La re eS ar a aa ge ak eS N52 a a a eA (283) fF Q2z(OA: Pa)+On( PaO)} Qu (OA: P4)—O2n( P40) okt? |—-Q2 (OA: Pc) +0On2( PcO) PU PXCR aaa “ Qu(OA: Pd) + Ou (—PdO) —Qn (OA: Pe) —On (—PeO)| L Se, Saco.) \ Where, according to the foregoing Conftruétion, AO fhould p a Pp —_—— _— — | : Seats 5 Ok=: fra? andPO=2z : but fince each z 3 Term in the Area, when actually divided by the common Divifor C%, will be affeéted both in its Numerator and De- nominator by one fingle Dimenfion or Power of Lines exhi- bited in the Circle, whofe Ratios do not at all depend on the Magnitude of that Circle, it matters not, whether AO, O&, and PO be taken exadtly equal to thofe Quantities, or to others in the fame Proportion, as p “zh n iy. tisk Wiel ‘ ‘a, if, and a 2a. PORCr. at and Tat » provided the reft of the Conftruction be retained. The like will] hold in the Area of the Curve whofe Abfcifla is z, and Ordinate — 227" , which by proceeding ath fab—' gh 427? in the fame Manner, from the fecond Cafe, will come out r Te PHA Pa? iy — =e ear eek 22% 2 in Ch +. RrXz a + —SsXz" oes Sc. eX Chae Mowe Pas Ton “till the Denominator becomes nothing or negative, Mm oe Se recs oneal Dahesh ies ca RE Fetes ies a tee ie area eee i RY i (134 ) | (On Ot 2). Ont wer O )> | Qn(OP: Ps) —O7( 6PO}) DOr er. Pry + Oe | cee) 17h (OP PY) + On (PPO) \—Qz (OP: Pe)—On (ee PO) Lo ee: | ey Hence, to find the Area ABCA of a Curve, whofe : ‘ . . nm bt Abfciffa is z, and Ordinate me? = , or the A- Bieatin® bua. 4 en zi z rea BFGD of that whofe Abfciffa is z (= A B) and Ordinate irr ates ay Pott 4 Pang 2p ftru€tion. Fromthe Centre O of the Circle ACB, whofe Ra- diusis = 1, take the Point & in the Diameter AB, towards A. (fee the foregoing F7g.), we have this Con- or B, ar asthe Sign of fa’—' z? is —or +, fo that O# may =~, and draw the Perpendicular Cé to the Point & in that Diameter, meeting the Circle in C; make Aa to ACGvasa ito”, and ACO to. AC, as r to #, and CR, RS, ST, &c. each equal to AC; and, beginning at the Point az, let the whole Cirele be divided into as many equal Parts as there are Units in #, as a6, be, cd, Ge. take va ee OP = =|”, and each of the Arches Q.0 5... 2.05 &c. = the Arch a6 ~ into the Remainder of r divided by 2, and draw Pa, Oa, Pd, O84, gc. and the Perpendiculars R r, Ss, Tt, Ge. Qa, Q2, Qn, &e, to the Diameter AB; then . (135 ) then will the Areas be refpeétively as above exhibited. And it muft be obferved, that this Solution holds in all Cafes where /f is lefs than 24, andr and 2 whole pofitive Numbers, : Note. That (OP.: Pa) is put to denote the hyper- and (@PQ) the Meafure of Pa OP? the Angle aPO in Parts of Radius or Unity ;. the like. is to be underftood of any other. cll tm bolical Logarithm of 9 ClO L414 0 M. H E above Solution being fomewhat intricate, others by infinite Series, where they will convergé, may be thought preferable; but as the greateft, if not the only, Difficulty in what has been here delivered, lies in finding ; M—1 .. n-t-m—1 i the Fluent of ——~—_—*—_ , or -~_—__—,, it may 1—2bxe7+ x 2” 1—2b6x7+x 27 be proper (before any thing is offered on this Head) to add a different Method whereby the faid Fluent of Se may, in the Form it ftands above, be more eafily inveftigated. In order thereto, the firft Conftru€&tion of the Points C,R,S, Sc. a, b,c, Sc. Ge. being premifed, let the Sines of the Arches AC, Aa, Ab, Ac, &c. be called B, C, D, E, &c. and their Co-fines 5, ic, 3d, te, &e. tefpeGiively: Then, becaufe 1—cx4+«x is= Pa’, 1—dx we =P Sy rex exw = Pec2, Ge.-and. Pa? P52 ~ Pc*xPd?, co. = i —2bx" + x7”, it follows, that the Sum of the Logarithms of 1—cx-+xx, 1—dx+ xx, €c. muft be equal to the Logarithm of 1 —24x74 7” > 2xnx—cx zxxmdx is and therefore 7, + Wyma? &c.the Sum of their 2 we Kee ee} ree Fluxions = , the Fluxion of that Lo- ° 4 T—2b v7 + x garithm. Hence, by taking each Side of the Equation from 22x eet, and dividing the whole by x, we have —in 55 abe fe 9 + xax—dx-+1? ( 437°) Sas ee 2n Xx '—be?—? ° : eo, = A; this Equation be- r—- bin” 22” ‘ x ° vas b ing multiplied by x”, and the former by ~-, and the Produéts added together, there will be 4 into —==5 + 2x—d as 2—cx 9—dx Lee eet ee her +e PC Teese? Ec. + x into — us +. ree i— ax pK 5 tl on Btxt—! (=- eXznx — | aS d¥ PEs id 5 Wane tom reauce p— 2b 2 x 2” pii2hs2 x2” a fe ihe ; A—¥ hy (lL 2 2—-4* _ & Fo, to lower Dimenfions; 2 “ xx—cx--l a ax—dx+1? 4 R—Z : t= Be ir A— 4. ey CRN ; s t Let Ax + Bx + Cx tae te Re he eS cP EM 2 he Of T—axpxx |? KX—— OK i¢ wg et ° : ° : 4 Seat oa ai then, by bringing the Equation into one —=— 2? xX —= CX +1 Denomination, and equating the like Terms, we have Az=ic, B=cA—t, C=cB—A, D=cC—B, &Fe, and v= —s: But, thefe Equations, it is manifeft, (from a known Property of the Circle) likewife exprefs the Rela- tion of the Co-fines of the Arches Aa, 2Aa, 3Aa, &e. therefore (A = iC) being the Co-fine of the firft of thofe Arches, and B that of the fecond, the Values of e D, iD €3c. which entirely depend on thefe, mutt confequently be equal to the Co-fines of the reft of thofe Arches refpec- tively ; and therefore t equal to the Co-fine of xAa, and — y= that of m—1x Aa. Wherefore, if Ax"? +- f Noa B Siig alc Ia Sh ct alga RINE BE le TIES ee EEN Nee ie UF ( 138 ) ee f : | Bae ee ex Re sre pilehee., be, | a—\ ST in ke Manner, aflumed = —— --. x * 2— 82 and 2 xKx—dx+tt “ee fut A. fie (oa. ON ec, = — _': x 2 2 HK —eO xt) &c. &c. it follows, for the very fame Reafon, that A, ie oe ee v, will be the Co-fines of the Arches A}, 2Ab, 3Ab, nx Ab, andz—1xA4, and A, B, C, 7, ee v, thofe of Ac, 2Ac, 3Ac,n2xAc, 2—1rxAc., &e. &c. refpectively ; and we fhall, by adding together the : wee oe i Pause nave — ae ar ie Se, ee fa q ’ z x ee eeae Te aomdarbe S Ge =x" XA LALA, Be x 3x Ba By HB Gap xt XC 4. C+ ©, Ge. Be, 4 ete I—c xxx Es sett But, A, A, A, &c. being the Co-fines of the Arches Az, AJ, Ac, &c. and therefore the Roots of an Equation, expreffing the Relation of the Co-fine of an Arch, to that of another Arch # times as great, wherein the Sam Term is.always wanting (vzde p. 106.) their Sum muft there- fore be equal to nothing, by common Algebra ; which is evi- dent even by Infpeétion, when z is an even Number; for, then every one of the Points a, 4, c, &c. above A B hay- ing another Point, of the fame Conftruction, diametrically oppofite to it, the Sines, as well as the Co-fines, anfwering to thefe Points, muft be equal and contrary, and therefore | deftroy a —E ( 139 ) deftroy each other. In hke manner B, B, B, &c. being the Co-fines of 2Aa, 2A4, 2Ac, &c. or the Roots of fach another Equation, they muft alfo deftroy one another, it fc. €c. Hence our Equation is reduced to —ix ~~ x 4 4 a “a 2—Cx: 2—adx Ce tx—+u | tx--wv Hi — SS = SS 2 I—c xXx x 1—dx-pxx? ; Il—cxpux 1—dx+xx sascha €3¢, But fince ¢ isfound to be the Co-fine of 2X A a 1— exper C as E WH U H fl AR dries él ee [XZ ad 0 /D ( 140 ) or AC, ¢that of nx Abor ABC, 7 that of 2x Ac or ABCEC, &c, —v = that of n_-1 x Aa, —v= that of z—1x Ad, &c. we have t=t=/, Se. = Ok=b, andv=—BC iB c “un iv] x ——, v= — BD — =" eee. and therefore — = x Sek ae 2-— dx ke __ bx —BC—ibc bx—BD—tbd T——cxpax Ia © xx? 7 cx tax t—dxtux ars Nm 2—cx 2—adx 2—ex Co be OF Xx OS uae —- Piguet 1—extxx? EFe, _ bc +2BC—2bx bd+z2BD—2bx : ; = ea eae » &c. which being ° ° ® se By ail EX fubftituted, inftead thereof in x”~" Aree en 22B* x” z2> = _____ (as abovefound) and the whole divided by 1—2be"7 +477 C D E 25, we fhall have I—c xe x Py 1—d xb x ts I—e xx x F 2Bx?—! Bc =: and, confequentl oe 1— fu-tx x? & 1 — 2b? px 22? ; d y oo Bex Bae tees C xi™ By 4 Ex” So ee aes ee See Ilex pax a I—d xp x pape eer 3c. Letnow Ax *—? 42 Bee 3 + Cx%—4 2e.eede ps oe be aflumed = eee ; then, by Reduction, Gc. we fhall get A= sie B=cA, C=cB —A, D=cC—B, &c. &c. where it appears, from the a- fore-named Property of the Circle, that A; B, @ MG > — v, are the Sines of A a, 2 Aa, BONG, 5. 1 Rie Aa ( 141 ) Aa refpectively. Hence, if Ax oe eee a Cg 24 . j 5 tx fu b fi Dx” whe aye O48 ty Sb ces Be Put se, Me nifeft, that A, B, C....#, —v will be the Sines of AJ, 2Ab, 3Ab....mxAb and m—1xAb tefpectively, Ge. €c. Therefore, as it is evident from the above Reafons, that A+ A, @c. and B+ B, &e. and Ci ee ee muft all be equal to Nothing, we have pastor nL. 1—2hx% + 2%” x X Sine of m«Aa—Sine of m—1XA ay xX Sine of m ae 5 7 n—7 nope r. 1a at =, p. 12%. J. laft, for x “3. rr, Lately Publifbed, i. COURSE of LECTURES in Natural Philofophy, by the late RicHarp HeEtsnam, M.D. Profeffor of Phyfick and Natural -Pbilofophy in the Univerfity of Dublin. In: Odtavo. Price Five Shillings. Publifhed by BRYAN ROBINSON, ALD. To which are added, by Way of ppendix, feveral Curious PROBLEMS, by the Editor. 2. Exercitatio Geometrica de Defcriptione Linearum Curva- rum, 4to. Auctore GULIELMO BRACKENRIDGE, Ecclefie Anglicane Presbytero. Printed for J. Noursez, at the Lams without Temple-Bar. MATHEMATICAL DISSERTATIONS On a VARIETY Of PHYSICAL and ANALYTICAL SC Utheen TS: Containing, among other Particulars, A Demonftration of the true Figure |] A new Theory of Aftronomical-Re- which the Earth, or any Planet muft fractions, with exact Tables deduced acquire from its Rotation about an therefrom. Axis. A new and very exact Method for A general Inveftigation of the At- approximating the Roots of Equa- aang! os eS Surfaces of Bodies tions in Numbers ; that quintuples nearly spherical. the Number of Places at each Ope- A Determination of the meridional ration. Parts, and the Lengths of the feve- ral Degrees of the Meridian, ac- cording to the true Figure of the Earth. Some new and very ufeful Improve- An Inveftigation of the Height of the -ments in the inverfe Method of Tides in the Ocean. Fluxions. Several new Methods for the Summa- tion of Series. THE WHOLE In a general and perfpicuous Manner. ! By THOMAS SIMPSON. BAR EEE a A To a RT IR aT EO Ta LON DON: Printed for T. Woopwarp, at the Half-Moon, between the two Zemple-Gates in Fleetfirect. 1743. Martin Folkes, Efquire, PRESIDENT OF THE ROYAL-SOCIETY. od RR Could not have wifh’d for a greater Honour than your condefcending to receive thefe Sheets under your Pro- tection : As every Man is in juftice anfwer- able, both to his Patron and the Publick, for what he prefumes to print, I hope I have taken care that they may not be wholly ( iv ) wholly unworthy your perufal. If they fhall have the good Fortune to meet with your Approbation, I need. not be anxious about their Reception elfewhere. In the mean time, Sir, I moft earneftly defire, that this Addrefs may be underftood as an humble Acknowledgment of the Favours which I a Stranger, however undeferving, have received at your Hands; and which I fhall always remember with the fincereft Gratitude. I am, Sir, Your moft obliged Humble Servant, THOMAS SIMPSON. eee eon. I is fo natural when a Work of this kind appears in the. World, to ask What there is new in it? and the greater Part of thofe who fet up for Fudges, are fo extreamly bent to depreciate every Thing to which they can frame the leaft Pretence of an Objection, that an Author, without any Impu- tation of Vanity, may fometimes be allowd to fet forth the Me- rits of bis own Performance, in order to give bis lefs difcern- ing Readers a true Reprefentation thereof: And this I hope will be thought a reafonable Apology for what I have to offer in behalf of the feveral Particulars that compofe this Mzfcellany. The Firft,which is one of the moft confiderable Papers in the whole Work, 1s concerned in determining the Figure which a Planet, or an homogenous Fluid, muft acquire from its Rotation about an Axis; wherein the true Figure, under fuch a Rotation, zs not only univerfally demonftrated, but the particular Species thereof, according to any afigned Time of Revolution; in which it is proved that the Gravitation at any Point in the Surface, is accurately as a Perpendicular to the Surface at that Point, produced from thence to the Axis of Revolution ; and that it is zmpoffible for the Parts of the Fluid ever to come to an Equili- brium among themfelves, when the Motion about the Axis ts fo great as to exceed a certain afignable Quantity ; with feveral other Particulars never before touch’d upon by Any. I muft own that, fince my firft drawing up this Paper, the World bas been obliged with fomething very curious on this Head, by that celebrated Mathematician Mr. Mac-Laurin, 72 which many of the b (ame (vi ) fame Things, are demonftrated. But what Ibere offer was read before the Royal Society *, and the greater Part of this Work printed off, many Months before the Publication of that Gentle- man’s Book; for which Reafon 1 fhall think myfelf fecure from any Imputations of Plagiarifm, efpectally as there is not the leat Likene/s between our two Methods. ' The fecond Paper, contains a general Inveftigation of the At- traction at the Surfaces of Bodies nearly /pberical. The Third, confiders the Heights of the Tides in the Ocean. The Fourth, exhibits a very eafy Method for finding the Length of a Degree of the Meridian, and the meridzonal Parts. anfwering to any given Latitude, according to the true [pherotdal Figure of the Earth. The Fifth, includes the Inveftigation of the Curve defcribed by @ Ray of Light in paffing thro’ an elaftic Medium, whofe Denfity either refpects a plane, or {pherical Surface, and varies according to any given Law: Whence are derived fome praciical, and very ufeful Conclufions, relating to the Refraétion which the Light of the Heavenly-Bodies fuffers in its Paffage thro’ the Earth's Atmo- phere ; with exa&t Tables thereof, laid down by the help of very accurate Obfervations. The Sixth, treats of the Summation of Series; which, befides containing feveral Matters tntirely new, 1s much more general and extenfive than any Thing I have bitherto met with, for the fame Purpofe. The Seventh, exhibits a new Method for fimding the Values of Series by Approximation. * It was read before the Royal-Society in March or April, 1741, and had ‘been printed in the Philofophical Tranfactions, had not I defired the contrary. rT JE The Eighth, comprehends the Inveftigation of fome very ufeful sh Theorems for approximating the Roots of Equations in Numbers, ™ much more exact than any Thing hitherto publifbed ; whereby the Number of Places 1s tripled, quadrupled, or even quintupled, at each Operation; to which are added, fome eafy and proper Appli- cations, in tluftratton thereof. The Ninth, relates to mechanic Quadratures, or the Method of approximating the Areas of Curves, by Means of equidiftant Ordinates. ‘Ihis Method was originally an Invention of Sir -Tfaac Newton’s, fince profecuted by Mr. De Moivre, Mr, Stir- ling, and Others: However, as I here affume nothing to my/elf, but a Liberty of putting the Matter in fuch a Light, as I judge will be moft plain and fatisfactory to the Reader, I fee no Reafon why I may not be allow’d the fame Privilege as Others. The Tenth, is concerned in finding and comparing of Fluents, and contains a great Variety of new and ufeful Improvements, being one of the moft confiderable Papers in the whole Work. The Eleventh, 1s an eafy Inveftigation of the Paths of Shadows, on the Plane of the Horizon. The Twelfth, contains a Determination of the Time of the Year when Days lengthen the fafteft, according to any affigned Excen- tricity of the Earth's Orbit. The Thirteenth, fhews how much the Defcent of Bodies, is affected by the Earth’s Rotation. The Fourteenth, is a Demonftration of the Law of Motion, that a Body deflected by two Forces, tending to two fix’a Points, de- feribes equal Solids in equal Times, about the Right-Line joining thofe Potnts. ! The ( vill ) etal Force, may continually defcend towards the Centre, yet never fo far as to come within a certain Diftance; and in what other pie it may continually afcend, yet never rife to a certain finite Altitude a LUM e . The Fifteenth, foews in what Cafes a Body, acted on by a centri- 4 : P The Sixteenth and laft, comprebends an eafy and general Invefit- gation of all the principal Theorems relating to Compound Intereft and Annuities, without being obliged to fum up the Terms of a geometrical Progreffion. Thefe fix laft Papers, tho more fimple in themfelves, and of lefs general Ufe than fome of the preceding, may neverthelefs be look’a upon as entertaining Speculations, and therefore not prove unac- ceptable. EVR: R Ava A. ———————- pea TASS 20S PAGE z. |. laf, and p. 3. 1 2. for a—zx* read a—z| x p. 4. 1. 1o. for aaa), Kaha r. a— x) Xa®— x 5 p. 5. 1. 3. dele the Semi-colon ; p. 20. 1. 21. for whereof, &¥'¢. read whereof the Time of Revolution can be fo foort, as of that, whofe equatoreal Diameter is to its Axis, as 2.7198 to Unity; p. 38.1. 12. for paffing, r. paffing thro ; p. 41.1. 4. for Spherodial, x. Spheroidal; p. 42.1. 9. for Part, r. Parts; p. 47. 1. 26. for dy, r. bes p. 61.1. 21. for could, x. could have been ; p. 64.1. 17. for R, r. 43 p.70. 1. 14. for i, r. to; p. 73.1. 16. for z,r.—25 p. 78. 1. 8. after Num- bers puta Semi-colon; p. 81.1. 15. for + 292%", r. seh x9; p. 86.1 13. for ktom, r. g-+-2m3 p. gg. 1. laft, for a*, r. a*x™5 p. 105. 1. 9. for 613, r. 61.33 p- 122. lL. 20. for independant, r. independent; p. 128. 1. 8. for se tals EX, 2 ae » Fe. p. 136.1. 5. for Negative, r. negative Number; p. 145. 1. 2. I ; for pi-3r, r. p42r3 p 149.1. 12. dele the upper Vinculum; p. 151. 1. 5. for = 2, r= x3 p. 161.1. 19. for gravitate, r. to gravitate. Y.—I— sussseihreslne iguana apatiaincatiasaibmammametisaiaaiainasiaossemmmemseettenantetneeraemanens oceeeee sete Sei ~ 3 r z . oF ee - —- SS : == - = 7 A MATHEMATIC AL DISSERTATION FSU Be OF THE. BA ROaT ey LeMM aA “4, 5 St j vibod ee UPPOSING AC perpendicular to A B, and 2 eet, a Corpujfcle at C to be attratted towards every Nevius Point or Particle in the Line AB, by Forces in Seeotaccatd the reciprocal duplicate Ratio of the Diftances: To. find the Ratio of the whole compounded Force, whereby the Corpufcle is urged in the Diredtion AC. A | Let 6 = 2 clan _ Sigcencen eterna Meee [2] wb Let AC=d, AB=x, and “/\® Bh =x: Therefore B Ct — a? _ --x;. confequently Fee will be as the Force of a Particle at B, in the Direction B C; but a? x7]f d ni]: ep ae ee 5 CC? i ffi- ye A Ppt” Ft he E Py, eacy of that Particle in the pro- pofed Direction A C; wherefore xe is the Fluxion of the: | x AB h ] ee a aaa < whole Force; whofe Fluent di? ail’ 7 be BC is, there fore, the Force itfelf. Q.E, I. LEMM A II. Qe a Cuneus of uniformly denfe Matter, compriz’a . _) between two equal and fimilar elliptical Planes A DB EA, Aprv A, inclined to each other, at the common vertex ig, of their firft or fecond Axes, in an indefinitely fmall angler A B; to find the attraéton of the faid Cuneus, whereby a Cor- pufele at A is urged in the Direétion of the Axis A Bor Ar. ' Let DE and me be two Ordinates to the Axis A B, indefi- nitely near to each other, and let AB=a, BC—2z, Cc=— Z, C D =), and the Sine of the Angle formed by the two Planes, to the Radius 1,—d; and let the Equation of the Ellipfis be .y* ee — g2* (which will anfwer either to the tran{verfe, or conjugate, Axis, according as the Value of g, is taken negative or pofitive): Now, it will be, as 1: d::.¢@—2:,d xa—2, the Diftance of the two Planes at the Ordinate DC, or the Depth of the propofed Matter at that Ordinate ; which therefore drawn, av fz—z—g & 5 G@—-Z¥N g— z 1 fz— 2 —g 2 es eos e : the into the Attraction of the Particles ; in the furface DC ceD 6. Vf E— Zz? — |B the. preceding Lemma). gives for the Fluxion of half the Forcerequired. But when J — Zee f & writing 7-+2xa inftead of fi in-the faid Fluxion, it will be- dzizxi gl SAT gi dete 1 8? 382 BP 3.5 g3z3 er ae iT eign k Ste, 2407 2.42 648 aG= 2: g2* becomes—o, z will be — = AB =a; therefore by &c, the Fluent: whereof, wie z becomes:— 4, will be. adx1ttglz% : 7 ” sae - Bey which, becaufe ax. 1-Fe| i me £ ge erey B= fei +g = jintgls Sc, will be ail fy, 2 2.48 , 214.697 2-4. 68 93 = came ———_—___—— (jp, i ; g Pigs ase eT a oe LEMMA La} "LE MM A_ II. DHE Fluent of a®—x" wx—' , being given; tis pro- pofed to find the Fluent of a*—x\"t? yxmty—t , when a*— «| ** becomes 0; fuppofing p and vu to be whole pofitive Numbers.‘ Let Q==a* — x*|" 1" ew, and let E, F,G,H, &c. denote the Fluents of a — x" x xm! x, a ee | yx xmpa-t y 2 a— x2] xxmben—ty refpecively: Then Q being (—=rn x x et ie a — xt — mM -- Ix nx x2—Tyqg2— xO)" Cx =e rn xm — be a — x" x a*— xa} — mM cd i Tear eeT ne ae — x % x) Sf Nw a? — x" x NRT yo rt+m+t “4% 2X a? — x7|™ x xma-po—t x, if, inftead of 2 — xl™ * xm? y and ga — xal™x gm-fa—t x, their Equals & and 7 ‘be here fubftituted, we thall get Q—=r2a°k —;ymtixe -* Fy Whence'Q s¢& ra Eom -ftixn x F, and confex "Ek— : quently F = —— 2; which therefore when Pak ac ; nxr + mr 1 2 .£ ji —o, willbe 2% =, | or Q becomes == 0, ea. And it is manifeft, by Infpection, from the very fame Reafons that G is = en Cs n 5 a n rXrLix@E ren DX Py tye Be Fa S ee, o CG =? rbtmt2 ta mR rbmtixrtm+t2 rxrbixr#axarE H= eM eraC eat a and therefore the Fluent of aot es a x1 ye ym bem—t Sin’ this Cirumftance) putting r+ m ee : rxrbaixrboa...rtu— rxa"E i +I1=4% will ae ta > Now if this Fluent be denoted by P, and xt" py K, the Flu. ent Ls] ent of a"— xA|" x K.x* x will, it is evident, be — r-+o gtv which taken from a P, that of ga— xay™ xK a" x, leaves q—rxa"P__m+ixaP qu qv —a— xl” xK xx or its Equal, a8 —xaj"+" , K x; therefore the Fluent of a* — x*|""" x K is to the Fluent of a=—x"\" xa"P, for the Fluent of ga x™|"xK a x; io n : x K x, as a *“, to 1; and, for the very fame reafons, will g VU the Fluent of a—xl"+*,.K x be to that of pon xa] FIK x, m+2xa™ iy Te Onl Gc. whence it manifeftly appears that the eee JF isa cs ae : Fluent of a2 — x]? Kx, or of a2 — xf PP etn of net oe ‘ £ +r m+ 2 m + : Pei eet age : will be expreffed by a “q+4o ots * +u+F2 cc Bids jor byEa” *™ into Lrb2r+3..03..r--o—1 *7-+-u+p—1 q-9t1.9g+2. 9+3. G+4.9+5 xX Mad. mF ams m+ p oy See. Geen w rem oe Q EL C’O:R MLL Awe yin If E be taken equal to the Fluent of Pw *s3 or « Part of the Periphery of the Circle whofe Radius is Unity, and 1 — |=, and the Fluent of 1—s5s|* 5 x du*t sw ds « x U**s?", when 1—ss is—oand ¢ and w, whole I—ss}? Numbers be required: Then, by writing 1, fora; 2, forr; — 2, for m; 1, for q; ¢, for p; and w, for v in the general or of Expreffion foregoing, we fhall have dE y 233-722" —t 2. 4. 6. 8, 10, 12 TO FT pi Pca ‘ * BT ees —_- for the Fluent in this Cafe. 4... he iw $2 t, B COROL- [6 ] COROLLARY, Il. Hence, may the Fluent of ISS Ro»—+ © &e, when 1—ss iso, and # any whole pofitive Number, be alfo determined; for let the Value of w (in the laft Corollary) be fucceffively expounded by 1, 2, 3, Gc. and that of ¢, at the fame time, by 7, z—1, n—2, Ge, and ds - P qn etQ Y Wz s+. Figs ; ds ds then it is evident, the Fluents of x US, 1 yy yn? sf I—ss i— ful 5 . 3. 5.) ese mood XE fice mill coeihd pale J Eg te dE x 246. Oe Beto 3 . 1.3.9....2N—5XI.3- —. 5 dE x sins —— ; Se. BA. Os nt 24. 6 Fee 20 Te refpectively; which therefore, being multiplied by their pro- per Coefficients P, Q, R, &e. and added together, give d Ex P95 le 2 Re ere as po 2 BeZ OY, Ee 275... . eT SOS 2. Awe. .* 2aeek 2 2A, BeBe 2n— I 5G. So See ol ; aS es ra iy PL a ee i eee 2n h2 2n—I 2a—1IX2N—3 | 2u—1xX2N—IJK2N—§H &c, for the Fluent fought. LEMM A My, Uppofing PA SEBPO to be any Spheroid, generated by . y the Rotation of an Ellitis PAS E about its leffer Axis PS; to find the Attraétion thereof exerted on a Corpufcle, at any given Point Qin ats Surface. Let Q R Land C Br be perpendicular, and 7L parallel to PS; and let the Square of any Ordinate BC of the ge- nerating Ellipfis, be to the Square of the correfponding Or- dinate B v of the Circle PuI S f P, defcribed from the fame Centre O, about the Axis PS, in any given Ratio of 14-B > to i : | | >to 1; let QH be the Axis of a Section Q ¢@ Hd Q of the propofed Solid, formed by the Interfection of a Plane pafling through the Point Q perpendicularly to the Plane PAS O of the generating Ellipfis; and let PO=R, O R — 3, the Sine of the Angle RQH to the Radius 1, =f, its Co- fine—=7, Qr—x, its correfponding Ordinate ray; QR (—\/T-CB x R°—&) —a, and RT (=1+B x4) =A: Therefore fince QL is—gx, andr L—=px, we have O B=6 px, and Br==qgx — a; whence B.C?=R2—4 b—2 b px —p* x xj +R da—2Apx—px —p Bx, and confequently 4? (= BC+—Br’*) = 249x—2ApK— pe —-F xX? — Bp? x”, or, becaufe p? ++ 9? is=1, 9? is ag—Apx2x—-x?—BY x’; which Equation being only of ‘two Dimenfions, fhews the Curve Q. a Hd Q, whereto it pertains, to be an Ellipfis. Let ag [3] Let now a Plane be fuppofed to revolve about Q as a Centre, always continuing perpendicular to the Plane of the generating Ellipfis P Q S$ E, and let Q4 H and Q mk be two Pofitions of that Plane indefinitely near to each other; and fuppofing Q F to be a Tangent to the Ellipfis P Q S at the Point Q, QT perpendicular thereto, and F 4 an Arch of a Circle whofe Centre is Q, and Radius Unity; let G 4, the Sine of that Arch, be denoted by s, its Cofine (Wy —ss) by v, and, 4m, the Fluxion of that Arch, by e: Then, fince the Angle B Q R is the Difference of the two Angles B Q T, R QT, and the Sine and Cofine of the laft of thefe two equal refpectively to eck and ‘ee fhall have ae astAyv | Nee - =the Sine of BQ R, and a its Cofine, by the Elements of Trigonometry ; which Values being therefore fubftituted inftead of p and g in the Equation above found, it aU—s al’ a* t+ A» by writing 25 Va? A? inftead of f, e inftead of d, and Bx av—s Ay : a*-F A* becomes y? = 2 s\/ a + AX x Hm x?7— Bx? x : Hence inftead of g, in the Theorem at the End of the fe- 2. 43 cond Lemma, we fhall get 2¢5 x a A*| . ae pullrAp > a4x6: Baus Al* * Ppa areas aia + ae x Wroearce fhewing the Ra- tio of the Force wherewith the Corputfcle at Q is urged in the Direction H Q_or £Q, by the AttraGtion of the Cuneus of Matter included between the two Ellipfes, whofe Axes are Q H and Q 4; from whence, by the Refolution of Forces, the Attraction of that Cuneus in the Dire@ions QF and Q T will be had equal to 2esv x a7 A? z x => > eee, SR St ee es Ret ees eee 2.48 —sAl* ef GL! bh necnte 2 Zncegit LM, gv—sAl &c. and 2¢55-% at Art oe 2a av—s Al’ ic q; 5 a* + A? preflions are, it is manifeft, as the Fluxions of the whole Force, exerted by the Part QH P Q of the Solid, in thofe &c, refpectively: which Ex- —~F K Directions. Suppofe, now, another Plane Q K to revolve about the fame Point Q, and with the fame Velocity as the former, but in a contrary Direétion, fo as to meet and coin- cide with it in the Perpendicular QT; then v in this Cafe becoming —¥v, the Fluxion of the Attraction of the Part QA K Q, in the forefaid Dire@tions Q F and Q_T, (by writing —-v inftead of -|- v) will be—2esvx a 1 Al rm ce z Ad ae [ 10 ] and By og @se c Al. I — ——— x ————._ &e, and rie a oT A i" Rm a> + A? * 2 3 cA i — “oes AG yee eee ie tS &c. refpectively: Where= 3 3-5 a ee | fore, if thefe Fluxions be added to thofe of the former Part in the like Directions, and —av—sAt be changed to av-+sAl!, which is equal to it, we fhall have 2esv x Sp ee og Se he ata — Seoiens \" 2 2? A‘l? into 2.4 DB gO av +sAl— ates A 2. AD Be x Bub. Bg 3. 5.7 av — sAlt—av—oe Ar ; ee Sa ee, 2 6 es et... and B es*-4: ae Al into > TT | a 4A 3 2 2.4B avs Ae tee U—S ee .4.6B? av-sAl'+} +-av—sAl* % eS ee «om — x 3-5 ats Ons, 7 At &c. for the Fluxion of both Parts together, in thefe Direc- tions. But, fince the Triangles Q b G and mhn are fimilar,. e will be tos (== mn): as1:V 1—ss; , therefore, by fubfti- 5 tuting ‘Ri Se) inftead of e, the foregoing Expreffions will 2ussxa@ LA, 2°48 avtsAl—ae Al become ee, AnD tie es a 55 335 aA? : ae eal ee AMET SOLA Ae 2S SxG A 2 2412.4 Bi go sAle-au—s Al &c, and into ——— Toss aT ene a*-A* €c¢, The Fluents of which, when s becomes —=1, will, it is manifeft, be as the whole required Forces, whereby the Cor- pufcle is urged in the DireCtions QF and Q T: Therefore, in order to find thofe Fluents (which is by far the moft difficult Part of the Hee let r be put equal to the quadrantal 3 Arc, 2ussxa - Aa 2.4.6.... 27+ 2xB* Arc, F be, and let oe peas iss]? 3+ 5+ Jenene 2M 4-3 au-+s A A ; : yo”? which is a general Term to the a? Att firft of the two Expreffions, be affumed; then the fame, by expanding ZV +s Al"—av—sal” into a Series, &c. will a A 2.4.6.... 2"---2x B's ay 9 == into2na9—' x aA 3.5.7....22-+3x1—ssl | Ny 2H 2H—I 2H7—2 A 22 A y22 5% 1% —- x — xQ72—3 A3 y2n—2 54 lei te + I 2 i 3 a x 22 —— i 2-2 2n—3 | 2 — —— x —— 3 a XA S.AS Utes 68 ERE. 5 2 3 4. But it is evident, from Coro/. Il. to Lemma II. that the Fluent 4x@LA} ha boat oe of this Series (by writing ae x Beis 2A Tene a” B25 Gak se 2 2g 2 Hit ges oO become 4 x Dee for d; 7 fOr 3-2 ame A, fae Ps ee 3 ° a V3 e oer 2H— x a—3 A3, for Q, Ge.) will be as in ee * a Pa OR Se ae Axa Ak 2.4.6....22-—- 2B. 2n a a into 2”a°7—* A 1. __ a’ +A 30 G7. ert eee 28 I thi Be 2 Qi Rg EGR AEE a gi i 2 RE, 2 2 oT I 2 ' 2N—2 2n— 2nu— x 3 4. 5 2N—Ix2N—3 r Be , ——_—— oo ae ILO 2na7—*A+2nxn—!i x 293 Ai tan a Ae | Hi——tJT n2—2 71—- J] 4-2 — x xX i q7 news A3--2 nN %~-—— * ——. * 3 quae 4 Al ‘ x Fu i = oe 2 I 2 ec, [12 ] ~ 47rBx2nAc , : a ~ @ As = into a%—? 1 2—1 x ann Art Ay = j NI n—2 rB.xy2nAa * BS RS = gt tet eee x ga A? —I — I vs a \- AP? 4rAa@ 2B ===" Let x be now expounded aA 2n--1x x2213 rA by 1, 2, 3, &e.' fucceflively, then will ae ee ’ a> +1 A? 22 Br t 4rAa 2B 47rAa ———__________—. become oe) eee 2n-+-Ix2n-+3 Jee * 3.5 Va? + At 4.B*,4rAa 6B: 23.2, ——3 &e, for the Fluents of the od, 3d, 5-7 Va? + Ae 7-9 &c, Terms of the forefaid general Expreffion, refpectively ; .and therefore the required Fluent of that whole Exprefiion, or the Force whereby the Corpufcle is urged in the Dire@tion A Q F will be truly defined by ae drawn into 2B 4 Bt geBS Bee to Bs V EO 5.7 7°9 g.11 11.13 fame Manner, the Force in the Dire€tion QT will come out &c, And, in the oe ee ee ey, pee Vat+ A? 13.» 35 ca 7 -—r-9 Var + A? 4 z fs -{s ae m ie ~ Which Values, by Writ- 3 5 ing 1-+ Bx band 1 rE BI x ic Db), for their Equals A and 4rbx1+BxR—dol 2B 4B , 6B: a, will become Re Bee. 3:5 5:7 37.9 re Rh 2 ae Be. lik) Bee ES ¢, age: Seer yD had: Be CBP PRR 4 BF 7-9 &Se, east oereneendeenanatninemesianeneel snsiaatthhediiiansnstmesoarndemmemeetee: sexes Sg Se rgd 2 i * oa. Fe, -f- 4rbxT+B) x ; Bo 7 +. 3, &e, refpectively. « But, VY R?+BS? y ‘ 2B 4B? ‘ 6 B3 8 Bt 5 ie ele feeing i. a be Tight ae &Seorwg = : B . =i -hg Box B— > oe OC ne oe te BB, = LB x —BAL IT +B x Bi +, &c, where Bi = - Hee? -_— - &c. isa known Series expreffing the Arch (Q) of a tele whofe Radius is 1, and Tangent B? , the faid Forces will, it is manifeft, be truly defined by eS Li 2+ Ad | a BReese gmat RTE B05 Fe ——_—_—_—_ Se ae nD oie Per Ce eag ca Mines pce fo V a?+A? 5 Be a V a? At os Ba V a? +A e TB x Q—B* Te refpettively. Q.E.I. B ©OR OLA RYE ah Hence, if R D be made perpendicular to QT, then, QR being =a, RT =A, RD will be = yoy QD = aa AA é Vaae and DT = ae and confequently the Attraction a*-+-A? in the forefaid Diretions QF and QT, as (RD) x ee AE eee Coy and (QD) a ee 3 Mest) 7. Oa Galt RP gees * 9° B B BXQ—3B Gc. + (DT) x2 — + a © Beoras (RD) x PER : 2 Be D and [te 3], and (QD) x a (DT) x = ly. Therefore, fince B is conftant, it follows, that the Force whereby a Corpufcle at any Point Q, in the Surface of a given Spheroid, is attracted in the Direction of the Tangent QF, will be fimply as R D. — Q refpective- CO ROA RY If If B be taken =o, or the Spheroid be fuppofed to dege- nerate to a Sphere, the Attraction, perpendicular to the Sur- face, will become as 1 QD ie TD; or as — of the Ra- 3 dius of that Sphere. ‘Therefore it follows, that the Attraction at any Point Q, in the Surface of a Spheroid PAS EP, in the Direction QF, of the Tangent, is to the Attra@ion’ at the Surface of a Sphere of any given Radius, as (RD) x cianils Csoal dome to~ of that Radius; and moreover, that the 2 Bz 3 AttraQtion in the perpendicular Dire€tion Q T, is to the At- traction at the Surface of the fame Sphere, as (TD) x blH = oe + (QD)x ree er eh to 3 of the fame Radius ; ZN 2 B? Bz mes, Paes eS =e = of, becaufe (TD) x ar --(QD) xit8 “xo=* jo wed Be 0! 3+ BXQ—3B° : . (QT) x a re (QD) ae ee 2s this laft Quan- 2 2 tity, to~ of that Radius, J C O- [as] COROLLARY. Ii But when the given Spheroid is nearly Globular, B will be very {mall, and therefore all the Terms in the foregoing Se- ries, wherein two or more Dimenfions of B are concerned may be neglected, as inconfiderable; and then the Attrac- tion in the Directions QF and QT, after proper Reduction, 4bcB nod 10 R?-- 3 BR*-+-+Bd? ——f will, in this Cafe, be as eae ee {pectively ; from whence it is eafy to determine, that the Po- fition of the Line Q X, wherein the Corpufcle gravitates or endeavours to defcend, is fuch that Ox is every where, to OT, as 3 to 5, as Mr. Sterling has found. ‘ CO. [ 16 ] COR GoEERA RY “IV. Hence it follows, that the Attraction at any Point Q, in the Surface of a Spheroid, not differing much from a Sphere, is to the Attraction of a Sphere upon the fame Axis, as 10 R” + 3BR*+ Bd’, to 10 R” nearly. It alfo follows, that the Attraction of fuch a Spheroid in going towards the Poles, in- creafes or decreafes in the duplicate Ratio of the Sine-Comple- ment of the Diftance from the Pole; and that at the Poles themfelves (where in an oblate Spheroid it is the greateft, and in an Oblong the leaft) it will be to the Attraction of a Sphere, having the fame Axis as 4 Times the Diameter of the greateft Circle of that Spheroid, increafed by the Axis, to 5 Times that Axis; and laftly, that the greateft Difference of Attrac- tion, on the Surface of fuch a Spheroid, will be to the Diffe- rence between the AttraCtion at its Pole, and at the Surface of the forefaid Sphere, as 1 to 4 very nearly, P ROG OS ton ba Ne I. F a Fluid or Body of homogeneous Matter, whofe Particles are freely difpofed to move, and mutually attract each o-. ther in the duplicate Ratio of their Diftances inver/ely, re- volves about an Axis, and all the Parts thereof retain the fame Situation, with refpect to each other; I fay, the Form which that Fluid muft be under, to preferve this Equilibrium of its Parts, 1s that of an oblate Spheroid. For, let PS be the Axis about which the propofed Fluid PASE P revolves, and QT a Perpendicular to the Surface at any Point Q, making QR, and RD perpendicular to PS and QT, and F Qf, parallel to RD. ‘Therefore, fince the abfolute centrifugal Force, whereby a Corpufcle at Q endea- 3 PS vours pay) vours to recede from the Centre R; in the Direction Qu, is known to be as RQ, that Part of it by which the Corpufcle is urged in the Direction Qf, of the Tangent, or tends to flide along the Surface, will, by the Refolution of Forces, iS be as RD. Therefore, as all the Particles remain quiefcent with Regard to each other, the Attraction exerted on the Cor- pufcle in the contrary Direction QF, to preferve this Equili- brium, muft, it is manifeft, be in the fame Ratio of RD; but the Attraction of an oblate Spheroid, in this Dire@tion ap- pears, from Corel, I. to Lem, IV. to be as RD: therefore the Figure PQ.ASE A, is an oblate Spheroid. O. Fe. \ E PRO [ 18] PROPOSITION IL HE fame being fuppofed asin the laft Propofition; and the time of Revolution, the Attrattion at the Surface of the Fluid, when at Reft under a fpherical Figure, toge- ou ther with the Diameter of that Sphere being given; to find the particular Spheroid which the Fluid retains by means of that Rotation, and alfo the Gravitation at any Point Q in the Surface thereof. The foregoing Conftruétion being retained, let the time of Rotation be denoted by m, and let the given Attraction at the Surface of the propofed Fluid, when at reft under the Form of a Sphere, be fuch, that a Projectile or revolving Bo- dy may thereby defcribe a circular Orbit, whofe Radius 1s equal to the Radius of that Sphere, in a given Time 2: Put- ting PO=R, RO=4, the Attraction at the Surface of the propofed Sphere =f, the Semi-Diameter of that Sphere = 4d, and the Proportion of the Square of the equatoreal Diameter AE, to the Square of the Axis PS, as 1-+B to Unity. Then, fince the centrifugal Forces of equal Bodies, movin in Circles, are known to be univerfally as the Radii of thofe Circles, applied to the Squares of the Times of Revolution, we fhall have as 2: 22 ::: 42 x (RQ) the Force with which a Particle of Matter at Q, thro’ the Rotation of the Fluid, endeavours to recede from the Centre R, in the Direction Qn; from whence, by the Refolution of Forces, the Forces in the Directions QT and QF, arifing from the fame Canfe, will be — (QD) x $4, and — (RD) x ; Z refpectively. But the Attraction in thefe two Directions, ‘uppofing Q to be the Arch of a Circle, whofe Radius is 1, and [ r9 ] and Tangent B*, will be to (f) the Attraction at the Sur- face of the Sphere, whofe Semidiameter is d, as (QT) x Bice Sp ae. z (QD) x ROS to 2, and as (RD) x z 2 a 3EBx Q—3B* 2B ceding Lemma: And therefore the whole compounded Force, whereby a Corpufcle at Q_is urged in thefe Directions will be = + (QD) x to < refpectively, by Corollary II. to the pre- rightly defined by oe x (O.1) x SEBO (QD) x oe, and “fx (RD) x 2:5" a 2 Bz --(RD) x a . The laft of which Expreffions, that the Cor- pufcle may remain at reft, and all the Parts of the Fluid in Equilibrio, muft, it is manifeft, be equal to nothing ; therefore Eos a on Ea io = - , and confequently the Gravitation, 2h, 3 or abfolute Force in the perpendicular Direction QT, as (QT) x ESS xf, or barely as (QT) x 7525 from a z whence, by help of a Table of Sines and Tangents, Gc. not only the Value of B, but the Gravitation anfwering to any affigned Value of =; se is fmall, the fame Things may be effected in a more ge- mm 3--B x Q—3 B* ws 3 ET Tae ae” neral Manner; for then our Equation ="——~; a 2B? may readily be determined. But when 2B B? 6 B3 n* may be reduced to —— — cee Het CR eee —s where, . -7 Q the Series converging fufficiently fwift, B will be found = 6 2 5 2" 25x bn* 125X 37%" ae fe a reer -- iva &e. or ince) near I ly. [ 20 ] ly.. Therefore the Ratio of the equatoreal Diameter to the ° . . e cy Iz ° # Axis will, in this Cafe, be as 1 +- —22*>— “to 1, or, if —= 14m’ —30n" ™ be very {mall, barely as 1 ++ pa to Unity, the fame as Sir 4 Mm Jfazac Newton and Mr. Starfing have made it. Oust. CORO LTA RY OT | the Left-hand-fide of our foregoing 3--B x Q—3B Becaufe ae sat 2 2 Equation) as appears from the Nature of the Expreffion, can never (let B be what it will) exceed a certain affignable Quantity, it is manifeft that if S ceed that Quantity, the Problem will become impoffible, Wherefore, to determine this Limit, let — x, and the e 4 ny Bx =. Bz e e Fluxion of nag ae (= Hoon ) which is be fo given, as to ex- 2a Q+3+x*xi-pex . 3x3—oxi+3x73fe°xQ = oo TOE tees Sree tr ets Seren re be put =o; and we fhall get Qx-+-7 x3—1-x? x QE X* x Q==o, where x is found = 2.5293; whence the corre- {ponding Values of — and ./ 1-+ B, come out 0.58053, &c. and 2.7198, &c. refpectively. Hence it appears, that it is impoflible for the Parts of the Fluid to continue at Reft a- mong themfelves, when the Motion round the Axis is fo great, that —- exceeds 0.58053, Gc. or, that any Spheroid fhould be affumed whereof the Ratio of the equatoreal Dia- meter to the Axis is greater than that of 2.7198 to Unity. But if the Motion be greater ‘than is here {pecified, the Fluid will contract its Axis, and continue rifing higher and higher . - towards Seabed tie etnias sae Set eR EAS ES [| 2] towards the Equator, till, by increafing its equatoreal Diame. ter and Time of Revolution, the Parts thereof either come to an equilibrium, or begin to fly off. C08 0: L.A RY 2 i, If, inftead of the time of Revolution, the Quantity of Mo- tion of the Fluid about its Axis be given, fo as to be to the Quantity of Motion in a folid Sphere of the fame Mafs and Denfity, revolving in the forementioned Time x, in any given Ratio of r to s; then, becaufe the Quantities of Motion in equal Spheroids of the fame Denfity about their Axis, are to one ano- ther in a Ratio compounded of the direct Ratio of the Radii of their greateft Circles, and the inverfe Ratio of the Times of : ; da A their Revolution, we fhall have as a is .5 5 eee eam a rma confequently AO = — (PO x AO?) is = 4, and therefore 274 (=AO)= But (becaufe the Maffes are equal) nS dx 1-+-B®; whence man x :x#+8% 0g wt 3 2 Tele he : 3m 35” x 1+B3 which Value being fubflituted for a in the foregoing gene- = r 2 SA BH Ge ral Equation, we LARS ore we = Px and there- peer ee I incacaeeee ie I B aad is z ° ° fore SEBX Q—3 BY x BF eg ; from whence it will be Bz eafy to determine the Spheroid which a Fluid, whofe Parti- cles are at Rett among themfelves, muft affume when the Motion about its Axis is increafed:or decreafed in any given Ratio; becaufe the abfolute Motion after fuch Increafe or De- creafe is given, and will be no ways affected by the Aion of the Particles upon one another while the F igure of the Fluid is changing, . CO. [ 22] COROLLARY Ill ane eee ROT OR fo eee oe 2, 3 ° ° + But (fince SE BxQ—3B' x18" both when B is nothing and Bz infinite, will be = 0) it is evident that the Value of po A amen can never, let B be what it will, exceed B? oO e ° ° a certain finite Quantity; and therefore if the given Motion ss fag. be fuch that —— exceeds that Quantity, it will be impoffible Ss 7 for the Parts of the Fluid ever to become quiefcent with regard to each other: Wherefore to determine this Limit, let x be put==B’, and. the Fluxion, of 3% *Q-3*xtrr p. 3 taken and made =o, and the Equation, duly ordered, will be x4-++-24.x7-+-27 x Q—16 x3—27x==0 ; where x will come Gute 7.5 very nearly, and the correfponding Value of zi i == 0.92705. Hence it appears, that the Particles cannot ‘poffibly come to an Equilibrium among themfelves, when the Motion round the Axis. is fo: great, that ae exceeds 0.92705, but will either fly off or continue to recede from the Axis zz Infinitum. C.O:.R. © Lake AR Xp: IV: Becaufe the Values of B andQ, when — is given, are alfo I : iit) Br given, it follows that the Gravitation (Q.T) x ae at any B Point in the Surface of the given Spheroid or Fluid, will be, accurately, as a Perpendicular to, the Surface at that Point, produced till it meets the Axis of the Figure. Therefore the ; Gravi- * ‘ 4 i nen hes ar acing ck aeons pen en eo Sean Seebeck ean aaeten op inte ae tag he SI [ 23 ] Gravitation or Force wherewith a Corpufcle tends to defcend at the Equator, is to the Gravitation at either of the Poles, as the equatoreal Diameter to the Axis inverfely. COR.0O LL AY, Hence, if the Spheroid be nearly globular, then QT, which by the Property of the Ellipfis, is univerfally equal to 1+B' x R2-- BE , will here become +B! x R-+ ae near- ly. Whence it appears that the Increafe of Gravitation from the Equator to the Pole, is in the Duplicate Ratio of the Sine. Complement of the Diftance from the Pole very nearly. CO:R:.O Link Byes VI. Moreover, becaufe the Ratio of the equatoreal Diameter to the Axis, when the Spheroid is nearly globular, becomes nearly as 1 ++ 8 to 1, the Excefs of tnat Diameter above the Axis, will, it is evident, be to-the Axis as ~*~; to: m’, or (becaufe the Forces by which Bodies are retained in. equal Circles, are in the duplicate Ratio of the Times inverfely) as of the centrifugal Force at the Equator to the mean Force of Gravity. Therefore, fince the Ratio of the centrifugal Force, in different Circles, is compounded of the direct Ratio of the Diameter, and the inverfe-duplicate Ratio of the Time, it follows that the forefaid Excefs, in Figures nearly fpherical, will be as the Diameter dire@tly, and the Denfity: and Square of the time of Revolution inverfely. A { 24 ] A TABLE fhewing the Time of Revolution, and the Momentum of Rotation of a Planet or given Fluid, accord- ing to the Ratio of tts Axis and equatoreal Diameter. | i. 1,01 41,2867: | 0.08925 | oe Vj,O5) $3137 2.(0,1970, I Pood. “Boron, He, 5508s I 2 1,0142 | 0,6944 5 io. ae 1,810” | 0,8774 5 Ee. 10 253397 | 0,92165 foe 3,110” | 00,8728 tos A0 4,2752 | 0,80005 17.100 6,600” | 0,7033 s I: 1000 20,640” | 0,4845'5 Note. ‘The firft Column towards the Left-hand fhews the Ratio of the Axis and equatoreal Diameter, the fecond the Time of Revolution, and the third, the correfponding Mo- mentum of Rotation, 2 being put for the Time in which a revolving Body or Satellite would defcribe a circular Orbit juift above the Surface of the Planet or Fluid, when at Reft under ° a fpherical Figure, and s for the Momentum of Rotation in an equal Sphere of the fame Denfity, revolving about its Axis in that Time. Sat AO: dod UM. If the above Conclufions be made ufe of to determine the Ratio of the equatoreal Diameter and Axis, and the Variation of Gravitation at the Surface of the Earth, the Time ~ (in which a revolving Body would defcribe a circular Orbit about the Earth, juft above its Surface, by means of its own Gravi- ty) will, it is known, be about 847 Minutes, and the Value I of Sa ge a { 25 ] of m (one entire Revolution of the Earth about its Axis) 14.36 Minutes ; therefore by writing thefe Values in the Ratio of : Lak a ea : 1 (as above found) it will become as 35 2” z : : far the'Ranosn: 4 iatoreal 1.00435 : I, or as 231 : 230 for the Ratio of the equatoreal Diameter and Axis of the Earth. Wherefore, as the former of thefe is about 8000 Miles, it muft exceed the latter by 345 Miles, and the Gravitation at the Equator will be to the Gra- vitation at the Poles as 230 to 231. Hence it will not be difficult to determine how much Pendulum Clocks are ac- celerated or retarded from the Alteration of Gravitation when tranfported into different Latitudes ; for the number of Vibra- tions performed by a given Pendulum, in any given Time be- ing in the Sub-duplicate Ratio of the Force by which it is actuated, we have as 4/230 ? / 2313 OF as 460 : 461; fo Is the Number of Vibrations of any Pendulum at the Equator, in any given Time, to the number of Vibrations of an equal Pendulum at either of the Poles in the fame Time. Hence it will be as 460: 1: : fo is 86400, the Seconds in 24 Hours ; to 188, the Seconds which a Clock would gain per Diem (from the Caufe under Confideration) when removed from the Equator to either of the Poles; and therefore, fince it is proved that the Gravitation increafes as the Square of the Sine of the Latitude, the Time which a Pendulum will gain or lofe per Diem, by being tranfported out of any one given Latitude to another, is to 188 Seconds as the Difference of the Squares of the Sines of thofe Latitudes to the Square of the Radius. | The above Proportions, as likewife that of the Axis and equatoreal Diameter, are derived from a Suppofition that all the Matter in the Earth is homogeneous (or nearly fo;) but if the Parts next the Centre fhould be much denfer than thofe nearer the Surface, the Conclufions will be pretty much affected thereby, as will appear from the following Propofitions. G LE M- [ 26 ] LEMMA. N a Spheroid ASEPA nearly globular, whofe Denfity a- bout the Surface 1s every where nearly equal, but in the lower Parts thereof greater, according to any Law of the Diftances from the Centre, if the Excefs of its Quantity of Matter above the Quantity of Matter which it would con- tain, were all its Parts only of the fame Denfity with thofe near the Surface, be to this laft {pecified Quantity of Matter in any given Ratio of p to 1; ’tis required to find the At- traction at any Place Q in the Surface of fuch Spheroid. The foregoing Conftruction being retained, join QO, and draw OB parallel to RD; then, fince the Attraction which a Sphere, whofe Denfity at equal Diftances from the Centre is the fame, exerts on a Corpufcle above its Surface, is known to be as the Quantity of Matter in that Sphere apply’d to the Square of the Diftance from its Centre, it is manifeft, that if the Attraction at the Surface of a Sphere, whofe uniform Deniity 1s defined by Unity, be reprefented by 5 of the Ra- dius (as in the laft Propofition) the Attration of the I : fore- [27] forefaid Excefs of Matter, on a Corpufcle at Q, will be OQ” 3 reprefented by 2XOPXAO" or by —LE AEE Os 3x R°+BR?—Bs? pR by oe ia nearly. Whence, by the Refolution of Forces, the Attraction of the faid Matter, in the Directions QT and QF, will be ae + Oe and ~ p x (BO) nearly ; R which being therefore refpectively added to 10 Rt + 38 ee 30 B : : : ‘ and + x (RD) the Attraction in the fame DireCtions, of 2 the Spheroid confidered as homogencous, (fee Corol. Il. Lem. : Ae So AR pBb? R BR BB? IV.) there will arife a ot ee ae at eae ee 2B I and oh x (BO) + x (RD) for the whole Forces whereby the Corpufcle is urged in thofe Directions; but OB being to RD, as OT to TR, or as B to 1-+B, the latter of thefe Forces will be as(RD) x ae =~ very nearly. OEE PR OP'O'S 1 Gio WT. F a Fluid nearly globular, whofe Denfity about the Surface is every where nearly equal, but in the lower Parts thereof, greater according to any Law of the Diftances from the Centre, be revolving uniformly about an Axis; I fay, the Figure of that Fluid under fuch a Rotation, 1s that of an oblate Spherotd nearly. | The Truth of this is manifeft from the firft Propofition andthe preceding Lemma; for, fince the Attraction of a Sphe- roid, whofe Denfity varies according to the fame Law, is, in | the [ 28 | the Direction of the Tangent QF, nearly as RD, by the Lemma, what hath been proved in that Propofition, with regard to an uniform Fluid, holds alfo in this Cafe. EK OOF. O75 | Tab ON. TEV. g \LLE fame being fuppofed as in the laff Propofition, and i the Ratio of the centrifugal Force at the Equator A FE, to the Gravity being given (asr.: 1); to find the Ratio of the equatorial Diameter to the Axis of the Spherotd or Fluid, and alfe the Gravitation at any Point Q in the Surface thereof. Let the fame Conftruction be ftill retained: Then, fince the ab- folute centrifugal Force at Q, referred to the Centre R; is known to beas R Q, the Forces arifing therefrom in the Directions Qr and QT, will, itis manifeft, be to the Force of Gravity as (RD) x=, to 1, and as (QD) x — to 1 refpectively. Where- c . to. fore it will be, as (RD) x 4: 1:: fois(RD) x St the Attraction in the Dire@ion QF (per Lemma) to LR oe. See ++ a that in the Direction 3 3R 3 10 30 R QT’; whence by multiplying Extreams and Means, and re- jecting all the Terms where more than one Dimenfion of B is found as inconfiderable (becaufe the Spheroid is fuppofed nearly globular) we fhall get B— soot ; and confequent- 2+5 ly the Proportion of the equatoreal Diameter to the Axis, as Tp SOE Unity. Moreover, by fubftituting this Value ead oe Pea: of B, in the Expreffion for the AttraGtion in the DireGion 1+exR 57X49 pao R &* cy T' We RVG ee x 7 ih, oR” from [49] from which deduéting Be me oe Xx o-i(==QD x : = iF#XR | Gc.) the centrifugal Fores in the oppofite Direction, 3 ; 5 bam iop RR? 20pb* there remains 144*2 4 74-p x7 x Sole oc ORX2+57 for the Gravitation. 7. E:T. COROLEARY £ Herice it appears that the Gravitation, in going towards the Pole, increafes as the Square of the Sine of the Latitude, and that the greateft Difference thereof, at the Pole and Equa- tor, is to the centrifugal Force at the Equator, as 5 ++ 20p: to4--1o0p. It alfo appears, that the greater the Den- fity is towards the Centre, with Refpect to that at the Sur- face, the nearer will the Figure approach to a Sphere, and the greater will be the Difference of the Gravitation at the Equator and Pole ; and that if p be conceived to become in- finite, or the Attration to tend to the Centre of the Fluid enly, and not to all the Parts thereof as fome have fuppofed (with refpect to the Earth) the Difference of Gravitation at the Pole and Equator, will be equal to twice the centrifugal Force at the Equator, and the Ratio of the equatoreal Diame- meter to the Axis of the Earth, only as 579 to 578. , . H C O- SS NE SET BIT Sa a EN P > [ao] CsO RO VLA RY. If If the Ratio of the equatoreal Diameter to the Axis be given as 1-+ v to 1, there will be given 1+ SEAS r-by, and confequently p ==27—*" e 10U—Sr SCHOLIUM. The Ratio of the greateft and leaft Diameters of Fupiter is, according to Mr. Pouna’s Obfervations, as 13 to 12, and the centrifugal Force at the Equator of ‘fupiter, to the mean Force of Attraction, as 1 to 10; therefore, the Quantity of Matter in that Planet, will, according to the foregoing Hypo- thefis, be greater by juft one half, than it would if the Den- fity was not greater towards the Centre, than it is nearer the Surface. There might, indeed, be other Hypothefes affumed, that would bring out the Conclufions a little different, but as no Hypothefis, for the Law of Variation of Denfity, can (from the Nature of the Thing) be verified either by Experi- ments, made on Pendulums in different Latitudes, or.an a¢tual Menfuration of the Degrees of the Meridian, I fhall infift no further on this Matter, but content myfelf with having pro- ved in. general, that the greater the Denfity is towards the Centre, the lefs will the Planet differ from a Sphere, and the greater will be the Variation of Gravitation at its Surface. [ 31 ] A GENERAL INVESTIGATION OF ,C.o% ATTRACTION at the SURFACES of Bopies nearly {pherical. LEM MA. Uspofing the Planes of two Curves ABDEA, AprvA, Cte ae having both the Jame Equation y7=fx— x?-++ ¢x?-+ hx3-++ixt, Ge. to be inclined to eaeb other at their common Vertex A, in an indefimtely fmall Angle BAtr, fo as to include between them the indefinitely fmall Cuneus of uniformly denfe Matter A D BE pr vA; to find the Attraction of that Cuneus exerted on a Corpufcle at A, or the Ratio of the Force by which that Corpufcle ts urged in the Dureétion BA. Since the Equation of either Curve 1S 9? f KAP XP bxi-tint, €c. by putting f—x-+g xb x’, oa — o and } . j e Zz reverting the Series, we fhall get x =i + ae OF oe 2 Or thbf+t+rhf*g, Gc. equal tothe Axis AB, ft fa cane St ape nearly circular, and the Equa- tion of the Circle agreeing, in Curvature, with it at the Vier tex being fx—x x, the reft of the Terms g x7; 2x3, 2x4, in the given Equation, muft be {mall in refpect of the two firft ; and therefore all the Terms wherein two or more Dimenfions c Or 3 We ie a Wi i Hi: is ME Bh eh | a q ‘a ‘| b ) ligg . 4 i q ie. a ae | 4 | ae | a 4 + om NM ABH Ne aa Ne ae, | Ele ; a Be iy aa a 5 a a ie lta HB’ bart Va) ihe iy a ia oi 1 i rk b " BI a A} | = sa pe an eee E eee spa ea: — a Se Es = aes er Ss Le of the Quantities, 9, 7, 2, €c. are found may be rejected, as inconfiderable, and then A B will become =f+fg-th f+ tf>--kf*, Ge. which let be reprefented by z, and let DC and mc reprefent any two Ordinates indefinitely near to each other: Putting BC (a—x) = 2, Cc==%, and the Sine of the given Angle formed by the two Planes, to the Radius 1, ==¢; then it will be as 1:¢::a—-z : exa—z, the Diftance of the Planes at the Ordinate DC, or the Thicknef of the pro- pofed Matter at that Ordinate; which drawn therefore into eee expreffing the Force of tlie: Particles in the Sur- face DCemD (vid. p 1. gives soe for the Attraction o tomes? fay? of the Matter included between the Ordinates DC and mc, which, by writing fx—x?+g xb x3,268c. for its equal y, becomes ea x xf—etex phe bin, Ge i , A X fmaet gx bh x?+i x3, ee @—§a4—ha—ia—kat, Ge, be refpetively fub- itituted, for their Equals x and J; it will become cz ; wherein if a—2z and A 234 pe rep eet nc et et RTS a ren ez X a—z* Xa—ga—bha’, Sem abe exa—z-hxa—z!, Ge. : ——— a oe oe SSS TE a—x -a—xxa—ga—b a?, Ft. —abetgrXxa—xp-hxa—z ,&c.? ek hen Adee a xs pe A” iis eT kc BAO SLT Wi extmy bw BP OR Ger aK 3.4 A 77g Cee which, being con- a—gx%—h %X2a—zZ—1 BX 34a—3az+R, Ge? cee x. ° . : e z% verted to an infinite Series, at length becomes-————_ x 2 av eee ee 24+-2%—a x $+ Z—a x bx 2a—z+2—a xix 3aa-- ZdZ+-ZS, €?c. The Fiuent whereof, when z is == a, will’be ea x : 22h 82a ; — tae — 5875, — eet €c, where, if the Value of a, as above found, be fubftituted, there will arife ef x : = eae Bg. OS 2 OSB iP. 24 8-10 27" eF = 248 4 op ep ie 5 ah * as OL ee 3:54 799: Eke” OE | | PROPOSITION. CrUppojing PASEPO to be a Sokd nearly fpherical, gene- S rated. by ‘the Rotation of any oval Figure PAS, whofe “Equation 1s included im this general Form y’—=3?—Az—z?— Bo—C2—Dz2't, Sc. To find the attrattive Force of that Solid exerted on a Corpufcle, at any given Point Q in its Surface. 3 Let QR L and © Br, be. perpendicular, and 7 L parallel to the Axis P'S, about which the Solid is generated ; and let ‘QH be the Axis *of: any Seétion, Q¢H 6Q of that Solid, formed* by the Interfe€tion of a Plane paffling thro’ the given Point Q perpendicularly to the Plane PAS O of the genera- ting Curve: Putting RQ=a, R..Bis'z,; BC: sz 9; the:fine of .the, Angle RQH, to the Radius 1 ==, its Cofine =, Qr =, and its correfponding Ordinate ra =u, Then, by plain Trigonometry, we thall have Q L. = 7.x, andrL==px==z; I which el [ 34 ] which Value of z being fubftituted in the Equation of the given Curve, it will become 4? (= BC’) =a? — Ab x — p* °—B p* x*—C p3 x3, Se. whence wu? (—=BC*—B 7?) ame Apx—p? °—B p* x*—Cp3 x3, 8c. —aa+2a GX — G? x? = 229—A pxx—1-+ Bp x x? —C pi x3 — D pt x4, Be, Lee now a Plane be conceived to revolve about the Point. © as a Centre, continuing always perpendicular to the Plane PAS of the generating Curve; and let QhH, and Qmé, be two Pofitions of that Plane indefinitely near to- each other; and, fuppofing FA’ to be an Arch of a Citcle whofe Centre is Q, and Semi-diameter Unity, let 5m, the Floxion of that Arch be denoted by e : Then by writing 2ag—Ap for J; — Bp? for g, —Cp3 for b, &c. in the above Lemma, we « . 2 ieee Seales iene fhall have e¢ into 7%24%—-AP _- 2.4 Bp* xX 2a¢—Ap_ id? FP 3 3< 5: ‘2 6C Rm oa Ap lela Goethe a ee 24.6 CP x 209 ee ee &e. for 335+ 9+ 9 3-5. 76. , | the Force wherewith the Corpufcle at Q is impelled in the: Direction Q'H by the. Attraction. of the €uneus- of: Matter included between the two °-@tions Q Hand Qk. But, to re- duce this Expreffion to a more commodious Form, let QF be a ‘Fangent to the generating Gurve at the Point Q, and QT perpendicular. to. it, and let the Sine G4, of the Angle BQF=,s, and its Cofine QG=v0: Therefore,. fince. the Fluxion of the Ordinate BC, when BR or z is— 0, istothe Fluxion of: PB as =. to Unity, the Tangent: of the Angle e A e e A R QT; will; be- ==; — ; confequently. its Sine —-——A__. R QT. Wi 2a» q ¥ Vi 4aa+ A> and its Cofine == TEE :: Wherefore, as the Angle BQR is the Difference of the two Angles BQ T, R QT, the Sine of that. Angle will be aaa, sae eee J ond we Coline : (sake ae its Cofine ia (by the Elements of Trigonometry).which Values being there- fore E584 fore refpectively fubftituted for p and g in the forefaid Ex- ee preflion of the Force, it. will become =““-*""" 4a BAS 3 2.4¢¢BX2a0—sA 2.4.6¢8* Cx2zav—sAl° 3.5 xX 4a AAL ene faa PAS 3 .4.6.8es3D —s Alt : 2.4. 6.8¢0D x Sh2= | &. But, by the Refolution of 3» 5. 7°9 4aa-+AA'2 Forces, as 1 (Qb): s (Gb): : fo is the faid Force’ to ANS A AA: 2047 i 2.346 OF Cran: __ft__ 4 PeaseFAA EER Fao — 4/ 4aa--AA 3 3-5 3-5-7 —_——__—— z.4.637D —-—_,4 2av—sA — ae x2av—sA', &c,. the Foree in the Direction QT; and as 1 to ¥v, fo is the fame Force : to [ 36 J OM. 2x 4aa4-A* 2h Baty 24-640 et HE Se Ayam oe “seafaa 3 3h a 2 240—sA', ce. that in the Direction QF; which Quanti- ties are, it is manifeft, as the.Fluxions of the ~wliole. Force exerted by the Part OH PQ, of thé Sclid'in thofe DireGtions, Let now another Plane Q K, be fuppofed:to revolve about the fame Point’ Q, and with the fame V elocity as the for- mer, but in a contrary Direction, fo as ‘to meeteand coincide with it in the Perpendicular Qw; then v, in-this Cafe, be- coming Negative or — v, the F eon of the Part -QKAQ ee Ss ye x eds a* AF in the faid Dire€tions, will be a SEE x Ope Oasis =a [ a9 3 to ==; the accelerative Force of the Satellite at the Point — x Q, becaufe CQ and QR ‘may be taken as equal; and for the very fame Reafon, the accelerative Force at E will be es ; but sone ft sine, &c. the Difference of thofe two is, it is manifeft, as the whole Force whereby a Particle of Matter at Q tends to recede from AE, or to alter its Situation, with refpeét to the Body of the Planet. Now this Force may be refolved into two others, one in the Di- reGtion of the Tangent QF, and the other Perpendicular thereto; whereof the former, which is nearly expounded by | x RD, fhews how much that Particle, by the Attraction of the Satellite, is urged in the Direction QF: Wherefore, this Force appearing to be in the fimple Ratio of RD, the Attraétion of the Planet in the contrary Direction Qf, as it is every where equal to it, muft confequently be in the Ratio of RD; and therefore the Figure of the Planet a Spheroid by what is proved in Page 14. Let therefore the Square of the Diameter PS, to the Square of the Diameter A E, be now affumed as 1 : to 1-+ B, then the Forces exerted, by the Planet in the Directions Q T and —2B “a RD, as a 2° Q f; will be to one another, nearly as 7 : to appears from Page 13. Hence we have as Lt xRD::f:4= xRD; where- fore B= —*, and confequently OP—OA === x OP. @. EB. f. COR Gia Yer. Hence it appears, that the Forces of the Planets, or any remote Bodies, to produce Tides at the Earth’s Surface, are to [ 40 ] to one another as the Quantities of Matter in. thofe Bodies di- rectly, and the Cubes of their Diftances inverfely, or as their Denfities and the Cubes of their apparent Diameters, con- junély ; and this, it is evident, holds equally, whether the Barth be confidered as partly: covered with Water, or uni- verfally fo, COR Och LA Revo “TT: If d be taken = 60, m= —, and O P = 21120000 Feet, and thefe Values be fubftituted in the foregoing Theorem, there will come out 6.11 Feet, for the height of the Tides which would arife from the Attra@tion of the Moon, was the whole Body of’ the Earth quite covered: with Water: Hence it follows, that tho’ the Tides when forced up Rivers, and into-narrow Inlets, are found in fome Places, at certain particular Times, to rife to a height greater than 40 Feet, yet ~ in the Main Ocean, the greateft Alteration of the height of .the Surface of the Water that can poflibly happen, when the Forces of the Sun and Moon are both united together to pro- duce the Effect, and the-Moon is in its Perige, will never ex- ceed 11 Feet; nor can it be quite fo much, fince, even in the great Pacifick Ocean, it.muft be lefS than it would, was the whole Earth quite covered with Water 2 or ei ct het nts seit ———— (4 das. To determine the Length of a Degree of the Meripian, and the meridional Parts an- fwering. to any given Latirupe, according to the true [pberodical Ficure of the EarTHe. Let POS be the Axis, AO the femi-equatoreal Diame- ter, and PBAS a Meridian of the Earth; and from any Point B in that Meridian, perpendicular to the Tangent B Q, draw BT meeting PS in T; and upon the Diameter PS defcribe the Semi-circle Pua 5. making Ov parallel to TB, Br to PS, and vz, BC, and Qrd, ‘each to AQ; putting PO ROL, OC a5, Br (Cd) =x, d@=1+4, the Sine (O 7) of the Latitude of the Place B, to the Radius 1 = 2 Prey OFF 5 —s5, and the meridional Diftance anfwering to that Latitude, in Parts of the Axis PO, => Thet: by the Property of the Fllipfis, we have BC=d4/i1—xx, CT=d+x, and BT = ee but as BT : CT :: Ow(1) : On (5); whence x (O' Ci= OH eS) ad Br A of d—bs? L eee [ 42 ] Therefore, becaufe the Triangles Onv, BQr as Pah ik teal wee are fimilar, it will be as \/1—ss (mv): 1 (Ov):: Fe == BQ, andas oo (BG): oe a's SSS So I—45.2 X deeb sels id's (BQ):- ds oe —b 21h P—bs?*xX1—ss may be reduced to 42 — eee of which Fluent being : I-—SS iecerr (715. ac .302585d taken, we fhall have y equal to eee (Br): pie into the ( Brigean) z Log. of 242, — 7397585 into the (Brigean) Log.. of ae 2 db d— brs. Periphery of the Earth at the Equator, in Part of the Semi- axis PO, is to 21600, the Meafure of the fame Periphery in Geograp hical Miles, ‘fo. is this Value of y, to 3958 x Log. : But as 3.14159, &e. x 2d, the Meafure of the whole i 495852 ae ce 2 x. Log. ee the Value of y. in Geogra- phical Miles, or the meridional Parts. required. ds Moreover, becaufe the Fluxion (; ——————-——- Jof the- P—bs2 x 5 s't Arch AB, 1s to the Fluxion ( of the correfponding: of Iss a st dent that the length of that Degree of the Meridian, whofe 60 Middle is B, will be. to — the er of a. Degree of the circular Arch au, whofe Sine is 5; as t@ 1;.i€ is €vi- Circle P'S in the fame Ratio of ae to-1 very nearly, and [ 490.5 and therefore is equal to ai fuch Parts (or Miles) where= meee 9 § $2 of every Degree of the Equator contains 60. Ours, I. CORO DEA KY 2: If we confider the Earth as (it really is) nearly fpherical, d will be nearly.== 1, and confequently the Value: of 6 very’ z be fmall, in which Cafe ed x Log. + ~ becomes==79 1605: : d-—bzs + nearly ; and confequently y =.3958 x Log. , — 791645: x But if we confider:it as a. perfe&t Sphere,. then et x -Log.: de will be == 0, and therefore y == 3958 x Log. aie a—bzs ion which Value, it is eafy to prove, is equal to 7916 multi- plied by. the Logarithmic Tangent of half the Diftance from the remoteft Pole (Radius being 1) Therefore, if this Product, or the meridional Parts anfwering to the given Latitude, when the Earth is confidered as a perfect Sphere, be denoted by Q,. it is manifeft that. the meridional Parts anfwering to the fame Latitude, when the Earth is taken asa Spheroid, will be de- fined by Q—79166s, or Q— 68.55; becaufe 1+ 6" being to i, as 231 to 230, (as has been before determined) 79164s is == 68.55. COR'O Layee RY. He Moreover, becaufe the Earth is nearly {pherical, dim b 57 : 60 XI e 4 Be will be nearly = 12=6ox1+ — x1 —= x b—b 3, 1pb—bislz a b s* a Se. == 60x 1—b-+ 2, Ge. whence it appears that the length. [ 44 ] length of a Degree of the Meridian increafes, from the Equa- tor to the.Pole, in the duplicate Ratio of the Sine of the La- titude very nearly. a wee a MOP AR. Let it be required to find the meridional Parts an{wering to - 0° Latitude, every Degree of the Equator being fuppofed to contain 60 Geographical Miles. Here the artificial or loga- rithmic Tangent of (70°) half the Diftance from the remotett is Q ic itiply’d ih oi Pole is 0,438934, which being muitiply’d by 7916, gives 3474,6 for the meridional Parts aniwering to. co? Latitude, confidering the Earth as a perfe€& Sphere: But as Radius to the Sine of 50, fo is 68.5 to 52.5; which waken from 3474.6, leaves 3422.1 for the true Value required. The like of any other. ‘SO MOrig M. From the foregoing Conclufions, the Ratio of the Fquato- real Diameter and Axis of the Earth may be determined, by knowing (from Experiment) the Ratio of the Lengths of two Degrees of the Meridian: For if the Sines of the Latitudes in the Middle of thofe Degrees, be denoted by s and §, and the Lengths of the Degrees themfelves be to one another, as 1 to m; then, from what has been found above, it will be as 60d 60d OS} a Fe :2. —— : —__.. as 2}2 —_ EEE ic2| 2: Lia ial aS whence 2 x dd—8S dd—bs?\*, and therefore 23 x dd—bS?=-d d—/ s*, butd#—=1-++- 4; whence, by Subftitution, 1-++-d x 3—dS? x n3=1-+-b—b s*, therefores— aes eae : ni—t ; —- j and d(a4- 8" ae From 2 2 = = C2 2 a oF 2 4— 3-3 S?*— 5? 1—n3-+-73S*—s whence it appears, that the equatoreal Diameter will be to the N3—1 | ae to 1. But when the Meafures 1—#3—+-n38* — 5? of the two Degrees are nearly equal, or the Figure differs but little © oxtS OS Ti ae [ 45.) little from a Sphere, 2 will be nearly == 1, and therefore, if inftead of 2 we fubftitute 1++-m, we fhall have 73 = 1 -+- oye e (becaufe all the Terms 5 x S73 of the Denominator, in which m enters, may be rejected ~as inconfiderable) Therefore, in this Cafe, the required Ratio ed 2m nearly, and confequently b= ill b oe tO. Gas I “— to I, very whl beast -- —= ry axGentr i! BE A ae. nearly. A TABLE /fhewing the length of a Degree of the Meridian, in fuch Parts (or Miles) whereof every Degree of the Equa tor contains 60. | 4159,484|34] 59,779 /64! 60,105} 6 59,488|3 659574566) 60,127 8159,495|38} 59577 1/68/00, 147 10} 5.9; 503 14.0] 595797|70|60, 165 ee | ES | | A | or! 12159,513142159,923|72| 60,182 14|59,52414.4159,851|74] 60,208 16] 59,5374] 59,978176)60,212 18159,552|48| 59,905 |78 00,224 20}59,568}50| 59,932 80 60,235 22h59,586152) 59,959|82/00,244 2.4|59,005154|59,985|84 00,251 26|59,626|56)60,011|86'60,256 2.8159,648 |58|60,036 38 60,259 30} 59,671 |60 60,060190 60,201 yp NS M | A 7 SIE GE Sea gee a ; [ 46 ] DETERMINATION Di or H.E REFRACTION which a Ray of Licur fuffers in its Paflage to the Eartu. PROPOSITION Uppofing the Velocity of Light, in refpeét to the Velocity S Juppetent to retain a Body in-a circular Orbit about the Earth juft above its Surface, to be very great: I Jay, the greatef horizontal Refraction that would arife from the Attratiion of the Earth, will be to 57° 17' 44", as the Square of the latter of thofe Velocities, to the Square of the former very nearly, For, fince the Earth’s Attra@tion is in the inverfe duplicate Ratio of the Diftance from its Centre (O), the Curve DA Pp Cc o\e which a Particle of Light would defcribe thereby (fetting all other Caufes afide) will, it is known, be one of the Conic-SeGtions ; | and {47 ] and therefore, fince the Velocity of Light is fuppofed very great in refpect to the propofed circular Velocity, it maft be an Hyperbola ; whofe Semi-Tranfverfe, and Semi-Conjugate Axis (AC and CP) if the Ratio of the faid Velocities be 2x AO ; AO ; é put as z to 1, will be ———— and a re{pectively (as is proved in Page 153 of my Book of Fluxions.) Therefore, ; et ee AO if the Afymptote CB be defcribed, it will be as ie = to A e e , Or’as 1 to =) fo is*Radius; to the#fangent of 2u—Z PCB, the total Refragtion of the Ray A D indefinitely pro- duced. But fince z is here very great, I ° Sell. Dearly ss ny/nn—t y ——; therefore, the Tangent of a very: {mall Arch being Un nearly equal to the Arch itfelf,’—— ‘will be the Meafure of the Angle BCP, to the Radius 1; in Parts of that Radius, I . hence we have as 1 to ——, or as mm to 1, fois 57° 17° 44" the Degrees, &c..in-an Arch equal to the Radius, to the Re- fraction, or Degrees, &c. in the forefaid Arch, whofe length is I Q. E.D. e 42 SG HO,.LIU.M. It is found, both from the Periodic Time of the Moon and from Experiments of Pendulums, that the Velocity fufficient to retain a Body in a circular Orbit about the Earth, juit above its Surface (fetting afide all\Refiftance, Gc.) muft be fuch as would carry it uniformly over a Space of 4.95 Miles per Second. Therefore, if Light, according to Obfervation, moves thro’ a Space equal to the Semi-Diameter of the Mag- nus Orbis in 8 Minutes time, and the Sun’s Parallax by 10 Se- conds of a Degree, the Velocity of Light muft be to the I | Velocity i 4 | nt P Bf AE! Btls We oe j a hat a e i a = i i R 5 CMT + SS eos [ 48 ] Velocity above named, nearly as 34090 to 1: Hence we have as 340907 ; 1%: 679-17) 44" 3 35" 18"', for the Horizontal, or greateft Re tp aGtion arifing from Gravity. Whence it ap- pears, that but a very {mall Part of the Refraction obferved in the Sun, Moon and Stars, can be owing to the forefaid Caufe, even fhould the Velocity ° Light, in reality, be euch’ lefs than it is at prefent fuppofed And therefore in all Practical Enquiries, about the Reuice of the Heavenly Bodies, the Contideraten of Gravity may be entirely neglected, as altogether too minute to caufe any fenfible Alteration. PUR O..P O08: 1 TeI-O.N- -H; To invefigate the Curve, which a Ray of Light, or any mo- ving Body, will deferibe by any given Force, €ontinually urging it perpendicularly towards a given Plane. Let EGL be the given Plane, AE the required Curve, Hand wv any two Pont tench indefinitely near to oh other ; and let the Force by which the Body or Particle is urged towards E.G, be reprefented by the. Ordinates BD, Gc. of any given Cue SDL, whofe Axis AG is perpendicular to E oe as Ui EG; drawRHB, vrknand QAS parallel to EG, and put AB=x, BH=y, BD=Q, AH=z, Hr (=Bék) =x, Ur na [ 49 ] vray, the Area ASD B==s, the Sine of the Angle GAH to the Radius 1, —4, its Cofine ==¢, and the Velocity at the Point A==g: Therefore as 1: b::g: 6g, the Velocity at A in the Dire@ion AQ; which, becaufe the Motion in the DireCtion of the Ordinate is not at all affected by the Force acting in the Dire@ion Hr, muft alfo be the Velocity at H in the Dire@ion HR; wherefore that in the Direction Hr will be “£5, whofe Fluxion =e making y conftant, will therefore be as. Q x Je, that is,“ as the Force by which the Motion is accelerated at H, drawn into the time of de- bg5! == 25 wehduel£ 2% : J: bg | Bi ==Qx(=BDzk) =s,. and confequently, by taking the Fluent on both fides, aS == 5 ++ fome conftant Quan- tity d; which to determine, let B coincide with A; then s fcribing Hv: Hence, by putting p p> z x? 5 “ Z being = 0, —2— will become = ¢, but —- being there 27 . ms aye = +, @will be = —_, and confequently a x 25 a bx uate eet. Wherefore 7 = sand & ——=*; from whence 2 ce ; oe v. ee fe te when s is given in Terms of x, the Values of y and z will be alfo given. O. Fat CORO LARRY. 4 Becaufe the Value of 5 at all equal Diftances from the given Plane EL is the fame, and (5). the Sine, of BAH is to b ‘ oe G ah — é) the Sine of 7 H-v, as4/ 1 += foun it 1 ae ees mn od follows, that the Sines of Refraction, or of the Angles; which N any ‘Ray of Light fhall defcribe a given Curve: For if 5 be I, [ 50 ] any two Rays AE, AK ‘(having the fame Velocity at A) make with the Perpendiculars FE, TK, at entering the given Plane or Surface EL, will be to one another as the Sines of the given Angles EAG, KAG. Therefore if the Refraction in any one Cafe, or anfwering to any one Angle KAG, be given from Experiment, the Refractions in all other Cafes will from hence be given, let the accelerating Force be what it will. 1 G OFR 0 LA RY» a. But if the Force whereby the Particle, in its Paflage be- tween AQ and EG L is accelerated, be the Attraction of an interjacent Medium, whofe Denfity in going from QS increafes according to fome given Law, not only the fame Thing, but the Curve itfelf willbe had: For, let B& be fuppofed con- ftant, or taken every where the fame; then (BD) the accele- rative Force of the Medium, or the indefinitely little Area BDz&, will, it is evident, be as the Difference of Denfities in B and &,. and confequently the Sum’ of all thefe indefinite —jittle Areas, or the whole curvilinear-Area AS DB; as the Difference of Denfities in A and B. Therefore fince s is as this given Difference of Denfities, the Nature of. the Curve will be readily had from the Equations foregoing. And hence it appears, that, if the Denfity in QS be nothing, and that in EL given, the Refraction will alfo be given or remain invariable, let AG, the height of the Medium, and the Law of Denfity be what they will, and therefore is the fame as it would be, was the Ray to be refracted immediately out of a Va- cuum into the faid given Denfity. COROLLARY Ii. Hence may alfo be found the Law of Denfity, whereby a taken [st] taken == 1, fo that A may be the principal Vertex of the Curve, ¥ will then ‘become barely = re and therefore s == 2s gx x “—- which is as the Denfity required. Zy E XSAN PEE 1: Let the given Curve bea Circle: Then y being =\/2rx—xx, . 2. e Y Komen 6 2 22 -K2ZrX——x** J an and s (= g \= = ao 2. kehere= a 27 %*——x ay" 2Xr—x fore the Denfity is as et or as the Square of the Tan- gent of thé Diftance from the higheft Point. . EXAMPLE Il. Suppofe the Denfity to increafe uniformly; to find the - Curve: Here by writing x inftead of s, in the former of the . two Equations, in Cor. II]. we have y = = and therefore *« your J 2X3 which anfwers to the common Parabole; the like of any other. . PROPOSITION Ii. To find the Curve which a Particle of Light or any moving Body will deferibe by any given Force, continually urging it direttly towards a given Centre, Let O be the'Centre to which the Body or Particle is urged, AR the required Curve, v and z any two Points therein in- definitely near to one another, and AF, v T Tangents at A and L3e7] and v; to which let OF and OT be perpendicular, and from the Centre O let the Circles ASH, 2x, and Bv be defcri- ‘bed: Let the Velocity at A be reprefented by Am, and that at v by Av, and let the Force whereby the Body is urged to- wards the Centre O, at any’ Diftance O B therefrom, be de- fined by the Ordinates BD; BD of any given Curve SPB: Putting AO=—4¢, Aum=g, Ary, BD=Q Ov=x, Pu==x, uN=z, the Area ABDS=s, and the Sine of RAB or OAF to the Radius 1, = 4. Then by the Refolution of Forces, it will be as 3: x2 Qe Pe, the Force acting in the Direction v7, whereby the Motion at v is acce- Jerated or retarded, ‘which therefore drawn into *. the Time UV of deferibing xv gives — = (—v) the Alteration of Ve- | locity [ 53°] locity in that Time: Hence we have vy =—Qx(= BDP&) =—-s, and, by taking the Fluent on both fides, 2 = & —s$; whence v= \/g?—25. Wherefore becaufe 2 2 © the Velocity, be the Law of Force what it will, is known to be inverfely as a Perpendicular falling from the Centre of Force to the Tangent, we fhall have d@ (FO): g (Am): : GSE oli addi Bee Ee CP ae tg VJ g2—2s (Ar): -OT == vT= ) as ; But as ae AY: xim—2s xb? a2g?? and as pees OP.::4m 2 2'1 (Radits}to-- rat ca ae the Fluxion or Decrement of the Angle Aon ; from which, when the Re- lation of x and s is given, the Curve itfelf will be given, Q. EL COROLLARY LIL If the Curve AR be that formed by a Ray of Light in paffing thro’ an elaftic Medium, and the Refraction be requi- reds then OT being = 7282, its Pluxion will be CH.Qb!l U M. The laft Conclufion will be found to afford a fhort and very ufeful Theorem for determining the RefraGtion which the Light of the heavenly Bodies fuffers in pafiing thro’ the Farth’s Atmofphere, by the help of one Obfervation only, in all Cafes where the Zenith Diftance is not very great: For let AE, &c. reprefent the Surface or a great Circle of the Earth; then, becaufe the Atmofphere at a {mall Height A B, above that Surface, in Comparifon of the Semi-Diameter AO, mutt be extreamly rarer than at the Surface itfelf, the Refraction beyond fuch Height will, at moft, be but very {mall, and therefore the Curvature, which any Rays RwvA, Cc A, fuffer below Bev, may be confidered as their total Refractions. But thefe Refractions being found by Experi- ment to be but fmall, the Angles vAB and AvO will be neatly equal, and therefore, if not very large, their Tangents will likewife be nearly equal ; from whence, and what has been faid in the laft Corollary, it plainly appears that, let the Law of Denfity of the Atmofphere be what it will, the Re- fractions of the Sun, Moon and Stars, at all Altitudes except very {mall ones, will be nearly as the Tangents of their appa- rent Zenith Diftances drawn into the refpective Denfity of the Atmofphere, at the Places and Times, for which fuch Re- fractions are to be determined ; and therefore if the Denfity be the [55] the fame, are fimply as the Tangents of their Zenith Diftances, But now to eftimate in fome fort, how near this Proportion | comes to Truth, and how far it may be relied on; let any two convenient Altitudes, z, ¢, that are neither very fmall nor very near each other be aflumed. Suppofe one of 20, and the other of 40 Degrees, and let few L, &c. bea Circle, or fphe- rical Surface dividing the Atmofphere into two Parts fo that the Denfity at that Surface may be equal but to half the Den- B \C PR fity at A. Now the Height of this Surface above A E, the Sur- face of the Earth, from the known Properties of Air, and Ex- periment made on the Tops of very high Hills, cannot be more than about 5 Miles, or —- of the Earth’s Radius. Therefore we have, as 1 -+ a (Ow) : 1 (AO): : the Cofine of (RAE = 20) the leaft given Altitude to the Sine of 69° : 48’, which being 96°] being increafed by 1 Minute on Account of the Curvature of the Ray Aw, gives 69° 49', for the Angle OwA very nearly. And in the fame manner the Angle Oe A, corre- fponding to the other given Altitude, will be found 49° 55’. Now it hath been proved, that if the Angles of Incidence OwvA, OcA, continued every where invariable, or equal to themfelves, the Refraétions would be to one another exactly as the Tangents of thofe Angles; therefore, becaufe the Diife- rence of the Tangents of vA Band OwA, Sc, 1s but little, and the Refraction above and below the Surface few L, nearly equal, therefore may 69°: 49’, and 49°: 55, be taken as mean Incidences, and then the Refractions, anfwering thereto, will be to one another as the Tangents of thofe Angles, or as 1 to 0.4372; which Proportion being much nearer the Truth than that of 1 to .4338, arifing immediately from the Theorem, the Error, in the confequent Term of this laft Proportion, cannot, it is plain, be much greater than (.0034) the Diffe- rence between .4372 and .4388; which, fhould it be even double that Quantity, would {carce caufe an Error in the Re- fraction itfelf of a fingle Second. Nor is it in this one parti- cular Cafe only, that the Rule anfwers fo exactly, the Error here being nearly as great, if not greater, than it can be in any other Cafe, where the leaft of the two propofed Altitudes is not lefs than 20 Degrees, as is eafy to fee from the Reafons foregoing. Hence it appears, that if by any Means we can come at the true Refraction correfponding to any one given Altitude, not lefs than about 20°, the Refraction at all higher Altitudes, for the fame Denfity of the Atmofphere, may be bad from the forefaid Proportion, and that to a fingle. Second. And this is to be the more relied on in Praétice, as it does not depend on any particular Hypothefis, for the Law of Denfity of the Atmofphere. : The Refractions in fimall Altitudes, which remain to be confidered, are not fo certain and eafy to come at, nor indeed, to be computed at all but by Virtue of fome Hypothefis. If 3 the [57 the Denfity of the Atmofphere, in going from the Earth, be fuppofed to decreafe uniformly (which Law will be found to anfwer better to Experiment than the commonly received one, founded on the Elafticity of Air) and 4 be put for A B the height of the whole Atmofphere, in Parts of the Earth’s Radius, and & be affumed equal to the greateft Value of s, correfpond- ing to this Height, then will s be to % as & to 4, and the Fluxion of Refraction (found in Cor. I.) will become C s_l, abgkx abgx HY 27 072 607-27 b> g* I e to-, or, if abe a ee and iS therefore to bX gr—25X 4/g? x? —25x*—a"b*g the Fluxion of the Angle AOvy, as bxXg*—2s “—**; which, becaufe x and 1—2s are always nearly the fame, will be as & to b very nearly. Wherefore, a the Angle vHB is equal to taken ==-1, and g==1, as k tod x [ 58 J to both the Angles Ov H, HOv, the Fluxion thereof, or that of the Refraction, will be equal to the Fluxions of both them two, and is, therefore, to the Fluxion of —OwH, in the conftant Ratio of £ to h-—&; therefore the Fluents them- felves (corrected by their proper conftant Quantities) muft be in the fame conftant Ratio, that is, the Refraétion will be to the Excefs of OAF above OwH, as £ to h—k. But, fince ; ; bag b , Qe) is foumd above to bée’s—' +; or _—==— °( becaute Vf g°—25 Yi—2k : ae & b @—==1, g=1, Se.) the Sine. of OvH is given = ———=———— 98 : ) 8 1ths/1—zk (=e oe) = 6x 1—b--&, very neatly. Therefore it will be as 1 to 1—/-+2:: fo is the Sine of any apparent Zenith Diftance, to the Sine of an Arc, the Difference between which Arc and the Zenith Diftance, multiplied by Be will sive the Refraction fought; from which Proportion, the Re- fraction may, in any Cafe, be determined, when 4 and & are given from Experiment; both which may be had from two Obfervations. But if the Altitude be pretty large, then the Difference of the two Arcs meafuring the Angles OA F, OwH, being nearly equal to the Difference of their Sines into Radius, applied to (c) the Cofine of the former, the Refra€tion will be barely b bxb—k a xk ( = Value of & may be found from one Obfervation only. For an Inftance hereof, let us fuppofe the Refraction at the Alti- tude of 30 Degrees to be given from Experiment, == 1' 30°2; then the length of an Arc of 1' 30'%, in Parts of the Ra- 2 x —) and therefore in any fuch Cafe, the dius, being = .o0044 and - the ]) 7 |p) BY > [pl wy eel & ls 2) 2 ie) & | o / a fe | / Cat J IN Q CAs. O25. 1 2 013.5.30125 1]23.50|26) 1.47 1125.30 |26 2|17.4.3/27| 1.42 2118.51/27 3113.4.4/28) 1.38 3) 14.36/28 4, tT.0$|29) 1.34 4111.43 |29 5} 9-10/39] 1.30 5] 9-44.30 C] 7.49/32] 1.23 6} 8.18 32 7 6.48/34) 1.17 7| 7-12134 8] 5.59/36] 31.12 8} 6.20/32 9| 5.21/38] 1.07 9} 5+39)38 10] 4.501490) 1.02 10] 5.07|40 11] 4.24/42] 0,58 Il] 4.40|42 12] 4.02/44] 0.54 12} 4.16/44 13) 3-43146) 0.50 13) 3.56|46 14) 3.27/48] 0 47 14| 3.39/48 TS} 3+13)59) 0-44 15) 3:24|50 EOL 3-67 |521 0.41 Loge TL ice 17] 2.50/54) 0.38 17} 2.59154 1c] 2.40/56) 0.35 13} 2.48/66 [9] 2.31/58] 0.32 19} 2.39/58 20] 2.23/60] 0.30 20] 2.31/60 21] 2.16}65} 0.24 21 2.23165 22] 2.09|7O] 0.19 22| 2.10)70 125) 2.040701, 14 23) 2.09/75 #4 1,6°77/80| *0,% 24| 2.03 |80 The Numbers whereon thefe Tables are grounded, were deduced from Obfervations communicated by Dr, Bevis; which, by their near Agreement with each other, feem to be taken with great Care and Exacinefs : And the only mate- rial [ 62 ] tial ObjeGtion (that I forefee) the Tables are liable to, is their being founded on a Suppofition, that the Denfity of the At- mofphere decreafes uniformly ; which is not only very diffe- rent from what hath been hitherto commonly received, but {eemingly contrary to Experiment, whereby it is proved, that the Denfity of Air decreafes as the comprefing Force: But it may be anfwered, that, tho’ this is allowed to be true in Air containing the fame Degree of Heat, yet it cannot be fuppo- fed to hold in the Earth’s Atmofphere, fince the upper Re- gion thereof is known to be much colder, and confequently the Elafticity there much lefs than at the Earth’s Surface: But, a convincing Proof that this Law of Denfity cannot obtain in our Atmofphere, 1s, that the mean horizontal Refra¢tion com- puted therefrom, according to the known refractive Power, and fpecifick Gravity of Air, will be found to come out no lefs than 52 Minutes, which is greater by almoft+ of a whole Degree than it ought to be; whereas, if the fame RefraGtion be calculated from the Hypothefis of a Denfity decreafing uni- formly, and compared with Obfervations, the Difference will not be near fo confiderable. This fhews the Tables to be much exacter, than they could had they been computed from the common Hypothefis ; I mean, in very {mall Altitudes; for the Refractions in high Altitudes, it has been proved, will be but little affected by different Laws of Denfity, and therefore come out very near the fame, compute them according to what Hypothefis you will; even fo near, that if the RefraGtion at any Altitude not lefs than about 7 Degrees be truly given from Experiment, the Refraétions, computed from thence, ac- cording to the two Hypothefes forenamed, for any higher Altitude, will never differ from one another by more than about 2 Seconds. From whence we may infer, that as the Hypothefis on which the abovefaid Tables are founded is much the exacter of the two, the Error arifing therefrom cannot in any fuch Altitude amount to more than a fingle Second, QU Vs OI Ha) Vite | tai) biFihipy tL Oniit ith i eH af ng a fi i , Baie Ais i ij | pO OP OO Oe ara Te I LS ee a a need [ 62 J OF THE SUMMATION of SERIES. PoRAO-P. OrSeler td O' Nex F WR an + ban—ix+can—2x*-+-dan—3, &c. be any Power (n) of the Binomial a-+-x, either whole or broken, pofitive or negative, and the Terms thereof be.refpectively multiplied by any Series of Quantities p,q, t, 8, &c. and the Differences of thefe Quantities be continually taken, and the firft Diffe- rence (q—p) of the firft Order, be denoted by D, and. the fio (c—2q-Ep) of the fecond Order, by D, &c. I fay, the Series pa® + q ban—! x-+rcan—2 x*, &c. thence arifing, hall be =p x apex! - Dbx xa-pxl Dex? x aspx +-Ddx3 xa-+xl >, &c. For, let Pxa-xl" + Qxxapxl +Ratxate exe x ate &c. be affumed = pa™ + gnar—ix rn x — QM—2 x71 5 x “—- x= an~3x3, Ge, (= pan + gba—ix+rcan—2x*+ sdar—3x3, Gc.) ; then, by con- verting the feveral Powers of a+ x to fimple Terms, and tranfpofing pa®—+-gnar—i x--rn x an—2 x*, Gc, we fhall have : -+-P { 63 ] Tiet=1I “bh Pan-enPanar x “2x — Panwz x7, &e, n=—TI * + Qari x + —— Qan—? x”, &Se. = - + Rar—2 x*, Ge, & * * ec, J nt —pan we gna} Yume PHN X = qu—z 2, esc. From whence, by equating the homologous Terms, there will be P=p, Q="xq—p, R=2 x — xr—2g9-+p, S= not It x —grhaPoP, Se. But wis = 6,0 x 2a xX n—t!I =—c, &e. q7—p=D, r—29-+p=D, €?¢, and confe- ccontly pat 9b ana eet rea eee aaa Gee == p x ste EPO Pe 2 ae 2 oe Dd ih geal + Dex xaexl + & c. where it is evident, that the Value of Coat pate ; &c. will be always had in finite Terms, when the laft Differences of the Quantities p,7,7, Sc. are equal. ee. D. CGi0 RAG LsA Roy, I. Hence, if the Values of ~,9,7,5, Ge. be refpectively ex- pounded by the Terms of any Arithmetical epee ction k, k--n1, Wt ite kam, k-+-3m, Ge. then p being —h, D==m, and D, 3h, &c, each = 0, we fhall have & gre p+ mx bane ee es can—2x7, Fc. barely = me x a--xi + mbx x a+xl < GF Rx a-Pxi + mn x % ax CO. a [ 64 J COROLLARY If But if the Values of pf, 9, 7, &c. be defined by 7 7s : oe ay Se. (the Reciprocals of that Progreffion) then D being —m # tH, 2M al —mM.2M.34m —= Ehbm D= hktm bk 2m? 1) =a et beam fil! Mh. 2M.3m.4 Mm __ gn Doi we Ghee Sienny aay &c. we fhall have =- eas se cary x* rages: os 2 ee a a+ x| _ ae eo os kom At3am”? Bie __ mbexafes| 1.2.1 Cx” Xapa| __ mam. 3md x3 xapex| oan bm kkt-mk--2m hiktem.k4-2m.k4- 3m mM. 2M.3m. 4mex*Xa-x|- ee ee = 2-2" pee BAe x hk} mh-2m.kt3m.k--4m k-l-4m . ye k k+m me w—1XB me a—2XC “Lee __, 4-—-3xD a+ ébam a+«x k4- 3m a+x k--+4m cee where A denotes the firftt Term, B the fecond, C the third, and fo on. COROLLARY 9 Ub Therefore, if 2 be taken —— 1, g==1, and x==2™, then gm . I é being a eT Her H I, é=— I Gc. we have 5 sii pes 23M 24m wary eee 2 ge ee &Fc. ——- eo a v kt 2 h4-3. me A k4-4m? kX 1 x™ a k+m % ke 2B m x" gC m zm 4D m2™ pane baal 4 ie” “Fae . I-27 aw, F44m 1 pe phe! Ab ae AQ z2BQ 7 OREse er oo ape te + Tfam ee &c, by putting Q = ——_ E43 m fo, pa [ 65 ] GO RALUARY¥Y 2D Moreover, when # is taken ==— 1 and a=1, we fhall i u p Dz D x* ae A aamaee 3 4. a fe == "See to have p—gx-trxt—sxi-pint, Ge. == (ys ee Tea — 2 &e. and confequently (by writing — x inftead of rl ate 2 43 eS rors P -- ln Dx? —- De X\ PE OX-ET XP 5X3, (aS Ta el aoa &c. fhewing the Value of the Series p--gx-trx?, Sc. con- tinued 22 znfinitum. 4 GORD Le Bn ae But if the Value of only a finite Number (7) of Terms be wanted, let the remaining Part of the Series be repre- fented Pxa-+-Qoant1+Raxntz,. and the Difference of the Coefficients P, Q, R, &c. be ‘continually taken ; and let the frft Difference of the firft Order be denoted by E, the firft Difference of the fecond Order by i, ce. Gc. Then, for the Dx ioraic very fame Reafons that p+-gx--rx*, Ge. is = i. + i N Ex E x? ec. will P+-Qx-+-Rx?, &e. bess +a yp S, T—-x| ie €c. and therefore Pxe+ Qanti+Raomtz, Ge. = — t n+1 ht n+-2 t E E | ae Laie af = &c. which taken from i ear x 1 I—x| l= , 1 n-br tt nbz poPx Dr—Ex D x? —E x { ey. } 3 3 &Fe, equal, i—x|- I— | to the firft’Terms of the Series propofed, R C O- &c, leaves i—v* sere ee =< ae ain ieee enneantieetreyptesanehitticneereremasiene-saemenememrrenear-os Soe a ——— a= Ee OE RRR eh [ 66 } CORGEL A RY! ‘VI. oe may “ _ of my Number (7) of Terms ofthe Series 7. ae Fine eee 7 ot Sc. where z is indetermi- - nate, be aie * alee for fince px-+gx*+rx3, &e. is nr ! Le nia, p—Px on Sate D 2-H 4 Ll oesnnnenntnemeenened e amie 5 ES dabomat dad ee eee ee aa It I— x| reek Be ye | x Jet 2 be written therein inftead of 1 and it will become. “ ? Pe fl WY Pe et Sie RM pene eet ss aie 3 Se. Se eee perc z—1| z—1| equal to the Value fought, which therefore, when z is infi- - nite, or the whole Series. is taken, will, be barely. ==,.—— Ral; os ah = “tS ee ge. xe 8 az ck &c. in the general Equation, we have 9 a? ae 2 [67 ] 36 «3 z gee oc -, Se. = 9x a+x-+- PEK ae SP 16 az z was to be found, ee 3 #7 xX arew| *. ae which EXAMPLE H: Where x being lefs than 1, ’tis required to find the Sum of | the infinite Series 1-+-2x-+-3x*+-4x%, &c. In this Cafe, p—=r, Gee. Jorn D5" Ta &c. and therefore (by Cor. IV.) I-+2x-+-3x7, Se. = —— —- Same = a: In like Manner. it will be found, that; 1-+-4x-+-9x?-+-16%x3, Gc, is ie ee 3x1 2 a Sout th = and that 14-8x-++-27x?-+-64x3, Ge. ae: —* —x'|- ; ; 7% 12%” 6x3. SS a eee eee. Sec, I—x . I—z| 1—x| I—2| cae — ————— SC HOLITUM. The foregoing Conclufions are not only ufeful in finding the Values of Series, which are.in their own Nature exactly fumible; but may alfo be ‘applied to very good Purpofe in the - Quadrature of .Curves, and in approximating the Values of fuch Series, whofe exact Values cannot be determined, Let it, for example, be required to approximate the Value of the 2 AKT 2x2 2x2 2.3%2 2.3.5%°5 2.3.5.9" 2 Series 3 Wkclpu aig o4l6-e aaa era: ea re exprefling the Area of the rectangular Hyperbola, whofe Ab- fciffa-is.x, and principal-Diameter Unity.. In order to effeét which, let.a few of the leading Terms (fuppofe the four firft) be collected into one Sum, and let the Differences of the Coefficients of a few of the firft of the remaining Terms, ° ° > 2. 3. « 3. 5- 7 3.5.7. which (in: this Cafe) -are te Ee etiat etietert> Fc. or or 0.0142, 0.00841, 0.00546, 0.00379, &c. be continually taken (as in the Margin ) PR oskit oocab eae Then, the firft Difference ==.00579—-.00295—.00167, Oe. of the firft Order being -++.00284-+-.00128, Se} —-o,00579, of the fecond a _00158, Sl Order. -+-0,00284, Of the third —o.00156, &ec. if in- Wf Mh mead Ot p,' 9, r,s, Sc: D, D, D, &c. the above Values 0.01420, 0.00841, 0.00646, 0.00379, &c. — 0.00579, 0.0284, — 0.00156, &c. be refpectively fubftituted in the gene- i Dx D x? i oe 25 43 = ie — eta ral Equation p—qx-+rx?-—sx3, &c. = ce tua os ra &c. (as found in Corol. IV.) we fhall get 0.01420—0 o0841x 2.3.5 g.§°5.7% “of 0.0054.6x* ee 0.003 79x3, &c, E628. U1 4. 6. 8 10. 13 28.5. Sao es co )\ cei420 = OO 0.00284x* 4.6; 3. terre. 12° cy I-bx rx] 1 x|° 0.00156 x? 263.5% 2 2.3.5.7" 2 a . and confequently — 735 ine al me s+ x" p &c. an q % 4.6.8.1 4.6,8.10.13 2a 5.7.90" 2 Set 2 ae SEX 0.01420 x 0.090.579 x* 4.6.8.10.12.15° ob Ta , Ir 1+ x| 0.00284 «3 ; — hy a) hue 3 + ———-,, &c, which added tox, /ex £4. FW 2 aieails BAP go renga nae the Sum of the four firtt Terms, will give the Value of the whole propounded Series; which Value may now be eafily: found in Numbers, that of x being given; for let x==1, then ; — 0.01420% 0.00579 %* — x4 a ——-, &c. ==— 0.0090, &c, will —x* /x xo ob Spyet 0.0090, ee tae andx,/x x4 + = — =~ + a 1.6896, and therefore the Value of the whole Series will be 1.6806, which is more exact, than if 20 Terms of the original Series had been taken, Again, 3 let [ 69] | - _— z ome NEL X XX Cae Ox Pe et x be taken = then will — x*,/ as Fw 0.00284 x* re sf 0.00156 xe ee E/E x 0.01420 mas 0.00579 Ion, 16 3 9 0.0028 0.00156 eR ie ts es ts lal &e, == 0.000243, and. x Aik x 4 Bee 3703 d conf tl th V lue of the ticle Serics = 0. 5 oe very nearly. And in the fame Manner may the,Values of other Series be approximated ; but the Advantages of this Method in many Cafes are much more confiderable ; for let the Series propounded be J enti -2m-p1 eom+t atm oi oe eae ee ———, &c. and let $a 2 m+} m1 a 1 hee of the firft Terms iarcar be collected (as above) and let the Denominator of the firft of the remaining Terms be denoted by 4,and then it is evident,from the Law of Continuation, that the true Value of thefe Terms will be rightly defined by oe at alii eae _k+2m ee at Vp ee Os Seton rir ae ea 2m 3m m ge aoe aii nce, ee ae os hom —— 3m? ec, But fince Z hem + ieban > &e, (as appears from Corollary II.) is univerfally equal to ee 1g eee 2BQ 4 3CQ 1 4DQ SEQ BX 1-z™ ay am Py re ot h-f-3 mt we k--4m Pay k4t- Sm ? 3c. where Q ftands for ——., and A, B, C, &c. for the firft, fee fecond, third €c. sath it is plain that F 2* x [eS +75 + as ob ed , Ge. will be equal to + z* x mm oom 3m u— ie —~ iis » Se.’ the Ufes whereof thallim- S mediately a. | ena ara ne —_—_—_—__—_———— ye") mediately be fhewr. And firft, let e==1, and m==2,-fo m+! 2m-—-1 : i that 2 Fu ae . — = aT Gg af ee &c, may become 1 a ai &c. (exprefling the length of ; of the Periphery of the Circle, whofe Radius is Unity). then the Value (0.7440117) of Cap —- ‘ - : -- : — | the fix firft Terms thereof being ‘colledted ee will remain. = — a &c. Therefore, feeing the Value of & (the Denominator of the firft remain- Pe : : UEeirhan a h2 ine, erty) is Were == ry, and (= eee by writing thefe Values in the above Equation, have — — — 5 2 13 15 E, ae I A 2B. SAY? ea ia ee tae oe ree &e, == 0.0384615 “+ 0.0025041 -+ 0.0003016 -++ 0,0000476 -+ 0.000009] ++ 0,0000024 +- 9.0000007 -F 0,0000002 -+ 0.090000 = 0.0413873; and this added to 0.7440117, giveso. 785399 fot the-Value of the whole Series; which 1s true in ‘the laft Place, and more exact than if 100000 Terms of the origi- nak Series. had “hecmy taken.<.Acain, let *=_ 1, and°m—= 1, fe I, we fhall, fo that the propounded Series may be - = —t Sp ta 3% (exprefling the hyperbolical Logarithm of 2) then the Sum (0.634523809) of the 8 firft Terms being taken, the remain- ing Part of the, Series will be ce —-— — = + = 1Or- II —o p Se. Therefore & being here = 9g, and Q=~, we fhall sc SO 8 AQ 2BQ 3CQ ae hs + kam 4 k+-3m ? Se. += f= ap op 2D a G p+ ~F6==0.055555555 + ae ‘ee Zk x far] -- 0.002779997 -- ©,000252525 - ©.00003156r ++ 0.000004856 -+ 0.000000867 -+ 0.000000173 ++ 0,000000038 -+ 02000006009 -+ 0.000000002 =x 0.058623367; and confequently 0.693147176 equal to the whole Value required ; which errs but 4 in the laft Place, and would have required, at leaft, 1too00000 Terms of the original Series. But after all it may not appear why a few initial: Terms are always firft to be taken, feeing the Series, . om x 2 £ + k Z —— Les 5 be for the Value.of “F 2k. xz ‘nd + ae holds uni verfally, let the Value of & be what it will; but the Reafon is this, the more Terms there are firft taken,-the-fafter will the Series exprefling the Value of the remaining Terms con- verge, fo that by firft collecting a proper Number of initial Terms ‘(which will. be greater or leffer, according asa great- - er or leffer Degree of Accuracy 1s required) the fame Con- - clufion will be brought .out with a great deal lefs Trouble, | than if the Value of the whole Series was to be found by. this Method; as upon Trial will plainly appear. .. PRO -F Ot TT Og, te Uppofing a® +-.ba®—! x-- can? x? --- dan—3 x3, &c. to be S as in the laft Propofition, and + any whole pofitive Num- ber, and that Sis equal to the n-+-r Power of the Binonual at-X, decreafed by the x jirft Terms: I fay, the Sum of the a" bat! » Ca? dav 3x3. QC, Pay ee SE re, CMM Ee Degman Mii repeat SE Ae aig trae meres. 7 33a. or + 2.3 .4eee? fol a FeGenP pa 4.5. Onnar 3 Suits &c. (whether finite or infinite) will be a x aes ee For the m-+-r Power of a-+-x being ant: + 2-Er x anti—ix denetr x SEZ x at? x2, Be, if from the fame the r firft tf i y ; bt 92) —e mmm Lose met. Hs "Terms be taken, there will remain beaks X apes. kay eee X ahx® EMS tice t oe nr 7X art ate 2QN—V x Fool ses n-Lr x et rs fA2e8-.. Tk ~2.3....7-+-2 Mets n n—I SAAT XN... MEEK x x 4% ae 6.) Stak) Maa ; 2-3 608 1.2.3.4....7-1 12 yee Ta 27 > He i. ; | we aa a a eh ge ide Ere 36 74-2 1.2.3... tmt n-—TI zX Oh ee te ne 2 , ic. == °S; therefore 2.3-4Pop 3-4e5-rb2 S n ni—I , e a 7a x“ f a TS Se ae ——$ $< —_______ ° a oe X UZ X U3. eer KX act 1s 273.000 = 2.3 dacat-Pl ; Se a pe ee QED Bes Zrooet “23-4earel Teg ieagas. = g Bae CORO L LA RY¥ 'T : . ——-++-1 feb. yin On Nias ‘Therefore-it follows that apa | att fee a ee n-+-1 Xx 7 i n-+2 fal". Bc, that 22 i = ot arp axe tts is ar gee ‘ mt=t X redbe2 X x7 “ n—t N22, § natin ES ba = ebb. gh cee ee ae. | and ..that Lt2 Ss 3.4 Fie : PSEC 2 2 I atx| Be gt Tae 4a bh? x23 X- r es, o IS = m1 xX ab2xXnt3xrs T.2.3 Nn—2Z n—I okies ete Ore 2 ee 3+4-5 Lary CO-R @EBA R-Y:. It. ‘Hence may the Value of an--ba0—'x--can—2x*+-dan— 3x3, Se. ( —a-+xi) when the Terms thereof are refpectively di- vided by any Series of figurate Numbers (as 1, 3, 6, 10, 15, Ge. or as 1,4, 10, 20, 35, &e.) be alfo eafily derived: n nf NZ 4 For, fince it is found that —2— +. 24 *_ waren bE ok 4.2.3-007 2.3.4..741 34S ese7mh2 Ese ; s a —__ t ioe oe a? let the whole Equation ° > 7 a be multiply’ by 1.2.3....7, and we thall have —— p Tee T n—2 _y 13 a PE es ee eg a ee ri rz : r+2 r43 rap LX St A ge 12.3300 KS \ S AS Loxmeper kel eet ode 1 : naked X ne X mB. steer Kx + eo ens 3 er : ri-z where i3 7---F;*7 -1'x 4) Gere known to reprefent univerfally any Series of figurate Numbers. EXAMPLE I. ki fn 't ; 4 2, : / nies 5 Let there be given => -F Bn le Ge, == -c—zl to 3 c Zee 8 ¢? nia rnd the exa@ Value. of —- + —= + she = Se. Then, ioe Zac Sec? Nye : ‘ ¥ ‘pecaufe 7 is == 1, a= ¢, ¥ ==2, and n==—>, we fhall in —n-pr Pe Bs SE rr —ti-e this Cafe have 2% cee pe ct. ere ee which was to be found, T E X- f 94, } EKRAMPEES I. "4 ° 5 i“ 3 4.2 Let it be required to find the Value of — +b age ee 10 &c, or of the fifth Power of a+-x, when the Terms thereof are re{pectively divided by 1, 4, 10, 20, 35, &c. Now, in crder that the above general Series. (for figurate Numbers) may agree with (1, 4, 10, 20) the particular onehere given, let the Value of 7-++1 be fo taken, that the fecond Terms of both Series may be equal, and “then the reft will be fo of Courfe; therefore rv, in this Cafe, being -—=3, == 5, and > he three firftt Terms of a--x! Fy cl | _ expanded 1 in a Series —a® ia 28. a'x*, we have S-a-+-xl®—a®—8.a7x—28a® x, and —— reer Ss i a+x|° —a*°—9a7x—28 2° x? hi ota ea Se Se ee fr, the Bret, aaa x 56x I s r Value that was to be found, Pwae.C P.O Ga ETO NO, ae axP-EbxP Ft pcxPten yg dxetst &c. fo be any Series, finite or infinite, whofe Sum A 1s given; and the Terms thereof to be refpectively multiply d by the Terms tr, rtn, r+a2n, &c. of any arithmetical Praga, whofe common Difference 7s n; to find the Sum (B). of all the Produéts, or the Value of the Series raxP-+-r+n x bxPt® --r--2nxcxPt*", &c. zhence arifing. Becaufe axP-+-dxPt™+cxPt™™, ec. is given== A, there will be given axtebott® pet", Ge, = Ax'™-P; whofe Fluxion being taken and divided by x*~P~"x, we have ra x? rn x bx 4 ran xcxPt, 8c, = r—px A+25 I ae ims =-B; where, becaufe A is given in finite Terms, A will be always had in finite Terms, and confequently the Value of B alfo, OLE. Tr: COROLLARY. Hence, for the very fame Reafons that (B) the Sum of the Series raxP-+r--n x bxPt "4-742 xcxPt", Be, is e- qual to r—p x A+ ae will (C) the Sum of the Series rsaxP+rtn.s+n.bxPt +p 2n.stan.cxPt™ &o, be —=s—p x Ree, and (D) the Sum of the Series 7.5.7. axP+rpn.sn. tn. bxP 4 ran. span. tan.cxP bn, . &e, == t—Pp x C + ee tr. Gc, Oe. WPxAM POE se Let there be given the Sum of the Series x — = = —*, &c. exprefling the Arc of a Circle, whofe Radi is I, and Tangent x, and let it be required io Bae the Sum (B) of the infinite Series 3 «— 5S ob vel — — + = Be, Becaufe, in this Cale, 2o= a) naz, aS and. se for the Value we have B (==r—f 7p xA—24)=2A+—* +e fought. EXAM PE al. Having given the Sum of the infinite Series x — = + = 7 , ce, equal to (A) the hyperbolical Yéeerienta of I-FX 5 Le) 4+bx;\'to find (C) the Sum of the infinite Series = ‘I r 3.gx7 4.4%3 Soot ie an Ma: eae ee 6c. Here.p.beine =-41, Aeae yo=2, s==2 (vid. Coro/.) and A= “* we have B (==r—+ x ae = —— A ne a eae Ban a B= re aoe and confequently C (=s—px B-+- ae) = A me a which was to.be found. LEMMA. / it ‘To divide a Compound Fraction, as Bie scent ae into as b+x x bx xb+x, Se. many fimple ones, as there are Factors in its Denominator d W fuppofing b, b, b, &c. to be unequal Numbers, anda, a, a, &c. any Numbers, either equal or unequal, and that the Number of Factors in the Numerator 1s lefs than the Num- ber of Factors in the Denominator. ‘Let A, B,C, D, &c. reprefent the unknown invariable Nu- I ! ° : ° . —j— 3? ‘ : 5, merators of the required Fractions to which sx alae 1 Sanson Es bx x b+ x b+, &Fe, may ‘be reduced, or, which is the fame in effect, let aie ; then by / " ‘B C oo; atx Xa-bxxatx, Se. ae . + 8 oo. = ae eT ee He Lie ete babe x bbe x bbe, Se. multiplying the whole Equation by 6-+ x, we fhall have Me a4 ok I : oe ec re RS EFc- A ak Sowa om But bx : bx bfx btxx dbx, Se fince the Equation holds univerfally, be the Value of » what it + [or 4 it will, let x be taken —=— 4, oF b +- x ==0, and then, all the Terms affeéed by 4-+-~ vanifhing out of the Equation, / “ —b Xa—b Xx a—b, Ee. we fhall have A = “~**— "=== for the Numerator b-bxb—bxb—b, Ge. of that of the required Fractions, whofe Denominator is b4-x: From whence the Numerators anfwering to the reft of the Denominators or Factors, b--x, b-Lx, €c, may be found by Infpeétion, and will be had (as the foregoing is) by taking the Quantity conjoined with x in the Facter propofed, and fubftituting the fame, with its Sign changed, inftead of x, in every other Factor of the Fra¢tion given; and fo we have atx oak, ee. ‘ i a been dee Efe. % z @ pockicbu bb te. @R* > bebgb dxdt ie bbe aes ie Wee a I stje x keane RK aX a-48s EF. 1 Be acl UP * il 3 &e. = ! WW Hl . b—b x b—bxb6—b,8¢. bx bee Xb xX bx x b-4+%, Oe. Q. E. I. COROLLARY. If the Number of Faétors in the Denominator of the pro- pofed Fraétion be greater by two, at leaft, than the Number Sf thofe in the Numerator, it will appear, by conceiving ‘ / KR - an 5 Jot = ret Bs Fc, =< bane te Stee bo th wth lp ebb xxtbxx+h, Ge. (== 0) to be reduced to one Denomination, that A4+B-+-C--D, é&c. will be the Sum of the Coefficients of the higheft Power of «in the Numerator, and therefore equal to nothing. U | PRO- Lae | PROP @seiTION. [V. flaving given (S) the Sum of the Infinite Series = + gm-fn ie 2m--2n + amhse 2m-+3n He aie wt 22 hi a m+ 32 44 ge im a df pigve oS Fluent of —-—* (which may be always had from the Qua drature of the Conic-Seétions)’ tis propojed to find the Sum of the Infinite Series ato x 6-0 X fo, Fe. p. aabn x ben x cm, Ge. p0X g+0x 7-0, c. — penx gpaxrfa, Se. &c. or the De azn x b--2n Xc4-22, oS an e232 X F432 x 34, See 4 adopt wil Fe, * ees ens eae. Fe &c. fuppofing —, ——“, ——, &c. to reprefent any une- gual whole Nahe: a, b, c, m, n, any Numbers equal ar unequal, and the Number of Faétors in each Numerator of the Series to be the fame, and lefs than the Number of Factors in the Denominator, this laft Number being aljo fuppofed the fame in every Term. Mase: = eee eee! 5 uns shied sees eee pe ee ee then, ie the preceding Lemma, it will appear that xe are) oe Eo &e, are the frmple Fractions to which the firft Term of - Series 1s eee that is eee will be — =. +2 4 + + =, &e. And in the fame Manner it will appear, that. the ae. um 4th, &¢. Terms of the faid Series will be equal to oe oa i 79 ] : fn yp nn n i Az Bz Dz. Se ee oe ee ag ae , &e. Ag2t B228 C e2n D2" Be 2n p+2n es g-+2" at ee a ro a a oe, 3n 2n a. Az Bz Cn? Dx3" Fc, — p+3n ee q+3% = rh 32 ae “532 — eee Therefore the Sum of all thefe, or the whole Series, c n Zn 32 4n wx i + = aes c: pb pn O ppan pe2n —- p+3% a p+47 n 2h «30 4n 4 B x - P+ e fe feet eo. Ec. q ~~ q+n a gpan —— 7432 phan Cxi+ = ge ey Bic. L 7 — r+n + r+ 22 Bis T+ 3% on ran &Se. Bat Yee es ah 21?" ego. is given = S, a ep ha m+-2n eae oe 4c. may be redutedsto —{- «x and Ax 5 spa hppa y 2 Oe aP 2Pr® gPt2n Bcueat. aa 5 ad -+- poe? €&Fc, it is evident that A x oe pace 2n A aS ean z& 1 ; = S peers % + - &c. will be equal to p a ee ® tet ———Th e ° ca ae A ae a ; and in the fame Manner will m-+- 27 p—n B ea Loop 8 Gt: -appear- cqual to sg, * 45 ae gtt54+4e PP 4 zi m2?! nm ,m-—n et x se sip Nee ial he , ce. and therefore the Sum mtn q—2 7 of the whole Series propounded is A ae S ae ma —_< eeten TET anes @©¢@¢6¢ 8&8 @ 8 oe a" zP wt m+n m+-2n p—n 7 ee RB et mn m-2n _g—n za x eee ae ——s “mtn ——- mton esee#s-e —r. on" > Cc 2m gman : emtz2 n wor j — x S — ———— Ss Le ead —_ — @ 6 8 6@ ¢ ¢@ eae > eS, Cz ot mn | 2 mt-2n r—wN When all the Signs thereof are given affirmative, and equal to Rea m+n mt+zn p—n S&S a eS oe. ae e wo m —— mtn m++-2n j—* a fo ae m+n _ ,m+2n q—n Bayt wl ae : a a — S M1 Fae m+n a m+-2n ot g-—? _m m+n ,m+2n. r—n ¢ : So eoceg ene oe =f+ = 7 ESe, —— m — m+n m2 1 when they are given affirmative and negative alternately ; in which Cafe, the laft Term of -every Line in the above Ex- prefiion, muft be taken affirmative. Ovk. 1 Note NG P—2 Oe 9's Ge. happens to be nothing or negative, the firit Term ($) of the correfponding Line is only to be taken. COR OGLA Re Tf: axbxexd, &e. mXxmt+-nxm+ton, Se. atuxbtaxctn oe ae atznxbtenxct2a, Ge. TH pdaxmpinxmyse ue Fban x mba mpg, OR aperture. 23°, &c. then p becoming = m — m+32xm+4ux m5 2, &e. ’ g=m-+-n, T= -+ 20, &c. the Sum of the whole Series 2 will Hence if the Series propounded be an Cc n an Tn oas & x an x = I mt — mt an COROLLARY: ci, But if the Series to be fummed, be comprifed in this Form t ee I zZ 0 PX p+2x p22, Ge. ps prea x p22 x p+37, Ge.’ &e. and the number of Factors in the Denominator of each Term be denoted by I gor v-+1, we fhall have A cal eck Pee ee, B=-—vA, C=vux-— A, D=—4 xx 24 A, Feo, 2 2 3 and therefore the Sum of the whole Series equal to ar rn y _—— el am gmtn gmp 2n _2P—2 —————————— ee x S ——— = ee OS eecese B62. 3-4000U X 2S? mm mtn m+-2n pon ==) es —=2n —nN eee I VR U1 Zz & yU—I ee -——-U X ea UX x i 1.202 DX RY ? 2 p a gait ae 2 —3n —2n —n eee ee ot. tee ny Fe, when. all> the Signs, of . se os n ae Peas S the faid Series are given affirmative ; but equal to m m ot n em + ‘a n gpa wiht + ey S—* 4 et ehb2 ages AK BSP m+n m--2n p— I ve val een gr? seemed acres etal” 5X +UxXx Xo Vx 1 .25:3-4..00X 2° p 2 p ps oh —3n —2n —n : ree a * __, @c, when they change p-p2n x : alternately [ 82 ] e e : e Wea ee alternately, in which Cafe the Sign -- or — before 1-+x—"| cbtains according as is an even or an odd Number. C OF 0 LEA RK Y¥cgelll Laftly, let = be taken = 1, and the Series propounded be axbxckd, Se. , atnuxbtnaxetn, Se. at2nxb+tanxctan, &e, px qx rXsy Fe. be pax gtn x rn, ec. paz x q+ 2nXr+a2a, EF, &c." then, fince the Sum of all the Coefficients A+ B+-C ED, &e. will be = o- (dy the Corol, to the preceding Lemma) when the Number of Factors in the Denominator of each Term Is greater by two, at leaft, than the Number of thofe in the ao B CG. Numerator, we fhall, in this Cafe, have = + 4 + >, @&c. x _m _m--n _p—n : °— z esa alfo equal to nothing ; and there- mn m-+-n p— fore, by expunging this out of the general Expreffion, and fubftituting 1 inftead of z, we have + OEE te eee Et , P ois p22 TP wer a@aXbxXxexd, Fe. I I I I es px OX rxs, Geo>- Tae GN aac op an —;. @@ eee > Cx ? ot pa eee r—p an x b-}-2 x ca, ER, : I I I I p+2zxqtenx r--n, Fe, a ee | EP. Ee, pon Ao Po, eT: ° . xz x3 ‘ 5 2 . Let there be given the Series -_— ye =, &c. ° : 2 > er the Arc of a Circle, whofe Radius is 1, and Tangent z, pac let the Sum of the Series SS : as Ge eo ea [ 83 ] be required. Here, by comparing thefe with the general Expreffions, we have m==1, n==2, @=2, p=3, 9=5; 2—3 1 2—5 \ p—7=1, J—Nu=3, A (ee 23) ——, (5 5 ' ae co Sa ae _ tl Te et ees Zs &c.-and confequently — Hob Acts ee eg Se 2 Zoe 225 4 zyS or 3+ x o—3 A equal to the Value fought. EX A dMeP-Loae-. Il. Let there be given the Sum of the Infinite Series x oF x x* : ’ Sie ee ae ~ Gc. expreffing the hyperbolicLogarithmot I +>; find the Sum of the Infinite Series 2 —* + has 34; panies a3 Here m= 1, 72 Se ge ee 15 ee a m9 3.4.5.6 — 4.5.6.9 a 5.6:7.8° g=22, T= 3,54, t=—0,> Se. therefore (by Corol. II.) | —tI! pam at —lI oo——2 I—x I 3x < x —3%x ie ORE tia3 I x 3 Te at I 2 sed eet “ rape It 7~—_-oOoOoo E ——! — oF ar ye Ae Beatie I Om that was to be found; which therefore, when x==1, or_the ae is the Value 12%" propofed Series is - 5 i + ri ae ae Zr Se. will be barely . Ae ‘9-49 ‘ -)° Stee Lg eee | | equal to 56 = gyi Ol gs BX‘AM PoE iit. Where it is required. to find the’ Sum of the Infinite Series : Bbivcigy Ash mio ee, In’ this-Cafe, ‘we have = —— 1.2.4.7 2,315.8 3.4.0.9 4.5:7-10 tee] , 7 ¥ mon] have po=t, G2, r= 4, S=7, B (se) == lm 2X 4-—2 X J—2 “10? I I i \ ee ce aks ae a) Tyg? B Spee gy go” E=0, teeny : fF ee fe -. Ge. and therefore (by Corol, III.) mp Ae ; Bee aN gy hp. Ta Tipe ues te, eg Wl cee oe ak is the Value required, EXAMPLE IV. Let there be given (S$) the hyperbolical Logarithm of —, to find the Sum of the Series 19 ONG es oe I—z : so 1.2.3 2.204 3.4.6 te Se oi Ca &c. This Series may be refolved into 4 4x 4x7 4x* Sir and wae 4.6% 1.2.3 Le 2.34 ay Hgts + 4.6. D> | i oF 9k 2 ‘ Pee en ee 57"; €8c. that is, into 4 x = — Se a Bid Be 2 4 Eazad ~ ZG. 3.408 : and — a 2 + 4 , &c. Now the Sum of the former of thefe, by proceeding as in the foregoing Examples, will be iced! ieee aac @ ae =, and that of the latter equal to & & z & 4S 3 XS—z pA oe zB ; which Values therefore, added together, give eed OS ae — — for the Sum of the whole Se- x3 & 2 = & ries propounded. In the fame manner may the Sum of any -other Series be determined, when the Denominators thereof come under the general Form in the Propofition, and the laft Differences of its Numerators are equal, provided the number of Differences before you come to the laft, be always lefs than the number of Fators in the Denominator of each Term, 3 E X- [ 85 ] EXAMPLE V. sD Let there be given the Sum of the Series % Sues as z i si €3e. (which is equal to half the Sum of ‘ale fy: perbolic Logarithm of - pa and the Length of the Arc whofe Radius is 1, and Anca z) and let it be required to find the Sum of the Infinite Series —-— ae Gy" pag ER gaye ee reas. ee le ae | “hb yay ty? &e, Put x+y, or z==y* ; then becaufe the Numerators a be reduced to = * apex By 8 x 12-24, = x 12x 162°,~ = x 16x 202%, &e. the Series ae ne 4.8 8.1224 12.16%° will be changed to = x ai ea ce vas =~, Se where m being = ‘ Te pos a= 5: he {3 -o==4, 5—8. we fhall have A=-7, b= 3 C= Ss and there= 2 a 32 32 z% cif ane tes fore oy into < 5 Sate -xS—z Se Het : 23S Ld eat Se ales ees sae 1024. «3072 1024%* 1024%° eT ‘teas | MEER ees ie = ae KIS eG es 1024 a j was to be found. ¥ PRO- [ 86 ] PROPOSITION Y. Uppofing the Sum of the Infinite Series bx oxi tt k--2l k+-31 —- dx + —--ex +3 , Sc. when the Terms. thereof are refpectively divided by thofe of any arithmetical Progreffion, whofe common Difference is m, to be given ;, "tis propofed ta ! I Jind the Sum of the Infinite Series 2£2X4+°% 240,88. x bat P+0, 9-4-0, rio, &c. off atm Xx Ee Katim,Se. x ar ans atom Xx pice Xx el amntere: Xx gal p+axgimxrtm, &e. pH2mxgtomxrbLam, Fe. Se. fuppofing p, q, r, s, Bc. to reprefent any unequal Num- bers, anda, a, a, ec. any Numbers equal or unequal, and that the Number of Factors in the Numerator of each Term 2s lefs than: the Number of Factors imthe Denominator. bales ba oo dx tet Let the Values of the Series > tae too ane per det ?l EFo bec ext ty dx t2l goepbe eta” ee Gee Ea (which are fuppofed given). be denoted. by P, Q,R, &e.. re- Ee. &Se; / i I 4 : — 2-4, Fe. yy a—4.a—4.a—g, Fe, {pectively ;. and let A — oa Pa ae ee ic. Then it will appear from the foregoing Lemma, that. B A GP hte <. Toe Preragaahs Vane > &c. are the fimple Fraétions into which. ato.ato.eto, ore: “ppo.gto.rto, Chie. dent, that the firft Term of the propofed’ Series will be equal. k k Kk ea oR k to Abx Béx Céx €Fco. or Ab x rn Béx Re C bx : eFr. may be refolved; and therefore it is eviz Re Fee eee eee and in the fame Manner it will appear, that the fecond and 3 third: £87] kL] I | third Terms, &e, will be equal to Act) Beskt ext Ut, kp! kal 4 oy io €c. and AdSt?! | path? Caer. &c. &c, There- p+2m g+2m r2m fore the whole Series is 4 k k ps yk eee 2 OO eT Bea &e. p q r s Acngts Beet! eee! Dix! EF, = ptm qm rf m sem 4 Ade t21 Ba a ie Cd Bay ms Daxtt 21 &Fc pPozm gt2m rom. san? ; me Se. &e, &e, Se. : , k k+l kt 2} Which, becanfe the Value (A x > 4 & : . p ptm pt2m. exits! nrg &e.) of the firft Column towards the Left-Hand is Me = AP; of the fecond, = BQ, &c. will therefore become AP+BQ+CR-+DS, &. QO fed, COROLLARY £ r k k+l If the Series propofed be Bi. of en pe ‘ ; Pg. 7 Ae pfag+mirtm, Fe. S y.) ‘* , We aks ae &c. then will A be ==—___*___ p+2m.9+-2m.r--2m, Se. 9—P.1—p.5—p Se, B=——_! &c, and the Value of the Series. as a. p—7.1—9.5— 9, Ge. bove exhibited. COQ. [ 88 ] COR SLs hy IL a ki pp tm.pt2m, &e. , ce. and the Num- But if the Series propofed be okt! dxekt21 “p-Em pom .p+-3m i, p42m .pt3m.p--4m ber of Fagtors in the Denominator of each Term be denoted by z-+1; then g becoming =f-+-m, r==p+-2m, s=p+-3™, &Se. I I re (hallhave Aes ee ee U0 «211 3 At 20 ttle <7 M1. ZI. ZT. woe u—t X wt I I C—- te Ce o A= ss me 2111 Il Zh ono I= 2 K M2 I 2034.02 XM A n 2——1 2 t—t B= — —, C=— x A, D=—-— * x I I 2- I Z a A, €c. and therefore the whole Series n—tI t—I 4 —2 Xx S, ie. 2 P—2zQ+ ~ x — 2 R—-—x 2. I Ex «APP Led a 3 5 7 Where the Sum of the Series = — — + Se Se 2 Oe i a) (exprefling the Arch of a Circle, whofe Radius is 1, and Tan- gent x) being given, it is required to find the Sum of the Series 3 a x *° . ee. eG, En ths Cale: we have po=1 13.5 3-5+7 5°7°9° ogee 2 x? tg ~ qQ==3, T==5, U2, M=2, Be Sor ok “al Se. Q= — 3 5 oP 3 5 — 2+ 5, 8. (== La Bo Ee. x (=e) and therefore by Cor, II. Ce — ig) [ 89 | xP | Ps ee 4x3 P—2% A te = a Ebel ape" x Popes) equal to 3 8 a the Value fought EXAM PL EvwHl. ps a8 phe i Let the Sum of the Series x + = —. = + =, Gc. =the 3 / Hyp. Log. of /!-+41 be given, and let it be ela > to find : x3 6x -7 the Sum of the Series ae . os 5.7) iy 7-9 he Doak : ap &Se. oe Dele 2=3/8y "Pee, =? | Pog A fe ea po BH) =H 1, eC (ae eS) SH P—-aXr—@ oa Pa Pee pe a Soe ES rere, (=) a i ee #° and Rae 5 + 7 ae 9? &Se, + _— oe —, &c. (=AP+BQ+CR) equal to 7 —x3 ee “-; which therefore, when x1, will x“ become barely = 7 Pea Ost oo) |. 1.0 Ni VE Uppojing n—r to reprefent any whole pofitive Number ; ‘tis required to find the Sum of the Infinite Series fe 2px ve AML -P »p- pt. x ae a.n+t-1.n-+2.p.p-1.p pz. Seta re 1.2.7. ee — 1.2.3.r.r--1.7t2 ° I ae xt Vac ig ee ten eS Eee eSex aphere te ond 1.2.3-4.7.7-- 1 .r2z.rt 3 x denote any Quantities at pleafure. The Solution of tais Problem is eafily derived from Pro- pofition I. For the propofed Series may be aca as ge- Z erated [ 90 ] nerated by a refpective Multiplication of the Terms of 14x! expanded in a Series, by thofe of the Series 1, nom y EI 2 yy BTI Y 242 ef. therefore if the Diffe- Peoer rti? + r—-t r+z rences of thefe laft Quantities be continually taken, according to the Method there propofed; then the firft Difference of the firft Order being —= aon” “Of the eon =e f r r+ i CPi ee —1—2 ae oe of the third =“ — x S = rot B12 3. it is evident, from I r+z2 what is there proved, that the Sum of the whole propofed Series, putting ~—r==qg, will be truly defined by raped: —jp—1 Ss 2 = —}-2z 5 pie he G—1.p. ae a. 9b Tp a yes ppti.x?.17 x| Pe or — 1.r ‘ 1.2.re7-- Be Pp x I ri ae I yune ae 2 1 «| ro rt | I, z.r.r+t.tta| 4:9 VG 2p PTL PI-2. 25 ical 1.2.3.r7fr.rb2.t px" is a whole pofitive Number, will always terminate in g-- x Terms; and therefore in all fuch Cafes, its exa@t Value will from hence be obtained. One + , Sc. which, when g, of 7—r Note. When two Signs are prefixed to one Term, as above, the upper takes place when all the Signs of the propofed Se- ries are given affirmative ; but the lower when they are given + and — alternately. C O- for] Cm RO LIA RY: AL * Therefore if p be taken = r, we fhall have 1 +--+ = a = pcabuate geo gb te, + SE ; ° + td tho rer i. 9:4—1 .9—2.%3 G:9—1-9—2.9—3-%* er ee Ps; Se. oo rrr rp2.1La| rerf2sr P2731 x COROLLARY But if p be taken =, and r=1, then we fhall: have 2 2 —— eee n™ rte — ius x eo? i ‘ n+2| x x3 is 4 4 9 a eioils 7 I ae ate tis Oe 7h 31 x xt, Se. 2e-—=—> inte dt mugs eas, Ee 4 9 26 ix I z, 2 .N— 2.0 —1 es oot 2. N—3 2? iB mm h e = x px ee ay o 1.4.9 7 ri n.N—A 1 2 —4 2 =—G xt 1.4.9.16 1 x| CLO R, Oo DAR Y ally Alfo from the foregoing ae py the Sum of any Et um p Infinite Series as 1+ = x = += x aie * i x ame ae nlm n--2m ? pty p2v 3 th x pg tae 7 ae ee Sceoh Wher "=" is 4 whole pofitive Number may be eafily derived: For 2 iL ; ° ° a Xx mn this Series may be changed to 1 += x e KF 3 e é ei} I 7m co : p L 92 | B , = nt Z | F a Pe | — X , Sc, and therefore by writing 4 — and — inftead # wm m I of p, 2 and r in that Exprefiion, we ‘hall have eh x Ta I + eee PD Se > PB VE Te He jx RT Oy 2 eae a 7-9 — .9—~2m.p. pHv.pt2zv.x , Be et : a r.rtm. PHP 2M.V.29.30.7 7 x| eee — ae a Te: nn} mp .p+-V.x? Bi 1.utem.nti2m.p.ptv.pt2v.x3 ar vs rhm.v U.2V a rirbm.rt2m.v.2v.3u , ‘ EXAM BD ee Let it be required to find the Sum of the Infinite Series I+ 27 x GES X20 tae xO x ox3, &e, or Kiel ax ~+ 947-1643, &c. then by comparing this Series with that m Corollary; TI. we. fhall ‘have «7’== 2, and confequently ae pte aimee ete equal to the Value fought. I—.+| [te Oe I—-x| EX AVE RSL E abe Let it be Das to sb the Sum of the Infinite Series f.10%* .10.13 2+ 1.3.5-10.13.16 2° i ree Be aes ae 2:4.6.4.7.10 35-7: 10-13.16.1 19% TH ie AGS 8A Fadia d we rae es Bee end 9 (2 —7~) —xOM we Fall, , Se. then p being =1, v==2, n=10, by fubflituting thefe Values in Corollary III. have o = x I eee : 6.1 z* 6.3/1 28 z* 1 — x ST pea aie — a x I 4.2 Iteg 4.7.2.4 1tx?| Tele coe zt 2 ze Let op a 8. Se forthe Valuesthat was to PR;O- Bos] P RG OS Ter Tt OM vil TO determine the Sum of any Infinite Series, as 1 +- aaa x n.n+-1.p.pHt 4 nnt-i.nt2z.p.pti pz ee ees ia rir t2z.v.0+1.0--2 * Se. where (v) one of the two Divifors t, v, 1s a whole pofitive Number, and the Difference (n—r) between the other (rt) -and one of the Multiplyers (n) a whole pofitive Number alfa. Dae ere eae era x fT db x I 2 v sat st d=e, a xe= oa x fag 3 x g==h, Ge. and let S=1+ ~F xx ee x x7, Fc. and then, the whole - Equation being multiplied by dx*~', we fhall have dS x¥—" ee n vi, mntt a.nternat2 | ied gE ae tC Fe te te ee te eT where it is evident that dx—?, ex¥, fxYt', gavt?, &e, ex- prefs the Terms that remain of the Binomial Pag a oe expanded, after the v—1 firft Terms are taken away. Where- fore, to deduce the true Value of the Series dxv-'t +- “ x ex’, €&c. (according to Prop. I.) let the v—-1 Terms of the Binomial, which are not above expreffed, be denoted by A+Bxe+Cx?+Dx3..... bx¥—3- ¢x"—2, and let -the. Series 2: I 2 ° d x¥—* 4“ x ext x fxvt!, &c,. be continued down- wards by the fame Law that it is continued upwards, fo that all thofe Terms may be taken in; and then, from fuch Continua- : : ° poammyndt r—v-}-2 r—v4-3 Who sae tion, there will arife Ee ok: ee a [ 94 ] t—Z2 Pom I ‘omen J A seco ~ pee * ey x b xXVv—3 - as x C xv Zz ‘ equal to O24 OR rer. Oc e.... r— I = r—v-t r—v+z r—vu-+-3 V—Z past tags SG ea pave ce. Wee pe mr X ¢x—?, by writing Q= ao coe a; T— TI] Hi ae 2 Which being therefore added to both fides of the 2 Equation, we fhall have dS xv! +. Qx A+ — i 2—v1 n—v+tLz2 6 n—v-+-1 + pak eae Cx, Sc. =QA ere x QBx ~f- st x —_ x CR. eh — X CXVO2Z Ld yv—t Ee et aE x fx*t!, Bc, which laft Expreffion, on the Right-hand-fide, may be confidered as generated by a re- {pective Multiplication of the Terms of the two following Lines Raee Bx Se me) 5 Cag meee Oe et) he ee, whereof the former is the Binomial 1—x' ?—"T” expanded, and the latter one regular Infinite Series, continued through- out by the fame Law. Let, therefore, the Differences of the Terms of this laft Series be continually taken (according to the forementioned Proepofition) and then, the firft Difference of the firft Order being —T x Q, of the fecond, —vv Ym P , Hf mF memen F Tedieeet 0 OF wethall die Se a, 1—-+| mr Be Nm! , t——p— J C x? oo ee Xe ements. * ae &e. equal we Pees &c, continued to v—1 Factors, in which laft nwt} p— p—v+-3” Value the Sign — prevails only when the Signs of the propo- fed Series are +- and — alternately, ana @, at the (ame tinic, is an even Number. OR OLE A Ry 2. Moreover from hence the Sum of any Infinite Series as ot ES a : A es in pe Saks Ni A A ha tl ae rerm.v.e-p-w nn. rt2m.v .vw.u-b2w I—wr : x3, €c. where = and —— are whole pofitive Numbers, may be eafily derived; for me this Series may be reduced to 2 a p ee — 414 —.—-1 pe vs Se x. Ae Bic let, —, =, ae we ae 2a ea : m° mm? w mH ; UW Ww and — be ick — for 2, r, p and v, in the preceding Corollary, and let »—r=g, r+-m— — = 5, p—vtw=t, and 0 — —— ce ks then i wall bx Twi sce bx ™ get * q-g—-m.t.t-bw cS 7 41S Ss + een Oe ee ee 5. I=* s.sbm.w.2uy I= Isp x| a w.t2w x3 ——- k 9-J—M.9—2M.t £~-W.t- eS é Silhewe SE Sg ne ere ide w fore $8. sf 2m.).2W. 3 | . is=x]” Se. bx oe 5.W ge ap. See Lint ay. Fah ezine tt Pew x x3, Be. s.stm.w.2w 5.5m. 5s4-2m.W.2W.3W be the true Value required ; where the firft Series is to be con- tinued till it terminates, and the fecond to ——~ Terms (in Cafe W Wis it does not terminate before) 4 being equal to = x 2W.s--m 3w.st2m ° a iy ee ° ee ‘ae &c. continued to —— Factors, in which Value, the Sign — obtains only when the Signs of the propofed Series change alternately, and.— is, at the fame time, an éven Number. EXAMPLE, Let it be required to find the Sum of the Infinite Series 2.10 2.3.10.13 2.3.4.10.13.16 2 x —_—=_—_— 2 SJ 3 - f.32 1.2.1Z.15 ma 1.2.3.12.15.18 x x8, Oe Then, by comparing this Series with that in the laft Corollary, we fhall have 2==2, rt, p10, v= 12, m—=1, and w——3; therefore — = 4, and pofitive Numbers is an Indication that the Series is exactly fu- mible: Therefore let all thefe feveral Values be now fubfti- tuted above, and we fhall have ga=31, so==—2, f==1, k= —1, {|= I—p == 1; which two laft being whole ve 6X— 8ix—3 and h (— 3 mod x XC) —*t; and therefore ese IX—I 4X0 7X1 7 7x 1-bals 3 ia es 2 x6 x , 1.1 a x Six me Td: alge 7 ee ——2.3 I “ —2.3 1403 X1-+4]3 dae ees is the true Value required. 14% 4 Bb E X- [ 98 ] EXAMPLE It. Where it’ is propofed to find ue Sum of the Series ot 3°5 Ese 504 (587) 5550 eat 9 as 6 eee ae ee %e Pabeeeke pas 5-6.5.7-9-11 Cea rf : tos am a caus yt, Gc. This Series may be re- e— x @ 3 or Bee as ee gh Ye x ake aor: tO 1.6 a “1.2.6.8 arte a x I+ ioxx pote x x?, Gc. by writing x = —_. Therefore, by comparing this a nitertle 2, We, we have 1=3, r= 1, p= 5 2X——I o—-6, wil, w—=2, @==2, So —-15. SI, p=. b( x IX! Xo 8x0 8xoXxx XI x 4 ) = * , and therefore ex ee ee 2X3 6 6X 1—-4|z —Loee —_ 2XIX1X3 me 8xoXxx eae IXI1 eet ee a a ee eee , —IXOX2X4 yx” ee a moe ep 8 xx -b ae ae ce x x*, &e. which laft Equation I—x|2 being multiplied me and the Value of x reftored, we fhall have ee as a > —"y, &c. which was to be found. J z Rh ore In this Example (s-+-m) one of the Factors in the Value of being = 0, it may feem, at firft Sight, as if both the Expreffions multiplied by 4 would intirely vanith ; but upon confidering that in the former of thefe Es ae , and a —o; and therefore P= : brn a—bh xn a nearly. CASE [ roo | CASE QW. HE Series propounded being as above, let eo wt ——7 7p," be affumed as nearly equal thereto; then, by proceeding as in the laft Cafe, we fhall have ax porn. eat te: & ec.) € pe gPx@t® 4p Px™t2" e.$ = 0; whence, by wax Ant * comparing the like Terms, P===-, A= 4—-—- ~, and there- ™ ac m--n : pen oe m-+-n ax” b——— x ae fore a ae) — =e 1 . x ab+bb—acX x® ee eee e bmme xe © ALA SE, ae Pe poe Qn Bech TP pict ge. (the Series firft propounded) then by folowmne the ee Method of Operation, there will | BT ot past be affumed as nearly equal to axe ad bd--co bxac—bb+axbo—ad will come out “<5 ——5 = P, =O: ar ®) n Oe a a == A, and therefore pos osm axac—bbxXx"taxbc—ad+bxac—bb Ke a. at ax ™+b-LaP Xx apn ee aman Sew e? Se ac—bbh+bc—ad X x"4-bd—ce x xn 1 Pa®+Qx7n where P and Q are as above fpecified, C O- [ ror ] COROLLARY: Hence it appears, that the true Value of the Series ax" jot th et ae "E80. is nearly equal to a more nearly equal to x™ x chante and {till nearer equal to omowmel XX amt ibaPxxit® op Zeambbx x axbomad bxae—bbx xT ™ "ap Px Qe ac) b4-bc—ad dx 41bd—ce x x20 E-X Ae ie TF: Let the Series x 7 + = — — =e exprefiing the length of the Arc whofe Radius is 1 and Tangent x, be propofed ; then, in this Cafe, m being == 1, m==2, a==1, b=—-, c= -, &c. the Value of the Series, by writing thefe Values in the fecond of the foregoing Expreffions, will come out equal to sae oe é 15-+4x7 (2x8) ex Bee nearly, 3 EXAMPLE UH. mI f 7 t— a x 3 xt, Bo, 4. 3 to be the Series propounded ; ai a bees 1 Bie =e ca x ae &c, the Value of the Series will be 2 2 i Suppofe x more nearly, n 2 ~ + ot z xt-Z x nearly, o1 a Xx A——Z 2 x ~ 6 Cé i Xe ig 2 ——_ feee | EM AM PG Boll. a x? x? Laftly, let — ge a aa Soa {2 2.9.4 Zz. 3 shh oO 2.3405 6.798 exprefiing the verfed Sine of the Arc x to the Radius 1, be propofed; then the Value thereof, by proceeding as above, ‘will Came Guan foe nearke. Wr x 2 602%? Y> 2 __153120—600.2 "15120-+. 660x7--13 x* more nearly. Of the Roots of Equations by Approximation. Let. there be given the Equation ax+-bx*+-cx3-++-dx4, Ge. =Y, where ax-+bx? +cx3-+-dx*t, Gc. reprefents any Infi- mite or _conterging Serzes ; to find the Value of x. T appears from the preceding Pages, that 9 Value of the propofed Series will be nearly defined by — but more nearly by tee and {till nearer ey ue 5 wherein P and Q ftand for! ; atid 77 tefpectively ; ; there- fore, if thefe three Ee afeGions be. a fucceffively, equal to y, we fhall have, firft, a’x—=ay—byx; and therefore c=, = se ape eh Secondly abx-+ bb—ac:x x*= a by—cyx, or b—tix tax +9 = y; whence, by putting b—< =A, and = + > =B, it will be Ax?+2Bx —as ee ot = x3 Laftly 2z—lI NI—2Z — x 2, 3 foo x So? cae which Values being fubftituted above, \z2 t=, bl a—t ae 3x4 Fe, ==; thea will, a1, oo ra? we thall have A", Bm tee ky Cool hee Gr a 2 12 > D—2t-2 “J, and x equal to oe —> nearly, but more nearly 4 I ao —y equal to ye a and {till nearer equal to JR 4+5—2. Hence we Wc a ready Method for finding the Cube, Biqua- drate, &c. Root of any given Quantity ; for let Q reprefent that Quantity, and let & be taken nearly equal to the required Root thereof, and let 2 be the given Index, that is, if the Cube-Root be required, let 2 be 3, if the Biquadrate, 4, &c. and let the true Root be ses by 4++-z, that is, let ae k+2z) be= a: Therefore I +5 io res we haver-+m x 7 and, by won x28 = es Q. there- 4 ay u—l z a—I Nome Z x3 _ QF fore oe ee, ease at te Pe e a £1 and, by putting 7 oan; CNEL ae a = 3, we have x + —— [ 165 ] x3, Ge, the very fame as above, and conf{v- yeep et tom? quently the Value of x is as there determined; whence the required Value of £+-kx(k+-2) is alfo given. & | BXAMPLE ‘I. i et it be required to find the length (z) of the circular Are, whofe Radius is 1, and verfed Sine ~: Here, by writing ~ inftead of y, in Corollary I. we fhall havé z equal to a :. > bs Ps Co Sidi ee ae (v o SY, 29 AD ad oe ) 1.0472 nearly, or 9 ( Se ¥ 723 iD 1.047198 more nearly, There- fore, fince the Arc, whofe verféd Sine is , is equal to + of the Semicircle, it is’ manifeft that the length of the whole Semicircle, according to the foregoing Numbers, ought to be 3.1416 nearly, or 3.141594 more nearly, Now the true length. of the Semicircle is known to be 3-1415926, &e, therefore the Error in the former of thefe Values, is lefs than —— Part of the whole, and in the latter lefs than — 40000 , é / 2000006 Part. And if the verfed Sine firft taken had been lefs, as for Inftance, that of 15° Degrees (== 1 — ,/ 3 — ed the Con= clufions would, ftill, have been much more exact, and true, at leaft, to g or 10 Places,. whichis { very near, that I believe it fearcely poffible to find out more ealy and exact Approxi- mations for the Arc of a’ Citcle, than thofe above given, Dd E X- ee ee f £06) EXAMPLE IL Where fuppofing the Value of an Annuity for 10 Years to be 8 Years Purchafe, ’tis required to find the ‘Rate of Intereft. By comparing the Values here given with thofe in Coro}. II. we have 2==10, s==8, therefore y (2) rr © Ce aXn+t-1 110? 15 sty el Pate 9) S41) (Eg ames) Ue 228 pe 20 ee 550” 2 5 7 act eh 2! en ON aes aie 9 mop) Cas Tee and confequently 1 —G + Jf BY ee == 1.042775 for the Rate of Intereft required very nearly. Now according to Dr. Halley’s Theorem, the Rate will be 1.042798, which is alfo very near the Truth, but not fo ex- act as the former, which is right in all its Places. EXAM PLE AAMT. Where it is required to extract the Cube-Root of 10. Be- caufe it appears that the Root required is a little greater than 2, let the Value thereof be reprefented by 2-++2z, or the Va- lue of 4, in Corol. III. be taken =-2; then, being = 3, y will be =—, C= and D=™%®; therefore (by the firft 12 72 48 of the Approximations there given) x will come out (-—) = .0768, &e. nearly ; or (by the laft) equal to (\/ 3 72 15 FOU. 14 077217 ftill nearer: Hence (£+-2x) the Value fought, will be 2.154, of 2.154434 more nearly. EXAMPLE: Ty. Let it be required to extra& the firft furfolid Root of 125000. Here, as the required Root appears, by Infpection, to be fome- thing greater than 10, let the fame be denoted by 10 +2; that { 107 ] | that is, let 1o-+- zl == 125000; then, by proceeding as in the laft Example, we aihall Have-y = 0.05, A==r, Bi=-o.82r, from whence, by the fecond Theorem (in Corol. III.) the re- quired Value will come out 10.45636, which is very near the Truth, but if the laft Theorem had been ufed, the Anfwer would have, ftill, been more exact. EXAMPLE-V. Where there is given the Equation 23-+ 2*-+ 2 = 90; to find the Root thereof. Since the Value of z, it is eafy to per= ceive, is not much greater than 4, let it be denoted by 4-+-x, and let this Value be fubftituted inftead of z in the given Equation, and it will become 84 + 57x + 13 x*-++ x3== 90, or 57x -+ 13 x?--x3=6; which being compared with the ge- neral Equation ax-+-6.x?-+-cx3-+-dx*, Gc. = y, we thence have a==57, 6—=13, c—=1, do, Ge. and y==6; where- fore, by the firft Approximation, the Value of x (=a45 ) will be 0.1028, and therefore that of = == 4.1028 nearly : But if a greater Degree of Exadtnefs be defired, then, according to the laft of the three Approximations, laid down in the ge- ° bc— bd— neral Propofition, we fhall have P (— =) =— -*,Q pee j aca-bb ° ° a—P , =, C (6+aP—Qy) = 7, D (<=) = 29 and 2 112 therefore x (/f e + 4 — z) = 0.10283235; which is true to the laft Place. EXAMPLE VL Let 300% — 23 be given = 1000; to find a Value of z. Here, as it appears by Infpection, that 3002— 23, when %==3, will be lefs, and when z==4, greater than 1000, let z be put = 3.5-+x, and then by writing this Value inftead of [ ro8 ] of 2 in the given Equation, we fhall have 263.2gx-—10,¢x* —X3=2— 7.125, or 2106n—84x*—8 x3 == — 57; therefore a being here = 2106, 6==— 84, and y=_— 57, we have y ee 1B ) ==—.002703633,and confequently z == 3.472963 very nearly. Note. When the Root of any high Equation is fought ac- cording to this Method, it will be convenient, and fhorten the Operation very much, to negle& all the Powers of the con- verging Quantity x, which, in fubftituting for the true Root (2) would rife higher than the 2d, 3d, or 4th Dimenfion, according as you would work by the rit, 2d, or 3d Theorem, or as a leffer or greater Degree of Accuracy is required. Note alfo, That if, after the Value of the Root is once ap= proximated, a greater Exactnefs be {till deemed neceffary, the Operation may be repeated till you arrive as near the: Truth as you defire, as will appear from the following, BX. A MLB LBs oVEb Wherein. 25+-2 24-4 323-+-42"+-5 2 being given: == 54329 ; ’tis required to find the. Value of z, according to the firft. Ap- proximation. Here, becaufe it. is eafy. to perceive that. & is greater than 8, and lefs than 9, write 8-++x—2.,; and then by involving 8+-x, and negleting all the Powers. of x above the ad, we fhall have 25=32768-+ 20480x-+-5120.%%, 24==4096 +-2048x4-384.x, B3==f 12+ 192X4-24.K7, B?—64-+ 1OX-EX?, and therefore 42792-+ 25221x-- §904.x*== 54321, that~is, 25221K-+ 6964x*== 11529; which, by ftriking off two Fi- gures in each Term (to fhorten the Operation). will be 252 x 2 , eo -+ 59x*== 115 nearly, confequently x (- ries ety ae ) == 0.41, and =58.41 nearly, Let, therefore, 8.41-+x be I . now [ 109 ] now affumed =z; then, by repeating the Operation, we fhall have 30479x-+ 6876x7==13¢.92 : whence (according to the forefaid Theorem) we have ere 479 X 135.92 30479] +6876 135.92 (a) \ == .004454 for a new Value of x; which, therefore, added to 8.41, gives 8.414454, equal to the true Value of = very nearly. “pet tlie gietB eg e Of the Arzas of Curvus, &c. by Approximation. roR O 2 OS 01 gay wT. Suppofing abc to bea {mall Portion of any Curve ab fi, and A a, Bb, Cc, three equidiftant Ordinates ; to find an Expreffion in Terms of thofe Ordinates, and the common Diftance A B, that Shall nearly exhibit the included Area AC cbaA. | ET acommon Parabola, having its Axis parallel to the given Ordinates, be defcribed thro’ the three Points a,b,c, of the propofed Curve, or rather, to avoid confufing the F igure, let that Curve itfelfreprefent a Portion of fucha Parabola; join A and C with a Right-Line, and make S$ 6T parallel thereto, produ- cing Ag, and Ce.to meet $4'T in S and T, and drawing vm from any Point v, in the Parabola, parallel to AS. Then vm, by the Pro- perty of the Parabola be- ing to Sa, as bm, to OS, or, in the duplicate Ratio | of 6m, the Space 42S b, included by the Parabola and the Right-Lines Sz, A oR ee D-H -R 6G aie and { 10 | and §4, will be 5 of the Parallelogram braSd, for the fame Reafon that a Pyramid, whofe Sections made by a Plane pa- rallel to the Bafe, are in a duplicate Ratio of their Diftances from the Vertex, is known to be - of its circumfcribing Prifm. Wherefore, feeing Bd x 2AB1s equal tothe AreaofACT4SA, and Aa--Cc x AB to thatof ACcraA, the former of thefe Quantities muft exceed the Parabolic Area ACcdaA by jutt half what the latter wants of it; and therefore twice the former added to once the latter, will be juft three times this Area, and confe- quently the Area itfelf equal to Aen eres x AB; which Quantity, fince a Parabola admits of infinite Variation of Cur- vature, fo as to nearly coincide with any Curve for a {mall Di- ftance, muft be equal alfo to the Area fought very nearly. Q.E, I. COROLLARY. Hence may the Area of the whole Curve be alfo nearly found ; for let the Abfcifla be divided into any even Num- ber of equal Parts, at the Points B, C, D, &c. according as a leffer or greater Degree of Accuracy is required, and let Bd, Cc, Dd, &c. be Ordinates to the Curve at thofe Points; then, for the fame Reafon that Aer seres x AB is the Area of ACcéaA, will ee rater es x CD, be the Area of CEedcC, wad feratseee x EF, that of EGgfeE, @c. &c. But the Sum of all thefe Areas taken together, or AB Aa+4Bb-+-2Cc-+-4Dd+2Ee+-4F/+2Gg, Ge. is the Area of the whole Curve: Hence it appears, that if to four times the Sum of the 2d, 4th, and 6th Ordinates, &c. be added the Double of all the reft, but the firft and laft, and the Sum be in- F223: | increafed by thefe two fingle Ordinates, and multiply’d by = of the common Diftance, the Produét will be the Area fought, | Bae PL: Bi. Suppofing AQ a@ to be a Quadrant of a Circle, whofe Ra- dius AQ is 8, and Aa, B4, Cc, Dd, Ee, five Ordinates thereto, whofe common Diftance AB is Unity; to find the Area AaceEA. Here, by the Property of the 7 ee Curve, Aa being = 8, Bd=— ap OR se 7.09726. eg == ar 00 sae. 74 500, Dd==,/ 5 6a A162, He 7 ae = 6.9282, we «havé 4.x Bb-+-Dd-+-2Cc-+Aa+Ee x a == 30.6113; which, by the foregoing Corollary, is the Area a Ep oe fought. From whence the Area of the whole Quadrant may be eafily found ; for, taking 13.8564 (= Ee x 3 AE) the A- rea of the Triangle AcE, from 30.6113 there remains 16.7549 for the Area of the Sector Aae A; the treble whereof, fince AE is=EQ, will be the Area of the whole Quadrant, which therefore is to its circumfcribing Square, as 0.78538, &c, to 1’ nearly, the fame as it is known to be‘by other Methods. Note. When the Area of any Part of a Curve near the Vertex, where the Ordinates are very oblique to the Curve, is propofed to be found, the Solution by this Method will be the leaft exact: Therefore, in all fuch Cafes it will be conve- 3 nient [ xx2 | nient, inftead of fuch Ordinates, to make ufe of Lines parallel to the Axis, as in the following. WX AW PoE AL There abcde being a Semi-Hyperbola, whofe Abfcifla Aad is 10, Ordinate AE 20, and Semi-Tranfverfe Oa 20; tis required to find an Expreflion for the Area AgcE A, in Numbers, that fhall be true, at leaft, to 3 or 4 Places. Sp CO p Firft, I fuppofe AE divided | : into 4 equal Parts, at the Points B,C, and D, and B4,Cc, Dd, &e. parallel to the Axis AO, produced to meet. the. conjugate Axis of the Hyperbola in the Points P, Q, R, S. Therefore, by the Property of the Curve, AE? (400) : E S*—— Oa? (500) 7: OP* (25) : P#—Oa?:: OQ? (100): Qc —Oa?:: OR? (225): R#—Oa?; whence Pb==20,766,Q¢= 22.913, ED G B A = Rd=26200; therefore Aa = 10, Bb= 9.234, Cc=7.087, Dd== 3.900, Ee = 0; and con- fequently 4 x 9.234-+ 3.900-+ 2 x 7.087 + 10-40. into 2 = 127.8 equal to the Area fought. EXAMPLES TH. Suppofe EG DHLE to be a Solid, generated by the Ro- tation of any Conic-Section about its Axis DA, vzz. either a Cone, Sphere, Spheroid, or Conoid ; and let the Content of any Frutum EF GHK LE of that Solid be required. Here if p be put for the Area of a Circle, whofe Diameter is Unity, and a Curve as a 4c be fuppofed, whofe Ordinates Ad, [exa3°] Aa, Bd, &c. fhall every where be as the Areas p x EL?, p x FK?, &c. of the correfponding Sec- tions, then the Area of that Curve will, it is manifeft, be as the required Content of the pro- pofed Fruftum : But this Curve is always a Portion of the com- mon Parabola, except in the pa- rabolic Conoid, ‘where it dege- nerates to a Right-line, and therefore its Area, fuppofing AB —BC, will be, exactly, equal to Aa+4B6+Cc x AC, and con- fequently the Content of the Fruftum, equal to HE L?-+-4FK’ + GH? x zp x AC; which is therefore to EL? x p x AC, the Content of the cireum{cribing Cylinder, as E L?+-4 FK?-+-+-G H? to 6EL*. This Propor- tion, if AC be, fuppofed = DC, and the whole Solid EGDHLE be taken, will become as EL?---4GH to 6EL?; where 4 G H? is equal to EL?, 2EL’, or 3 EL’, according as the Solid is a Cone, parabolic Conoid, or Semi-fpheroid. Hence it appears that a Cone, a parabolic Conoid, a Semi- fpheroid and a Cylinder, having the fame common Bafe and Altitude, are to one another as 2, 3, 4.and 6 refpectively. \ [ x14 ] Le MMA; If in any Series of Quantities a, b, c, d, e, &c. there be taken A==b—a, B==c—z2b--a, D=d—3c-+3b—a, E=e— 4d +-6c—4b--a, F=-f—se+-1od—1o0c-++-sb—a, &e. fo that the Uncie of the Values of A, B, C, D, &c. may be thofe of a Binomial raifed to the 1ft, 2d, ad, Ath, Be, Powers; fay, the Value of any Term in that Series, oe va a Se the jirft is denoted we Rm i ne Se, For, fince ba is =A, ¢c—25-+-a=>B, &e. we hall, by Tranfpofition, have b—=-a-+aA ¢cm2b6—a-+-bB d=3¢—36+a+C em4d—6c+4b—-a+D fF =5e—i0d+10c—sb+a-+E, Wo. 4 &e, EF. where, by taking the Value of 4, as found in the firft E Equa- tion, abd fubftituting it in the reft, there comes out cap at2A+sB d= 3¢—3A—2 2a+C e=4d—bo6c¢ +4A+4 34+ D Me od cea ee Sc. ce: Cen in which the Value of c, here found, being fubftituted, we next have a= tars | = @+3A+3B+C e =4d—3a —8A—6B+D J. =5e—i10d+6a+i15A+10B+E, Se. Se, Se. In like manner the Values of ¢, f, &c. are found to be a+-4A+-6B+4C+-Dand a+-5A-+-10B-++-10C+5D-+E, &c. ‘Where the Unciz, in the Value of each of the Terms 4, c, d, &c. are, it is manifeft, thofe of a Binomial raifed to that Power, whofe Index is equal to the Number denoting the Diftance of that Term from the firft in the Series ; therefore the Value of that Term, whofe Diftance from the firft is de- noted by 7, will be o+-n No x = x BY Ge. Os D: a Re OOS a1 Tt Onn ale Suppojng abcde, Gc. to be a Curve of any kind, and Aa, Bb, Cc, Dd, Ee, &c. given Ordinates thereto, at equal Difiances, but not very far from each other ; to approximate the Area of the Curve by means of thofe Ordinates: Ec Ags, Bb). f gq Cec: dsr. 2 = a A Bese C, Seve. and C Pare Y the number of given Or- dinates, Aa, Bd, &c. e- qual to x-++1; putting b—a=A, c—2b--a==B, d—3¢+-3b—a=C, e— 4.4d+-6 c—46+-a—=D, &c, Then that Ordinate, whofe Place from the firft eG Dp eR OG Diftance is denoted by ”, or whofe [ 116 | Diftance from the firft is 2 times the coreg Ditence, will, by the foregoing Lemma, be a+nA+-2x B--n x oe CE aS “xt eee es Ge. Wherefore. if 2p, Zz 3 the Ditlanee of this nas . the Point A, be put = x, and — me, Ce hei above inftead of its equal (7) we fhall have a4 ; CS ae = x = Lf om x rae &c, equal to that Ordinate, ce correfponding Abfcifla is x; which reduced to fimple Terms, will be a+=* She a bio : ; Cx3 Cx? Cx Dx* Dx3 11 Dx? Dx Pep ae ap zag ae $e 4g ea! €3c. Hence it is manifeft that 2 — S — a ce, == y, is the Equation of a arene Cartes which, be- ing icicribed to the Abfcifla A G, will pafs thro’ all the gI- ven Points a, 4, ¢, 4, &c. ‘Pherefore the Area of this Curve, hich: by the common Methods is found to be x x Ax Bx” Be Cz Cx* Cx Det Cora k 6p° 4p + 2463 6p? 6p al “T20p* een RT CC ee Dx} 11 Dx? PSP Ex> ee ee ae met + SEPT ca &c. mut be equal, very nearly, to the Area of the propofed Curve AadcdgG. QE. 1. COR OLEARY. Hence, if x be = 2, or the Ordinates given be only a, 4, and ¢; then A being = ae B=c—2 6-+-4, C=o, &e. the included Area x x ot a ai &§c. will be a eK But, if'x=23 p, or 4 Diodes a,b, cand d be eel C will be 0B ES c-+36—a, D=o, Ge. and therefore the included 3 Area Ltt7 i Area, equal to seh ies x x : Moreover, if 5,6, 7, Sc. be the Number of Ordinates given, then the required Area, by proceeding in the fame manner, will come out yaqeg2b4i2ef32db7e ) 4 Wet 7s obsoctsodt se riot > > 90 288 7 7+. 216 and, *tee2ter are h 2728 per BT EE X x, Ge. relpec- 840 tively; where x, in each Cafe, denotes the Diftance of the firft and laft Ordinates.. Note. When the Equation of the given Curve is compre- hended in this Form, viz. y =3 Q+ Rx + Sx?-+-Tx3, Ge. where Q,R, $8, &c. may fignity any determinate Quantities what- ever, the Curve defcribed' thro’: the given Points a, 6, c, &e. (as above) will be the very Curve given: Therefore, in this Cafe, the required Area may be exactly had, by making ufe only of as many Ordinates as there ate “Terms-in the Value of y, or as there are are Units in the Index of the higheft Power of x in that Equation, inéreafed by one. EXAMPLE 1. Let O be the Centre, 2 the Vertex, and OD an Affymp- tote of the equilateral Hyperbola abcd; and, fuppofing OA, Aa, and AD, each equal to Unity; let it be required to find the Area comprehended by the Curve, the Aflymptote, and the Ordinates:A@ and Dd. Here, if only three Ordinates be’ ufed, then @ being = 1, b= .6666, c==.5, by the Property of the Curve, we fhall have ae x1==.6944, &c. for the Area fought. But if four Gg ) Ordi- [ 118 ] oS be taken, then @ being == 1, d= 75) 6 oe 0, dm SEREIEA 1 = 6937, Be. will be the Area fought ; A RIB CG OD which Value is fomething nearer than the former, the true Area being .69314, &c. EXAMPLE II. The fame being fuppofed as in the preceding Example, ’tis required to find the length of the Arch ad. Let RP (y) be any Ordinate to the Hyperbola, O R (x) its correfponding Abfciffla, and Rg==x; then y being = + ~ by the Property of the Curve, we fhall have y= == and. AEE aeeia . (Styx fit =; equal to the Fluxion of the Curve. Wherefore, if we now fuppofe another Curve HGd/m, whofe Ab- STs {cifla is x, and Ordinate ./ 1 + => the Meafure of the Area of this Curve, will, it is manifeft, alfo exprefs the Length of the required Arch ad, the Fluxions of both being the fame. Hence, if AD be divided, as in the foregoing Ex- ample, into any Number of equal Parts (fuppofe 3) by the I rdi- { 119 ] Ordinates B 2, C/, &e. and the Values of thefe Ordinates be fubftituted in the foregoing Corollary, there will come out 1.134 for the Arch required. By proceeding in this Man- ner, to find a Curve whofe Area fhall be as the Value fought, not only the Lengths of Curves, but any other Quantities, whofe Fluxionsare given, may be approximated, even when the given Fluxions are fo complicated, as to render a Solution by Infinite Series very troublefome, if not impraéticable. Of Quapratures and the Comparifon of FiuuEnts. PROPOSEYVAIQON SE Uppofing 4+-cz" equal to x, I fay, the Fluent of ¢+-cz*|" x dzt"—' z, or the Area of the Curve, whofe Abfcifla is Soa ek n—tI : dx™ pn z, and Ordinate a-+e2l" x dz? » will be a cz" Mm. M—t een m.Mm—1.m—2 cf gin an — —— xX —— eee —_—" oes ———— LL IL en ae er eee M.M—1.m—2.m—3 ctzth bra ro era Nineteen ae ed x ar ee ES¢, ppt pep +3-2F4 a? For a-Fez"|” x dz?” —" 3 reduced to a Series, will become Seer ne m mM—I 2 wm M—l M—z2 Ane Sa oe ha be oF Sx ae Cae IG, ui— n Ma log WM .M—t! in of which the Fluent is dz” x pee -t~ a ae, aS Fs ——2 2 a M—1 n a” 77” , Gk m.m—t1 — ck i puton n : by 2 =—=2 2 2n a” 2 2% a. TP &c, (by the common Method) But the Series $2 > { 420 | Mt Mal on og tee, &e. is (by Corel. i. Prop. \. of the Summa- p I ‘ ; eat atex |” 27.6% tion of Series) equal to LEAR tae ee et | p ; Pi xXapex” Hl. ti—=1 Keer a . . v 5 “7 0 a ae a ee fo ey 6) ee pt aan ean? p i ue 1 ah ee OE me, me Tem Ses cog t a zen a ag tye ee CO ang thererore x p+ sae ptt. pz. p+3- eae? Fae WwW {pote d x? ae ee r/pd c x” Mm. I 4 Z Ca oC) re ~ < . ae denen eo 5 A eae Gy a , [ : Ne Se. eee ee, > 4 p p+i Site fo pn I p+! pe x Hr pes cage 5 io, EF c. OQ. Ff: D. EB Xe APM Poleky el; } ze, 20% Let it be required to find the Fluent. of a=ezl x mate or the Area of the Conic-Section, whofe Trantverfe is: a, Ce jugate 6, and Abfcifla z. ‘By comparing this with a-He2\" x dz!" the foregoing general Expreffton, we have === 1, ees I 3 Bit oe? = mA=o=I, a= Pp Bes = and therefore de Xx Ie 2 2 3a 5% a ite aes x” Fes Ze I : z ES wz” ee fae ie or ———— X¥ — == — — ee 5. 78 5 +7: 13 1.3.5% 3.5.7% 3 4 [ eee — ree OC, co (v) and all the reft as in the pre- ceding Corollaries, C/O RAO: LT ARR OY VAIL If c be negative and p, p+ v, m+-1 and tr 1 affirmative, or c, p, and p-+v affirmative, and m-—- p and I [ 129 ] m-+-p-+-r--v negative, and z be fuppofed to flow till 2+cz" becomes nothing or infinite, or tll pees the Ae of the whole Curve, whofe Ordinate is atoz| x dee yit is evident, from Corollary 1V. that the Area of the whole Curve, whofe Ordinate is a-Fez"\” err. geht tot lobe truly de- fi pb pti 3 pte m4+1 es aoe m+-3 ee eee menial: 5 x vps es =f a where s is = p-+-m-+1, and g==p+-m-t-v, and where v and r may reprefent any whole Numbers pofitive or negative, under the forementioned Limitations. ‘Therefore if 7 be bien =o, and c==—ZJ, then the Area of the whole Curve, whofe Ordinate. is a—bel" x dz? japde ae when the Values of b, p, and m-+-1 are all pofitive, will be equal to £ x ae. pi Pace = (v) x = : From whence it appears, that the Aten of rite whole iGaive, whofe Abfciffa is z, and Ordinate g—bz" xdze"—* x A+Bz°+C2"+D23", &e. will be truly reprefented by Q x 5 + -—- A a io. P ie 3 p-P+1-p+2-Dat ue ae cae , &c. where A, B, C, a ftand for any de- terminate uh atities, COROLLARY IX. Hence if ¢ be put to denote any Number at pleafure, and A+Be2+C2"-+D23", &c. be taken == 1—t/2 eo om aoe x [222% * xt a x [3338, ce. I+/z1, we " fhall then Sage Ane 1, B=—fi/, &c, and there- fore the Area of the whole Curve, whofe Sas is 7 abe x dab : x I+-/21, or ST , will be Kk Qx a—oz x dzit=! ———E = _ — ESET ASE a EI BEE ET Tit a —— g, ttt 7J> Rte Lee 2D. pte 3 O ea ams Fx Tr a oe ee ee Xs =< 122. Fa : 1.2.3.5.s+1. 5-2 é &e, ach, Value, if w be ah t—-s, will be truly ex- : bP eo ae A gee id prefled by == Qs __ wp pine: al oo We W—=- lp. ptt ” . b+al| are 1.5 b+al 2.5.54 er ae as ay a x teal? &ce. as appears trom. Propofition VL of the Summation of Series ; and there~ es fore in all Cafes, where w or ¢—=5 isa whole pofitive Num- ber, the’ Area from hence may be exactly obtained: And hence alfo may the Area of the whole Curve, whofe Ordinate is a—bz| x gd OTe kTx"/' ST eedanit k b : : reduced to &t x1 eel let be fubftituted inftead of J, in the forefaid general eeeiion’ and the whole be divided by ge zh? ; l Rk‘, and then we fhall have — Linto poe xiant , be eafily derived; for, fince 24-/z"| may be bk+ all? ins bktal ar ae W -B— lp. pti Pi a® ]* W. Wee l.W=—2.p.pt1.pt2 | 1265.51 Eads 3: 1.2.3.8.S5+-1.5-42 sil &c, for the true Area in this Cafe; s being (as before {pecified) =p-+m-+ 1. COROLLA, RY Xx. Alfo from hence the exact Area of the whole Curve, a—bn"| x gba! 7x"\* and ¢ are both whole pofitive Numbers, tho’ the latter fhould not happen to be the greater, as is required in the laft Co- whofe Ordinate 1s , may be deduced, when s rollary: For the Series Qx 1 — aaa x a Be ¢. (univerfally exprefling that Area) may in all fuch Cafes be fummed, and by [ 131 ] by Corol. I. Prop. VII. of the Summation of Series, is equal to eet a et x al a m.m—tL.t—1.t—2 all L.s-—2 b+al 12.52.53 ia aes 2 Jr o : 7 or ae ER eg PU EE 6+al| ala aaa .2.5——2.5— al z-» Gc. w and P being as hereunder fpecified: From which Area that of the Curve, whofe Ordinate is a—be| X att) bp Lal’ ceeding as in the laft Corollary, will come out as follows, ee a a POS bial apmtt al ek m.m—1.f—1.t—2 Pot s a i a Ee cae {135 ] S, will, in this Cafe, be = t.t+-1.f+2.7+3(w) x me nkt Ga=-T ropa] xx tex) Px xt Ne multiply’d by the Fluent Obes cas ee ee dp. Sz (w) 38 1.2.3.4 (w—1) Bt I W— 1X Wo] 0a ee W—1. — xe d D+-1.3 te 2.d-42.)5” 2.3.d-+3.33 , Se. taken in the forementioned Circumftance, when x from no- thine b at ing, is become = TT OX. A MP 1ok.T: ————_ i 2 x %% being given, ’tis required The Fluent (Q) of 4? 271 to find the Fluent of 32%? x 25%. I. : a : ° ST a Here, by comparing 2? 27|* x 25%, with ate2"| xdz "sz : ‘ I (Vid. Corol. I.) we have a0, c—=1, n=2, m—=-, d=, and en—1I or 2e—1==5, whence e-=3, v=2, os and atcz|” m1 of BU x dz et t (e+m) = Z and confequently — tcn =~i—t{ So a4 2 4 ac —* + QU x - x ~equalto the Value fought. But fince p is here a whole potitive Num- ber, the true Value may be had independant of Q, being (by eS i eee en— oe ee ae atez"\" 1 xdz E——I a re Coral. IIT.) equate tema ee icn Ge . "1°. : : Bios continued till it terminates; that is, equal to —t-_ 407% 4.26% ‘ Bape’ |* xX 15 x4 — 12.27 B74 854 ao 5x 5-3"? 105 |=-— E X- [ 136 } EXAMPLE IL ree ESS = ss Where it is propofed to find the Fluent of 63—x3| 7x *x. vie In this Cafe we have a==$3, c—=>—1, 2==*, N==3, M==— i I -, d==1,en—I=—=— >, whence Cah: and ¢ (=-e-+-m)==-—1; 3 2 which laft Value being a whole Negative, indicates that the required Fluent may be exactly found in finite Terms, There- fore let thefe feveral Values be now fubftituted in the fecond general Expreflion in Corollary III. ‘and we fhall have X2* * for the true Value fought. EXAMPLE Iii. — a HS, ; ‘ Let (Q) the Fluent of gt-+-x*\* x x 3 x be given; to AT B x Xx 5 a, i \& find the Fluent of g#+-x4 Thefe Expreffions being compared with thofe in Cor, II. A : I coe. WE have Gg = 8", c=, ai aes or 72 —4, d= r; m1 én shed I er y iN 13 atez | Xdz Nant; Peo) ee, ¢ = —~, and —— é { gins” gttxt|* xX x ta xt ’ — oe. —i4g* * te + ae * 8 > ee a 9 yx —& equal to the V . air Z ig equal to alue re quired. EAA MP LE {v. Where (Q) the Fluent of x3——B3|? x 24%, or the Area of a 2 the Curve, whofe Abfcifla is z, and Ordinate 23—43|* x 3 at aan. | z* being given ; “tis required to find the Fluent of z3—43|/3 x zz or the Area of the Curve whofe Abfciffa is alfo z, an as Ordinate 23—d3|? x 27°. By comparing thefe Expreffions with thofe in Corol. IV. 13 3 there will be cm=1, a—=—43, d=1, 2=3, m=, po, 3 11 I 12 2 U==2, r== 4, C= a fs go £=7F a FRR and therefore, by writing thefe Values in the laft of the E- is ° e rt oF pS ————SSSSSG quations there given, we fhall have = ** Bl x ieee il 1353 xz3—5)| ot. 13 X 100° Xe —e 13X10 x 789 ZI 21X18 ZUX18X15 ae 4 13X10X7X 45°? X x3 —h3] F ec Sarks AX7X10X13X5X8b77Q + ZAX2ZIXIS X15 X12 ee aN 9 _ QXIZ2X15X18X 21X24 for the true Fluent or Area fought ; which therefore, when z id Shr8 ea F\ is barely: te OA Ho CORTE K1IGR 13 K2UNEA EXAM BELLE V; P| Let there be given (Q) the Fluent of I—wxl* x? x, when t—x becomes = 0, or the Periphery of the Circle, whofe Diameter is Unity ; to find the exact Fluent of low 2 Uk, x *x, when 1—x becomes = 0; or the Area of the whole 7) . ar ee uz Curve whofe Ordinate is 1—x| *xx %, : I In’ this Cafe g==1) C== —1, 2 =x, 21, 2S 2 P= =; §(p+-m+1)—=1, g (p-+m-+-v) =v ; and therefore, by Corol. VII. we fhall have 1.3-5+7+9) fe. tov Factors, into 1.3.5.7.9, &c. to r Factors. 2.4.6.8.10.12.14.16,18.20.22, Ge. to rw Factors, the Value fought. x ©, efor Mm | Ek [ 138 ] EXAMPLE VI. The Fluent Q_of 4*-+ xl" * x x being given, ’tis propofed a to find the Fluent of 42?-- x7|7— > x x*x, when 47-4 x? be- comes infinite, or the Area of the whole Curve, whofe Or- eS dinate 1s 4?-+-x71 * x x4, Here we have a=h?, c=1, a 1==2, M==—-, d=, z ; whence, by oa VIII. there will be 4 x an - (2) x OF) x BE (a2) x 2 ee vitigc, 5 pr=p ’H=— 2, V=2, S= 2g=< ole Sa a ehh g—l Lag go 11 — ame (2)x = = 12g a. Ree x + x a, an to the exact Area which was to be found. Note. The Area of this fame Curve, or of any Part of it, may alfo be found by Corollary VI. EXAMPLE VIL Where eS the Area of the whole Curve, whofe Ordinate is fone parse being fuppofed given; tis required to find the a. Area of the whole Curve, whofe Ordinate is ——— 1 SSS tele Firft, to determine the Area of the Curve whofe Ordi- ——__—__—_—_—1 nate is #3—23|? x 23, which is requifite for finding that re- 3 quired, { 139 ] quired, let a-cz"|” x dz?" and ace tt » dzt?bor—' (as expreffed in Corol. VIII.) be compared with /3—<3]~ * and fi—zi|* x 23, and there will be a=—/3, co=—1, n=3, m=—~, m-r=-, d=1, pn—1=0, and pu-t-un—i=3 ; whence r=1, p=-, v=I, s (p--m--1) =, & (p+m-+v) 9 eS i4 pti m+ ree e. = %, and therefore ¢ * Pah }& chi ee (7) into aetr a , 3 of cj isequal to the exact Area of the Curve, whofe ; a 535 ae Ordinate is ‘Ag a x 23, Let thererore O.be pute— oi 5 fi —2z3 fie? x 2 xX 23 a—bz | xdzb™—! and lee be now compar’d with Bele _ ER (Vid. Corol, IX.) and we fhall have a==f3, 6-1, a= 3, Sees d=1, p=% kab, fe i ae Sites 8 3b 4b—t oe w (t—s) =o; and terete (2 el be-all, ° 6K ; aren oer =, is the true Value required. ss hXb fs EXAM Pt BE VAur. The fame being given as in the preceding Example, let it be required to find the exact Area of the whole Curve, whofe Ordinate is#—* ee Here, by proceeding as above, we ee no) thal hehaget. 2 ERS ae ets, for os exact Area EXIT X 172A 29 of the whole Curve, whofe Ordinate is / 3 3)? ZX) 298 WED let be denoted by Q, and then, by comparing foal x bimme | o : with Sea = . ee [*a4o. | with 2 Fa oe We hal have Vals Soe ees, kJ x4 3 10 1 18 oe ai, a ts k=b3 Soar 1-5 t— I= 35 and w 2° 3 2 6 6? (¢—s)=1, and confequently ~2 = Q <1 oh ee he 3 7x hb3—Ff3 i h3 are 3 _ 86 K x 7h? Lees ‘ : . oS a Sia) equal to the required Area in this 6236458 a xXBfle Cafe, EX ADMR LE IX: Where the Area of (Q) of the whole Curve, whofe Or- s Reece 3 Ym} g * i Z dinate 1s /°—~2"|2 . being given ; ’tis required to find i é ea ale aa! that of the whole Curve, whofe Ordinate is “=*"! ae e s d Z z . . I 5 In this Cafe we have Gao": baa. m=, d= L,.2 = k==g", /==1, f—=1, and s(p-+-m+1)=—3: Therefore, s be- ing here a whole Number greater than ¢, let thefe feveral Values be written in Corol. X. and we fhall have w=2, a eee n 4 Zz n n 8 go P=(—4 \—8, and — os eee ora — = % a ates ee Be 4 bY —8 x grb pi Bal Pe xr Ox phe 9 Te equal to the exact Are required, BoA WEP LX. Let it be required to find the Area of the whole carves whofe Ordinate Peel a, x R, fuppofing R= y—z"] 7 re Thefe [ r41 ] Thefe Expreffions being compared with thofe in Corollary XI. we have Ga b==1. mw=—2 tt ee a= es k==1 3 > > 2? 3 2? 3 a i I d=1, t=5,9=5, P=, and s===; therefore Q (the Area & S ee ir ar are aL UF cscs of the whole Curve whofe oe is a—bz"|_ x dz?” ) i. L will here be ==1, and eee — x ; +f x4, At J. pt I bh r= y eee Sp = +- —, &e. ‘het to the true Value fought. But Mihi Area of ase cee whofe Ordinate is i—zl* x R, is alfo equal to the Fluent of 1—2? 1247 & x R,or RR, that is = =; where R, in the propofed Cir- cumftance, is equal to 7 Part of the Periphery of the Cir- cle, whofe Radius is ee Boe it appears, that - Sum ef the Series 1-+- ~ -—- — + oT &c. is equal to 5 = ect of the Square of the ee of the Circle, i Diame- ter is Unity. But the Sum of the Series 1 +- i -+- : + — T ere is @&c. is juft = Part of the Square of that Pe- tiphery, becaufe ; + 16 of the Reciprocals of the even Numbers, is = + = &c. the Sum of the Squares xI- Ale 4. + a Coe. “tat. is. a r of the whole Series, Ao : EXAMPLE XI. Where R being =< tl 2 x z, tis aoe to or | the Area (S) of the whole Curve, whofe Ordir iS a xecR, i Na dicre Ne ne [ 142 ] | Here, by proceeding as in the laft Example, we fhall have 2 : 1 2 : Cs Cai 1=—=2, CUE. 5 “= > ie Woe? fie 1 te Bee 2 ae ae paca a age f= 2, 9 = 5 pl, sj, and Q==c¢*); whence, accord- ; QH H ing to Corol, XII. w (#—s)==0, and $ Q — i= 2 a t—q A 2 nike”. al ; o> ae. eae Br ad H being the Fluent of 1—y| *y *¥, when y is = rae or twice the Arch of the Circle, whofe Diameter is Unity, he and verfed Sine ae Therefore the Value here fought, will be exactly equal to the Meafure of the faid-Arch, S-C'H.O Eh UM: Tho’ the chief Defign of the preceding Propofitions is to exhibit the Relation of fuch Fluents, as can be expreffed in Terms of each other, and algebraic Quantities; yet from thence a Method may be derived for finding Fluents origi- nally, by Infinite Series, much preferable to that commonly made ufe of. Let the general Expreffion a-ez?|” x dz°"— % be propofed, in order to find the Fluent thereof. ‘Take uv equal to any whole pofitive Number, and let p=e-+v, t= e-+-m, x—a--ez", ahd Q= the Fluent of cz" x dz?” | &, which, according to Prop. I. is = ¢=" #4" x y¥——*_ x= 1 pn ati wo ee eh a = » Sc. ‘Then; byProp. 11..Corol.. 1: | ta a en m+3 1 ired FI ill be truly defined by “2 * the required Fluent will be truly defined by ——_— x Be a) ee z pay hee Pete. AZ ee ee QcY — p-em 1 Lr, fe ee ee a> (v) = ak a ee ae ee ae ence, to find the Fluent of p—-2 P93 3 { 143 ] pe Bee tt mT atcz | xdz 2%, let there be taken yg B= ena A x ee Oa oz | ORR Te t+3 * ce Bx a * {ee D= C Ra ez? , and fo on to any Number of Terms (v) and let the latt of thofe Terms be denoted by Q; then take R-==-—Q x pn x ap cz eee Gee m—t ce opt ae a Rx>j*x> T= SX ios ae V=—T x i &c, then will A+B+C+D..... +Q+R+S+T+V+W, &c. be the true Value fought. Now the chief Advantage of this Method confifts in this, that, as v may be any Number at Pleafure, the Value thereof may be fo afligned, or fuch a Number of Terms, A, B, C, éc. of the frft Series may be taken, as to make the fecond Series R-+-S-+-T, Sc. converge exceeding faft, when the Se- ries refulting from the common Method diverges, or conver- ges fo flow, as to be intirely ufelefs. But this will appear better by an Example or two. Let it be required to find the Fluent of 1+ 27]_' x %, when z==1 (exprefling the length of - of the Periphery of the Circle, whofe Radius is Unity). Here 1-271” ° x %, being compared with Gace. CimT ° / ‘ x dz » there will be. a1, c=1, 72}: m==—1, d==1; Bol, eN—I, or 2¢—I==0; whence, if wv be taken = 6, I we {hall have p (e+-v) = ~, —==—-=, == 2, A=1, B= a2 ,C=,D=—,, E=<, oe R= 58 ra, v= it a =~, Ge. and confequently 15” 19” A-+B-+C.. \O-+- Rae Cpa 10: 785398: which Num- ber, found by, taking only 8 Terms of the Series R4-S--T, &c, is right in the laft Place, and would have required, at | i waleatt, > — at oe EE Se EE Ee Ee EPR aT PEE er aera [ 144 ] ° I r leaft, 1000000 Terms of the common Series 1 — = : mes eFe, Again, let it be propofed to find the Fluent of 1+ 24? x % In this Cafe, taking v4, we have a=, ‘T? €2r) tas, m=, aT, C=, oe 7 = ? A=2x", > 7Ax* C = Be ae ce, TaN 11 Bx* Br ero) ee Pte te 1902? eS Tae Vis 6T x* Sou 10 Vet Tyee eee 20x. yo ne 2g? Te 33% ce. and A+B4C+QER+S, &c. equal to the Fluent fought; which if z be taken==1, will become 1.08942, in Bains rahenor no more than 6 Dicms of the Series R-- S+T, Se, are requifite ; ; and if z had been taken fmaller, the Capdleaca would have been ftill more exaét.—Befides Per Ules of the foregoing Method, another Confequence may be derived therefrom, ist altogether inconfiderable. We have Tee e - proved that the Fluent of a-ez"l" x dz?” " %, is univerfally equal to aig I ep eG i pn pee ett pp pe? a Se Therefore if a@ be taken 0, c==1, d=1, then x being arer bt mn eee [en pz we fhall have FS) ee ee et Se es Spo SE. equal pe mnt pn? 5 whence nt 4/4 Mmm mm 77] —_ i he p41 x p42? Fe, Wi ee From which the Sum to the Fluent of 23)" x 2?”7"% that is — of any Infinite Series as = - = x ere + > xo x x See &c, where m, p, and 7 denote any Quantities at Pleafure, is eafily deduced ; for, fince this Series may be changed to mn 7 a sae p oat E x a eee - 5: (Se, let ee 1, and — = be fubftituted . rT for [ 45 ] for p and m, in the laft Equation, we fhall have = oe cM ie amr on oS mar mr mba a eK eer ae 2 ptr ~ phar on per “tar ars" ce. == —“—. Hence we ie eather, that if the P+3r p—T—m § Terms A, B, C, D, &c. of any Infinite Series, be fo related that 2x m+-r mor r B= A, C= pee x B, Dae _ ce the Value of sah Series, will be truly defined OF gs sine & fb 5 avhich therefore is finite or infinite, according : p is greater or lefs than m-+-r. Example: Let it be required to find the Sum : ‘aes Lie3 345. * Xo Koa oe eg, oo age ata y ha Gita &c, Then p=4, m=1, r==2, and —— ——— = 1 = the Va- due fought. Again, let the Series roraied be — + — 2.3 “2, Ge, then will A — a B= = A, C=7B,&c.m=1, =3, r==1, and — + —, &. et Ayo, Or, 1 : : m~m+-r.m—-2r(n) mr. m-2r.m+3r(2) 3 mfar.mb3r.mear(ay &c, and the Value ; | thereof will come out Ge teets ae which laft Ex.. preffion is the fame with that in Page 92 of my Effays, but found in a different Manner. more univerfally, ie the Series propounded be Oo PRO- [ 146 J PROMOS TT 1.0 NUIT If x and y be two variable Quantities any how related to each’ other, and the Fluent of yx —" x be. taken, and multiply’d by x°—" x, and the Fluent of the Product be again taken and multiply d by ore x, and the Fluent-of this laft Product be alfo taken, and foon continually; and there be made i=, Wo G s-+-h=—t, t+i=v, soe... A == : s—rt—r.v—r(ny 5a ee fen ed aes ee 2 Pan S.ta—5.0—s, (2)? C r—t.s—t.v—t (n)? D= pas v.00 (ae &c, and the Sum of all the Indices $£+-¢+h-+i, Ge. be denotea by p; I fay, that Fluent whofe Place in the Pro- greffion is denoted by n-+-1, will, when x becomes equal to any given Quantity a, be equal to the Fluent of Bx et es Dx’ ae ee ay ary x Ax = XxX Ex “(na ) a For let Px? Q ft -R ee a Gee oe , Fe. (which is a general Expreflion for any Quantity whatever) be aflumed — y; then will yx’ x, or xl x be =P alt ee Qt eo Rae x, €&c, and therefore its Fluent pxft* baat ar He Rae ftrbem aie ae Ge aie Shak ame &c. whic pn ~ Cm ° aad 5 tiply’d by x° "x, and the Fluent taken, we have a oer m RAtst2” wet. pci in aes =" Ora ae Te ne a pat iB ieee) eae 7a, Capitan -- reacts &c. From which Me- thod of Operation it is evident, that the Fluent propofed, or that whofe Place in the Progreffion is denoted by 2-41, will be 2 { 147 ] p+9 p+y+m™ be. trade “aeined baie ask bya eee ¥ y q+r.g-s. gin) cg grt. gsm .g ttm (2) nebo? + ee gtrt 2m.g--s-2m.g+-t2m(z)’ of the Summation of Series, is equal to €&c. which Value, by Prop. V. os Lp ea pl 9 oo qubtar2 RPtytem * gtr : gir--m. gtrtzm ? 7 Gag prt? orbit R beat 2m Ax Bisa eet ns iS a 2 qs a8 q+stm ar gtst2m ? Jo en eT TT Cri ae Cx pyet?2 a Qfiria s Rf tit2™ Pig? ce Ce oF re Pe me ec. EFc,. Begone hs Oe ek pyPt7 ee nxltor2™ bial u Se Sass as Rall ee tog inthe pro. - But A x ae + ue ee pro pofed Circumftance, when x is =a, will be equal to the Fluent ef Aa «xP ~eee ee ER aR eer ec. or of Aa’ ’xyx’ — x (fuppofing x andy variable.) And in the fame manner it will appear that the fecond Series ee ae Oe ey: Pte” s & mM , ce, is equal to the — Box ae epee eee j Fluent of Ba xyx &, Ge. Ge. Therefore the Sum of all thefe Series will be equal to the Fluent of Aah” x Ae a x Seah xy mente xyx x7 8c. oF of an x ee eget ee fe ES 2 (ght): Q. E. D. a a ; is Note. That all the Fluents abovementioned are fuppofed to be contemporaneous, OF generated in the fame Time. C O- ‘T 48 | COR OUELARY v1 _If the! Fluent propofed be reprefented by K; then, fince the | Pex a Ae ae . AG eRe Fluent of —= x — POC soe GY ee hee ra‘ 5a a Be &c,— the-Fluent of a’y x a Ax : ; Bx Cxt ye Se See ee ta* - this laft Value, on the Right-hand-fide of the Equation, will. -alfo be equal to K; and will be found more.commodious .than the former, when the Relation of x and -y cannot ‘be exhibited, but by the Meafures.of Angles and Ratios, .&c, COROLLARY IL “Tf all the Indices f, g, b, &c. be equal to each other, then Will == er. if or, Ur bet E xr, OA Cae I - a 1t—t As es So a ee B==—wzA, C—=—x=— A,D == coy xi ee Gs Aste wg U7" I 2 I i : omm J I——Z ae ° 5 ; A, &c. and confequently K in this Cafe, equal to mtr . Pak thy NO GT Me ie Lig Se y the. Fluent of reg we © 3 7) ee ae oa 2 PL wee 1 fou] . Bak ° ° e Benya ; which Equation, if r be iste. oe ar taken —1, will be the fame with that delivered by Sir J/aac Newton, ~in the eleventh-Propofition of his Book of Qua- Gratures. (n--1) or to that of TXAM PELE. Let,y = 2—x'l*, and f, g, b, Bc. each = 2, and let jt be required to find the third Fluent of the -Progreffion, 3 gene= ? 4 ; a i a a i he Ti nt 2 ee ee ee ee a ee [ 149 ] generated while x, from nothing, is increafed to a, Here, accord- : ot tol ees ° eet 4 DSC Re ing to Cor. II. we have ee | S98 ee eee 2.28. ee § —— 2 whofe Fluent, univerfally expreffed, is— =F ne in which taking x==a, according to the foregoing Prefcript, we have - for the true Value fought. Here follow fome ufeful Theorems, extracted out of the fore- going Propofittons. 3 THEOREM =— — x — I 2V 2U=<—=2 eae 2U— 4 (r) and the reft as in the preceding Theorem. Then’ 5 — as a 2h—2.% 2r7—I Xuax* re SKE el ar 2r—3.a 2h—2.2h—4.x goa 2U—1.a Se ee Oe Rh n ate, Meas - 2r—3.27—5 a” (7 ) oe a ii roe 2u—2.cx® 2ZU—1.2U—3 0" a pees. . ee ook 3h eee ; a aaa ; But when “is greater than v, F will be =o, and shetePties in that Cafe, S is barely 2dz" $$ OT as A 2h—2.x 2h—2.2b—4.% 2r—I Xnax* . MXR Lhe oe 2r—3.4 ote Zim 3.27 5.4” Se. continued till it terminates. THEOREM IIL. tn—Un—1 : Saget x dz? Ze Let b=v—r, Fes x be ar 2r—z e (rn), G=— x ie 4 Bi— : ag (v), and x and Q as Eis 2dx* —_—— nae Qo .. S—— Se eK SKS xX I— 2h 2.E€S 2U——1 X 7ax"" 2U—3.2a ee a econ 2 h—2.2 b—A. ¢7%*0 dGew 2r—i.@ Be 2U—-3. 2U—5 a” (v et rnav & fez" + a Zr—2.x% 2r—-1.27— a” eke ° e het oe ar ney x ¢Y: which, when v is 2P— 2.2 rm 2dax" ereater i r, will be barely —=— ——— Xf XZ" x 2U—I1 XK nax's I " : [ x5r ] 2 hmm 2 .cz" heen ta? &ce, continued till it terminates, becaufe then G9. THEOREM. IV. 2:9 ee =I} Hr 72m] o S =a-c2z"| x dz Ze Put v--m=—e, and a+¢2°=z. eee et ar ae dt} vU—i.czo U—1.U—-2 6" &* Then S=— > x1 St we continued till it terminates. CTHEOR-EM #y. we a Un~a=I - Sa-+e2"! xdz z. Let v-em=e, and a-+-c2z"=x«. LT! DOE op VOB U—1.a —I.U—2.a” ren Gee ee oe enc e—— 1.6%" €—I,€—2.¢ 3°" €?c. continued till it terminates. Note. In the two laft Theorems ™ may reprefent any Num- ber whatever, whole or broken, pofitive or negative, THEOREM VI. = yee L — — . Spee ede ee 2% 2U—tI 2U—3 2U—5 d (7),G= 1345 (v), «=a+cz", and Q equal to —— x 26 ny @ ; open? — a2 ; Hyp. dogger OO ye Arch, whofe Radius 1, : apex |? + a? na and [ 152 ] and Secant ,/5=, according as the Value of a is pofitive or —24 negative. ae - 27—1.4 2r—1.2%—3 a” Then § = ~ ek i + er (r) d¥Fa' x ron 2ZU—1.62%" 2U—1.2U— 3.07% 79 v) Un Zin 2U—2.8 ZVem2.20—4. a" ( U =FGQc a”. THEOREM VI. as. ee e S=a+cz"1 2 Lis Po I _ 2v1 20-43 20-5 = Let b=r-+v---, Bi=cy *- joa. G=-x' >x2(v), and x and Q is in the precedent. x pn SS a SE Ey 2dx 2h—2. 2h——2.2h—— z Then S = 2 .7——3,.17—4. 123 oa . 3 (eae1 ) 2 r, will be barely = 7—I Xnak till it terminates. THEOREM IX. a un-Lint - : ES ; £ being any Integer not lefs aplz than 7-+v-+ I. Ae io = cogs 1.3.5 (0) X 4.5.5 (/) Let p=0-+ =, SSr-+-+1, wt—s, H = “a Ga and Q= Periphery of the Circle, whofe Diameter is Unity. Then the required Fluent, when a—éz°==o, or the Value of S generated while 42", from Nothing, becomes==a, will da x HQ. nk —? x bial], ele Freee eee Perr Sor ahich 1.2.5.s--1.béal| 1.2.3.s5.st1.st2.bkal| Series will always terminate in w+-1 Terms. w.p.al i ee ee Ee ea 1.5. bk52a2 be accurately == Tire OR PM x, Peart un+in—! : i ; Sat el| Xe =; ¢ being any whole pofitive kal =| Number, not exceeding r--v, Let p, s, H and Q be as in the precedent, and let m==r—~ ttf 1. t4e2 (ew) X w.avt-1.wtz (t—1) Mh » M-—~1. thom 2, item 3 Thee), (sommei) Qq lat =, Wassert and P= = ;1n which SSF55 a SSS ee Hc" —. and as bd—c SAG = ee 6 bx bd-bep ; ay ie bx boca 1 ae zhdp whence AH = “2. x pet se ae 3 bd+cp 7 Bi=—cpe, (beep 2hdp miata 2hdp eo a a eS SS = oe eet b* d’—c* X1—dd bb--ce X dd—ce — d?, and 4*+c?==1). Moreover as d: p: ce ; (Pe)?: ef — Uc or We wherefore, ae the Property of te Curve, will be — (em) 1: EE (ATH) b? p Sas : st = boas Hence it appears, that if d be greater than c, or the Declination greater than the Complement of Lati- tude, the Curve defcribed will be an Ellipfis, whofe tran{verfe dh b and conjugate Axes are 2 > and aed refpectively, and its ate d?—c>|= : Parameter equal to le : Therefore when Hae Declination is equal to the Complement of Latitude, then ee becoming infinite, the Ellipfis will degenerate to a eta. whofe Pa- rameter Is Be ; but af the Declination be lefs than the Com- plement { 157 ] plement of Latitude, then ee will become an Hyperbola; whofe Tranfverfe and Conjugate being negative, the Curve bpd 2ph : 4 Axes are =-. and—2/2__. and its Parameter == 7"” except co —d o/c? 4? a when d==0, or the Sun is in the Equinox, in which Cafe, b ; é é ’ the Parameter i becoming infinite, the Hyperbola degene- rates to a Right-line. A Determination of the Time of the Year when Days lengthen the fafteft, according to apparent Time, and to any afigned Exxcentricity of the Earths Orbit. : Let AvPOA be the Orbit of the Earth, AP its principal Axis, C the Centre, and S$ ar the Sun, in one of the Foci; and,v being the Place of the Earth at the Winter Solftice, let O be its Place at the Time required: Draw Sv and SO, and alfo Oz, perpendicular to AP ; put- ing PC=2g, SC==-, SO==z, the Sine-ot 4, | vSP, to the Radius 1, =m, its Cofine ==m, and the Cofine of vSO, or the Sine of the Sun’sDiftance from the Equino@tial Point at the required Time =x. Therefore, the Sine of vSO bein g —=,/I—xx, and the Angle PSO equal to the Difference of the two Angles vSO and vSP, the Cofine of PSO will be—=nx | + m/i—xx, by the Elements of Trigonometry ; wherefore it will be, as 1 (Radius) : 2x--m/1—xx:: 2 (SO): Sx —=nx2z-+mz/1—xx, But, by the Property of the Curve, GO & anne won en sia Gowaan ; whence (Sz)== — mel KS Rr C2453 dezi Ca= 258 | ote ¢ ++ M%.fi—xx, confequently z = : v : d y atnextme /1—xx 7 ——— z EE SRS SEE ° therefore * = | ae tnex-+tm e/i—xxl ; which laft Ex- preffion, fince the Alteration of Longitude, or of the An- gle PSO, in a given Particle of Time, is inverfely as the Square of Radius SO, will, it is manifeft, be alfo as the Alteration of Longitude, in a given Particle of Time, or in one whole Day very nearly. and 2 This being now obtained, let AC denote ~, the Ecliptic, AB the Equino@ial, CB a Me- ridian, and C and r thofe two Points of the Ecliptic, wherein the Sun is at rifing on the A B two required Days, when the Difference of the Hour and Minute of his rifing is the reateft poffible, and let Cz be the Difference of Declination in thofe Points; putting Cr=-y, and the Sine of CAB, the Sun’s ereateft Declination == d: Then as 1 (Radius: d: : x (Sine of AC) : dx = Sine BC; therefore its Cofine = /1—d? x? ; Again, as ./1—xx (Cofine of AC): 1 (Radius) : : ¥ ied aan (Co-Tang. of A): a == Langent of: ACB, or 7Ce I—“*¥ therefore its Secant = a Wherefore, beeaufe the Tri- angle Cru, by Reafon of its fmallnefs, may be confider’d as reétilineal, it will be as YI=42*" 2 rs: y (Cr): Cn me PV afi—xx f 1— d*x? equal to the Alteration of Declination, from Sun-rifing to Sun- rifing, on the faid two Days very nearly. Let this Alteration, therefore, be now reprefented byO4, fuppofing HRO to be [as9 J be the Horizon, PH the Meridian, PR - f\P and PO Complements. of Declination, at the Times abovementioned ; and let the Cofine of the Latitude (PH) be A denoted by 4: Then it will be as / 1—a?x? (Sine of PO): 1 (Radius) : : : R Ed J/ +e (Sine of the Latitude ) : ae — Sine of POH ; therefore its Cofine, or the Sine of 1—dx? | OR/is = vo —#*" therefore it will be as Ve iV ee f 1—d" x* of 1—d* x? : of 1—d* x? SLE i Bie es ‘bb Eig dy of 1—xx Ob): dys REY BeeOe = Rb; but as ier Vine x (Pade dy fi—xxxY1—bb dy f1—xx Xf 1—bb Jim darx (Pada PK / Bd x? — the Arch of the Equator, meafuring the Angle RPO, or the Difference of the Semi-diurnal Arches of the Sun on the two Days above fpecified. This Difference therefore, fince y, by the former Part of the Problem, is found to be as o/ wie eh wb dione fe Idx X of bd x? where, if the Fluxion be taken, and made equal to Nothing, the required Value of x may, it is manifeft, be determined, let (e) the Excentricity of the Orbit and the Latitude of the Place be what they will. But the Exceftricity, as given from Obfervation being fmall, the greateft lengthening of Days, if the propofed Place be not very near the Frigid Zone, muft be near the Time of the vernal Equinox, and the V alue of x but {mall; therefore, if the forefaid Expreflion be converted into a Series, and all the Terms wherein more than two Dimentions of e and x are concerned, be neglected as inconfiderable, it will be Jf 1— ax? ¥ tee —_—_——_—_—_—r—rn ooo |? : atnex+me/1—Xxx; will be as [ 160 ] 2,2 d* be reduced to aa--2amemm=* x I—2d*—— -+ 2aenx, where, by taking the Fluxion, @c. x comes out == 22! aX 12d Note, From the Equation foregoing, the greateft lengthen- ing of Days at London, will be found to be about 7 Days be- fore the vernal Equinox. 4 Determination how far a heavy Body, freely defcending from Reft, falls from. perpendicular, by Means of the Earth's Rotation. ¥ KO, P.O.Sal Fi ON 4 I Uppojing the Earth to be perfectly Spherical, and that a heavy S Body defcends from a given Point above its Surface in any given Latitude ; to find how far it will impinge from a per- pendicular, let fall from that Point to the Surface, thro the Caufe above fpecified. Let the Axis of the Earth be confi- dered as abfolutely at Reft, and let EA be the perpendicular Height from whence the Body is let fall, and by the Force of Gravity and the Motion acquired by the Earth’s Rotation, begins to defcribe the elliptical Area ACFA, in the Plane of the great Circle EFC, about C the Centre of Force, while the Point E is carry’d by the Rotation of the Earth, in in its Parallel of Latitude EaS from E to-=- (FG) towards a; let F be the Place where the Ball falls, and FS the Diftance of that Place from the faid Parallel ; and let the Point a be the Pofition of E, at the Time when the Body impinges on the Surface at F. Therefore, fince the Velocity acquired by the Rotation of the Earth, and the Attraction at the Point A are both given, the Ellipfis AF will be given both in Mag- nitude and Species (by Page 23 of my Effays) whence EF and Ea will be given, and confequently the required Diftance Fa, But when the Height AE is fuppofed fmall in refpect to the Earth’s Radius EC, as in the Cafe propofed, the Solution may be, otherwife, more eafily inveftigated: For then Sa be- ing {mall in refpect of FS, the latter of thefe may, without fenfible Error, be taken for Fa; but FS is to the verfed Sine Ep, of the Arch FE, as the Tangent of the given Latitude to Radius nearly. But FE is given from the Time of Defcent, whence FS will be given alfo. Q@.E. 4. PR ey Gs big a Ne Te To determine the fame as in the laft Propofition; fuppofing all Bodies gravitate perpendicularly to the Surface of the Earth. Let ACDR, &c. reprefent the. Earth (whether under a {pherical or an oval Figure) AB, &c, its Axis con- fidered as abfolutely at reft, and RpDnxR the given Parallel of Lati- tude ; let CRS be perpendicular to the Surface at R, RS the given Height, or Diftance defcended, and SO the Direction in which the Body would fall was it not for the Earth’s Rotation. Then, as the Attraction, exerted at S, ats in the Direction $O, the Body, upon its oe. Sf leaving ( 263°) leaving S, will begin, thro’ that AttraCtion and the Motion re- ceived from the Earth’s Rotation, to move in a Curve Line Sz, that may, without fenfible Error, be confidered as Part of an Ellipfis, formed by the Interfection of the Conical Surface CRzDp produced, and a Plane paffing through S and O; and will continue to defcribe the fame Areas, in equal Times, about the Point O or C, as it did before its leaving S (fetting afide what arifes from the Alteration of the Centre of At- traction, &c, which is too minute here, to require a particular Confideration.) Hence if the Point m be fo taken in the gi- - ven Parallel of Latitude, that the Area of the Sector CSrmC, may be equal to the Area CSzC, then will the Point m, it is evident, be the Pofition of the Place R, at the time when the Body impinges on the Surface at x. Now the Height RS being {mall, when compared with the Diameter of the Earth, the Curve Sz may be taken as a Semi-Parabola, whofe Vertex is S, and Rz as a Right-line; whence the Area 2SR~z is found = §R x =Rz, and therefore SuvS=SR x : Ra; which is alfo the Area of the Sector rCnv, becaufe CSrC being equal to CSxC, let each of thefe be taken from CSvC, and there re- mains rCwv equal to SzvS: Therefore will xm be = eC nearly, and fo much will the Body fall eafterly of the Perpendicular. N. B. The two foregoing Propofitions might be of Service in proving the Motion of the Earth, by the Defcent of heavy Bodies, provided the Experiment could be made with fufficient Accuracy. ( 163 ) A DeMonsTRATION Of the Law of Motion that a Body deflected by two Forces tending to two fixed Points, will deferibe equal So- lids in equal Times about the Right-line joining thofe Points, Let A and B be the two propofed Points, and C any Place of the Body, and let the Dire@tion of its Motion, at that Place, make any given Angle with the Plane ABC, or with any Right-line drawn in that Plane ; and fuppofe the Body, upon its leaving C, to be impelled by any Forces whatever, tend- ing either to the Points A and B, or to any Parts of the Line AB, and let Cv be the Right-line, which afterwards, by its com- pound Motion, it will proceed to defcribe, and let the motive Force, before the Impulfe at C, be refolved into two others, one in the Direction of a Right-line lying in the Plane ACB, and the other perpendicular thereto. Then, fince the laft of thefe is not at all affected by the Impulfe, a¢ting in the Plane, the perpendicular Diftance of the Body, from the Plane at the end of a given Time, will, it is manifeft, be the fame, let the greatnefs of the Impulfe be what it will, and therefore in different Times, direétly as thofe Times. But ACBvw, the Solid defcribed about the Line AB, being an oblique Pyramid, is known to be as the faid perpendicular Diftance, and therefore mutt likewife be as the Time: Hence it appears, that whether the Body be, or be not impelled at the Point C, the Magnitude or Content of the Solid defcribed about A B, will be the fame, and proportional to the Time in which it is defcribed: There- fore, feeing no fingle Impulfe, however great, can affect the equable Defcription of Solids about AB, it is evident, that no Num- ( 164 ) Number of fuch Impulfes can, nor any Forces tending con- tinually to the Points A and B. Q.E. D. ° N. B. The Propofition would have been equally true, and the Demonftration the very fame, had there been fuppofed ever fo many Forces tending to the fixed Line AB; andif in- {tead of the Solids defcribed about’ that whole Line, thofe de- {cribed about any given Part of it had been taken. A DETERMINATION in what Cafes a Body, acted on by a centripetal Force, may continually defcend in a fptral Line towards the Centre, and yet never fo far as to approach it within a certain Diftance; and alfo in what Cafes it may continu- ally afcend, yet never rife to a certain affignable Altitude. Mr. Mac-Laurin, at the End of his Treatife of Fluxions, has found, that if the centripetal Force be as the sth Power of the Diftance inverfely, a Body may continually defcend to- wards the Centre, and yet never fo low as to come within a certain Circle, or may recede for ever from the Centre, yet never rife toa certain Height; which remarkable Circumftance had not been taken Notice of by any preceding Authors. But the fame Thing will alfo happen in an Infinity of other Cafes. For let C be the Centre of Force, and let the Body proceed from P in any given Direction Pg, with a Velocity, which is to the Velocity whereby it might defcribe the Circle PBS, in the Ratio of pto1; let R be any Point in the Trajectory; and make Cg perpendicular to Pg; putting cP=1, Cg=s, CR —x, and PBS==A. ‘Then, if the centripetal Force be fup- pofed as any Power (w) of the Diftance we fhall have A= a eet; and: the Velocity of the Body Vien eo Se en} ‘i n-y-I n+l at R, will be to the Velocity whereby it might defcribe a Circle at [ 165 at the Diftance CR, in the Ratio of ./#*+- ~~ pon ge 1 tt net tor, as is proved in Page 31 of my Effays, ‘This being pre- 3 be now taken =o, mifed, let x Y/p?-+ —p* ee 2 and f/f? aay * pt FI =1; and then, the Equations : soe being duly ordered, we fhall have # = ape , and #5? a3 a\a--1 mo Ebetixe aoe oe et, Wherefore, with this Value of x, a3 a Radius, conceive the Circle AH to be defcribed, and ‘let the Velocity at P be fuch, that p* s* (when poflible) Tt may : [ 166 ] a+-3 2+ 24-1x p? ee may be = ee | 3 then if AC be greater than CP, and the Body upon its leaving P begins to afcend, it will con- tinue to afcend ad infinitum, and yet never rife fo high as the Circle ADSH: For it cannot begin to defcend before it ar- rives at its higher Ap/e, which (if it can properly be faid to have any) will be in that Circle, becaufe AC will be the Value of x, when ./p?-+ a XX? —f? 5% at is equal to Nothing : Nor can it ever rife fo high as the Circle AJH ; for if it fhould, its Velocity there being juft fufficient to retain it in that Cir- cle, it would continue to move therein, and not defcend again in the fame manner it afcended, which is abfurd. By a pa- rity of Reafoning it will appear, that if AC be lef than CP, and the Body upon leaving P begins to defcend, it will conti- nue to defcend for ever, but never fo low as to enter within the Circle AER. It therefore now only remains to find in what 2-3 seb Cafes the forementioned Equation, #7 s* = a Hera a3 is poflible, and in what Cafes it is not. a3; HOU Sas _- set pett In order to which, let the Fluxion_ of -auceeit n-3 x a (== 5?) be taken, -confideting p as-variable, and ‘you ‘will i , 2 el LMA Xie a 18-EF ’ eens ae have PO? y 42s APEERP | which “Flexion ‘will ‘be ‘pofi- aS Pe a+ 3 _ tive or negative, according, as ay IS politive-or negative; be- 2 ; ; tne . - caufe a | mhoft -be pofitive, -elfe ‘5? cannot! be fo. But [ 167 ] nto 3 apni xp [e | a3 x == 1; which laft is manifeftly the greateft or leaft Value, poffible, of that Expreffion ; that is, the greateft when 2+-3 js negative, becaufe then the Fluxion, while p is lefs than 1, will be pofitive, and afterwards negative; but the leaft when +3 2tntix p? (re n+-3 But when f is =1, aan will be == 0, and nm-+-3 is pofitive, fince then the Fluxion of x ms is firft negative and then pofitive : In the former of which Heat a See ea x—- ==s?, can be n+3 poflible, feeing s, by the Nature of the Problem, mutt be lefs than 1, or Cg than CP. Therefore fince it appears that the forementioned Circumftances can only take Place, when the Value of m-+-3 is negative, or the Law of centripetal Force more than the Cube of the Diftance inverfely, let —m—3 be fabftituted inftead of , in order to reduce the Equation toa Form more commodious for this Cafe ; then we fhall have ——I . wt Fax"? == AC, and ahmxp— |" i 2 Cafes only the Equation pis e where ‘tis evident, from what has been faid above, that the Root /, let s'be what it will, has two pofitive Values, one of them lefs than Unity, the other greater 5 whereof the former (which gives AC greater than AP) muft be taken when the Body afcends, ‘but the latter when it defcends, Q. E. 1. An [ 168 ] An eafy and general Woy of Inveftigating the common Theorems relating to Compound-Intereft and Annuities, without being obliged to fum up the Terms of a geometrical Progreffion. Let R be the amount of one Pound in one Year, vz. Prin- cipal and Intereft, P any Sum put out at Intereft for any Num- ber. z of Years, a its Amount, A any Annuity forborn x Years, mits Amount, and vits worth in prefent Money, for the fame ‘Time. Therefore, fince one Pound put out at Intereft, in the firft Year is increafed to R, it will be as 1 toR, fois R, the Sum for- born the fecond Year, to R?, the amount of one Pound in two Years; and therefore as 1 to R, fo is R2, the Sum forborn the third Year, to R3, the amount in three Years: Whence it ap- pears that R*, or R, raifed to the Power, whofe Exponent is the Number of Years, will be the amount of one Pound in thofe Years: But as 1/. to its amount R’, fo is P to (a) its amount in the fame Time; whence we have P x R°=a, - Moreover, becaufe the amount of one Pound in z# Years is R°, its Increafe in that Time will be R°—1; but its Intereft for one fingle Year, or the Annuity an{wering to that Increafe, is R—1,; therefore as R—1 to R"—1, fois Atom. Hence AxR'—1 Rea Pound ready Money, is equivalent to R®*, to be received at the AxR*—1 R—1 (the Sum in Arrear) tov, its worth in Ready-Money ; whence we get =m. Furthermore, fince it appears that one Expiration of 2 Years, we have as R® tor, fo is A B4 I recat R—1 the various Queftions relating to Compound-Intereft, Annui- ties in Arrear, and purchafing of Annuities, are, refpectively, refolved. It R* =v. From which three Theorems, or Equations, FEN Teo, » i ¢ , A hg wb, fp Aime fi Me LG eu *?F. 3 fr & f FY f iA, AEH O%;ls Si YO! : te i MIS@ EIS ANE OUS T Rea 'C Pos ‘ON Some curious, and very interefting SuBjECTs kN Mecuanics, Puysicat-Astronomy, and SpEcuLa- TIVE MaTHEMATICS 3 WHEREIN, The Preceffion of the Equrnox, the Nutation of the Earru’s Axis, and the Motion of the Moon in her Orsit, are determined. By THOMAS SIMPSON, F.R.S. And Member of the Rovat Acapemy of Sciences at STOCKHOLM. LONDON, Printed for J. NouRsE over-againft Katherine-ftreet in the Strand. MDCCLVIL, SFT ee EE OE Se IY Oe ee RBCS TO THE RIGHT HONOURABLE THE KARL oF MACCLESFIELD, &c. PRESIDENT of the RoyAL SOCIETY. My Lorp, rE YHATEVER Luttre, to the pub- VY lic Eye, Works of Learning may derive from the Patronage of the Great, it is to your Lordfhip’s perfonal Acquirements, and extenfive Knowledge in the Mathemati- cal Sciences, that my Ambition of defiring 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 with, that any thing thefe theets 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. A 2 Were DEDICATION. Were your Character, My Lord, lef confpicuous and diftinguifh’d, the Obligati- ons I have to your Lordfhip’s. Goodnefs, would, alone, be Motives fufficient to make me gladly embrace this Opportunity of pub- licly expreffing my warmeft Gratitude, and of teftifying the perfec Efteem, and pro- foundeft Deference, with which I am, My Lorp, Your Lordfhip’s | moft Obliged, and ever Obedient Humble Servant Thomas Simpfon. PURE Br Ae “HE Tracts, or PAPERS compofing the Worx bere offered to the Publick, were drawn up at feveral, diftant times, and upon different occafions ; either, with a view to clear up, or fettle fome dif- ficult or controverted point in Aftronomy, to fhew the conformity of Theory wth Obfervations; or to extend and facilitate the analytic- method of computation, by fome improvements and applications, that have not at all, or but lightly, been touched upon, at leaft by any Eng- lifh Author. The firft of thefe Pavers, which is one of the moft confiderable in the whole work, is concerned in determining the Preceflion of the Equinox, and the various inequalities thereof, with the different motions of nu= tation of the Earth's Axis, arifing from the attraction of the fun and moon ; wherein the late important difcovery of Dr. Bradley, relating to an apparent motion of the Fix’d Stars, unknown to former Aftrono- mers, zs /hewn to be intirely confiftent with the Theory of Gravitation. —This piece was drawn up about five years ago, in confequence of another on the fame fubjec#, by M. Silvabelle (a French Gentleman) then delivered to me, for my opinion, fince printed tn the Philofophical Tranfactions.—Tho’ [have particular reafons for mentioning this cir- cumftance, Iwould not be thought to infinuate here, that my opinion had any weight with Thofe to whom the publication of that paper was owing : Ihave, indeed, no reafon to believe it. —Tho the author thereof had gone through one patt of the fubject with fucce/s and perfpicuity, and though his conclufions were found perfectly conformable to Dr. Bradley's ob/er- vations, He neverthelefs appeared (and till appears) to me: to bave greatly failed in a very material, and indeed the only very difficult part, zhat is, in the determination of the momentary. alteration of the pofition of the earth’s axis, caufed by the forces of the Sun and Moon; of. which forces, the quantities, but not the effects, are truly inveftigated. The Second Paper, contains the inveftigation of an eafy, and very exact method, or rule, for finding the place of a Planet im its Orbit, from a correction of Dr. Ward's circular hypothefis, by means of cer- tain Equations applied to the motion about the upper focus of the ellip- jis. From whence that table of Dr. Halley’s, entitled, ‘Tabula pro expediendo calculo Aiquationis centri Lune, may be very readtly con- frruéted. Poe er oAL E. ftruéted.—By this method, the refult, evenin the orbit of Mercury, may be found within a fecond of the truth, without repeating the operation. ~The Third, /bews the manner of transferring the motion of aCo- met from a parabolic, zo az elliptic Orbit; dezng of great ufe, when the obferved Places of a (new) Comet, ure found to differ fenfibly from thofe computed on the hypothefis of a parabolic orbit. The Fourth, 2s an attempt to foew, from mathematical principles, the advantage arifing by taking the mean of a number of obfervati- ons, im practical Aftronomy ; wherein the odds that the refult, this way, 1s more exact, than from one fingle obfervation, 1s evinced, and the utility of the method in practice, clearly made to appear.— A part of this, and of the 7th paper, 15 inferted in the xL1x volume of the Philofo- phical TranfaCtions ; but the farther improvements bere added, will (T hope) be a fuficient apology for my printing the whole again, in this work. The Fifth, contains the determination of certain Fluents, and the refolution of Jome very ufeful Equations, 2m the higher orders of fiuxions, by means of the meafures of angles and ratios, and the right fines, and verfed fines of circular arcs. The Sixth, treats of the refolution of algebraical equations, by the method of furd-divifors ; wherein the grounds of that, method, as laid down by Sir Mfaac Newton, are inveftigated and explained. The Seventh, exbibits the znveftigation of a general rule for the refolution of \foperimetrical Problems of a// orders, together with fome examples of the ufe and application of the faid rule. The Eighth (and laft) Part, comprebends the refolution of fome ge- neral, and very interefting problems, in mechanics and phyfical Aftro- nomy ; wherein, among other particulars, the principal parts of the third, and ninth {eCtions of the firft Book of Szr Itaac Newton’s Prin- cipia,are demonftrated, in a new, and very concife manner.— But what, I apprehend, may beft recommend this part of the work, ts the applica- tion of the general equations therein derived, to the determination of the lunar Orbit: In which I have exerted my utmoft endeavours to render the whole intelligible even to Thofe who have arrived but to a tolerable proficiency in the higher geometry. The greater part of what 1s here delivered on this fubjett,was drawn up in the year 1750, agreeably to what is intimated at the conclufion of my Dotrine of Fluxions, where the general equations are alfo given. The famous objection, about that time made to Sit Ifaac Newton’s gene- ral Law of Gravitation, by that eminent mathematician M, Clairaut, ; I of Pao MPraA G & of the Royal Academy of Sciences af Paris, was 4 motive fupicient to induce me (among many Others) to endeavour to difcover,, 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 inverfe ratio of the fquares of the diftances. The fucce/i was anfwerable to my hopes, and fuch as induced me to look farther into other parts of the theory of the moon's motion, than L firft intended: but, before I bad completed my defign, £ received the honour of a vifit from M. Clairaut ( ‘yuft then arrived in England) of whom I learned, that he bad a little before printed a piece on that fubject ; a copy of which I afterwards received, asaprefent at bis hands; wherein L found moft of the fame things demonfirated, befides feveral others, to which I had not then extended my enquiry. Upon this, L at that time defied from a farther profecution of the fubject ;. being chiefly diverted therefrom by a call then fubfjhing for a new edition of another work, in which fome additions feemed wanting. But I cannot omit to observe bere, in juftice to M. Clairaut, that, tho be indeed fell into.a miftake, by too haftily inferring a defect in the received law of at- traétion, from the infuficiency of the known methods for determining the effect of that attraétion, inthe motion of the moou’s apogee, yet be was bim/elf, the firft who difcovered the true fource, of that miftake, and who placed the matter in a proper light: T hough there are {ome * who have, both before and fince, undertaken to give the true quantity of that motion, from fuch principles, only, as are laid down in the ninth {eCtionof the firft Book of the Principia: but that thefe Gentlemen, however they may have made their numbers to agree, have been greatly deceived in their calculations, is very certain; fince.a confiderable part of the faid motion depends on that part of the folar force. acting in the direction per pendicu- lar to the Radius-vector, which is by them, either intirely difregarded, 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 fuccefs. Since M. Clairaut’s prece fir made its appearance, the moft eminent mathematicians, in different parts of Europe, have turned their thoughts that way. But tho what [now offer on the fame fubject, may, perhaps, appear of lefs value, after what has been already done by thefe great men, yet Lam not very folicitous upon that account, as it will be found, that I have neither copied from ee * Vid. Walmfley’s Theorie du mouvement des apfides (tranflated into Englifh) and Vol. 47. Ne x1. of the Philsfophical Tranfactions, their PR’ BF AS C1E. hake thoughts, nor detratted from their merit. The facility of the me- thod T have fallen upon, will, I flatter myfelf, be allowed by all, who are appriz'd of the real dijiculty of the fubject ; and the ex tenfivenels thereof will, in Jome meafure, appear from this, that it not only determines the motion of the apogee in the fame manner, and with tbe fame eafe, as the other equations, but utterly excludes, at the fame time, all terms of that dangerous fpecies (if Imay fo exprefs myfelf) that have hitherto embar- raffed the greatefi Mathematicians, and that would, after a great num- ber of revolutions, intirely change ‘the Jgure of the orbit. It thereby appears, that all the terms, or equations in general, wtil be expreffed by fines and co-fines, barely, without any multiplication into the arcs Fs: From whence this important pefeeee 1s derived, that the mean motion, and the greateft quantities of the Jeveral equations will remain unchanged ; unlefi diflurbed by the intervention of fome foreign, or Uecedentad caufe. In treating of this fubjeé#, as well as in moft of the other parts of the enfuing work, I have chiefly adbered to the analytic method of Invef- tigation, as being the moft direct and extenfive, and beft adapted to thefe abjirufe kinds of fpeculations. Where a geometrical demonftration could be introduced, and feemed preferable, I [ have given one: but, tho a pro- blem , fometimes, by this laft method, acquires a degree of perfpicuity and elegance, not eafy to be arrived at any othcr way, yet I cannot be of the opinion of Thofe who affect to fhew a diflike to every thing performed by means of {ymbols and an algebraical Procefs; /ince, fo far zs the fyn- thetic method from having the advantage in all cafes, that there are mnumerable enquirtes into nature, as well as in abftraéted ference, where it cannot be at all applied, to any purpofe. Sir Ifaac Newton bimfe Uf (who perhaps extended it as far as Ke man could) has eveninthe moft fi mple cafe of the lunar orbit (Princip. B.3. prop. 28) been obliged to call in the affftance of algebra; which be bas alfo done, in treating of the motion of bodies in refifting mediums, and in various other places. And it appears clear to me, that, it 1s by a dtligent cultivation of the Mo- dern Analyfis, that Foreion Mathematicians have, of late, been able to pufh their Refearches farther,i in many particulars, than Sit aac New- ton and bis Followers bere, have.done: tho it muff be allowed, on the other band, that the fame Neatnefs, and Accuracy of Demonttration, 18 not every-where to be found in tho/é Authors; owing in some meafure, perhaps, to too great a difregard for the Geometry “of the Antients. I A D E- Ceca cae teoecac tec ET IK IG I RR AR RRR KR RR aK i a aig gay ith yy YOK ¥SK 9S IA eK SEEESESRERRRRER ERE EERE E RES | PP Pap PPA Pp Ap EpEpypPApApAPApIP PAPA PAPEPARI AAR A DETERMINATION OF THE PRECESSEON- OF THE BE OUINGA, And the different Mortons of the Eartu’s Axis, Arifing from the Arrraction of the Sun and Moon. *XOCAHE Precession of the Eguimox, whereby the fix’d e , ftars appear to have changed their places by more jw xox? than a whole /gn, fince the time of the moft ancient Aftronomers, is phyfically accounted for, from the at-= traction of the fu and wioon on the protuberant matter about the earth’s equator; whereby the pofition of the faid equator with refpect to the plane of the ecliptic is fubjeCted to a per= petual variation. Were the earth to be perfectly fpherical and of an uniform denfity, no change in the pofition of the terre- ftrial equator could be produced, from the attraction of any re- mote body; becaufe the force of each particle of matter in the earth, to turn the whole earth about its center, in confequence of fuch attraction, would then be exaétly counterbalanced by an equal, and contrary force. But as the earth, by reafon of the centrifugal force of the parts thereof, arifing from the di- urnal rotation, muft, to preferve an equilibrium, put on an ob- late figure, and rife higher about the equatoreal parts than at the poles, the action of the fun on the faid equatoreal parts will have an effect to make the plane of the terreftrial equator to coincide with that of the ecliptic: which would a¢tually be Bes brought Of the Preceffion of the Equinox, brought to pafs (neglecting other caufes) was the fun, or earth, to remain fix’d in either of the /o//fices, and the diurnal rotation at the fame time to ceafe. But, though both the motions of the earth contribute to prevent an effect of that fort, yet, in confequence of this a¢tion of the fun, a new motion of rotati- on, about that diameter of the equator lying in the circle of the fun’s declination, is produced ; from which the preceffion of the equinox and the nutation of the earth’s axis have ia rife. The effect of the oOo} as it is much more conti derable than that of the fun, fo is it likewife lable to fome inec qualities to which that of the fun is not ft fubject, Were the inclination of the lunar orbit to the plane of the e ee t@ remain, always, nearly the fame, like that of the earth, the fame calcule ations that anfwer’d in the one cafe would alfo nhiWer in the other ; but that inclination is continually varying; and, when the afcending node is in the beginning of Aves, is greater by above 4th part than the mean value; and therefore, as the force of the moon to turn the earth about its center (other circum- {tances remaining the fame) is found, hereafter, to be as the fine of the double of the inclination, i it is manifeft, that, in the faid pofition of the node, the motion of preceffion will go on much quicker than at the mean rate ; and confequently that an equation, depending on the place of the node, will neceflarily are. The determination of which, as well as of the o:her motions of preceffion and nutation arifing from the attraction both of the fun and moon, I fhall now proceed to fhew: but in order to pave the way thereto, it will be proper to begin with premifing the fubfequent Lemmas. 7 EVV? I Suppofing all the particles of a given fpheroid A’PapO to be Jollicited parallel to the axis Pp, by forces proportional to the di- ftances from a plane PAOpa pa fae | yy the faid axis, im fuch fort that the two oppofite fomi- _{pheroids, "h Pp, Pp, may thereby be equally urged in contrary directions; 4f 1s propofe d to determine the whole effect of all the forces to turn the ftherotd about its center ° Let and the different Motions of the Earth's Axis. ( a == femi-diam. OA’ (perpend. to the plane PAO; | A= atea of the ellipfe P AOpa, Let¢ of al. , » == force acting on a oe at the remoteft point A, Lous e ditt. of any f gees DQNEQ fi om PAOfa: Then, this {ection being alfo an ellipfe, fimilar to PApa, we fhall have, by the dl of the elif, as A’O° fais A‘O* — ON’ (act se re ADOLS AON : the area, PAQDA (4) 10 aa— Xx ise the area DONEQ = 4 x ——— (by the property of fimilar SS \ y\ i y te }? = B=— KM x figures). Hence it is soe fet ye the fum of all the forces whereby the particles in the ellipfe DQE' Q are urged parallel to the axis Pp of the fpheroid ; which quantity, drawn into (x) the length of the lever ON, will, confequently, exprefs the effect of all the faid forces to turn the {pheroid about its center: and fo the fluent of a . . 2 AX ——— x =X Axx y, which is Ax==X y (when x =a) will truly exprefs one half of the quantity fought. CORO LA kX If the mafs, or content of the fpheroid, which is 4 x 24 Xo be denoted by S; then the force 4x "id x y, whereby the fpheroid tends to turn about its center, will be truly defined by 4S xa@x¥y, which therefore is juft +th part of what it would be, if all the particles were to act at the diftance of the remoteft point A’. LEMMA IL. 3 Fisch. Suppofe a body to revolve in the circuntference of a circle AFaF, Fig. 2. whilft the circle itfelf turns uniformly about one of its diameters Aa, as an axis, with a very flow motion ; it 1s propofed to deter- mine the law of the force, aching on the body in a direction per- pendicular to the plane of the circle, neceffary to the continuation of a motion thus compounded. B 2 Let Fig. 2. S > MC <= TK Of the Preceffion of the Equinox, Let AFaF and Afaf be two pofitions of the circle, indefinitely near to each,other, and let R and 7 be the two correfpond- ing pofitions of the body; let alfo the planes RDz and mac be perpendicular to AFaF and to the axis AOa; in which planes let there be drawn Ruz and mvc perpendicular to DR and din, neeting the plane Avaf (produced out) in 2 and ¢; and let there be drawn xv, parallel to the tangent Rtm, meeting mc in ‘vu. Ifthe velocity of the body along the circumference be ex- se prefied by Rm, the velocity in the perpendicular direction Ry, arifing from the motion of the circle about the axis Aa, will be reprefented by Rv. And, if the body were to be fuftered to purfue its own direction from the point R, it-would, by the compofition of thofe motions, arrive at the oppofite angle v of the parallelogram Raum, in the fame time that it might move through Rm by the motion Rm alone; and fo would fall thort of the plane by the diftance cv*. It therefore appears that the required force, neceflary to keep the body in the plane, mutt be fuch as is fufficient to caufe a body to move over the diftance cv in the aforefaid time ; and that hrs force muft, therefore, be to the centrifugal force of the body in the circumference (whofe meafure is ef) as ev to ef; fince the {paces deferibed in equal times, are dire¢tly 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 fuppofed as r to unity; then, the latter of thefe celerities being reprefented by Rw, the former will be defined by rx Rm; and confe-— DR OF Moreover, becaufe of the fiilarity of the triangles DRz and dmc, it will be, as DR: Ra (rxRm i) =: ae (DR +- sm) evi Ei sg Ram sm oe quently the celerity (Rz) in the direction Rz, by rx Rmx on : from whence, taking away Rm X sm hws the value of mv or Ru, we get cu = rx — : which is OF * The lineola cr, lying in the plane of the circle, muft be anfwered by a force tending to the center of the circle; with which we have nothing to do in the prefent confideration. In and the different Motions of the Earth's Axis. 5 in proportion to the meafure of the centrifugal force ef, or it's mal 2OR’ angles ORD and Rsm, as 27 x OD to OR or OA. Hence it is evident that the body, to continue in the plane of the circle, muft be conffantly a€ted on, in a direction per- pendicular to the plane, by a force varying according to the co- ine of the diftance AR of the body from the extremity of the axis; whofe greateft value, at A, is to the centrifugal force in the circle, as 27 to unity. Q,H. 1. CO.8 Oh RAY i If, inftead of one, a great number of bodies or corpufcles, fo as to touch one another and thereby form a continued ring AFaF, were to revolve at the fame time, and to be acted. on in the fame manner (that is to fay, by forces in the ratio of the diftances from the diameter FF perpendicular to the axis Aa), it is evident that they would all continue in the fame plane. And this will alfo be the cafe, when a number of con- Fig. 3. centric rings ERGeG, &e. are fuppofed to perform their revo- lutions together about the common axis AEea. For, affuming B to denote the centrifugal force of a corpufcle in the outer- moft ring AR’FaF, the centrifugal force of an equal corpufcle equal as rx sm to ~Rm, or, becaufe of the fimilar tri- (R’) in the ring ER’eG, will be equal to Bx = : whence, by ; 5 OF « the foregoing proportions, 27 x Cx aa will be the force act- : : : D : ing perpendicular to the plane at E: and ar x 6 x or x are : OD (= 2r x Bx a will be the true meafure of the force acting ft? on a corpufcle at R’; which, as 7, G, and OA are all of them conftant, is evidently as the diftance from the diameter. FF. Whence it follows, becaufe the diftance below FF becomes negative, that the forces above and below that diameter muft have contrary directions. : COROLEARY IL Whatever hath been faid in the preceding Corollar holds ? P 5 e ; equally, - Steal SL A SES oa 6 Of the Preceffion of the Equinox, equally, when the line or saxis Ag, about which the plane is {uppofed to turn, hath a progreffive motion, or is carried uni- formly forward, parallel to itfelf; provided the angular celerity about that axis continues the fame; as is evident from the re- folution of forces. . Hence it follows, that, if a circle EF’EC&, confidered:as, compofed of an indefinite number of concentric rings, be fuppofed. to revolve uniformly about its center C whilft the center itfelf and the right-line OC (which, to help the imagination, may be taken -as the axis of a cone E’O¢, whofe bate is E’Keé) move uniformly in the plane-PépA’ about the point O; I fay, it follows that the forces neceffary to keep the particles in the plane, under fuch a compound motion, will be the very fame as if the circle was to turn about the line Ee (perpendicular to the plane PpA’) at reft, with an: angular celerity equal to that of the center C about the point O: becautfe, the angle OCE’ being ahways a right one, the angular celerity e mes S a of the moveable circle about the line ECe (which remains every-where parallel to. itfelf) 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 neceflary to retain the particles in the plane E’E¢e, will be, every-where, as the diftances from ‘the diameter. E’Cé, or the plane PzpA’, let the diftance of the plane E’Ede from the center O be what it will. CiO.R°O LL AR. Y.;-IL Conceive now OAPapA’a to be an homogenous fluid, re- volving uniformly about the axis POp, under the form of an oblate fpheroid * ; whilft the axis itfelf is fuppofed 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 eguzibrio. among themfelves, mutt be fo- _ lierted parallel.to the axis, sby forces that are as the diftances from the plane PaépA’; fach,: that the force acting at the re- moteft point A may be defined by 278; where @ (by Corol, I.) * ‘That the particles will remain in equilibrio, under the form of an ob- Jate fpheroid (when the. axis is at reft), is demonftrated in Part II. Se& 9. ef my Doctrine and Application of Fluxiens. ) reprefents and the diferent Motions of the Earth s Axis. reprefents the centrifugal force in the circumference AdaA’ of the greateft circle, and r the meafure of the angular motion of the axis itfelf, that of the rotation, about the axis, being denoted by unity. But it appears further, from Lemma J, that the efficacy of all the faid forces to turn the {pheroid about its center (making y here = 27) is truly defined by arBxtSxOA. Whence it is plain, that all the particles of the body will remain in equilibrio among themfelves, under the two different moti- ons above explained, when the whole force producing the mo- tion of the axis, is expreffed by 2x@x 48x OA. And, when the forces refpecting the feveral particles are fuppofed to act ac- cording to a-different law, the effect produced: by them will be the fame, provided their joint efficacy, to turn the body about its center, be the fame: fince the fame force muft be an{wer- ed, or fatisfied with the fame kind and degree-of motion in the whole body ; if we except only, the exceeding {mall diffe- rence that will arife from the alteration of the figure; which figure will not be accurately a fpheroid, in this cafe, but nearly fuch, as the motion of the axis and, confequently, the forces pro- ducing it, are fuppofed very fmall. Neither will the axis con- tinue to move in the fame 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 (Aq) wherein the whole perturbating force may be conceived to act, as by a lever, to turn the body about its center. Laftly, it may be obferved here, that the time of revolution about the axis will not, in this cafe, continue accurately the fame; fince a change of the figure mutt neceffarily 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 fun and moon upon the earth, is fo extremely fmall, as to be quite inconfiderable, even in comparifon of the very flow motion of the axis above fpoken of. LE VENA... HIT. Suppofing all the particles of a given ellipfe MF Nf to be urged from a right-line GG coinciding with a given diameter M N, by forces 7 Se i ee u Of ibe Preceffion of the Equinox, forces proportional to the diflances from the fad line, fuch that the force atting at a given difiance a, may be exprefed by a given quantity y; 1 1s required to find the whole efficacy of all thefe forces, to turn the ellipfe about its center OQ. If BC be fuppofed parallel to GG, interfeCting 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 exprefled by — x Vw, or 2 x OD; and con- fequently it’s efficacy to turn the ellipfe about its center. by ~xOD x Ow, or = xODxDV. Let there be taken Cv — 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 former particle at V, gives o xODxDC. Therefore, feeing the joint aétion of any two particles in DC, equally diftant from the middle one I, is exprefled by the /ame quantity ~ x OD x DC, the efficacy of all the particles muft confe- quently be equal to that quantity drawn into half the number of the particles; and fo is truly expounded by = xz0DxDC’. By the fame argument, the force of all the particles in the line BD to turn the ellipfe about its.center, the contrary way, will be mae 40D xBD*. Therefore the difference of thefe two values, = x tOD x BD?— CD’, is the whole force of all the particles in the line BC, to turn the ellipfe about its center (downwards) ; which expreffion, if Ff the conjugate dia- meter to MN be drawn, bifecting BC in E, will become ~ x :0D x BD-+-CD x BD —CD = - x OD x BC x DE. Put, now, OF ==c, OM =—d, FH (perpendicular to MN) = f, OH =g; and let OE and OD, confidered as variable, be de- noted by x and y, refpectively. Then, by the property of the and the different Motions of the Earth's Axis. 9 the ellipfis, it will be, cc : dd:: cc—xx: yy ie ea eau : ce adV co xx and confequently BC = . Alfo (by fimilar trian- ghs\c:firxty=Z; and.¢ +4 i+01 DE ===. Pence c c our expreffion =x OD x DE x BC, derived above, by fubftitut- itg thefe values, becomes = x ef x ce — xx|' x x": and there- Zz fore the whole fluent of = x — x ce — xxl xxxy, or of its equal 4 d e 2 Ex . | . =x ou x CO— xx) XxX i will be the force of all the parti- cles in the femi-ellipfe MFN. In order to the finding of this fluent, let 4 be taken to denote the area of the femi-ellipfe, or, which is the fame, the fluent of ax (ose HE ; then, c € by comparifon, the whole fluent of 24x cc — xxl x xx J, when € c x==c, will be found to be 4x tc*: whence shat of our given expreflion, % x “£2 xce— xxx x’x/, muft confequently be - a c ¢ Ss Me pls RN eee ee ee ee = 2 xEx CA 7X GA 7% +FH x OH x 4;,. the double of which, or = x {FHx OH x area MFN/M,, is there- fore the true meafure of the whole force whereby the ellipfe tends to move about its center. Q,E.L COROLLARY I. If the fame value be required by means of the angle AOM included between the diameter MN and the principal axis AOg (fuppofed to be given) ; then let POp and FR be drawn per- pendicular to.OA, and’ TF to OT, meeting OA produced, in i fuppofe L to be the interfeGtion of AO and FH; and let Fig. 6. the fine and co-fine of the faid given angle AOM (to the ra- dius 1) be denoted by ™ and 2, refpectively. Becaufe FL is perpendicular to the tangent TQ, we have, by the property af Pas | C ! the IO Of the Preceffion of the Equinox, the ellipfis, as AO* : AO?—- OP’ :: OR: OL:: OR x OQ (AQ’): OLxOQ; and confequently AO*—-OP*=OL x OQ. But. 22.7... OL: OF 3 and ‘To. 500 : OF (FH); “whence, by compofition, 1 : m2 :: OL x OQ (= AO*— OP*) : FH x OH = mn x AO*— OP’: and fo, by fubftituting this value above, we get ~ x cs x AO*— OP* x area of the ellipfe, for another expreffion of the required force. COROLLARY U. Hence may be eafily deduced the force by which the fphe- roid, generated by the rotation of the ellipfe about its leffer axe Pp, tends to turn about its center, when all the particles are urged from a plane GG paffing through the center, by forces proportional to the diftances from the faid plane. For, as any fection of the fpheroid, parallel to the middle one ApaP, is alfo an ellipfe, fimilar to it, the area of that fection will be in pro- portion to the area of ApaP (which I fhall denote by 2.) as the fquare of its greater femi-axe, to the {quare of the greater femi-axe OA of the given ellipfe PApa: fo that, if OA be denoted by 2, PO by 4, and the diftance of the faid {ection from the center of the fpheroid by w, we fhall have, aa : aa—uu (== {q. greater femi-axis of that fection, by the property of the circle) :: 2: Qx aq uh the area of the fec- aa tion. Moreover, by reafon of the fimilar figures, we have aa — bb aa: aa—bb:: aa—uu: Xda—uu, the difference of aa the fquares of the greater and leffer femi-axes of the fection. Therefore, by fubftituting thefe values in the above general aa—bb . ~ aa ise aA == Un X da—uu X QxX -—— expreflion, we get — x = x (en 2 x SxS a 2x eet for the force of all the a. aa aa particles in that fection to turn the body about the common aXis of motion ftanding at right-angles to the plane PAga. This quan- tity, drawn into 2, will, therefore, be the fluxion of force of the : femi- and the different Motions of the Earth s Axis. Il femi-{pheroid in which that fection is; whofe fluent, when u=a, ; —bb will be found = =x ne x a xQx = : the double where- 4aQ of, or = X mn X aa— bbx == muft confequently be the re- quired force of the whole fpheroid : which force, as Q x i is known to exprefs the content, or maf{s of the fpheroid, will alfo be truly defined by . x= x aa — bbxS; S being put (as in the preceding Lemmas) to reprefent the faid content of mafs. PROBEEM' L To determine the efficacy of the fun’s attraétion, on a corpufcle, any where in the body of the earth, to turn the earth about it's center. Let CDHE reprefent the earth, C the center thereof, 5 that Fig. 7. of the fun, and GCG a plane perpendicular to the line CS joining the centers of the earth and fun; let D be the place of the corpufcle, and upon the diagonal SD let the parallelogram QCSD be conftituted; producing QD to meet GCG in K. If F be taken to denote the fun’s abfolute force on a particle at the center C, his force on a particle at D will be f’x a which may be refolved into two others, the one in the di- reGion DC (which has no effeét at all to turn the earth about its center); and the other in the direction DQ, exprefied by Fx Sc Xa from which the force F, in the parallel di- rection CS, being deducted, the remainder f’x sae be the true meafure of that part of the force in the direction DQ, whereby the particle at D tends to change its pofition with refpect to the plane GCG. But this value is reducible ST to Fx OO ae which, as SC — SD (by reafon of the great diftance of the fun) is nearly equal to C32 DK, will 12 Fig..7. Of the Preceffion of the Equinox, DK x 38D* DK DK, will become = F x x Apxy. But the time wherein the fun’s longitude -sS is augmented by the particle z will be T x ai which being wrote in the ‘ é room of 7; we thence have 360° x = x ? X xyz: and this, by x fubftituting “1 — xx, and = inftead of their equals y I—wx* and , will be farther transformed to 360° xX me x 2 X XX. Whofe fluent, 360° x re =e is confequently the true quan~ tity of the nutation that was to be determined. Again, with regard to the preceflion of the equinox, the in- creafe thereof (by the Corol. to the precedent) being in propotti- on to the correfponding decrement of the inclination of the equator to the ecliptic, as the tangent of =A to the fine of 4%, or, in fpecies, as — to p, it therefore appears (by multi- I— XxX : plying the fluxion of the nutation by —f—) that 360° x Pp I—K*Xx a x 4 —X*— will be the fluxion of the quantity under con- 4 e I — xx za 1 Xx —_——_——_—_—— fideration ; whofe fluent, which is 360° x os x = x A is therefore the preceflion itfelf. 2, £. J. COROL. and the different Motions of the Earth's Axis. COROLLARY et When the fun arrives at the folftitial point D, the value of x being = 1, and that of z — +e, the quantity of the precefli- 3 kg 1 on becomes barely equal to 360° x GT 3 whofe quadruple, 360° Mo x kq, will be the whole of the annual preceffion, depending on the fun ; which, in numbers (by making ¢= 1, T = 3661, g = .91723 —'co-fine of 23°28", @ = 231, aa — bb 2 fl = ) = ai) comes out 21° 7". But it will appear from what follows hereafter, that this quantity, derived on the hypothefis of an uniform denfity of all the parts of the earth, ought to be reduced to abowy142", to agree with obfervations. b= 220, and=k(== COR OLD #RY* Ib Since the preceffion during ith of the annual revolution is found to be 360° yee we have as i¢: z:: 360° x 34 x 40 4 40 a (360° x x wo x 2) the mean preceffion during the time of defcribing the arch: which being taken from the true fp ere preceffion, 360° x = = XZ—x/1—xx, the remainder, 360° x x s x—x/1— xx, will confequently be the egua- 4 é tion of the preceffion; which therefore is to the mean pre- ceffion, as —x/ 1 — xx 2 Z OF aS — 2n/ I — xx 2 2%; that is, as — fin.2z:2z. But the mean preceffion, in the time of defcribing the arch z, is 21” 7’"x = (or rather 142"x 2) 2é 2e by Corol. I. ‘Therefore the equation correfponding will be — 2? x fin. 22 = — 1742” fin. 22, when the denfity is 4e taken as uniform ; but when taken to correfpond with the obfer- ° . : 142" vation, it will be — *2_ x fine 22 ——1"10'’x fine 2z. Hence 4e D it 18 Of the Preceffion of the Equinox, it appears, that the greateft equation of the preceflion (when the fun is in the mid-way between the equinox and folftice) is 1710”; and that the general equation (which is fubtractive in the firft and third quadrants of the ecliptic) will be in pro- portion to the faid greateft equation, as the fine of twice the fun’s diftance from the equinoctial point is to the radius. COR OAL AR YY... a. Furthermore, becaufe the quantity of the nutation is, univer- kpx? 22. poffible, when the fun is in the folftice and x is the greateft pof- fible : after which it will decreafe, according to the fame law whereby it before increafed; ’till, on the fun’s arrival at the other equinoctial point, it intirely vanifhes, and the inclination is thereby reftored to its firft quantity. It is alfo evident that the quantity of the nutation will, in all circumftances, be in , it will therefore be the greateft fally, equal to 360° a x _ proportion to the preceffion, during {th of the fun’s revolution, as PE to f, or as £* to <, that is, as the product under the {quare of the fine of the fun’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 go degrees. According to which pro- portion (taking the faid preceflion = jth of 14+”) the greateit nutation comes out one /econd, very near; the inclination of the two planes decreafing from the time of the fun’s leaving the equinodtial points, to his arrival at the folftices, and that in the’ duplicate ratio of the fine of his diftance from the faid equi- noctial points. It may be'obferved that, in order to aveid trouble, the quantitiés p and q are taken as conflant ; the error, or difference thence arifing {carcely amounting to +; ;- +. known to exprefs the mafs or content of the eae} 4 ° . ° ° x 1") a a) —— 23 x b*yy — y3y in the room of its equal — x3x, our fluxion is is found equal to fpheroid, the momentum of rotation of any {pheroid about its axis appears, therefore, to be juft the fame as would arife, if 2ths of the whole mafs was to revolve at the diftance of the higheft point (A) from the axis of motion. PR O:BILEM . Vv. To determine the alteration of the pofition of the terreftrial equator, arifing from the aétion of the fun on the whole mafs of the earth, during an tnftant of time. “ ct 22 Fig. 126 equator or ring, will be 6 x Of the Preceffiion of the Equinox, Let OAPap be the earth, under the form of an oblate fphe- roid ; let AlaL be the plane of its equator, and HICL a plane paffing thro’ 5 the center of the fun, making '‘right-angles with the plane of the meridian HAPC and with the plane GG; It is found, zz Prob. I. that the force whereby a particle, at any point E in the equator AELaI, tends from the plane GG, is in proportion to that refpecting the higheft point A, as the di- {tance EF to the diftance AK, or as ED to AO (fuppofing 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, muft 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 cofine of the arch AE to the radius. But this, dy Corol. I, Lem. II. appears to be the law of the forces under which a ring of particles AELaI, detached from the earth, may continue zm eguilibrio, in the fame plane, under a twofold motion about the center O, and about the line Ag as an axis. Imagine now this ring to be exceeding denfe, fo that its momentum of rotation about its center O, may be equal to that of the earth itfelf, or fo that the two bodies may equally en- deavour to perfevere in the fame ftate and direction of motion, in oppofition to any new force impreffed. ‘Then it is evident, that, were the forces whereby the two bodies tend to turn about the line LI, through the fun’s attraction, to be g//o equal, the ame effect, or alteration of motion, would be produced in both ; and confequently, that the effects produced, when the forces applied are unequal, will be in proportion direétly as the forces. Now the force whereby a particle at A is urged from the plane GG, is found to be 6 x = x as (by Prop. I. Co- rol. I.) ; which, in a dire€tion perpendicular to the plane of the ini Naa oi en x 3 x mn. TT* AQ ,AO ot. Therefore, the force acting on a particle at E, in a like di- reCtion, being exprefled by 6 x ae X IN Xam the effect thereof to turn the ring about the line IL will be exprefled by and the different Motions of the Earth's Axis. 23 by B xo x mn x ; which being as the {quare of the di- ftance ED, it follows (from the Corol. to Lem. IV. ) that, if Mbe taken to denote the mafs 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 fun on all the particles, will be truly defined by Bx =e xmnax+M. Again, becaufe Bx ip x mn is the force acting on a particle at A, in a direGtion perpendicular to the plane of the ring, it is evident, from Corol. I. to Lem. II. that the ring will, in confequence of that force, have a motion about the line Ag as an axe; whofe celerity will be to the ce- Pilerity of the other motion about the center, in the proportion att : ee ee a of r to 1, or of orp Xm tor; becaufe, BX aa X mn be- ing == 8x 2r, 7 will here be — ae x mn. ‘Therefore, if N be afflumed to denote the fun’s force to turn the earth about its center, we fhall (from the above obfervation) have, tt I : ‘ ° eee Bx Tt x mna x ” which, therefore, will be known, when the relation of z, W, and D’is affigned. If w be fuppofed conftant, or, which is the fame; if all the firata are conceived to be fimilar to one another, then our ex- BiiRi ce HU w* x fluent of D X flux. 2° unity) will be truly defined by refiion will become WO TT Bey Se iat Pha ee . 5a ae 1 w xX fuent of D x flux. 2° gmntt ~ w— w* O17 Pe RE 3tt. mn x aa— bb b St ye ee SK IE SS SX ecaufe 2TT Ww 2tT bk aa ( b ; i 10 <= 20) :. which conclufton appears: to be the very fame with a \ that found when the denfity was fuppofed uniform.. From whence it is evident that an increafe or decreafe of denfity, in going towards the center, makes no fort of difference here ; provided the furfaces of the feveral_/frata are all fimilar to one another and to the furface of the earth. If indeed the /rata ate diffmilar, the cafe will be otherwife ; as will be feen by the following example: which ought not be looked upon as a matter of mere fpeculation ; fince it will appear, in the fequel, that the preceffion of the equinox cannot be accounted for, fo as to agree with the phenomenon, .apon the fuppofition. of an uniform denfity of all the parts of the earth; the refult, this way, coming out about +d part greater than the real quantity, determined by obfervation. Let then, as before, the greateft femi-diameter of any ftra- tum be denoted by z, and let the leaft femi-diameter. (lying in the axe of the earth) be in proportion thereto as 1— Az? to unity; alfo let the denfity be fuppofed to increafe, in approach= v ing the center of the earth, in the ratio of r—az—1x—; fo U a as to vary according to fome power (z”) of the diftance, and that the meafure thereof at the center, may be to that at the fur- face, in any given ratio of sto1. Then, by taking a, , and v, as conftant quantities, and writing 1 — 22’, and *— m—I1 x eer oe ad av inftead of their equals w and D, we ‘thall here have D and the different Motions of the Earth's Axis. ee U D x flux.ws}— wixs = —a7—Ix-ix O-+ ex 2rzr+H43, a. ‘ nearly (becaufe, to render the calculus lefs laborious, the terms involving 2° and 43 may be here neglected as inconfiderable) ; the fluent of which expreffion, when z = a, will be found Bh ane eae P+5 s —77x 27ers — BT DRO ARGS Jae Feo rire Ts, 2rarts, P+ots D thedks Moreover we have, in this cafe, Dx fluxion of wz5 — a ee z ee v x 5243 — 5 + Ox azetis = # - 2-1 x — a x 5242, nearly (becaufe 5 + @ Xazets, as the earth is nearly {pherical, is inconfiderable in refpect of 52*%); whereof the Cee 7 v a fluent, when z =a, will be had == wa8 — 7 PSE PFT S yg, “at tao ie ake Now let thefe two values be fubftituted in the general expreffi- & I gmntt x fluent ef D x flux. wx3—wiz5 2TT fiuent of D X flux. wz? mutt i comes = x Pes Cae Mas A v+o+5xur+5 b ; 9 (ae -} will be == I — aa’, and w* == 1 — 2Aa?, nearly; on ; by which means it be- x 2aa*. But, when z= = a, 2 b — bb whence we have 2ag? = 1 — w —=1 —= os ffene ; and fo, by fubftitution, our laft formula becomes gmntt vr+o+tsxvt5. aa—bb 2TT ! x Ba es Coe Re SNe ee an requited motion, in this cafe, is to the motion, when the den- fity is fuppofed uniform, in the proportion of VATE BT DMs ty | PP | | Pe vfe-+sxur+s to unity.—From this proportion a great number of Corollaries may be drawn ; but thefe will be, more ‘properly, confidered hereafter. Whence it appears that the E 2 PRO- 29 28 Fig. 13. Of the Preceffion of the Equinox, PROB LE Me Vis To find the quantity of the preceffion of the equinox, and alfa that of the nutation of the earth’s axis, caufed by the moon, dur- ing the time of balf a revolution in her orbit. Let fFNcE be the orbit of the moon (on the furface of the fphere) interfeCting the ecliptic = Wy in N; and f{uppofe F:DE/ to be the pofition of the equator, on the moon’s paffing it at F, and féDea the pofition thereof when fhe re- pafleth it again at e: let moreover the quantity of the annual preceffion arifing from the fun (given by Prob. IV.) be denoted by 4; and let the ratio of the denfities of the moon and fun be exprefled by that of m to unity: then, taking r to reprefent the given time in which the moon is moving from F to e, the mean quantity of the preceffion, arifing from the fun, in that time, will be a x A: and therefore, fince the perturbating forces of the fun and moon are as the denfities (by Prod. I. Corol. II.) it is evident that the preceffion (Ee) caufed by the moon, in the fame time, with refpect to the plane of her own orbit, would be truly expreffed by m x = x 4, were the or it’s inclination to the equator to be always the fame as that of the ecliptic to the equator: but, fince the magnitude, as well as the pofition, of the angle E varies, with the place of the node, the faid quantity mx x4 muft therefore be dimi- nithed in the ratio of the co-fine of E to the co-fine of ¢P (as ap- pears by the faid Prob. IV.) and then we fhall get ue pe es co-fin. p” for the true value of the preceflion Ee, caufed by the moon, with refpect to her own orbit. But now, in order to refer this to the ecliptic, it will be re- quifite to obferve firft of all, that, as the inclination of the earth’s axis, at the end’of every half revolution, on the return ef the fun or moon again: into the plane of the equator, is re~. ftored to its former quantity (by Corol. III. to Prob. IV.) it fol- lows, feeing the angles E, ¢, F, f are thus equal, that the triangles; and the different Motions of the Earth's Axis. 29 triangles DEe and DfF will alfo be equal and alike, in all refpects ; and fo, DE + De being = DE +- DF =a femi- . circle, both DE and De may be taken as quadrantal, or arcs of go° each : whence, if VER, the meafure of the angle 7, be fuppofed to meet 2ED in 7, it will be, as_ fz. ED (radius) mtA _ co-fin. E mtA — fin. E Xco-f.E : fin. e(E) <: He (may oo) BD? ert alfo, as fin.a(P) : fin. PD (co-fin. PE) :: ED? =a S= mrA fin. E x co-fin. E x co-fin. rE ts fin. y x co-fin. y xX rad. preceffion : And, as fin. r (radius) : fin. DR (PE) :: EDe(RDr): Rr A fin. -fin. E x fin. rE j ‘ atmriy, fin Ex cota = N= hie correfponding quantity oF , the required quantity of the man x a co-fin. y» x rad. the nutation, or the decreafe of the inclination of the equator to the ecliptic. Q, 2.1 COROLLARY. It is evident from hence that the quantity of the nutation is fin. yE , co-fin. rE eceffion, a aes to that of the preceffion, as —{— to ———-, or as fin. to rad. x co-fin.vE . that is, as the fine of ¢P to the co-tangent of fin. YE E. It appears moreover (becaufe fiz.E : fin. PN :: fin.N : fin. PE, p. fpherics) that the former of thefe quantities is ; A fin. N x fin. WN -fin. E 5 alfo truly explicable by a x Se te eee ere : which co-fin. y x rad.} expreftion will be of ufe in the following Problem. PROBLEM VIL To determine the preceffion of the equinox, and the quantity of the nutation of the earth's axis, caufed by the moon, during tbe time of half a revolution of the node of the moon's orbit. Things. being fuppofed as in the preceding Problem, let the Fig. re- diftance (PN of the node from the equinoétial point be denot- ed by 2, its fine by x, and its co-fine by y; let alfo the fine of the angle N°PE be put = 4, its cofine = b, the fine e 30 Oj the Preceffion of the Equinox, N = c, its co-fine = d, the femi-periphery T 84 =e, and the time of half a revolution of the node = R. If °Q be fuppofed perpendicular to NE, it will be, p. /phe- rics, as co-fin. NY (y) : radius (1) : : co-tang. N =) : tang. NYPQ= . ; let this be denoted by 4, then the fecant of the | h ; oneney ae -1tS fame angle will be = /1-+ 6, its fine = i-- bh T s co-fine =? whence, by the known rules for finding 1-+-)p the fine and co-fine of the difference of two angles, the fine of EQ will alfo be had = oe and its cofine = pe V1 bh VW 1-+ bp Whence, again, p. /pherics, we have, as fin. NPQ: fin. EPQ :: co-fin. N : co-finuE SAAS a bd —% —bd—acy; alfo, as.co-fin. N°9PQ_: co-fin. E7Q:: co-tang. Nop cs : > COLane.. Oi ee Doe 14X% = ad + on becaufe ).—= “ x cx But, by the Corol. to the laft Problem, the quantity of the nu- . 3 : fin, Nxfin.yNxco-f.E tation for the time 7, is exprefled by lay en Ss — 5 ; T ~* co-fin. NvE x rad. which, in algebraic terms (by fubftituting the above values), will become oe x 1 Se But the time +, during which the longitude (z) of the node is increafed by %, being to R the time. of half a revolution of the node, as 2 to e, its value will therefore be expounded by Rx, or its equal R xX —=——=: é EV I —— KX and fo by fubftituting this value, and writing /1 — «x in the room of y, our laft expreffion will be reduced to a xe x T eb pe bdxx , : w — acxx; whofe fluent “48 x © x dd bd/ 1-xx - Lacxx /1— xx ce eb and the different Motions of the Earth's Axis. (=a— — x = x bdx verted fine z— ae Xverled fine 22) is srt nthe the meafure of the nutation, or the decreafe of the inclination of the equator to the ecliptic, caufed by the moon, from the time of the node’s coinciding with the equinoctial point °P, to its arrival at the pofition N. Again, with regard to the preceflion. of the equinox, the in- creafe thereof being in proportion to the decrement of the in- clination, as the co-tangent of YE to the fine of NPE (é4y A Aes the Corollary. to the precedent) or, in {pecies, as ge (or ' cx ad +b be 1 x8 I x%x pet - to a, its fluxion will therefore be had by mul- tiplying that of the nutation, given above, into irk! Beat cx and fo is found to be 7 mAR. 1. abd? x — oT EXT = TH — aa x cd — abc? tin i — xx 5 whofe a which is ns a Xp x abd? 4+-bb— aa x cde — + abe’ 2— abe" xf 1 — xx eat oA ation BOLE age aE RS) muft confequently be the preceffion itfelf. But,. at the-end of half'a revolution, when the node N ar-. rives at the other equino¢tial point 4, both this, and the ex- preffion for the nutation, will become much more: fimple, x being then = 0, and z =e; whence the nutation will be = mAR : 7 _ mAR 2¢cd OX Gp ee oss op Ra R —— eaten = x dd — £e¢ = oe xX 1 — icc (becaufe’ ce + dd — r). Q) BL. ; and the preceffion equal to COR OL LARRY «1. It appears from hence, that the mean preceffion of the equi- nox, arifing from the action of the moo n, is In proportion to what it would otherwife be, if the moon’s orbit was to coin- cide with the ecliptic, as 1 — 2cc to. unity: whence. the true alue: Qi Of the Precefion of the Equinox, value thereof is to that depending on the fun, in a ratio com- pounded of the ratio of the denfity of the moon to the denfity of the fun, and the aforefaid ratio of 1 — ice (or 0,988) to unity. COROLLARY 11. It appears likewife, that the whole quantity of the nutation, . in half a revolution of the node, is to the correfponding quan- tity of the preceffion, as a to dd — icc, or as unity to é x X=; that is, as the radius to the excefs of the co- tangent, above half the tangent of the orbit’s inclination, drawn into (1.5708) the meafure of half the periphery of the circle whofe diameter is unity. This proportion, in numbers, fup- pofing the mean inclination of the orbit to be 5° 8’, will be found to be as 10 to 174, very near. COROLLARY II. Moreover, feeing the preceffion in half a revolution of the node is nak xdd— *¢c, we have, ase: % 3: a x dd — 500 : nla See —-cc, the quantity of the mean preceffion é during the time in which the node moves over the arch 2, or oN. This being fubtracted from the true preceffion, found above, the remainder ——— ke O_O mm se x = x bb — aaxcdx — tgbeix/1—xx will confe- quently be the equation of the preceffion, or the excefs of the true above the mean: which equation or excefs, if we neglect the term — Labe*x/ 1 —xx (whofe value, by reafon of the {mallnefs of c*, never amounts to ith of a fecond) will evi- dently be at its greateft value at the end of ith of a revolu- tien, on the node’s arrival at the folftice ; when it becomes R See * : : ae x ~ x bb —aa xcd; and, is therefore, in proportion to mAR ., 2¢d =; % —>» the whole, or greateft quantity of the nutation, during half a revolution of the node, as bb — aa : 2ab, or as and the different Motions of the Earth's Axis. 2ab Bieta: the inclination of the equator to the ecliptic. aS ad. s , that is, as the radius to the tangent of double aa \ COR GLE AR Y...1V- Furthermore, fince the value of c (the fine of the orbit’s inclination) is but fmall, the laft term of the general ex- preffion for the nutation, as well as that for the excefs of the true preceflion above the mean, may be rejected, without producing any confiderable error ; whence the nutation is re- AR Srnec ae = x @ x }—/1— xx, and the preceffion to. é duced to meee he bb aa'x eden Vente at appears that the de- a abe creafe of the inclination, from the time of the node’s leav- ing the equinottial point 9, will be as the verfed fine (1 — /1— xx) of the node’s true longitude ; and that the excefs of the true preceffion above the mean, will be always as the fine (x) of the fame longitude. SCHOLIUM. The quantity of the annual preceffion of the equinox ari ing from the force of the fun, is found in Prob. IV, to be 21” 7”; upon the fuppofition of all the parts of the earth be- ing homogenous, and in a {tate of fluidity. If, therefore, this quantity be taken from (50°) the whole, ob/erved, annual pre- ceffion, arifing from the fun and moon conjun@ly, the remain- der 28” 53°” will confequently be the mean annual preceffion depending on the moon; which being increafed in the ratio of 1000 to 988 (according to Prob.VII. Corol.I.) gives 29°14”, for the quantity of the preceffion, if the orbit of the moon were to coincide with the plane of the ecliptic. Hence it will be (by the fame Corol.) as 2.1'°7""is to 29"14”, fo is the denfity of the fun to the denfity of the moon, according to this bypothefis. But it is evident from experience (whether we regard the proporti- on of the tides, or the accurate obfervations of Dr. BRaptry) that the denfity of the moon in refpe&t to ‘that of the fun, cannot be {fo {mall as it is here affigned. t ss Of the Preceffion of the Equinox, It is true, there is no way of knowing the exaéd? ratio of the denfities of the two luminaries ; fince theory, for want of fuff- cient data, fails us here. And as to the method, dy objferving and comparing the fpring and neap tides * (whether we regard the quantities or times of them) it cannot be otherwife than very precarious ; confidering the many obftacles and intervening caufes by which hey are perpetually, more or lefs, influenced and difturbed. Upon the whole it therefore feems to me, that the beft method to fettle this point (as far as the nature of the fubject will allow of) is from the obferved quantity of the nu- tation itfelf; agreeable to what has been hinted on this head by that celebrated Aftronomer, to whofe accurate obfervations we owe this important difcovery. Let us, therefore, take g to denote the greateft nutation of the earth’s axis, as given by obfervation ; and then, if f be taken to reprefent the mean annual preceffion, gven im lke manner, it will appear (by Prob. VII. Corel. I[.) that a= “4 at x g is the part of the faid annual preceflion depending on the moon ; whence the remaining part, owing to the fun, muft neceflarily be fae axe (= tee Therefore we have (dy Pro-. 93 Zt blem VIL. Corol. I.) as BPE Uih50P agi dif or 58g hte ¢ oe a 16°8 3 3 — 58¢ IGO0O “983° the fun; which, in numbers (making g — 18”), will come out as 2,09 tor. But if the value of g be fuppofed only a fecond or two greater or lefs than 18”, the refult will be fenfi- bly different, as may be feen in the annexed Table ; wherein, befides the ratio of the denfities, are alfo exhibited the mean quantities of the annual preceflion, depending on the forces of the fun and moon, refpectively ; together with the greateft equation of the faid preceflion, as given by Problem VII. Co- rollary III. x to 1, fo is the denfity of the moon to the denfity of * SirIsaac Newron, by this method, makes the proportion to be a8 44 to1; and M, Danizu Bernov.ut, only as 21 to 1. Greateft and the different Motions of the Earth's Axis. Ratio of the} Annual precef- | Mean annual } Greateft equa- Greateft| denfities -of | fion caufed by | preceffion cauf-| tion of the pre- nwtation.| the fun and | the fun. ed by the moon.| ceflion caufed moon. ; by the moon. Seconds Seconds | Third: | Seconds | Thirds] Seconds | Thirds 16. 1 P PGE] 20" 4 8 29 57 140) 58 rye E77 ees 21 340 15 64 18 4 yes B69.) 10°" 19 BQ NGT 16 50 19 FE L2; FO) 14: 27 4. 35 038 17 46 Bo” | 4 PROTA 19, 2 BF B7IE Ios 18. 42 Were I to deliver my opinion which of the different cafes here put down anfwers beft to the phenomenon, and the gene- ral law of gravitation, I fhould, without hefitating, fix upon that preceding the laft; which, upon the whole, will be found to agree better with Dr. BrapLey’s obfervations than any of the others: befides, though the obfervations on the tides cannot be relied on to any great degree of exactnefs, yet, by them, it is fufficiently evident, that the perturbating force of the moon cannot be to that of the fun in a lefs proporti- on than of about 2; to 1. From the greateft nutation, and the greateft equation of the preceffion, grven above, the quantity of the nutation and the equation of the preceffion, correfponding to any given pofition of the lunar node, may be very eafily determined : for, fir/f, it will be, dy Corol. IV. As the radius is to the fine of the node's diftance from the neareft equinottial point, fo is the greateft equation of the preceffion to the equation fought. , Which muft be added to the mean preceffion when the node (viz. the afcending one) is in any of the fix fouthern fiens ; but fubtracted, when in any of the fix northern ones. Secondly, it will appear, by the fame Corollary,. that the de- creafe of the inclination of the equator to the ecliptic, from the time of the node’s coinciding with the equinoctial point 7, is proportional to the verfed fine of the node’s prefent diftance from that point: whence it follows, that the faid inclination will be at its mean value when the node is in the folftice ; and confequently, that the difference between the mean, and true FE 2 values, 35 36 Of the Preceffion of the Equinox, values, will be as the difference between the verfed fine-of the node's prefent diftance from P, and the verfed fine of go de- grees, that is, as the co-fine of the node’s diftance from 7. Therefore, to find the nutation at any given time, it will be, As the radius 1s to the co-fine of the node's diflance from the neareft equinoctial point, fo 1s the greateft nutation to the nutation fought. Which, to have the true obliquity of the equator to the ecliptic, muft be added, when the node is in any of the fix afcending fiensW, wt, #,, 5, IL; but, otherwife, fubtracted. The following Table, fhewing by infpection, as well the equation of the preceffion, as that of the obliquity of the eclip- tic, is computed from the proportions here laid down ; upon fuppofition that the greateft quantity of the nutation is 19 feconds. The Hquauon ot the Preceftion ot the |] ‘ihe &quation of the Ubuquity or me Equinox, Eclivtic. p’s 8 | Sie. o | Sig. I | Sioa TE L-Subtre pls shes e So. Ot py oig. da ote. b Ad from ¥ | Sig. Vi {Sie VIl|sig.VIL| Add || from v | Sig. V1 | Sig. Vii [sig. VL] Subtr. Deg. | Seconds] seconds Seconds Deg. Deg. |Seconds|Seconds|Seconds} beg. o ©,0 ° 8,8 reve 30 ° 935 8,2 457 30 5 1,5 10,1 16,1 25 5 94. 7,8 4,0 2m 10 3,0 L146 Fes 7, 20 10 933 753 $49 20 15 455 i236 oe 15 15 Q>2 6,7 2 sh. 15 rb ZO 6,0 13,6 E58 10 20 9,0 6,1 1,7 10 as TR CARS AAI BS S87 UAY Bh AGO 5 30 8,8 he. 2 1757 ° 30 8,2 Ae 0,0 o Subtr. | Sig. V | Sig. LV ]Sig. Ui} y’s 8 f] Subtr. | Sig. V Sig. IV] sig. LL] »’s Q | Add |Sie. X1] Sig. X Sig. 1X | fromy |{ Add Sig. XI] Sig. X [Sig 1X]from + I eR RS SS ESTE SRE SNIPES 2 EET ED Fig. 15. To place what has been delivered above in another view, fuppofe PE to be equal to the mean diftance 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 greateft nutation ; and about AB, as an axis, conceive an ellipfe ACBF to be defcribed ; whofe 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 reprefents the mean place of the pole of the equator, the true place will always be found in the circumference of the faid ellipfe. And if, on the diameter AB, a circle ADBG be alfo defcribed, and the angle APS be made equal to the diftance of the node from the equino¢tial point 7; then, I fay, a perpen- dicular SRP, falling from the point S upon the diameter AB, will and the different Motions of the Earth's Axis. will interfect the circumference of the ellipfe in the point where the pole of the equator (f) is, at that time, pofited, For, firft, it is clear, from what has been already remarked, that AE and BE will be the greateft and leaft diftances of the two poles, as being equivalent to the refpective inclinations of the two planes, the equator and the ecliptic: from whence and Corol. IV. it is manifeft, that ER or Ep will be the true diftance of the . faid poles, when the verfed fine of the node’s diftance (APS) from 9? is AR. Moreover, éy con/truétion, CP : AB :: 4 the co- 2 fine 2EP : co-fin. EP; that is, in fpecies, CP : AB: al : 6. And, p. fpberics, tang. PEC +: tang. PC (:: PEC = PC, nearly) :: rad.(1):a@. Therefore, by compounding thefe two proportions, we have PEC: AB:: CeO gh 5 aaa: 2 2ab: which proportion, for finding the angle PEC, is the very fame with that determining the greateft difference of the mean, and true longitudes, as given by Coro/. ITI. Whence it eafily follows, that the angle RE will exprefs the difference of the mean and true longitudes, at the given pofition of the node ; fince, as the radius : fine APS (:: PD.: RS :: PF : Rf): : the angle PEC : the angle RE#, as it ought to be, dy Corol. IV. The ratio of CF to AB is here determined to a geometrical exactnefs, as no-ways depending, either, on the denfity of the moon, or on any other phyfical hypothelis. Having now laid down the general proportions for the nu- tation of the earth’s axis, and the preceffion of the equinox, I fhall here fubjoin the neceflary rules for determining how much the declinations and right-afcenfions of the ftars are af- fected by thofe inequalities. | 1°. For the alteration of a ftar’s declination, and right-afcen- fion, arifing from the nutation of the earth’s axis ; it will be As the radius is to the fine of the ftar’s right-afcenfion, fo 1s the nutation (or the given alteration of the equator’s inchnation to the ecliptic) to the alteration of the ftar’s declination, caufed by the nutation ; And, as the co-tangent of the ftar’s declination is to the co-fne of its right-afcenfion, fo is the nutation to the alteration of the frar’s right-afcenfion, correfponding. 2, ae He ia itt 1 Ha { i ae ne eee en ae a Ne Of rhe Preceffion of the Equinos, 2°, For the alteration of the ftar’s declination and right-af- cenfion, arifing from the preceffion of the equinox’; it will be As the co-fecant of the obliquity of the echptic is to the co-fine of the fiar’s right-afcenfion, fo is the preceffion of the equinox (or the alteration of the frar’s longitude) to the alteration of the fiar's declination, caufed by the preceffion ; And as the co-fine of the flar’s declination ts to the co-tangent of its angle of pofition, Jo is the alteration of declination, found by the loft. proportion, to the alteration of right-alcenfion, anfwering thereto. Any one, but little acquainted with the fphere, will eafily fee when thefe equations are additive, and when fubtractive : nor will it be at all difficult to comprehend the réafons upon which they are founded ; they being nothing more than fo many particular cafes of the general relation fubfifting between the fluxions of the fides and angles of a fpherical triangle *. It will not, however, be improper to remark here, that, when the quantity of the preceffion, in the fecond of the preceding cafes, amounts to fome minutes, it will be neceflary, in order to have the conclufion fufficiently exact, to make ufe of the mean right-a{cenfion, at the middle of the given interval; which, from the given right-afcenfion at the beginning of the interval, may be eftimated near enough for the purpofe, in moit cafes, without the trouble of acalculation: but in other cafes, and when the utmoft exaétnefs is required, it will be neceflary to repeat. the operation. It may not be improper to obferve likewife, that, befides the equations depending on the pofition of the lunar nodes, com- uted above, there is a fmall motion of nutation and preceffion arifing from the moon’s declination ; whereof the greateft quan- tity is to the greateft quantity of zhat depending on the fun, in a ratio compounded of the ratio of the denfities of the two bodies, that of their periodic times, and that of the fines of the inclinations of their refpeCtive 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 circumftance, amount to more than about +th of a fecond; a quantity too {mall to merit attention in the practice of Aftronomy. soe * See my Dodtrine of Fluxions, Part I. Se. I. Remarks and the different Motions of the Earth's Axis. Remarks on fome Particulars in the preceding Theory and Calculations; in order to explain and obviate certain difficulties and objettions that may thence arife. FT may be obferved, in the firft place, that we have, all along, confidered the effects of the fun and moon fepa- rately; and, confequently, have fuppofed them to be no-ways influenced or difturbed by each other. This may feem too bold an affumption ; efpecially, as it is known that the tides, which are produced by the very fame forces, depend upon, and are greatly varied by, the different pofitions of the two lu- minaries. To remove this objection, let PSM reprefent the plane of the earth’s equator POs, its interfection with the plane of the ecliptic, 9S the right-afcenfion of the fun, and 7M the right-aicenfion of the moon; and let the forces of the two bodies to turn the earth about its center, in thofe pofitions, be reprefented by fand F, refpectively. | Thefe forces may be confidered as acting perpendicular to the plane of the equator in the points S and M, and will be equivalent to, and have the fame effect with, one fingle 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, exprefled by f+ Fx ‘al or f+ fx as (fuppof- ing NQ, PR, as alfo SB and MC, to be perpendicular to POR): ) But the quantity of the preceffion, during a given moment of time, is known to be as the force, and as the fine of the right-afcenfion, conjunctly (by Prob. III.) ; from whence the two quantities arifing from the fun and moon, confidered fepa- rately, are expounded by fxSB, and Fx MC, refpectively. But, fuppofing both bodies to aé& together, or, which is the’ fame, fuppofing one fingle force, exprefled by f+ Fx ey to oF Fig. I7e Of the Preceffion of the Equinox, ta act at P, the quantity of the preceffion will then (by the very fame rule) be truly defined by f+ Fx oe x PR, or its ee equal f-+ Fx NQ; which quantity, by the property of the center of gravity, 1s known to be equal to fx SB + Fx MC. Hence it is manifeft that, whether the forces of the lumind- ries be joined together, or treated apart, the refult will be the fame. The next difficulty, relates to the excentricity of the lunar orbit, and the inequality of the motion in that orbit; which . may be thought fufficient to occafion a fenfible deviation from rules founded on a fuppofition that pays no regard to them. In order to clear up this point alfo, imagine ADBE to be an ellipfe, in which the moon is fuppofed to revolve, about the center of the earth, placed in the lower focus F of the ellipfe : let AB be the tranfverfe axis of the ellipfe, 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 {mall (given) angle DFd with each other. The perturbating force of the moon, at the diftance DF, will (dy Prop. I. Corol. IT.) be, inverfly, as the cube of that di- ftance ; and the time of defcribing the given angle DFd will, it is well known, be directly as the {quare of the fame diftance. Therefore, by compofition, the quantity of the moon’s a¢tion, during the time of defcribing this angle, will be in the fimple ratio of the faid diftance, zzverfly. Hence it appears, that the fum of the forces employed, during the times of defcribing the oppofite angles DFd, EF’, will be truly defined by TD ta FE + FD Upon AB let fall the perpendiculars DN and EM ; fo fhall. PE — FH: FI(PH)—FD:: FM: FN (6p. elipfe)_:: FE : FD (p. fim. triang.) > confequently FE x FD —FH x FD = FH x FE — FD xFE, or 2FExFD — FHx FE + FD: therefore, as it appears from hence that ae Ss the meafure of the {aid forces, is, every-where, equal to the conftant quan- tity or its equal and the different Motions of the Earth's Axis. tity — it is evident that the excentricity of the orbit and the pofition of the apogee have no effect on the motion of the earth’s' axis. An objection may, perhaps, arife, with regard to the addi- tion of the forces employed by the moon in oppofite parts of her orbit; which ftep may be looked upon as arbitrary: but the reafon upon which it is founded will be clear, by confider- ing that the moon’s inclination to the plane of the equator, in oppofite points of her orbit, is always the fame ; and that, therefore, the very fame effect in the alteration of the pofition of the equator will be produced, whether the whole force employed during the defcription of the correfponding oppofité angles, be equally, or unequally, divided, with refpe to’ the {aid angles ; fince the faid force ats with the fame advantage, or under the fame circumftance of declination, in both cafes. Another difficulty that may arife, is in relation to our having made the effect of the fun’s force to be about } part lefs than the quantity refulting from calculations founded on hydroftatical principles and the hypothefis of an uniform denfity of all the parts of the earth. But, that the phenomenon cannot be truly accounted for, upon this hypothefis, appears from the concurrence of all experiments in general: for, whether we regard the menfura- tion of the degrees of the earth, the accurate obfervations of Dr. BRADLEY, or the proportions and times of the tides, the cafe is the fame, and requires a much lefs effect from the action of the fun than refults from, or can confift with, the faid hypothefis. But if the denfity of the earth, inftead of being uniform, is fuppofed to increafe from the furface to the center (as there is the greateft reafon to imagine it does), then the phenomenon may be eafily made to quadrate with the principles of gravitation ; and that according to innumerable fuppofitions, refpecting the law whereby the denfity may be conceived to increafe. Thus, conformable to the hypothefis laid down in the Scho- lium after Prob. V. the motion of the equinoctial points will be in proportion to the motion of the fame points, when the denfity is fuppofed uniform, as eo r, that is, Ut? TF SXUR ES G as Al Of the Preceffion of the Equinox, as I — pene es is to 1: therefore, by making 1 — ULOt5SXUt+ 5 pe a (agreeable to what has been above vtotsxumts 3 obferved), we fhall have ———-~"—--—— == —: by means v+ots5xum+5 3 of which equation, the relation of x, @, and v may be fo af- figned, as to give the true quantity of the preceffion, and that innumerable ways. As one inftance hereof, let us fuppofe == 2, or that the denfity at the center is juft double to that at the furface; and let the value of © be fuppofed very great, or, which comes to the fame, let the /rata in the lower parts of the earth, be fuppofed very nearly fpherical, or orbicular : then our equation will become se dete i oe eh ibh: v+te+5x2u 3 Sa 8 becaufe @ is fuppofed very great, will be = => near- - Sic ly ; whence v is given = 5: {fo that, it eine to ‘fe hypo- thefis, the decreafe of denfity, in going from the center of the earth to the furface, will be in the quintuplicate ratio of the diftances from the center. No One can imagine that we pretend here to afcertain the ftructure and denfity of the interior parts of the earth: all that is attempted, is to fhew (which indeed is all that can be done) that the preceflion of the equinox may be truly accounted for upon the principles of gravitation, though not in the hypothefis of an uniform denfity of all the parts of the earth, unlefs by affuming the difference of the leaft and greateft diameters much fmaller than it is found to be, either, from hydroftatical | principles, or by an actual menfuration of the degrees of the earth’s meridian. There remains yet another particular that I cannot avoid tak- ing fome further notice of ; which is the wide difference to be found between our conclufion, in Prob. IV. Corol. I. and thar brought out by Sir Isaac Newton (77 Prop.39. Book IH. of bis Princitia) from the very fame data. lam and the different Motions of the Earths Axis, Iam fenfible that this is a delicate point to touch upon; but then I know likewife, that I might leave my Readers diffatisfied, were not I to endeavour to point out the caufes of the faid dif ference.—At firft I had, myfelf,.a ftrong fufpicion that I had, fomewhere, fallen into an error; which put me upon attempt- ing the {olution by different methods, as the most proper way to arrive at certainty, and to difcover the miftake, if any fuch had crept into my calculations. ‘Two. of thefe methods I have given; the others feemed unneceffary. ‘The exact concurrence of them all, firft made me think, that it was not impoffible but there might be a fault in that Author’s folution; and occafion- ed my looking into his method with a more particular attenti- on than I had before regarded it with. What, at firft, feemed moft doubtful to me was his hypo- thefis, that the motion of the nodes of a ring would be the fame, whether the ring were fluid, or whether it confifted of a hard rigid matter *: this, I fay, did not feem at all clear, at firft; but upon recollecting the demonftration of my fecond Lemma (wherein this point is fully, though not direétly, proved) I was foon convinced that the fault (if fuch there was) muft be owing to fomething elfe. | In the next place, his third Lemma did not appear to me fo well grounded as the two preceding ones. In this Lemma * The celebrated mathematician M. D’ALEMBERT, who has with great fubtlety expatiated on Sir Isaac NewtTon’s folution of this Problem, repre- fents the above hypothe/is, as ill founded ; and fays, that, when the ring is in a fluid ftate, the particles, or detached moons will not have their centers in one and the fame plane (i/ ef? certain que des lunes ifolées n’aurcient pas toujours leurs centres placés dans un méme plan). Now if, by this, we are to underftand, that the deviation from a plane is fomething fenfible in comparifon of the nutation in queftion, what is advanced is repugnant to what is demonftrated in our fecond Lemma. But if an exceeding {mall deviation (depending on the fecond term of a feries) be only intended (and fuch it mutt be, if any thing at all), fuch a fup- pofition will make nothing againft our Author’s aflumption; as, in phyfical fubjects, a perfeét accuracy is not to be expected. This learned gentleman him- felf allows, that, the confidering of all the particles (or the ring of moons) as being in the fame plane, produces no error in the conclufion : from whence it might, with fome reafon, be imagined, that the hypothefis itfelf could not be otherwife than true, And it feems farther plain to me, that, whatever lights that Author’s overfights in the folution of this Problem are capable of being placed, his real miftakes are two only. 2 he a} 44 Of the Preceffion of the Equinox, he determines, that the motion of the whole earth about its axe, arifing from the motion of all the particles, will be to the motion of a ring about the fame 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. ‘This propor- tion is, indifputably, true, in the fenfe 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 perfevere in its prefent ftate of motion, in oppofition to any new force imprefied. Now it feems clear to me, that it is this laft kind of momentum that ought to be re- garded, in computing the alteration of the body’s motion, in confequence of any fuch force. And here every particle is to be confidered as aéting by a lever terminating in the axis of motion: fo that, to have the whole momentum fought, the moving force of each particle muft be multiplied into the length of the lever by which it fuppofed to act: whence the momen- tum of each particle, will be proportional to the {quare of the diftance from the axis of motion; as it is known to be in find- ing the centers of percuffion of bodies, «which depend on the very fame principles. _ Now, according to this way of proceeding, it will be found, that the momentum of the whole earth (taken as a fphere) will be to the momentum of a very flender ring, of the fame dia- meter, revolving in the fame time, about the fame axe, in a proportion compounded of the proportion of the matter in the earth to the matter in the ring, and of the number 800000 to the number 1000000. Which proportion, therefore, differs from that of Sir Isaac NewTon, given above, in the ratio of 800000 to 925725: fo that, if his refult, which is 97 20"; be increafed in this ratio, we fhall then have 10” 33", for the quantity of the annual preceflion of the equinox, “arifing from the force of the fun; allowing for the above-mentioned dif- ference. It appears further, by perufing his 39th Propofition, that he there affumes it as a principle, that, if a ring, encompafling the earth, at its equator AlaL (but detached therefrom) was to tend, or begin, to move about its diameter LI with the fame accelera- tive and the different Motions of the Earth's Axis. 45 tive force, or angular celerity, as that whereby the earth itfelf tends to move about the fame diameter, through the action of the fun (at S); that then the motion of the nodes of the ring and of the equator would be exactly the fame. Now this Fig. 8. swould indeed be the cafe, were not the effects of thefe 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 fphere, let the direction of its rotation be which way it will, that is, let it move about what diameter it will, has al- ways the fame momentum, provided it has the fame angular ce- lerity: but the momentum of a very flender ring, revolving about one of its diameters, appears (4y 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 fame motion of preceflion and nutation with the earth’s equator, ought to tend to move about the diameter LI with an accelerative force double to zhat whereby the earth itfelf tends to move about the fame diameter, through the action of the fun: fince, in this cafe, the quantities of motion, or the momenta, generated in the two bodies, during any very {mall particle of time, would be exactly proportional to the refpective momenta of rotation, whereby the bodies endeavour to perfevere in their prefent {tate and direction of motion, in oppofition to any new force im- preffed. Hence it follows that all conclufions, relating to the change of the pofition of the earth’s axe, drawn from the prin- ciple above {pecified, muft be too little by juft one half; and confequently that the quantity of the annual preceflion of the equinox, arifing from the action of the fun, ought to be the double of 10” 33”; which is 21” 6”, and agrees, to a ¢hird, with what we before found it to be, by two different methods. fone md —s US Leal MUM Me Se Me MIL ML ML SHG HLM St ODF EEO EOE MOLE A very exaét Meruop for finding the Prace of a PLANET in its orbit, from a Correction of Warp’s hypothefis, by means of One, or more Equations, applied to the’ motion smiadit the upper focus. . Bs Lest E T ABPC be the ellipfes in which the planet revolves aA , about the fun in the lower focus S; let F be the up- eA f dM and ma ene laces of the pla- sexe per focus, an nd m any Pp Pp net indefinitely near to each other ; and let FM, Fw, SM, Sm be drawn, as likewife MN, perpendicular to the greater axis AP: from the center F, with the radius FD —1, let the circumference of a circle DEe be defcribed, and from its inter- feGtion with FM draw EH perpendicular to AP: put AO (= OP) =a, OB (= OC) =4, OF (= OS) —c, FM=z, Pix ti = y, Di 2 2, and Bese =: then SM being a which, becaufe yz ==— x, and xz= y, will te reduced to 1H % + a xX — xy — - S ~; whofe fluent will therefore be truly exprefled by 40°z te ae: 3 a or 167% fe a & 3B Ky — an eee the area DHE —22 DE 2a x DF — {HF x EH = jz — xy). This expreffion, when M coincides with P, and z is = the femi-circumference DEK (=p), will become a + pote ~ (fuppofing d= 1 + 4aa i o) the area of the femi-ellipfis ABP. Therefore we have, 2aa Od; bd: Le cs b? SEY, EN oe ae Seely (== the area ASM) :: ~£ (= 2 2 Aaa 2 34,3 the length of an arch of 180°) : z— —2 ae fae the 2a°d 3a3d length of an arch (A) exprefiing the planet's mean anomaly : ah exy 2c3y3 Co from which equation, 2 + —— a + —— eros A+ ar eee 3 fin. 2z + x fin. z]’ (becaufe xy, or co-fin. zx fin.z = 3a3d ;fin.2z): where the two laft terms being very {mall in com- parifon of the others (and, therefore, z nearly = A), we may, inftead of fin. z and fin. 2z, fubftitute fin. 4 and fin. 24; by e 3 which means we have z= A —*_ x fin. 2A + —_x fin. Al’, 4a°d 3a°d From whence it appears, that, in order to have the angle AFM at the upper focus, the mean anomaly (A) of the planet at the ee given, muft be increafed by the quantity, or correc- > x fin. 2A + = = > x fin. A). tion But to exprefs the value of this correction in feconds of a degree (which in practice is the moft commodious) it will 48. A very exadi Method for finding will be, as 3,1416 (the length of an arch of 180 degrees) is to 648000 (the number of feconds in that arch) fo is € _xfin. 2A + 2 xfin AP to 51567 x x fin. 2A + 4a°d 3a°d avd 137513 Xe x fin. Al’ = the number of {econds in the {aid correction: the logarithm of the latter term of which will, therefore, be = 5,1383 — log.d + 3 log. 4 3 log. fin. A; and that.of the former = 4,7123 —log.d + 2 log. — +- log. fin. 2A. But, to render thefe expretfions {till more convenient tor practice, the log. of d, by reafon of its fmallnefs, may be, either, intirely neglected, or elfe {fo aflumed, to be nearly a mean of what. it is known to be in.the planetary orbits. Ac- or 8 cordingly, by afluming the excentricity c = of the mean diftance (which is a {mall matter lefs than the excentricity of Mars, but fomething. greater than shofe of the Moon, Saturn, and “fupeter), the value of d (= 1+ —.) will be = 1,0032, and its logarithm == 0,0014._ Whence the log. of the former part of our correction, by fubftituting this value, will be 551369 ++ 3 log. — + 3 log. fin. A= 3 xX 1,7123 -plog.— + log. fA ; and ¢bat of the latter —= 4,7109 + 2 log. — + log. fin. 2A: which, exprefled in words at length,. give the following Praétical Rules. 1°. To the fum of the conftant logarithm 1,7127 and the log. of the excentricity in parts of the mean diftance, add the log. fine of the. mean anomaly ; the fum (rejecting the ra- dius) being tripled, will give the log. of the firft equation (in feconds) to be added to the mean anomaly. 2°. To the fum of the conftant log. 4,7109 and twice the log. of the excentricity, add the log. fine of twice the mean anomaly ;. the {um (rejeting the radius) will be the log. of the fecond equation; to be added or fubtracted, according as the mean anomaly is lefs, or greater than go degrees. Here the Place of a Planet in its Orbit. Here, and in what follows, the anomaly is to be always reckoned from, or to the aphelion, the neareft way ; in which the feconds may be omitted, in computing the propofed cor- rections. Which corrections being made, the true anomaly, or angle at the lower focus, will be had from the common proportion, by faying, as the aphelion-diftance is to the peri- helion-diftance, fo is the tangent of half the mean anomaly, thus corrected, to the tangent of half the true anomaly fought. As an example of the ufe of the method here laid down, let the excentricity be fuppofed = 0,048219 (being that af- figned, by Dr. HaLey, to the orbit of Jupiter) and let the mean anomaly be 45°. Then, log. excent. . . 2,6832 2: log. -exeent.. 2953668 -+ confit. log. s . 1,712 + confit. log... . 4,7109 == log. for 1ftequ. 0,3955 = log. for 2d equ. 2,0773 log. fin. anom. .°. 9,8494 log. fin. 2 anom., 10,0000 0,2449 adequs = EIQ, 2,077 firft equ. = 52" 057347 From _1,978537 = log. of perihelion-dift. 0,95178 1, fubtr. 0,020452 = log. of the aphelion-dift. 1,0482109 ; therem. 1,95808 5, willbea(3d) conft. log. for this orbit: towhich add 9,617596 = log. tang. + cor.anom. 22° 2) 2a fo fhall 9,575681 = log. tang. } true anomaly 20° 37’ 40”: oer double, 41° 15°20", is therefore the true anomaly re- uired. 3 The fame excentricity being retained; let the mean anomaly be, now, fuppofed 120 degrees, Then, log. fin. anom. 9,9375 log. fin. 2 anom. . . 9,9375 log. for firft equat. 0,3955 log. for 2d equation 2,0773 29,3330 ad equat.== 103; 2,0148 firft equat. = 10” 0,9990 Hence 120° - 10" — 1° 437" = 119° 58’ 262” = the cor. anomaly. | Therf. log. tang. + cor. anom. 59° 59/132”. . . 10,238331 + third conft. log. for this orbit... . 1,958085 == log. tang. + true anom. 57° 32’10” 10,196416 Whence 49 50 A very exatt Method for finding Whence the anomaly itfelf is given = 3° 25° 4’ 20”. If the excentricity be affumed — 0,066777 (being the ereateft that Dr. Hairy gives to the lunar orbit) the two conftant 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 faid anomaly be taken = 50° (at which, according to the Doétor’s Table, pro expediendo calculo equationts centri lune, the whole corretion is a maximum), the former part of the faid correction will be found = 182”, and the latter part = 225+": therefore the fum of both is 244”, or 4’ 4”; agreeing, exactly, with the quantity given in the Tab/e. And in the very fame manner, the proper correCctions correfponding to other anoma- lies and excentricities may be computed ; the error never amounting to above a fingle fecond in any of the planets, ex- cept Mars and Mercury: in the place of Mars, the greateft error.will be two or three feconds; and in that of Mercury, about as many minutes.—As to the Earth and Venus, the {e- cond equation, alone, will be fufficient to give their places to lefs than a fecond. To obtain a farther correction, which will be necef— fary when the orbit is very eccentrical, we may (inftead of the two firft only) make ufe of a greater number of terms of e T 2 Lind Zand a I the: feries — — = ee (= == }; by a a* a y ae e ae cx} a which means the fluxion (£4°z i of the area MSA at cx} defcribed about the focus $, which is proportional to the time,. will be here reprefented by ibrib ib xe ye— ety ae +f gety rE — gery nts Be. (fhtp- pofing e = +). But it is well’ known that y* (fin. z]’) —1—" co-fin. 22; wh ‘ei Xco-li —1—- ! co-fin.22; whence yx (= y x co-fin..z) == 3 co- fin. z — + co-fin..2z x co-fin. z = | co-fin. z — 4 co-fin. 3 — Fir ; Me I I f wh + co-fin. 32 * = + co-fin. z — { co-fin. 32; and therefore y'x * This, and all that follows to the fame purpofe, 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 fum of the co-fines of the fum and difference of thofe angles. os —— the Place of a Planet in its Orbit. == +-co-fin. z x co-fin. z — { co-fin. 32 x co-fin. ze = %-+- | co- fin. 22 — + co-fin. 22 — }co-fin-4z =; — { co-fin. 42: whence alfo y*x3 == 4 co-fin. — ; co-fin. 42 xX co-fin. == ¥ ¢o-fin. z — +, co-fin.3z — +, co-fin.5z. Which values being now fub- ftituted for their equals, our flaxion, above given, will become ibs +. 30" into e X [2 — + xX 22 €o-fin. 22 — SS — 405 xX 72co-l.z%— 7, x 3zco-f yz — sox gzco-L5z +e. and, confequently, the fluent thereof = ié°2 + 70° into e*xig— fin. 2z—2e3xife— Af 324 3etx 72— ffin.4z — 4e5 x tfinez — 2, fine 3z — ¢; fine 52 Gc. = 7b" into rte + ict x = — te3 4 105 x fins — Je fin. az + 463 4.425 x fin. 32 — yf Az + a fin. 5z (fuppofing all | fuch terms wherein e rifes to more than five dimenfions, to be difregarded). This expreflion, when z = p (== the femi-cir- cumference AEK) becomes 26° x 1 + te* + {e+ Ge. x p = 30° zb°p Ka 2” E as SRR RRA MIR EAE EE EES BE SASREA SPIRE Pep EE PHP PARE Re A DETERMINATION OF THE Difference between the Motion of a Comet in an Elliptic, and a Parabolic Orbit. “AK HET PNG be a parabola, and PBH a very excentrical * OL # ellipfis, having the fame focus S, and vertex P with the parabola; let moreover N and xz be confidered as cotemporary pofitions of two bodies, in thefe orbits, moving from the perihelion P at the fame time, about the fun ‘in the focus S. Make NBC perpendicular to PSO, and call PC, x; PS, c; and the greater axis of the ellipfis, @: then the lefler axis will be ==2WVcxa—c3 the parameter = se ; and the ordinate BC = ee oe (bythe ape aa ty of the ellipfis) = 2c*x* x :—4| “I —+\ = 2c*x* x Ga a y—+— =, nearly, This laft taken from CN (= 2¢°x*, 2a 24 Y by the nature of the parabola) leaves cx X= -- = == BN; which being multiplied by « and the fluent found, we thence i have cx x + = for the meafure of the area NPB, or NPv, very near: which fubtraéted from the area NSP (= CN x2PC — CN xiCs = 20K X2K — 2cix® x ix—ie = cx’ xc-ix), leaves c*x® xc tx — = — = — the area a vPS. Moreover, the parameter of ‘the parabola being 4¢, and | that Of the Difference of Motion of a Comet, &c. that of the ellipfis = 4“—*"*, we have 4c to 4ac e Hy a is the parabolic area NSP (c?x*® x e-+4x) to the correfponding, . s é EOE ieee or cotemporary elliptical area mSP = .f I ——Xe'x" XC4- 5x. — ox? x I= xc+4x (nearly) = cx? Xe in—f—S ,; ae 2a 3 7 “cae 3 2a 6a” . . ee ee cx anx which fubduéted from the arta vPS, leaves ¢’x* x = — 5 — ca == the area vSz. Let now NM be fuppofed perpendicular to the tangent NA, meeting the axis in M; then SM, SN, and SA will be ail equal to each other, by the nature of the parabola; and con- fequently the angle PMN equal to half PSN ; whofe tangent (to the radius 1) let be denoted by z : whence it will be, as Binie ise, MiGdacysEN. (2c’x*); and confequently « = cz". Which value being fubftituted in the area v5z, it will become et OP ee ie. oe - = (2 X— — yxy 1—2° — +24; andthis di- 2a 2a 5a 24 2 . ae, : STE EE ‘ Sy vided by + Su (= 4 XO xX SX! + zz ), gives — x 7 for the meafure of the angle vSz, in parts of the radius (1). Therefore, if m be put = 3437 = the number of ZX 1—z’—42z* Be Al oer A it is evident that © x mu will exprefs the meafure of the faid a minutes in an arch equal to the radius, and « = angle in minutes of a degree. To find now the ratio Sz to SN, which {till remains to be determined, we have (by Tri- fin.B fin. AFC Bx fin, M gonometry) Nv ==; x BN = Fags * ON = cots M x BN == tang.M x BN — XOX X ct * — exexx COBB ==—X! + zz. Alfo (fuppofing nb perpendicular to Sv) it will be, as 1 (radius) : (the arch meafuring the angle 75v) a I 2 | te 59 60 Of the Difference of Motion of a Comet ::¢xI- zz (== Sz, nearly) : nb = — X1-+ 22: whence, again, by fimilar triangles, 1: z (:: CM : CN) :: <= x I+ 2z CCZSB (710) "300 == am xitezz: this added to Nv = wn I+ 22, a gives Nb = “xu-+zx1-+ 22: which therefore is to SN a (cx 1+ 22) as =xu+ 2 to unity; and confequently the re- a : < CZ Fat? (eae: quired proportion of Sz (or Sb) to SN, as 1 — ~ Xu -+ z to unity. From this laft conclufion, and that derived above, exhibit- ing the value of the angle NSz, the following Table is com- puted ; whofe ufe is thus: Find, in the firft column, the co- met’s longitude from the perihelion, as given from the hypo- thefis of a parabolic orbit (cither by Dr. Hatiey’s Table, or any other of the like kind); againft which, in the third co- lumn, you have the logarithm of a number of minutes (ex- prefled in the fecond column) ; from which fubtrating the logarithm of the ratio of the greater axe of the ellipfe divided by the perihelion diftance, the remainder will be the logarithm of a number of minutes to be added to, or fubtracted from the aforefaid longitude, as the Table directs: whereby the comet’s longitude, for the fame time, in the elliptic orbit will be given. And if, from the logarithm found in the-fourth column, the logarithm of the fame ratio be alfo fubtra@ed, the remainder, abating 10, will be the logarithm of a quantity to be taken from the logarithm of the comet’s diftance from the fun, com- puted according to the aforefaid hypothefis. Thus, for example, let the greater axis of the ellipfis be fuppofed = 35,727, and the perihelion diftance = 0,582 %. (anfwering to the orbit of the comet of the year eighty-two) ; and let the longitude from the perihelion, according to Dr. Hatiey’s Table, computed for a parabolic orbit, be 44° 2.20): correfponding to which, the logarithm of the diftance from the fun in an elliptic and a parabolic Orbit. fun is given == 0,065838. Here, then, the log. of 2224 ss7 be- ing — 1,7877, this value is to be fubtracted fom) Bott the logarithms 2,9228 and 930555> {tanding againft 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 fubtracted from 44° 3°20", and 0,06 58 38, the required longitude from the perihelion is: given from thence = 43° 49 41", and the logarithm of the comet’s diftance from the fun = 0,063985. The fame Table, not only furnifhes an eafy way, for de- ducing the motion in an elliptic orbit, from the motion in a parabolic one, but may be farther ufeful in determining, in fome degree, the fpecies of the ellipfis which a new comet de- {cribes, when the obfervations thereon are found to differ fen- fibly from the places computed according to the hypothefis of a parabolic orbit. 7? 61 62 Of the Difference of Motion of a Comet fubtr. | fubtraét 1°, 30]1,4770 2| 60]1,7770 J | 199) 4903 4|1Ig9|2,0778 _5|149| 21740 61179] 2,2521 71208] 2,3178 8 |237| 2.3745 9 | 266} 2,4242 10|294 2,4683 11 |322| 2,5074 121349 | 25543" 13 |376| 2,5760 14|403| 2,6058 | 8,1074 151430] 2,6331 | 8,1666 161455| 2,658 | 8,2217 fubtract fubtr. fubtract fubtrack } 558200 1414 836 | 2,9220 | 9,0006 654220 ||42 | 836) 2,9229 | 9,0195 | 6,7735|143 | 836 | 2,9232 | 9,0378 7,0242 || 44] 836 | 2,9228 | 9,0555 | 732478 || 45} 835 | 2.9217 1.900727 752759 |146} 832 | 2,9200 | 9,0894 | 7,5095 ||47| 827| 2.9176] 9,1056 756250 ||48 | 8211 2,9145 | 9,1215 757269 1149] 814] 2,9107 | 9,1369 7,8178 1150 | 806] 2,9061 | 9,1520 728998 729747 8,0437 201552) 2,7420| 8,4116 21 [575 | 257594 | 854.528 221596) 2,7755 | 8.4921 617} 2,7906 | 8,5295 ||63 | 570] 2.7557 24 1637| 2,804.6 | 8,5652 ||64 | 542 | 2,73 38 25 |657| 2,8176]| 8,5994 ||6 26 16761 2,8297 8,6320. 27 |694| 2,8410 | 8,6634 710} 2,8514 | 8,6934 |/68 | 416] 2,6187 | 9,3674 29 1726 2,8610 | 8,7224 ||/69 | 381 | 2,5805 | 9,37 69 30 | 741] 2,8698 | 8,7502 755) 238779 | 857770 32 |768| 2,8853 | 8,8030 61} 621 | 2,7932 62) 596) 257754 331780] 2,8920 | 8,8280 4132 341791 | 2,8980 | 8,8521 |}74 954.218 B5ASOU 25834 | 5875511751340 | 211458 1934903, 76| 95) 0,977 | 924380 77| 481 1,6840| 9.4408 78iadd}| add |9,4546 1,6910 6 [40 [833 : | 2.0006 in an elliptic and a parabolic Orbit. add add fubtract 152 | 2,1827| 954777 206 | 253139] 9.4851 261 |2,4170 | 94924 318 | 2,5021 | 9,4996 5| 376 1255750 | 9.5067 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109g 110 III 112 113 114 IIs 116 Li 118 119g 120 1850 A35 | 2,6386| 9,5138 496 | 2,6951 | 9,5207 558 | 257470 | 955275 022 | 257936 | 955342 688 | 2,8373 | 9.5409 754 | 258776 | 9,5475 823 | 2,9152|9,5540 892 2.9506 9,5005 963 | 2,9837| 9,5670 1036 | 3,0154 | 9.5734 TI1O | 350455 | 955797 1186 | 3,0742 | 9,5860 1264 | 3,1016 | 9,5923 1343 | 31279] 9,5986 1423 | 391531 | 9.0050 1505 | 3,1774 | 90113 1588 | 3,2008 | 9,6176 1673 | 3,2236 | 9,6239 1761 | 3,2457 | 9,6302 | 32671 | 9,6366 Ig4I 3,2879 9,0430 2033 | 353081 | 9,6495 2127 | 3,3278 | 9,6560 2223 | 3,3469 | 9,6627 2321 | 353656 9,0694 2421 | 353840 96763 2523 |3,4019 |9,6831 2627 | 3,4195 | 9,6901 2734 | 3.4368 | 9,6972 2843 | 3,453819,7044 2954 | 3.4704 | 9.7119 3068 | 3,4868 | 9,7195 3184 | 3,5030| 9,7272 3302 | 3.5188 | 9,7352 34221 3,5345 | 9.7433 add 355499 335053 355805 355956 3,6105 3,6253 336399 356545 3,6690 326835 3,6980 357125 357269 337413 7557 357701 3.7845 3.7991 3,8138 3,8286 358435 358585 3,8736 23,8889 309044 | 399199 359360 329524 339690 329859 4,0031 4,0208 40390 40575 40766 450963 4,1168 4,1380 4,1600 4,1825 fubtract 95751 6 9,7602 97690 957781 957874 ror 9,8070 g,8172 9,8278 928387 g,8500 9,8618 9,8740 9,8866 ae Oo” Pos 929429 93957" Sn Bs): 929893 10,0064 10,0242 10,0427 10,0620 10,0822 10,1032 10,1251 10,1481 10,1720 10,1970 10,2291 10,2504 10,2792 10,3094 } 10,3412 10,3746 10,4098 10,4409 10,4861 | & bz JN ex Yes Ne ee Al aC Ft aye Labdeshtesdt $F ae SN Se TG Leable Ht Wah lesb eshte cae SPR OE EE EERIE EEE RE An Arremprt to fhew the Advantage arifing by Taking the Mean of a Number of Obfervations, in practical Aftronomy. eS HOUGH the method practifed by Afronomers, in - tT §® order to diminith the errors arifing from the imper- eT fection of inftruments and of the organs of fenfe, by taking the mean of feveral obfervations, is of very great utility, and almoft univerfally followed, yet has it not, that I know of, been hitherto fubjected to any kind of demon- {tration. In this Effay, fome light is attempted to be thrown on the fubje@t, from mathematical principles: in order to the appli- cation of which, it feemed neceflary to lay down the following eee . That there is nothing in the conftruCiion, or pofition of ae infiruiment whereby the errors are conftantly made to tend the fame way, but that the refpective chances for their hap- pening in excefs, and in defect, are either accurately, or nearly, the fame. 2. That there are certain aflignable limits between which all thefe errors may be fuppofed to fall; which limits depend on the goodnefs of the inftrument and the fkill of the obferver. Thefe particulars being premifed, I fhall deliver what I have to offer on the fubject, in the following Propofitions. PROPOSITION I. Suppofing that the feveral chances for the different errors that any fingle ober vation can admit of, are expreffed ) the terms of SUGLICIOR FOE oN oeeeaet “aif Bp ik sy Toye Ray Moore fs * is where the exponents denote the quantities and qualities of the re- [pective errors, and the terms kbemferves, the re[pective chances for their happening ; it 1s propofed to determine the probabslity, or odds, Of the Advantage arifing, &c. odds, that the error, by taking the mean of a given number (n) of . . . im obfervations, exceeds not a given quantity (=}. It is evident from the laws of chance, that, if the given {eries Sing aes pr3treprttprtrnrtr tril... 4-7", expreffing all the chances in one obfervation, be raifed to the nth power, the terms of the feries thence arifing will truly ex- hibit all the different chances in all the propofed (x) obferva- tions. But in order to raife this power, with the greateft faci- oe USS lity, our given feries may be reduced to -—* x —_____ (by the © I—r known rule for fumming up the terms of a geometrical pro- greflion); whereof the zth power (making w = 2v-+ 1) will be r7" x y —7*\' x 1 — 7)"; which expanded, becomes fart as ype tie gee eee St ee ee, a2 eee eae ° 2 x into 1-b ar + = SET,’ BOE ee eee ae ae ; ince re ee L323 7T 3,4 1 Ge, | 4 Now, to find from hence the fum of all the chances, where- by the excefs of the pofitive errors above the negative ones, can amount to a given number m precifely, it will be fufficient (in- ftead of multiplying the former feries by the whole of the lat- ter) to multiply by fuch terms of the latter only, as are ne- ceffary to the production of the given exponent m, in queftion. Thus the firft term (7—*”) of the former feries, is to be mul- tiplied by that term of the fecond whofe exponent is nv + m, in order that the power of r, in the produét, may be 7”: but it is plain, from the law of the feries, that the coefficient of this term (putting xv -+- m= ) will be 2 “Es AE? TES (9), qg being the number of factors ; and, confequently, that the pro- duct under confideration will be ee (7).% 2". Again, the fecond term of the former feries being — ur*~*, the exponent of the correfponding term of the latter mutt therefore be —w-+nv-+m(=g—w), and the term it- K felf, 66 Of the Advantage arifing by Taking the Mean felf, SURI TE? gq — w) x A"; which, drawn into I 2 Sr”, sce ye (¢—w) x nr” for the fecond be 3 | term required. In like manner the third term of the product whofe expo- nn+t n-+-2 ae ae 2 3 2 And the fum of all the terms, having the fame, given expo- nent, will confequently be nent is m, will be found; (g—2w) x ~. : n n+ n+-2 2+3 “A a a (q) x7 mets i eS ( —w) xr” EC 3 4 7 Sore re tg i — 24) xX — bie 2 4 I naon+t+-rnt2n+3 x nm n——I n—2 me et (7 me 2) K — 2 ——: c I 2; 3 4 (7 2 we I 2 3 &e. Se. From which general expreffion, by expounding m by o, +1, —1, +2, — 2,@c. fucceffiyely, the fum. of the feveral chances whereby the difference of the pofitive and negative errots can fall within the propofed limits (+ ™, — mm) will be found: which divided by the total of all the chances, or p-™ xT — rx 1—7\_’, -will be the true meafure of the probability fought. From whence the advantage, by taking the mean of feveral obfervations, might be made to appear: ‘but this will be fhewn more properly in the next Propofition ; which is better adapted, and to which this is premifed, as 2 Lemma. . REMARK. If + be taken —1, or the chances for the pofitive, and the negative errors be fuppofed accurately the fame ; then our €x- prefiion, by expunging the powers of r, will be the very fame with that fhewing the chances ‘for throwing #- ¢ points, precifely, with ‘ dice, each die having as many faces (w) as the refult of any one fingle obfervation can come out different ways. Which may be made to appear, independent of any kind, of a Number of Obfervations in praétical Aftronomy. kind of calculation, from the bare confideration, that the chances for throwing, precifely, the number m, with z dice, whereof the faces, of each, are numbered —v.....—3, —2, —I, —o, +1, +2, +3....+4, ‘muft be ‘the very fame as the chances whereby the pofitive errors can ex- ceed the negative ones by that precife number: but the former are, evidently, the fame as the chances for throwing precifely the number v-+1x2-+m (or 2+ q) with the fame x dice, when they are numbered in the common way, with the terms of the natural progreffion 1, 2, 3, 4, 5, and fo on; becaufe the number upon each face being, here, increafed by v--1, the whole increafe upon all the (w) faces will be exprefied by v-+-1x2; fo that there will be, ow, the very fame chances for the number v-+-1xm-+ m, as there was before for the number m; fince the chances for throwing any faces afligned will continue the fame, however thofe faces are numbered. PROPOSITION & Suppofing the refpective chances for the different errors, which : any jingle obfervation can admit of, to be expreffed by the terms of the feries 17% 4B 22th BG. cM BaD en oes Be + ore! + ° (whereof the coefficients, from the middle one (v--1) decreafe both ways, according to the terms of an arithmetical pro- greffion) ; it is propofed to find the probability, or odds, that the error, by taking the mean of a given number (t) of obfervations, exceeds not a given quantity ( oe i Following the method laid down in the preceding propofiti- on, the fum, or value of the feries here propofed will appear ae pea eat zi to be Re | (being the fame with the fquare of the 7 a oti gne rPOGT yz—iy 1 Pa ah = gn E j Wt oe : (2 —1) x ur : p— I p’—2 p'— 3 we n nT WRgen ea a ay es oe Keen er 5 : Lat hes (7 Paras 1) xX Be Jm os I 2 3 roe &Se. ee. Which expreffion is to be continued till fome 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 pf points, precifely, on 2 dice, having each w faces. And, in this cafe, where the chances for the errors in ex- cefs and in defeé are the fame, the folution is the moft fimple it can be; fince, from the chances above determined, anfwer- ing to the number p precifely, the fum of the chances for all the inferior numbers to p, may be readily obtained, being given (from the method of increments) equal to Poe A es) I Z 3 — Pt Pe 8 (n) x np EES) xo ipa p'—2 p'—3 n n—1.’2—2 fe te STE) oe yes 3 + &c. The dif ference between’ which and half (w*) the fum of all the chances (which difference I fhall denote by D) will confe- quently be the true number of the chances whereby the errors in excefs (or in defect) can fall within the given limit {m) : fo Le that will be the true meafure of the required probability, 2” zw : that the error, by taking the mean of ¢ obfervations, exceeds not the quantity = propofed. But Of the Advantage arifing by " aking the Mean But now, to illuftrate this by an example, from whence the. utility of the method in practice may clearly appear, it will be néceffary, in the firft place, to affign fome number for v, ex- preffing the limits of the errors to which any obfervation is fubject. Thefe limits indeed (as has been before obferved) depend on the goodnefs of the inftrument, and the fkill of the obferver: but I fhall here fuppofe, that every obfervation. may be relied on, to five feconds; and that the chances for the fe- veral errors — 5”, — 4”, — 3”, — 2”, —1', 0", + i, + 2”, + 3”, +4", + 5”, included within the limits thus affigned, are refpeCtively proportional to the terms of the feries 1, 2, 3, AS By 65.554, 3. ae Which ferics ts much better adapted, than if all the terms were to be equal; fince it is highly rea- fonable to fuppofe, that the chances for the refpective errors de- creafe, in proportion as the errors themfelves increafe. Thefe particulars being premifed, let it be now required to find what the probability, or chance for an error Oia 2, 0 4; or. 5 feconds will be, when, inftead of relying on, one, the mean of fix obfervations is taken. Here, v being == 4, and ¢== 6, we fhall have x fot) 12, w(==v+1)=—6, and p(> fututm) = 424: but the value of wm, if we firft feek the chances whereby the. error exceeds not one fecond, will be had from the equation “==+15 where either fign may be ufed (the chances being the fame) but the negative one is the moft commodions: from whence we have m (= — ¢) == — 6; and therefore p = 30, P= 3% P= 24, Se. : Which values being fubftituted in the general expreffion above determined, it will become 22.24.33 (12) — seo gi lig tT A fy iia5iR Re Figs oko is aad peal eee mane (12) x 12-4 Eig ert 2)-*(00 er iat \ ) Fad 299576368: andthis fubtracted from 1088391168 (= + x6"); leaves.7888 1480, for the value of D correfponding: thereiore the required probability that the error, by taking the mean ot fix obfervations, exceeds not a firiele fecond, will be truly mea- o x > ie) xX 220 8 i fured by the fraction TE and confequently the odds 1088391168 ” a wil © of a Number of Observations in practical Afironomy. will be as 788814800 to 299576368, or nearly as 21 to 1. But the odds, or proportion, when one fingle obfervation is : 8 taken, is only as 16 to 20, or.as <5 to I. To determine, now, the probability that the refult comes within two feconds of the truth, let - be made —=— 2; fo fhall m (== — 2t) == -—~ 12%. therefore p = 30,, f'==.24, p = 18, &c. and our general expreffion will here come out == 36079407; whence D = 1052311761. Confequently TOS2311701 1088391168 and the odds, or proportion of the chances, will therefore be that of 1052311761 to 36079407, or as 29 tol, nearly. But the proportion, or odds, when a finele obfervation is taken, is only as 2 to 1: fo that the chance for an error exceeding two feconds, 1s not ;‘;th part fo great, from the mean of fix, as from one fingle obfervation. And it will be found in the fame manner, that the chance for an error exceeding three feconds is not bere -;~ part fo great as it will be from one obfervation only. Upon the whole of which it appears, that, the taking of the mean of a number of obfervations, greatly diminithes the chances for all the fmaller errors, and cuts off almoft all poffibility of-any large ones: which laft confideration alone is fufficient to recommend the ufe of the method, not only to Afironomers, but to all Others concerned in making experiments, or obfervations of any kind, which will allow of being. re- peated under the fame circumftances. will be the true meafure of the probability fought: In the preceding calculations, the different errors to which any obfervation is fuppofed fubject, are reftrained to whole quantities, or a certain, precife, number of feconds; it being impoffible, from the moft exaét inftruments, to take off the quantity of an angle to a geometrical exatinefs. But 1 fhall now fhew how the chances may be computed, when the error ad- mits of any value whatever, whole or broken, within the pro- pofed limits, or when the -refult of each obfervation is. fup- pofed to be accurately known. Let “3 Of the Advantage arifing by by Taking the Mean Let, then, the line AB reprefent the whole extent of the given interval, within which all the obfervations are fuppofed to fall; and conceive the fame to be divided into an exceeding great number of very {mall, equal particles, by perpendiculars terminating in the fides AD, BD of an ifofceles triangle ABD, formed upon the bafe AB: and let the probability or chance whereby the refult of any obfervation tends to fall within any of thefe very fmall intervals Nz, be proportional to the corre- {ponding area NMyz, or to the perpendicular NM; then, fince thefe chances (or areas) ne frofa'the extremes A and B, increafe according to the terms of the arithmetical progreffion 1, 2, 3, 4, &e. it is evident that the cafe is here the fame with that in the latter part of Prop. II.; only, as the number v (exprefling the particles in AC or BC) is indefinitely great, all (finite) quantities joined to v, or its multiples, with the figns of addition or fubtraction, will here vanith, as being no~- thing in comparifon of v. By which means the general ex- preffion pee (n) — pe ts (n)xn + 2 3 I 2 p. Cn. p" pi pies : 1a) XM es ec. ) there. determined, - will bere become 2.2.2 (n) —2.£ 2 (n) xn, Gc. = 1 _ 102 G2.3 ES ay 15 2s:34(0) x pp? — np + 1. pl rep iage b's &c. (wherein prey mm, p aap boy p = p— 20, p’ = p— 3v, Se.) and therefore, the value of D in the prefent cafe, being iv” — tee —*.p— 20| 5 coe. at is Ye Bae oh evident that the probability (FF —) of the error’s not exceed- xp — 1p abe ing the quantity > ~ (in taking the mean of ¢ obfervations) will be truly defined by ey © x om? — fe 2ot xb aI, ee -which may be reprefented by the curvilineal area CNF E, cor refpondin g of alNumber of Obfervations in prattical Afironomy. refponding to the given value or abfcifla CN (= fy. Now, though the numbers v, £, and m are, all of them, here fup- pofed to be indefinitely great, yet they may be exterminated, and the value of the expreffion determined, from their known relation to each other. For if the given ratio of = to v, or of CN to CA, be expreffed by that of x to 1,°or, which is the fame, if the error in queftion be fuppofed the x part of the greateft error; then, m being = tux, p (= tv fm) will be == tv + ¢vux, and therefore - = ¢x14 x; which let be de- noted by y: then, by fubftitution, our laft general expreffion will become yo—ny— +2 fot 2 Fes cust bblidetiioen Se occa x > 122.3;(a) n u—I n—2 yo ay <" 2 e 3 “Jy — 3 3 Fe, which feries is to be continued till the quantities y, Vt; 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 fix obfervations, exceeds not a fingle fe- cond ; fuppofing (as in the former example) that the greateft error, that any obfervation can admit of, is limited to five feconds. Here ¢ being == 6, #2(=> 2t).— 12; and x <= e we have y (= txX1—x) = 4,8; and therefore the meafure of the 2 probability fought will be equal to 1 — Dae la 4,8] — 12 x3,8)'? + 66x 2,8)'*— 220 x 1,8)? + 495x0,01” = 0,7668, nearly: fo that the odds, that the error exceeds not a fingle fecond, will be as 09,7668 to 0,23323; which is more than three to one. But the proportion, when one fingle obfervation is relied on, is only as 36 to 64, or as g to 16. In the fame manner, taking « = = it will be found, that 3 :, the 13 7° Of the Advantage arifing from Taking the Mean the odds, of the error’s not exceeding two feconds, when the mean of fix obfervations is taken, will be as 0,985 to 0,015, nearly, or as.652 to 1; whereas the odds on one fingle ob- fervation, is only as 64 to 36, or as 1% to 1: fo that the chance for an error of two feconds is not -*,th part fo great, from the mean of fix, as from one fingle obfervation. And it will farther appear, by making x = 3, that the probability of an error of three feconds, here, is not —~.th part fo great as from one fingle obfervation : fo that in this, as well as in the former hypothefis, almoft all poffibility of any large error is cut off. And the cafe will be found the fame, whatever hy- othefis is aflumed to exprefs the chances for the errors to which any fingle obfervation is fubject. From the fame general expreffion by which the foregoing proportions are derived, it will be eafy to determine the odds, that the mean ofa given number of obfervations is nearer -to the truth than one fingle obfervation, taken indifferently. For, if°2 be put —=1 —-«) = =, and. § == =, then, y being = 7g, the quantity s xy'—n.y—tl tn. y— a, &c. (exprefling 1. 2..3(2) 2 the probability that the refult falls within the diftance z of the greateft limit) will here, by fubftitution, become 3 a ve a ee nla a ty e— asl, &c. which, in cafe of one fingle obfervation (when ¢==1, and 7= 2) is barely 2°, and its fluxion 222: therefore, if we now multiply by 22%, the product ? : " : n—TI 1 : = x 2s — n.z—s| e+. L— 25 .2z, Ge. Lg aiga(n) 2 will give the fluxion of the probability that the refult of ¢ ob- fervations is farther from the truth, or nearer to the limits, than one fingle obfervation taken indifferently. And con- e e " e np fequently the fluent thereof, which is cele inte 2 12. B47) n—-- 2 ae of alNumber of Obfervations in prattical Aftronomy. 76 a eee Zao re ” nmr, 28.z— a)" %—25\"* I n+ I n-+- 2 Le n-+-- I - n-+-2 &c. will, when z—=1, be the true meafure of the proba-~ bility itfelf. Which, in the cafe above propofed, where ¢ = 6, and 2 == 12, will be found = 0,245, and, confequently, the odds that the mean of fix, is nearer to the truth than one fingle obfervation, as 755 to 245, or as 151 to 49. Le | A DE- SEIS OCIOIO OO IOC SOD MAID GLOOM ERID A DETERMINATION | * OF Certain Firuentrs, and the Resonurion of fome very ufeful Equations in the higher Orders of Fluxions ; by means of the Meafures of Angles and Ratios, and the Right-fines and Verfed- fines of circular Arcs. OK N order to treat the matter here propofed with due g I x perfpicuity, it will be neceflary, previous thereto, to Wweose give a demonitration of the two fubfequent Lemmas. LEMMA. I. The double of the rectangle contained under the co-fines of any two arcs, fuppofing the radius to be unity, as equal to the fum of the co-fines of the fum, and difference of thofe arcs. Fig.21. For, let AB and BD (= BE) be the two arcs, and CG and Cn their refpective co-fines ; likewife let CH be the co-fine of their fum AE, and'CF that of their difference AD ; making nm parallel to BG. ‘Then, Dz being = Ez, it follows that Fm = Hm; and confequently that 2C# — CH-+ CF: but, by fimilar triangles, CB :CG:: Cz: Cm; whence CG x 2Cz — CB x 2Cm = CB x CH+CF. 2,£.D. LEM M.A’. Il. If A be any arch of a circle whofe radius is unity, and n any whole pofitive number; then will co-fin. ay at into co-fin. nA + n co-fin.n—2.A--n.— 2, m—I ’—-2 rage . -in.au—mA. A : -in.a—0O.A . cont. co-fin.n— 4. 0 : -CO-fin. 2 — © A+&c | mnued ae £8 The Refolution of certain fluxionary Equations, &c. to = = t, orin-+-1 terms, according as n 1s an odd, or ewen nun ber ; in the latter of which cafes the half, only, of the laft term 1s to be taken. | . For, by the preceding Lemma, 2 cof. A\ == cof A +A + cof A—A = cof. 2A -+1; which equation multiplied by 2 cof.A, gives 2* xcof-A|? = cof. 2A x 2 cof, A + 2cof. fA | A = cof.3A + or ee Tem. I.) == cof.3A+-3cof. A. Multiply, again, by 2 cof. A; fo fhall 23 x cof. Aj? = cof. 3A x 2cof A + 3 cof. A-x 2 cof A = cof. 4A + we ae + 3cof.0 (by Lemma I.) = cof. 4A + 4 cof. 2A-+ 3. In the fame manner we have, 2+ xcof. Al5==cof. 5A + 5cof.3A + 10cof.A, 25 xcot. A]®=cof.6A +6cof.4A +15 cof.2A + Io, 2° x cof. A\'—=cof.7A +7cof. 5A +21cof.3A + 35 cof A, | an Ce. Se. where the law of continuation is manifeft; the numeral co- efficients being the fame, and generated in the very fame man- ner, with thofe of a binomial raifed to the 2d, 3d, 4th, sth, &c. powers, fucceflively ; except in the laft term, when the ex- ponent # is even, in which cafe one half only of the corre-. fpondent (or middle) term of the involved binomial is con- cerned, Hence the propofition is manifeft. The fame otherwife. If the co-fine of A be denoted by x, it is well known that 4 — ——=_. Multiply the whole equation by W/—1, fo — n/ —1 —ixX—I x / 1 — xx oF ne Ge ee ie v ae whence, by taking the fluent, we have A/— 1 = hyp. log. x o/xx—1. Let N be the number whofe hyperbolical lo- garithm thall AY —1 = | 78 The Refolution of certain fluxionary Equations, garithm is 1; then, fince hyp. log. N4%=7? (<= Av/—1) = hyp. log.x + / xx — I, itis evident that NAY —t== «+ /xx—1,. or M8 = » + Wxx — 1 (by making M = N%—"). From. Ay yA which equation. x (the co-fine of A) is found = ete (and from thence (\W1— xx) the fine of A = /—1 x A —A aise ; but this laft by the bye). Now feeing that 2 cof. A = M4-+ M-A, we there- fore have 2 cof. A\’ = Mee M-4|" — M+ M-"4 f nx M34 M-*t-8 1g x M44 More A 1c, by expanding M* -+- Nive ” and uniting, in pairs, the corre- fpondent terms (vz. the firft and laft, the fecond and laft but one, and fo'on)... But M*4-+ M-*, the firft of thefe pairs, is the double of the co-fine of A; for the very fame reafon that: M4 -+ M~4 was found to exprefs the double of the co-fine of A. And thus, A*-?-4-+ A-*+.-A will appear to exprefs the dou- ble of the co-fine of »—2.A, &c. And our equation will. therefore be reduced to 2” x cof. Aj” = 2 cof. vA + an cof.n—2.A + an .=—— cof. n—4.A + &. or to cof. Ay = 2)" cof.zA + ncofin—2.A + 2.2 coin —4.A 2 +n. “oes cof.z—6.A + &e. 2 x where, when the exponent z is even (the number of terms in M* ++ M-A\’, expanded, being odd) there will be a middle. term (no-ways effected by M or A) which being an abfolute number, muft be taken fingly, and confequently, only the half thereof when the whole feries is divided by 2, as is the cafe in the conclufion. 2, E. D. COR OL- by Means of the Meafures of Angles and Ratios. COROLLARY, if Q be taken to reprefent an arch of go degrees, and the con ) of the arch A,be put = B; then, by fubftituting fin. B for cof. A, and Q — 5B for A in our ge- ee ft — eral equation, we fhall have fin. B|’ = =} into cof. zQ—nB c eam —I a ae ar Sh, +- 2 col. 2—2.Q—nu—2 B72. Col. 7-4 O27 ab 1 ao -+ &c. being a general expreffion for any power of the /ime of an arch (as the former was of the co-fine). But this ex- preflion may be reduced to a form fomewhat more commodi- ous, regard being had to the different interpretations of 7, with refpect to even, and odd numbers. ‘Thus, if 2 be ex- pounded by any term of the feries 4, 8, 12, 16, Gc. it is evi- dent that 7Q (in the firft term) will be an even multiple of the femi-periphery ; and that 2 — 2. Q (in the fecond term) will be an odd one, and fo on, alternately. But it is well known, that fubtraGting, or cafting off any multiple of the femi-periphery no-ways affects the value of ms fine, or co-fine ; except, that fuch value, when the multiple is an odd one, will be changed from pofitive to negative (and vice verfi). Hence our laft equa- u—TJ tion will be reduced to fin. B/ = z into cof. — 2B — ~ ie 7 n cof. — nu—2.B + 2x. —= cof. 2—4.B— Ge = n—TI =| into cofi.zB — ncof.mz—2.B +27. pe cof, 2—4'.B 2} 2, ee, And, for the fame reafons, the equation, when 7 1s in- terpreted by any term of the feries 2, 6, 10, 14, &e. will ap- n—I ‘ pear to be fin. Bl” = 4 into — cof.72B -+- x col 4— 2B nin 2" colin 4. BP Ge: Zz But, when x is expounded by any of the odd numbers 1; 5; 13, &c. we fhall then (by rejecting the multiples of the femi- O em 80 Fig. 21. The Refolution of certain fluxionary Equations, femi-periphery, &c.) have fin. Bl’ = = into cof. Q— xB — n col. OQ — 7-628 + 2.~— cof Q—n—4.B — ~ | t= i ° Boga Saree iF Ce. == 4] into fin.wB — zfin.n—2.B + ».—— fin. 2—4.B—&e, - Laftly, if z be expounded by any term of the feries 3, 7, 11, 15, &c. the refult, or feries, will be the fame as in the preceding cafe, only the figns of all the terms muft be changed to their contrary. But all thefe different cafes may be otherwife, more directly, inveftigated, by means of the two following Theorems; where- of the Demonftration is obvious, from that of Lemma I*. . The double of the rectangle contained under the fines of any two arcs, fuppofing the radius to be unity, 1s equal to the dz ference of the co-fines of the fum, and ae of thofe arcs. 2°. And the double of the rectangle ‘under the co fine of the one and the fine.of the other, ts equal to the difference of the fines of the Jum, and di ifference of the faid arcs. Hence it follows, that fin. B x 2 fin. B = — cof. BB + cof. B—B (by Theor. I.) == — cof.2B + 1: whence, multiplying the whole equation by 2 fin.B, we have 2* x fin. BI = — cof. oB x 2fin.B + 2 fin. B= — fin. 3B Eby Theor. I.) = — fin. 3B +- 3 fin. B. Whence, again, by equal multiplication, 23 x fin, B]* == — fin. 3B x 2 fin.B + 3 fin. B x 2 fin, B = + cof. 4B — ee 2B i 3colio (by Theo- vem I.) = cof. 4B — 4 cof. 2B + 3. In like manner, 2* x fin. BP == fin.5B— 5 fin.3B+-rofin.B; and 25 x fin. B]’—=—cof.6B+-6 cof. 4B — 1 5 cof. 2B+-10, &e. * By fim. A’s; BC(r): BG :: DE (2Dz) : Dp(CF —CH)=2Dzx BG: And BC (1): CG :: DE(2Dz) : Bp (EE DF) = = 2CG x Dn. oe Whence, by Means of the Meafures of Angles and Ratios. Whence, unzverfally,: fin. By = | : x i— ; -fin.z—4-.B4 &e.' + fin. “B+ 2 fin.n—2.Btz2. when 7 is an odd number ; and fin. Bl” ae . _ x i—— I] 2 when 7 is an even number: where the number of terms, in the former cafe, will be zs and in the latter ia -- 1; in which cafe the half, only, of the laft term 1s to be taken ; and is always pofitive, as well as the laft term in the former cafe: whereby the figns of all the other terms (as they change alternately) will be known. | cow 4. By Se. + cof. 7B + ncol.az—2.B42. TE a. By 95:0 > sce M be affumed to exprefs the terms (1, 7, n.—, n.~—* 7, €c.) of 1 +1 raifed to the wth power (M being the middle, or greateft term) it is evident that the fe- cond cafe of our general equation (wherein 7 is even) will ftand thus, fin. BJ’ = = into + acof. nA + B cof.n—2.A + ycofin—4.A F dcoln—6.A..... + iM. By the fame method of proceeding, an expretfion exhibiting the continual produét of the co-fines, or fines of any number of unequal arches, may be derived. For (by Lemma I.) cof.A x 2cof.B = cof. A +B + cof. A —B; whence cof. A x 2cofB x 2cofC = cof A+ B x 2cof.C + cof A — Bx 2cof.C (by equal multiplication a cof. A+B—C = cof: ALB4+C gol. A —B +e (by the Lena) ; cof. —A--+B+C whence, again (by equal multiplication and the Lemma) we M have SI 82 The Refolution of certain fluxionary Equations, have cof. A'x 2 cof. Bos. 2 cof. C'x 2cof D — cof. col AS BOSD ao oem es eee A-+B-+-C--D -- Car Ap Boca aad cof. ee go nap eaay LO As BaD from which the law of continuation is manifeft. PROBLEM I To determine the fluent of ae n being any odd affirmative i #K% number. If A be aflumed to denote the arch of a circle, whofe fine is x and radius 1, it is well known that A = : and, I— *¥ y the Corol. to Lem. IT, it alfo appears that x” == + a x fin. vA — nfin.n—2. A+ 2. hae oeat a) Hence we have —— (= xt A he eat ay x oS — xx AxfnA—nAxfin—2.A+2.2 A (2 . But the fluxion of any arch, multiplied by the fine (the radius being unity) is equal to the fluxion of the verfed-fine : therefore the fluxions of the verfed-fines of the arches 7A, n—2.A,n—4.A, &c. will be ~Ax fin. vA, n—2.Ax fin.2—2.A, n—4.A x fin.n—4.A, @&e. refpectively ; and confequently the faid verfed-fines, the true fluents of thefe fluxions: whence it is manifeft that the true fluent of our whole expreffion will be + a into ~ xverf. fin.zA — wm 2: -fin.za—4.A — — a ; 2 ee ee: A(24), Wherein, of the by Means of the Meafures of Angles and Ratios. the figns -+ and — before eS . , the former, or latter ob- ° ° n I ‘ tains, according as _ , expreffing the number of terms, is odd or even. PROBLEM. Il. To find the fluent of Bee n being any even affirmative T= X*% number. By the preceding Problem —= FES = A; and, by the Coral. T= xx oY es I ah to Lem. I, x* == + Be ne nee P a Fe GENE te Te ae cere eat gt oe yr Senecio ee ere a col.zA—@xcofmu—2,A+yxcofn—4.A..... +iM. edd. jo n—tI Therefor. == =a oe - x aA x coiaA — BAxcof. 2-2. A+yA x cof.n—4.A. IMA, But the fluxion of any arch, multiplied by he: co-fine, is equal to the fluxion of the tne: drawn i into the radius: hee it follows that mA x cof.nzA, n—2.A x cof.n—2.A, n—4.A xX cof.n—4.A, &c. are the fluxions of the fines of the arches A, n—2.A, n—4.A, Gc. refpettively ; and, Sonne that - an n— 4. A— eo seley. n—6., A + ete will be the true fluent fought : ie eon i, P=1y= Bix Tt, 3 = xt, €c. and wherein the fign 4- or —, before He » Obtains: according as in +1, coe the number of terms, is odd, or even. Me COR OF; 83 84 a divided by 2* gives —; = + The Refolution of certain fluxionary Equations, C-OR OL IL AR Ys I. Since the value (M) of the middle term of the binomial : s : R—Tt 1-—2 1 + 11", expanded in a feries, is known to be —x x : Peers ae *, by the law of the feries, it is evident that the +n net term next adjacent to it (on either fide) will be expreffed by M x = 2" or by Mx—*_; making m == +n. And, in the n+1 m+-1 fame manner, it will aay that the next term to this laft will be oe by Mx i xa and the next to that, by ys — 2 TM ae ax cas a x ao and fo on. Therefore, by fubftitut- ing thefe values above, and i inverting the order of the terms, the general — there givens will a be eee to i— . 2” Jj m—t Mmm Diy m mM—T M2. m—3 a Gases LOA a hei pale te xif8A €3c. where the feries is to be continued till it terminates; and where the value of the general multiplicator will be truly (and moft commodioufly) exprefied by = x oe S22 REY For n n.u—TI N——-2eut~=— cece a cn M being = TSE SEE if the numerator hereof be multiplied by 17 .22—1.in—2.in—3..-3- 2.1, and the denominator, at the fame time, by its equal 1.2 eee 1B eAveeeeet—2.in—t.in, we fhall then have M= ns NR He RS __1.2.3-4-5- BY}. ce'e othe B ° oe ee ea a a "3 1.2+3-4. TERI LGR ys Live See a +2-3+4-5.6.7.8. ee re 5. eee AI 7 k ag n CcOROL- by Means of the Meafures of Angles and Ratios. COR ODL ARC YH, Hence may the fluent of xx / | — xx be likewife deduced ; for this expreffion may be changed to os or to I—*% th tt? . ae eae already found — ~ x 2x 2 ge | 2 tie n whereof the fluent of the firft term is A— m : m m 7a an On PS gees making ==" =- 2; and: af =i! (= Te ), the fluent of xt are the fecond term — bs 5 Ea ey, in the fame / i — xx Vi — xx : ’ n— I 4 5 sae ™m m ++ 6 n+ 2 A — 22x fin, 2A +7 x Sees &c. Whence, penne A— Sdn 2A, Geta == ea by aie ig fluents of fan terms together, we have, after proper reduction eee Ps ee ee 2, AYO O fim oan n.n+2 A" 5h eee A ea oe a m + 2, Tare mas) 4 m+2-m+-3m-+-4 | xt fin. 6A - eS x F fin. 8A, Be. L a m+ 3. m+4.m+5 where the law of continuation is manifeft, the differences (6, 10, 14, €&¢.) of the numbers 1, 7, 17, 31, Ge. being in arith- metical progreffion. 0 See: II. Moreover, from hence the fluent of F ai # «efx -gxtt-bx® V1 — KK &c. may be eafily deduced: for, putting =x a 3x3 ne ee nu eee) * x tfin.4A, Se. And, by 8 5 86 The Refolution of certain fluxionary Equations, xe, is given, dy Co- == gs tire fluent of ihe firft term = Lo KX gE A hakadndaait es AA and that of the fecond term = x fx j=! =i : x gfx ka 6X rae == gex A—— } m AS Vin 2A + = coo oo eae tt ‘fin. Rte There- fore the fluent of the whole reuptelion will be found = gq eae afi HX ae x fin.4A &e. see — te xfin.2A + 2x Tx $fin. 4A Ge. &e. &Se. : which, by making 7 = ar 5-5 == Ta xh f= Tt xs, &c. will be reduced to Shes eie ts gr tea m-+-1 . m 2 = o - — Joe mas gst &e. x fin.2A m+ 2, 3 m-+-% m+2 m+1 ae m-+-1 eae oa ts sae wren a e Sexi f4A m—It m—2 m+-1 m } ‘ m1 "m+2 "m4" Oe ake L m4 fr + &e.x: {6A &Se. &Fe. SCHOLIUM. From the pees determined in the preceding Propofition, th ae gh TPT. thofe of = 5 Ja — bz? x gret— 33, and = Ng ee Ma hes? x e+ fe? 4- gz? + hz? Gc. (where m denotes any whole po- fitive number, and / any pofitive number. whatever, whole or broken) may be eau’ deduced, by means of a proper transfor- mation ; by Means of the Meafures of Angles and Ratios, ° ie 2 z ar mation: for, / a— bet being == af I— be let there a bal a be made en” y OFZP = x"s ; then 2% + #— x x20-b, and Gonfequentls, by taking the. fluxion on both fides, > a mp + 3p ip gre tib— 1% —— 2, X 2m + 1: xx, or err hPa mp + ate > mt-2 2 coe z ae | x He Therefore our firft expreffion, a— bz" will be transformed to — : x (fuppofing i MD I whofe fluent (dy Preble I. Corollary L.) will be given = BS Ry Frets seccc bane mI: 2a” . bs gs SSE aoc wie pene m Be Bl, m— tI teas 3 dase es sare aA m-+-2 maa 6A + &c. where A reprefents the arch whofe fine is ee —, radius being unity. In the fame manner the fluent of our fecond expreffion, JS a— bz x emer *?—1z, will be given ( by Corol. IT.) equal to Ve Shep sac cee eee POT Beak 2: Gs oe ae ni 2 oom M1 m.m— 2. WI—T.mM—tI oe fin. «Mina A Bae . A m+ 2 ae T ingea: mas a ae M-+-2.M-+-3.m+4 mM. mM—1.m—-2.m— ec aa a pare x i fin. 6A ae ae ee eat ¢fin. 8A — &e, pt ae — st Laftly, = == Sed et fee tg + gz? + bz? Sc. will be trans- a— bz formed to — x = —xe+Lx x? + xt Be, and the po ©? Si—xx fluent thereof (4y Corol. IIT.) will ahecarsed be given equal to 5a Hea + fr x + gsxe oh Be, x A porte * SU rarer a EEE wise PO , Oy eee yf oA a4 a or x5 eg bEsx Gx 2a woxt ok 87 88 Lhe Refolution of certain fluxionary Equations, 2a SRR Dn enn aa m m—-% ajm+y m : x ghee rex pe x {4A ESC. &Fe. iy et me n+ ° eee : (ed a ale n+ 3 nb 4 Se = ae, bz” : : . When — becomes equal to the radius, or unity, A will be an arch of go degrees ; and therefore, the fines of all the arches 2A, 4A, 6A, &c. being then equal to nothing, the mp ope z z fluent of will, in that circumftance, be barely = NM a— bz" BoBu oe ee ye A Moreover, -the fluent of Berl lets iasempe na—2 pont? eo ay I “ 5 esis 5. eal wie: uefa J/a— bz x 2m + 2% —'z will then become Praneienet yes DAs Ove Onivinc? s y's n+2 oat} SS tak ei Ea ee x eae and that of ee xe feet gz? + bz? a— va Be ee ee n+1 Ja n+1 n+3 ga n+ aes ia Co Se SS ae be era Ps be & Tas see na bP rite Eo Ee Oe. ixXA; where g = pont? BL Oa cise + m=—=o, q muft be taken = 1. 29a” x , and where, if “PROBLEM It. To determine the fluent of % x cof.mz x cof. nz x cof. pz &c. in which mz, nz, pz, Kc. are any given multiples of the arch 2 ; the radius of the circle being untty. Make A = mz, B = nz, C = pz, &ec. and find ( by the method on p. 81) a feries of co-fines of the multiples oF testo éxprefs the continual product (cof.A x cof.B x cof.C, &c.) of the co-fines propounded ; which feries let be denoted by ax cof.az -- cof. Bz-+-cof.yz-+- cof.dz Ge. (a, a, B, Be. being conftant quantities): then will our given expreflion become az by Means of the Meafures of Angles and Ratios. ax xX Cof.az + cof. Gz cof-yz-+-cof.dz Ge. and its fluent will therefore (by proceeding as in Prob. II.) appear.to be = a into fin. az fin. Bz fin. yz fin. d% a5 . a 3 -|- - + ; &Fe. Thus, for example, let the fluent of 2% x cof.mz x cof. nz be required ; then will cof.A x cof.B (= cof. mz x cofl.mz).== —x cof. A-+-B + cof,A—B = = Xcol. m+>- 2. 4+ M—1.%: c ° IT therefore, in this cafe, a= 3 em -+n, and @=m—n; fin. mn. % a 1 < and confequently the fluent fought = = into ma fin.m—n.z% Ti ——= "hg In like manner, if the fluent of % x cof'mz x cof-nz x cof. pz were to be required ; then would cof. mz x cof. nz X col. pz (= cof. A x cof. B x cof. C) = -, into cof. m +- + p.z at cof. m +nu—p.z + cof.m—n-+ p.z + cof.-m+n+p.2; fin.m—+n+p.2 ob and therefore the fluent fought = es into m+n-+p fin.m-+-2u—p-2% | Shere Phy Ba gs Ss eee m+-n—p mM—n—+p —nu+t+m+p By the very fame method the fluent may be determined when fome, or all of the factors are fines (inftead of co-fines). Thus, if there be given % x cof. mz x fin. uz, it may be wrote % x cof. mz x cof, go? — nz; which is = into cof. go° + m—n.z + coiimt+n.z— 90° = = into fin. 2— m2 + fin.z + m.z; and fo the fluent (by proceeding as in ; fed-in. ere Problem I.) will come out = J eto a 2 a—m verfed-fin. 7 + m.z ee = = 5 N PRO- 89 QO The Refolution of certain fluxionary Equations, PROBLEM TV. From the equation ay -+- 2 -|- 2 4. a -|- “. +. &, =o (wherein a, b, c, d, &c. denote confiant quantities) ; 1t is pro- poled to find the value of y in terms of &. Affume y = aM™ + @6M™ ++ yM’* -- IM &c. in which M denotes the number whofe hyperbolical logarithm is unity: then will y= mzaM” +- nzBM* 4+- pzyMrs Ge. § = mz aM + nz BM -+ p2'y Me Ce. jf = 330M" + 0323BM" -+- p323y~MP* Ge. ee, ESe. ‘Which values being fubftituted in the given equation, we have aaM”* + aBM™ +- ayMe* &e. bmaM”™ +- bnBM™ +- dpyMr« &e. cmaM* +- cn*CM”" -- cp*yMP* Se. diBaM +- dn3GM* -- dp3yMi* Se. ea. &e, =O ‘From whence, by equating the homologous terms, we have a+ bm + cm? + dm Se. = 0, a + bn + cn* + dn Ge. = 0, a+ bp + cp + G3 Ge. = 0, Ge. that 1s, the re- quired values of m, , p, &c. will always be the roots of an equation, a+ bx + cx? + dx} Gc. = 0, wherein the given quantities are the fame, in every term, with thofe in the flu- xional equation propounded. ‘Therefore, when thefe roots are known, the value of y will alfo be known: in which the coef- ficients a, B, y, 0, &c. may denote any conftant quantities at pleafure ; as is evident from the procefs. When fome of the roots of the equation @ -- dx -|- cx? +- dx3 +. ex* &c. —= 0, happen to be impoflible, the values of the correfponding terms of the feries aM -- BM” -- »M* -+- dM &c. will then be exprefled by means of the fines and co-fines of circular arcs. ‘Thus, for example, let the fluxio- nary equation propounded be y — — o; then we fhall have 1 — x'== 0; whereof the four roots are 1, —1, + /—1, and —/—1; and, thefe being fubftituted for m, , p, and qs by Means of the Meafures of Angles and Ratios. g, refpectively, y will here become = aM? -+- BM-* 4- yMte"— +. JM-*”=1, Now, to take away the imaginary terms yM2”— -- IM—*"—, we may write & + /=y, and k — 1=0; whereby the fum of the faid terms will be = hx MY te Moet Fo Meo! Mie’ oe 2k x con z+ var x fin. z (vid. p. 78): whence (putting 4 = l yaar we have y = aM* + 6M-* + 24 x cof. z -+- 2h x fin. 5 where a, B, 6, and & denote any conftant quantities, at pleafure. In like manner, fuppofing the equation given to be y -+ a == 0, we fhall have 1 -+ dx3 — 03; whereof the three roots Set ee er. Le ee are —d_t, d Lote if; 7 and d x3 = 3 which, if s be put==d—+, and ¢ == d—?x a ee will be more commodioufly expreffed by —s, ist t/ — ¢, and 1s——~f#\/_ 1: And thefe values being fubftituted in the room of m, n, and , we thall have y see! eM - AME te 4. yMie ee i oM—* + M** x GM . Ween ; which, by reafon- ing as in the preceding cafe, is reduced to y = aM—*-+- M** x 2b x fin,tz + 2k x cof. tz. PROGLEM . ¥. From the equation ay +- a a 2 is py Ge. = AMP 4. BM? -+- CM” &c. to determine the value of y3 fuppofing M to denote the number whofe hyperbolical logarithm ts bes a, b 6, &c. A, B, C, Ge. any conflant quantities. Affuming y == PM’ +- QM? + RM™ &c, we have Y= psPM + gzQM-+ rzRM* &e. j =p = BM + 9°2°QM* + 7’z’RM® &e, GPa. &e. N 2 which s 9 bo The Refolution of certain fluxionary Equations, which values being fubftituted in the given equation, it becomes aPM? + aQM? 4+- aRM”® Ge. bpPM’ -|- bgQM* + d7RM®™ &c.( == AMP +- BM -+- cp PMP +- cq’ QM -+- cr’ RM* &e, CM” &ec, Se. &e From whence, by comparing the homologous terms, we have A B Se eS eT ge ye ga TERE PTH Re QS Tate pee om FE a EF. whereby one value of y is known. But the value or fluent thus found, in order to render it eneral, muft be corrected by the value of y found in the preceding Problem, that is, by the quantity «M”* +- BM** + yMr* &c. wherein m; n', p', Sc. denote the roots of the equation a +- dx + cx’ +- dys Se. ==C, and a, B, y, &e. any conftant quantities. For, fince all the terms arifing from this laft part of the value of y, by fubftituting in the given equa- tion, do mutually deftroy one another, the other terms affected with P, Q, R, &c, will be no-ways influenced thereby, but remain exactly the fame as above determined, C0 R:0-Gas A ROY: If the equation given be my + & — AM +. BM? -- CM -+- DM €c. then (a being = m', c==1, and 4, d, e, A ete mm + pp ane oes ot m/—1t; and confequently 5 aM — == : oa pz 8 GM-m=" = _AM* 4. Ee ae = 2h x fin. mz -\- 2k tSr..each ==..0) ave bave P= Q=—_,, &e. min at pp — mm eta 94 x cof. mz + oat e -+- _ Ste (jee Ex. Tt, to Prob.d’.) Hence it follows, that, if the equation given be m‘y ++ (= AM™"= 4. AM~*"— 1. A'Me¥— + AMY &e. ) == 2AX cof. TZ ae 2A’x cof. ez Be. the value of y (by fubfti- tuting — 7 comely ot — 7 =F, —p =r, —p =S, feb, AC, A’ D, Ge.) will come out'== 26 x fin.mz -+- | by Means of the Mea/ures of Angles and Ratios. nz¥—1 wage ee + 2k x col maz 4 AM 4 AR hide ETT mn — TT 2A 2A’ + 2k xcof mz + ~— x cof. ez 4+ — MM —— TIE Ni ee Which equation (wherein 4 and & may denote any con{tant quantities) is of fingular ufe in determining the figure of the lunar orbit. 6c, — 2h x fin. mz x cof. pz Ge. In like manner, when the general equation propounded is of : by ay Pitts \n = this form, ‘ay +3 2S ed Se Ae > Bas aCe z& x > --+ Dz’-3 @e. the value of-y maybe determined, by affum- ing Per + Qe + Re? &c. = y3 from whence, by fub- {tituting in the given equation, and comparing the homologous ‘ij A B — vbP terms, there will. be. had P.= —, Q = — R= a a C-v-1.6Q—v.v-1 cP g Dae Rava 0-2 Qo9 971002 AP a Pkg a ; €3c. where the feries will always terminate, provided wv is any pofitive integer ; and where, if to the value of y thus deter- mined, the corretion or feries (a&M* + GM” --+ yM?* Gc.) found by Prob. IV. be added, the general value of y will be obtained, PROBLEM VI. To determine the value of y in any fluxionary equation of this form, = + is 5 -|- gy + dy = A; fuppofng A to repre- & Pe % Jfent any quantity expreffed in terms of x and known coefficients. 1°, Make y == M x flu. 2PM—* (wherein P denotes a variable quantity, and p a conftant one, to be determined) : fo fhall yx M-#* = flu. gzPM-?*, or (by taking the fluxions) yx M- — y x paMP = sPM-#; whence, dividing the whole by 3M-**, we have 4- — py = P. 2°. Make P == M? x flu.2QM-#; then our lait equa- tion will be transformed to 2— py x M~* = flu, 2QM~# ; & ‘ whence, 93 94 The Refolution of certain fluxionary Equations, whence, by taking the fluxions, + — py x M-# +1. ee py = 4 en gMr iOS, WL PTH tH =e by dividing the whole by 2M. : 3°. Make, now, Q = M** flu. zRM="*; then will = — p+9-2 = pay x M- == fluent: of zRM-*,- or Lp +gt+ryxM-"+2—pF9.24+ py x - rsM- = &RM-*, or, laftly, 2 —ptgpr. t+ pat+pr tor. — piry=R. 4°. Make, again, R = M* x flu. zSM—*, and ptoceed in the fame manner; fo {hall x —ptgtr at's petites. me PIP ERAGE LETS 2 Po Epes Ft prspgrs. = + porsy = 8: from whence the law of continuation is manifeft. rey Let, now, the feveral terms of the equation 2, —ptgtrts a 5 ee, == S be compared with the correfponding terms of the given one, Zt + 2 Gt. = A: fo fhall p--+¢+r &c. = — a, pa + pr+ps+ or &. = — 4, por + pgs &. = —c, &c. &. Whence, from the gene/is of equations, it is evident, that ~, 9, 7, &c. are the roots of an equation x* + ax3 +. bx* + cx +d = 0 (or, x* + ax’? + bx? Ge. = 0) wherein the given quantities are the very fame with thofe in the equation propounded. Therefore, when the valves of thefe roots are found (by any of the known methods) the values of R, Q, P, and y. may alfo be found, one from. another, fucceflively, 9. FL. The 4 J i é ! by Means of the Menfures of Angles and Ratio, 95 The fame otherwife. _ Let (if poffible) y = AM? x flu. zAM-* -- BM? x flu. zM—* +- CM™ x flu. 3AM~ &c. (A, B, C, &e. Pe Gar, &c. being conftant quantities to be determined): then, by taking the fluxions, we fhal! have 2 = pAMP*xflu.3AM-#-+ AA -+-9BM?x flu. SAM—#-4 Ge, Ser -® =p AMP xf.cAM-* + pAA-+"S4 eBMerx§ ZAM PE, 2 & 5 =p AMP x fu.zdM—* 4 p'Aa +28 4 BO 4 ayy, = pPAMPx fin. z4M—# pA +POS PAA 4 BS ee, Which values being fubftituted in the given equation, and the homologous terms being compared, we fhall thereby get p+-++ 93 +o’ + p¢+d=—0, ¢+a3 4 bf +oatd=o, &e. alfo en es +e tag + ee rar tbr+cx C+ star +os+e¢xD pe taptbx A+ 9 +aq+bx B+7?-+Lar+-bxC+s'+as-+bxD=o ptexAt+gqtaxB+r+taxC+s54+axD=0 A+B+C4+D= 0. Now, from the former of thefe equations, p+ + ap} -| bp* +ptd=o, ¢ + apt b¢ + c¢4+d=0, &e. it ap- pears evident, that p, g, r, &c. are the roots of the equation xt -- ax3 - bx? + cx + d= (or, more generally, of x" -- ax’ + bx"? Se. —= 0, m denoting the order to which the fluxions afcend in the given equation); which roots being therefore found (by any of the-known methods) the values of p> % 7%, &c. will be obtained. But to find from thence and the remaining equations, the values of A, B, C, &c. let the laft of thefe equations multiplied by @, be fubtra¢ted from the preceding one, fo thall p2-++ gB-+-rC-++ sD=o: moreover, let this new equation multiplied by a, be fubtracted from the laft but two, and from the remainder let JA +- 4B 4- 6C -+- bD = o be again fubtracted, whence will be had p*A ++ te — > ON Lhe Refolution of certain fluxionary Equations, 17°C + °D =o: and, in the fame manner, from the firft equation, will be had p3A + B+ r'C + 8D=1 (be- caufe r, and not o, forms the latter part of that equation). Now, from each of the equations (A + B+ C-++-D=o, pA +qB+r7C4+sD=0, PpA+gB+rC+sD=—09, DA + B+ 73C +- 3D = 1) thus derived, let the preced- ing one multiplied by , be fubtracted : fo fhall g—p.B +r—p.C +s5—p.D =0, g—p.9qB + r—p.rC -+s—p.sD =0, g—p-¢B+r—p.7rC4+s5—p.sD=1. Moreover, from each of thefe laft equations, let the preced- ing one multiplied by g, be in like manner fubtra¢ted; whence will be had _ ) r—p.r—g.C + s—p.s—q.D =09, r—p.r—g.rC+ Sp 9. sD, Again, from the laft of thefe, let the > preceding one multi- plied by 7, be fubtracted ; then wills —p.s—q.s—r.D I ==‘, and ’confequently D = =— whence it is s—p.s—q-5—Tr manifeft by infpe€tion (becaufe p is the fame with refpect to A, as s isto D, ec.) that A = ———"— 3 Bes Ae p— -p—r.&e. q—p.q—r. &c- Co is IS ey? Prom whence the value of y (— ; r—p.r—qg.Xc. AM?~x flu.sAM—-++- BM?x flu.sAM~? &c.) will be known, let the orders of fluxions in the equation afcend to what height they will.—Thus, for example, let the equation propounded. be = 4. m*y = M*™: in which cafe, a being = 0, b=—m’, ¢== 0, &c. our general equation, x" ax" A bx? + cxt-3 &c. = 0, will therefore become x* ++ m* = 0; whereof the two roots (p and g) are m/—1i, and — mf/—1; from whence A (= eae Hist and B (eat ) =— PAT am/— 1 Ear I s -——: alfo, becaufe A is here == M**, we have y = AM’ amv —I x by Means of the Meafures of Angles and Ratios. 97 : ' ache AM™ , BM™ x fl e WB— PB q® a BB— 9B —— ce ata ee u. 3M +. BM? x flu. zM pes mM” - (by fubftituting the values of ~, g, A, and B). But = in order to render the folution general, the value of y thus found muft, always, be corrected, or augmented by the quan- tity «MM + BM + yM”™ &c. (given by Prob. IV.) where a GB, y, 0, &c. may denote any conftant quantities whatever, pofitive, or negative.—Other inftances of the ufe of exponential quantities, and of the meafures of angles and ratios, in the refolution of fluxionary equations of different kinds, might be given; but I fhall conclude. here, with obferving, that A, in this laft folution, may denote any quantity wherein both y and z enter, as well as one in which z is alone concerned in- dependent of y. etcetera pe eas gky Bie BSAA AS ARRAS cae}, REAR RRII CREM RRRER REEF Ste Stee" ae See? Stee Sa See - Soren SER ep Lara ECS An InvesTicationofaGEn ERAL Ru Le for the Refolution of Hoperimetrical Problems of all Orders. LEMMA. A Hd slrswincg a, B, y, 0, €, Se. to be a feries of in deternunate quantities, Fake f 7 ' 9, Re a qT” | Qo” R", Mee t are any quanti- (and given and. tbat. si RS. m I” Cries one °d of | 4 ee ; | oe, Res s” qi | | &c. r | Bec. j It is propofed to find an equation for the relation of a, B, y, 6 Ge. fo that the quantity 24- B+ QW + VW’ QW” Be, Jeall be a maximum ot minimum, at the fame time oh the other quan= tities R ~\ R’ ob R’+- R” ~- Ie EF ec. S a ~ +5 ot ha hie +8" &e. and T4+-T' + T" 4- T° 4-F"" &e. are all of them given, or fuppofed to remain invariable, Let 2, R, S, T denote any correfponding terms of the {e- res’s Qt BQ’ + QQ" Ge. RR + RL R” ot R” Ee. Ss + S’ - ae -- S -- naked ESc, T + T' “+ TT” +I" 47" Be. re{pectively, ape in terms of uw, any one of the propofed quantities a, 8, Y, 9 ¢ Se. moreover let the fluxion of 2 (a being variable) be denoted by ga; that of R by ra; that of Q' by gh; that R’ by ¥ 2, &e. &e, It is evident that the quantity Q 4- 2° ++ Q’ + Qe +9” &ce. cannot be a maximum or minimum, when R -|- R’ + R” fn R” +. RY EFe, S + AY oh ae + on 4 SO &c. and T + qT’ +7" 47" 47” &e. are given quantities, unle(s the part + QAY + Q is a Mmaximum or minimum, when the parts- SR eT. 1. An Invefligation of a general Rule, &c. parts RRR RY 8 ES CS Ss and TT + 7’.1. 7” are given quantities; becaufe the terms in thefe parts may be alone made variable, while the other terms are f{uppofed to remain the fame, whereby the whole fums, R —. R |. R’ | RO |. aoc: &c. Ss ot S’ a Se -+ ie a ee &c. will remain the fame, and the quantity 2+ 9’ +. 9” + Q” 4+ Q'" &e. will be a maximum or minimum, when the pat 2+ 2+ 2 + Q" is fo. But when 2-+4+ 2-4 Q’ + QVis a maximum ot minimum, and R + R+ RB + R’, Ststs’+8", andT4+74 7747" are given (or conftant) quantities, their fluxions will be, all of them, equal to nothing ; whence we have thefe equations, qu + qa t+9B +q'y =o ri ++ ra + ret ry —=0 sa sa LB. sy =o ta I- ta + 7¢B4 ry =o ae In order now to exterminate the fluxions z, a, B, y, let thefe equations be refpectively multiplied by 1, e, f g, (yet unknown) and let all the produéts thence arifing be added together; whence will be had g -- er 4+ fs + gtxut q-er+fs-tetxatd er +jitgt x B+ er" + fet x5, =O. Make, now, g + e+ /s + gt=o9, dh: yates Soe ob er” fi gt = 0, From whence, there being as many equations as quantities; e, f, gs to be determined, the values of thofe quantities will always be given from thence, in terms of the quantities g, 7,5, ¢, G05 5,07 4 F955, bf excluave OL gos Sy. t,). Now, fee- ing all the terms of the equation g -+ er -- fs S-gtxute gtertpspetxatg + er fi pet « B+ per” 4 fi" ot’ 4 == 0, after the firlt (¢ er + fs + gtx u) do thus actually vanifh (by their coefficients being taken equal to nothing), it is evident, therefore, thatg ter + fs + gtmuttalfo be =o (or flux. Q + ¢ flux. R+ fx flux. § + ex flux.T =0); where O 2 é; 100 An Inveftigation of ageneral Rule e, f, g being quantities depending intirely upon q, 7, 5, ¢, &c. (exclufive of 9, 7, 5,4), they muft neceflarily be invaria- ble, or continue of the fame value, let g, 7, 5, ¢, ftand for which terms you will of the correfponding feries’s, g’+-¢""&c. “lt 44 ier / / / f Mf yur obec, becanfe they quantities g, 74, 4; Wy 7, Sa fe-9) r”, s’, t, (on which e, f, g, depend) have themfelves a deter- minate value each, in the required circumftance, when 5 / { 7] = . ee 942 41-Q &c. is a maximum, or minimum, PROPOSITION. Sufpofing y and u to be two flowing quantities, and that 2, R, S,T ec. are quantities expreffed in terms of y, U, ana given coefficients; tis propofed to find an equation, exprefing the relation of y and u (or of 2, R, 8, T, Se.) fo that the flu- ent of Qy, corre[ponding to a given value of y, foall be a maximum or minimum, and the fluents of Ry, Sy, Ty, Se. all of them, at the fame time, equal to given quantities. Let 2, 9’, 2”, 2”, &c. be the different values of 2, that will arife, when y is, fucceflively, expounded by the terms of a given arithmetical progreffion whofe common difference is the indefinitely {mall quantity y (a, B, y, 0, &c. denoting the re- {pective values of uz), and let REAR GARY RS we che the correfponding values of R, &c. &c. Then it is well known that the fum of all the quantities Qy + ey ae Q"y + 9"y' + 9°" &c. will be = fluent of 9; and the fum of all the quantities Ry’ + Ry'-- R’y+ R%y + Ry &c. = fluent of Ry, &c. But, dy the Lemma, it appears, that Q-+ 9’ + Q” +9” L &c. or Qy + By + Q"y -+- QM y' +- &c. (because ‘ is conftant) will be a maximum or minimum, and the quan- tities Ry’ + Ry’ +- Ry’ + Ry’, Be. Sy’ + S’y +e Sy 4 SY, &c. at the fame time equal to given ones, when the relation of y and w (or of Q, R, S, I, &c.) is expreffed by the equa- tion, flux.Q+-e x flux.R+/x flux..S-+ gx flux.T' =o: where e, SF, & .&c. denote (unknown) conftant quantities; and where, in taking the fluxions of 2, R, S, T, &c. the quantity is, alone, for the Refolution of Loperimetrical Problems. ror alone, to be confidered as variable;. becaufe the. fucceffive va- lues of y, entering refpectively into 2, 9’, 9”, 9", &c. are ‘conftant quantities , being (by hypothefis) fuch as fucceffively arife from the terms of a given arithmetical progreffion. Hence we have the following GENERAL RULE for the Refolution of Tfoperimetrical Problems of all orders. Take the fluxions of all the propofed expreffions (as well that re- [petting the maximum or minimum, as of the others whofe fluents are to be given quantities, making that quantity, and lkewife tts fiuxion, invariable, whereof the fluxion (as well as the quantity itfelf) enters into the faid expreffions; and, having divided every- where by the fluxton of the other quantity made variable, let the quantities hence arifing, joined to general coepicients 1, e, f, , Ge. be united into one fum, and the whole be made equal to nothing: rom which equation (wherein the values of e, f, g, Sc. may be et- ther pofitive, or negative, or nothing, as the cafe requires), the re- quired relation of the two variable quantities will be truly exhibited. To illuftrate the ufe of the rule here laid down by an example, Fig, 22. let x and y be fuppofed to reprefent the ordinate (PQ ) and ab- {cifla (AP) of a curve ADQE; and fuppofe AFRG to be an- other curve, having the fame abfcifla, whofe ordinate PR is, every-where, = ax” y"; "tis required to find the relation of x and y, fo that the area BFGC, anfwering to a given value of BC, fhall be a maximum or minimum, at the fame time that the cor- refponding area BDEC is equal toa given quantity. Here, dy hypothefis, the fluent of ax”y"y is to be a maximum or minimum, and that of xy equal to a given quantity: taking, therefore, the fluxions of both expreffions, &c. (making x alone variable, according to the rule), we thence get max"—'y"y —ey == 0: WiLenGe Ma ee —; and. confequently ax”y" (= PR) = , Therefore, feeing PR is in a conftant ratioto PQ), it is evident, that both the curves will be of the fame kind; and that they will be both Ayperbolas, or both parabolas, according as the values of the exponents m—1, and z (in the general equa- tion 102 Fir. 23. An Inveftigation of a general Rule on ah Are —) are like, or unlike, with regard to pofitive and negative. If m—1 be pofitive, the equation gives a mini- mum; \f negative, a maximum, but when m—1 =o, or when m — 0, the equation fails; in which cafes there will be neither a maximum, Nor a minimum. x23 For another example, let the fluxions given be = and x ; the fluent of the former (anfwering to a given value of y) being to be a minimum, and that of the latter, at the fame time, equal to a given quantity. Here (x being concerned independently, either, of its fluent x or fluxion 4) let the fluxions of both ex- preffions be taken, making x alone variable; whence, after dividing by x, we have a and 1: therefore, in this cafe —- --e==o0: whence xa? y—! y (fuppofing a== — +e); and con- I fequently x24! y? ; being an equation anfwering to the com- mon parabola. "The fame conclufion may be otherwife derived (without fecond-fluxions) by affuming : ==; whereby our two given expreffions will be transformed to yyu’ and ju: from whence, by the rule, we get.3u yy + ey =o; and therefore v j)= a: y+; whence x == a'y~*y, and confequently « = 2a*y*, the fame as before. If the abfciffa (AP) of a curve AQC be denoted by x, and the ordinate PQ by y, and p be taken to exprefs the meafure of the circumference of a circle whofe diameter is unity; it is well known that the feveral fluxions, of the abfcifla AP, curve- line AQ, area APQ, fuperficies of the generated folid (by a ro- tation about the axis 4P), and of the foliditfelf, will be, re- fpectively, reprefented by x, / xe yy, yo, apy f xx yy, and py x: if therefore, the fluxions of thefe different expreffions be taken, as before (making x alone variable) we fhall get 1 +. ex x 2 e Winiae +/+ TAS -+- hy == 0; being a general equa-. tion for determining the relation of x and y, When. any one of thofe five quantities (wiz. the abfciffa, curve-line, ‘area, {uperficies, or folid.), is a maximum. or minimum, and. all,, er for the Refolution of Tfoperimetrical Problems. or any number of the others, at the fame time, equal to given quantities ; wherein the coefficients e, 7, g, and 4, may be po- fitive, negative, or nothing, as the cafe propofed may require. Thus, for inftance, if the length of the curve only be given, and the area correfponding is required to be a maximum, our . . ex 3 equation will then become ———— —0, or ax = q Af xx +. +h 2 yy y’ x ae 90 + yy (by making a= +); whence x —= Jae and confequently » == a— /aa—yy, or 2ax —x*— 43 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 fup- pofed given) the abfciffa, atthe end of the fluent, be given likewife, and the fuperficies generated by the rotation of the curve about its axis be a minimum; then, from the fame equa- tion, we fhall have 1 +- pit ~ , ax DV I ) a; ay — —) xis found == ——2= : ae : Nar and: from thence x — gx hyp. == 0, whence (making ¢ = log. ob aol which equation, by being impoffible when y is lefs than a, fhews that the curve (which is here the cafe- naria) cannot poflibly meet the axis about which the {folid is generated; and confequently, that the cafe will not admit of any mintmum, unlefs the firft, or leaft given value of y, exceeds a certain affignable magnitude. When any, or all of the above-fpecified quantities are given, and the contemporary fluent of fome other expreffion, as xx yy] xy"y*—", is required to be a maximum or minimum, our equation (by taking the fluxion of this laft expreffion, and joining it to the former). will then be xa + yyl"" x anxyry— ex | Lyx : . a vas = +A+ Tai rE: + by = @: — iad m1, and“z==—1, will be that defining the folid of the leaft refiftance; and this, when the length of the axis, only, is fuppofed to be given (without farther reftriGtions) will be ex- prefled 103 & An Inveftigation of a general Rule prefled by «x Layl* x — axyy’ + d= 0, or 2yy'x = d X xx + yy\ ; being the cafe firft confidered by Sir Isaac New- TON.--If both the length and the folid content be given, the equation will be — axyy x xx tb yy\~* + d +- by* = 0; but if, befides thefe, the fuperficies is given likewife, it will then be — 2xyy’x xx yy” + d+ jens yO. Thus, in like manner, by afluming #m == — a and z= =} y Pa ex Lyx . epee ix gre ND hag panes hy —o-: ie have (gee pots? bere Vropgp a w being the general equation of the curve of the fwifte/t defcent ; which, when e, f, g, and & are all of them taken equal to nothing, will become : == d; which is the cafe confi- xx—+ty dered by many Others, aniweninte to the cycloid. When the length of the arch defcribed in the whole defcent (along with the values of x and y) is given, the equation will then be oe rae ——,> ., —— Jay tet ier —o, or ety 4*\ xx = d xxx+yy. And thus may the relation of x and y be determined in any other cafe, and under any number of reftrictions; provided that one of thefe quantities, only, enters into the feveral expreflions given. —When both x and y are concerned, as well as their fluxions, the confideration becomes more compli- cated; nor does it feem practicable to arrive at a General Rule, to anfwer equally in fuch cafes, JNeverthelefs, if the ultimate values of x andy are fuppofed given, or the required curve is to pafs through two given points, without being con- fined to farther limitations, except that of the maximum or minimum (which cafe is the chief, and the moft ufeful that can occur); then the method of folution may be as follows: Take the fluxion of the given expreffion (whofe fluent is to be a maximum or minimug.) making % alone variable, and, having di- vided by x, let the quotient be denoted by u. Take, again, the fluxion of the fame expreffion, making x, alone, variable; which divide by x: then will thts laf? quotient = u. ; From ee ae al tg ee ag I Oe i I rng A hE eh Cag jor the Refolution of Loperimetrical Problems. From which equation the value of w, and the relation of x and y will be determined. Thus, for example, if the expreffion propounded (whofe fluent, correfponding to any given values of x and y, is to bea atte —— ee ice : minimum) were to be f+ gxx2 : ; then the fluxion thereof, , and, when : ° ‘ ; Ss Qyx7 when « alone is made variable, being f/+-gx x === w x alone is made variable, equal to 2°, we here have f-gx x B= DD y =u, and ay == 2; the latter of which, divided by the former, wy ; whence hyp. log. z = hyp. log. f + gx\? + hyp. log. d (d “being any conftant quantity). Confequently w= dx f+ gx|?; which value being fubftituted in the equati- on f + gx x a =u, we thence have f + gx?x #7 — td’; Ses x ¥ = oy) (making co JS, and awa af? : u eos oIves — = —= 5 u ox confequently by taking the fluent again, we have — 2cy'; exprefling the general relation of x and y, iene them both to begin to be generated together. ae fad cd g = 0, the fluxion propounded will become (the fame as wy in the firft of the former examples) ; and here, x being = cy—*y, « will be = 2cy?, anfwering (as before) to the com- mon parabola.—But if f == 9 and ¢ —=1, then. our’ giver fluxion will become 2% ——, and the refulting equation will be X ¥ 2 o Sls . ae == 2cy?, or x® == wy’ (a@ being put = a )s which alfo. anfwers to a parabola, but of an higher order—The very fame conclufions will, in like manner, be brought out by making y and y, fucceffively, variable (inftead of » and ~). For, here, the two fluxions refulting (after having divided by y an 105 106 An Inveftigation of a general Rule, &c. and y) appear to be fen Be =u, and CR —u: | wience, dividing the latter by the former, we — ae =; and therefore — hyp. log. y? ++ hyp. log. a = hyp. log. -z (2 bei being any conftant quantity). Confequently, ay“? w= Lee , and fF ge}! x * == cy'y (c’ being put = | ad oan Henvé, by \taking the fluent again, we have ; Sere Bl ee Paling acy’, the very fame-as before. , 48 Of *s & Of the Rowe of nes ee by the Method of Surp Divisors; containing an Ex- planation of the Grounds of that Method, as it is laid down by Sir Isaac Newron in his Uni- verfal Arithmetick. AHOK HE reduction of equations by furd divifors, which is T & looked upon, by many, as a very intricate kind of ewe {peculation, is founded on the fame principles with the method of extragting the roots of common qua- dratic equations, by compleating of the {quare, with this diffe- rence only, that the fquares on both fides of the equation are, here, affected by the unknown quantity; whereas, in the com- mon method, the fquare on the’ right-hand fide is a quantity intirely known. What we, therefore, have to do, is, 70 Jeparate, and fo order the terms of the equation given, that both fides thereof may (if poffible) be complete fquares. Casz I. If the given equation be a brquadratic one, \et it be a fe px? Le gx® + rx etek s =o, and let there be aflumed sep pe PQ) — Zep BY ax! pp! 4 9x"-x-+4+5(=0); that is, let the values of the quantities 2, Mand B be fuppofed fuch, that the coefficients of the powers of x, when xx-- 3 px-+-Q, and 4x -~- B are {quared, fhall agree, or be the fame, in every term, with thofe of the equation given. Then, the faid quan- tities being a¢ctually fquared, our equation will become xb pri p 29 #8) + ips poe $2 te toe beets — AAx' —2 ABx—B* From whence, by equating the coefficients of the homologous powers, and putting a@=49g— pp, we have, 1. 224+ ip’ — A’ =, or, 22=A +a; 2 2 AB Ser; Or, ai 24B+r; O° ——B Ses or, Ve B+s. E23 Now, Ga N ¢ 108 Of the Reduétion of Algebraic Equations, ‘Now, if the value of 2, as given by the firft equation, be fubftituted in the other two, we fhall get 1p4° — 24AB = 8, and 14° + ta4° — B* = ¢, fuppofing B =r — tap, and @ == s—tae. In which equations the unknown quantities appertaining to the latter of the two affumed {quares are only concerned, and from which their values might be found. But as the refulting equation, when one of the quantities is exter- minated, rifes to the fixth dimenfion, and would, perhaps, re- quire more trouble to reduce it than, even, the original one propounded, little advantage would be reaped therefrom. In- {tead, therefore, of proceeding farther in a direct manner, it may be of ufe to try, whether fome property, or relation of thefe quantities cannot from henge be difcovered, whereby we may be enabled to guefs at chefalnes ; which may be after- wards tried by means of the equations here exhibited. Firft, then, it is evident, if both A and B are either integers or rational quantities, that the equation x* + tax + Q) — Ax B] (= x*-+ px’ + gx? + 7x) = 0 will, even after it is reduced to x + ipx-+ Q = Ax -+ B, be intirely free from radical quantities. In which cafe, the method of rational d:- vifors taking place, a reduction by means of furd quantities, or divifors, as they do not naturally arife in the confideration, cannot be of ufe. But the relation of the given quantities p> % 7, 5 (which we fhall always, hereafter, confider as inte- gers) may be fuch, that the values of A and B fhall be radical quantities, commenfurate to each other ; in which cafe, where the method of rational divifors fails, we may aflume \/ x for the common radical divifor, and exprefs the quantities them- felves by &\/ n, and 4/n; that is, we may make A= h/ xz, and B==//n; by which means our two equations, derived above, will be changed to 3 pk’ — 2kin = B, and 14*n* 4- tok’n — In* = @, or to 1pk° — 2k] = . and tk*n + ak* — 2l° = -, refpectively. Now, fince z is fuppofed to be an integer, it is plain from hence (confidering & and / alfo as integers, or the halves of fuch by the Method of Surd Divifors. fuch) that £ and “t muift be integers likewife, or, at leaft, the halves of integers; and confequently that » (whofe value we are here feeking) ought to be fome common integral divifor of 6 and 2¢. Moreover, with regard to & and / it is evident from the B firft of thofe equations (1p2 — 2/ = ) that the former (2) B ought to be fome divifor of 8 ; and that, if the quotient > be 71 E taken from 1p&, the remainder (162 — :) will be the dou- ble of /. 1 < It farther appears, from the equations 2 —= be oe a and Qo — 2 B’ +s, by fubftituting for A* and B* their equals 74” and ni’, that 2 will:be == Sates and # —= 22=*. From the frmeér 1 of which Q| will be known, when z and & are known; by means whereof and the other equation, / may be, a fecond time, found ; and the agreement, or coincidence of this value with that before determined for 4, will prove the folution in all refpects; becaufe then the conditions of three original equations (2Q = A*+ a, pQ = 2AB-+7, Q’ = B*-+ 5) will be all compleatly fulfilled—It is true indeed, that no immediate regard, in the conclufion, is had to the fecond of thofe equations ; we then it ought to be obferved, that the equation zph =; whereby / is, the firft time, found, is a confequence thereof, being derived from that, and the firft equation, conjunctly: and it is known, that, whatever values are difcovered for unknown quantities, by means of équati- ons derived from others, fuch values do equally anfwer the conditions of the original equations propounded. Seeing the method of folution, above traced out, depends upon the affuming proper divifors of @, 22@, and £, for the values Tog Ero Of the Reduétion of Algebraic Equations, values of 7 and , it may therefore be expedient, firft of all, in order to bring the work into lefs compas, to reje& fuch divifors of thofe quantities (if we can by any means difcover them) which we know are not for our purpofe. And this: may, in fome meafure, be effected, from the confideration of the properties of even and odd numbers. | In order to which Q = — being previoufly trans- formed to nk* (= 2Q—a) = 2Q —g+ ip) = 3p] —f (by putting g —2Q = /), it is evident,. from thence, that 7f Pp be an odd number, p* — 4f, and confequently its equal 4nk’, will likewife be an odd number; becaufe an even number (4 f) fubtracted from the {quare of an odd one, always, leaves odd. Therefore, feeing 44 x 7 is here an odd number, both # and’ 4k* mutt 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, becaufe’ 24] ‘is odd, that 24 mutt be odd too 3; and confequently. £.the half of an odd number. _ Now, feeing p; 2, and. 2& are all of them. odd numbers (when. is fuch) they may, therefore, be exprefled by 2¢-+-1,, 25-1, and 2¢-+ 1, refpectively; a, 6, andc being integers : in confequence of which aflumption. the equation 47k* = p* — 4f, will, by fubftitution,, be changed to 8dc* +. 84¢ +. 26. +- 40° + 4c ea == ga? + ga 1 — af, or 2be* + 2be tote +e=ata—f. From whence it is. mani- feft; as all the terms; but +4, are known to be integers, that’ +4 muft be an integer likewife: and fo, 4 being an even num- ber, it follows that x, or 26 -+- 1, muft be the double of an even number (ora multiple of 4) increafed by unity. There- fore all the divifors of 8 and 2¢ that have not.this property may be fafely rejected, as not for the purpofe. In like manner, zf p be even, the fame limitations will take place, provided that r 1s odd; which will be the cafe when Q is the half of an odd number (For, when Q is an integer,, Ai == tpl —f) and BY (= Q" — 5) being integers, their product A’B" will be an integer, and confequently the {quare soot thereof AB (being rational) will likewife be an integer; and 4 4 : k by the Method of Surd Divifors. and. fo, pQ and 2AB being both even numbers, their diffe- rence 7, as given by the equation AQ — 2AB + +, would be ‘even, and not odd). Therefore, feeing B*, or its equal ni’, is here equal to the fquare of half of an odd number (Q) joined to an integer (—s), in the fame manner as 7” was in the preceding cafe; it is evident, from the reafoning there laid down, that the value of x is fubject to the very fame reftriti- ons here, as there-—Other limitations might be pointed out, from the properties of even and odd numbers, were the thing worth purfuing farther. What is already delivered on this head is fufficient for the purpofe, and for the underftanding of Sir Isaac Newron: I fhall therefore, from the fevera] conclufions above derived, now lay down the fubfequent R.U LE Sor the reduction of an equation (x* + px? + gx*t-rx +s = ) of four dimenfins. Make a9 —'p*, B—=r— ap, and ¢=5-— ‘ae; then put for n fome common integral divifor of B and 2@, that is neither _ a fquare, nor divifible by a fquare, and which being divided by 4, feall leave unity, if etther p or r be odd. Put alfo fork Jame di- wifor.of = if p be.even, or half of the odd divifor if p be odd : take the quotient from pk, and call half the remainder J. Make Dawes cen and try if n divides QQ — s, and the root of the quoteent be equal to) ; if tt fo happen, then the propofed equa- tion, by means of the values thus determined, will be reduced to wx + tpx+ Q—+Y/nxkoAl. That the divifor 2 ought not shere to be-a fquare, is evident from what has been already remarked, fince both A and B would then be rational quantities ; and that z¢ ought not to be divifible by a {quare, will alfo appear, if it be confidered that & and / in the equations &/ m = A, and //n = B, are to be taken the greateft, and # the leaft, that the cafe will ad- mut of, No IIL 112 Of the Refolution of Algebraic Equations, No regard in this Rule is had to that circumftance, in which 8 @ happens to be nothing. Sir Isaac NewTon here directs, to take k alfo equal to nothing, The reafon of which depends on the equation ip" — 24/ = = which in this cafe becomes 2 pk’ — 2kl= 0; where one root, or value of & muft, necef- farily, be nothing. Therefore Q. being = ja, we have l/a(=VQG— j)=v ige—s; fo that, by a direct pro- cefs, our given equation is here reduced to x* + ipx +t ja = VJ tae — s, wherein wis given = 9 — 7p. The celebrated mathematician Maciaurin, who, in his Treatife of Algebra, has commented largely on the difcoveries of our Author, feems to reprefent this part of the General Rule, as not well grounded; laying down, at the fame time, two new Rules, in order to fupply the defect. Which Rules, I muft confefs, to me appear unneceflary ; fince it is certain, that the method of folution, as laid down by Sir Isaac NEwTon, is more direct and eligible in this particular cafe than in any other. Tt mutt be allowed, indeed, that the manner of applying the Rule, in this cafe, is left fomewhat obfcure ; but as to his di- recting, to take & ==0, when @ =o, it cannot, I am fully perfuaded, admit of any well-grounded objection. For, though ‘< it does not neceffarily follow that & muft be = 0, when GB =o,” yet the taking of & thus = 0, involves no abfurdity; feeing one value of & (at leaft) will be nothing. —The truth is, there are three different values that 2 may admit of (as ap- pears by the fubfequent note *) 5 all of which will, equally, fulfil the feveral conditions required, and bring out the very fame conclufion. ‘Thus the value of & in the equation * Tf the {quare of half the fecond of the original equations, 2Q—a— AA, pQ—r=2AB, QQ—s=BB, be fubtraéted from the product of the other two, there will be obtained the equation Q? — 4gQ? + Apr—s xQ+ + x as — irr == 03 wherein the unknown quantity Q is alone concerned ; which equation being of three dimenfions, the root Q., and confequently & (ms “a will .dmit of three different values.—From this equation it alfo appears, that Q muft always be a divifor of the quantity as — rr 5 which is a circumftance taken notice of by our Author. 4 xt i ; by the Method of Surd Divifors. x* 2x’ — 37x°— 38x-+1 = 0 (propofed by this peng, may be 0, 3 or 4; or, which comes to the fame, the equation itfelf may be reduced to x? +» —19 = + 6v Io, tex + 2 =tV5x3«+4, OLAO Xo Ye ae / 2X 4x fp, All which are, in effect, but one and the fame equation, as will appear by fquaring both fides of each, and properly tran{- pofing; from whence the given equation x* + 2x’ — 37% — 38x ae 1 = 0, will in every cafe emerge. The fecond of thefe equations is that brought out by Mr. MACLAURIN; but the firft, which is that found by our Author's Rule, is not only more commodious, but eafier to be determined, bein derived by a direct, and very fhort procefs.—And fo much for equations of four dimenfions. usec if the equation to be reduced ts of fix dimenfions, let it be x® + px’ + gx" rx! fsx? - tx + v= 0 5 and let there be affumed x’ -++ 7px 4+ Qx +> R) — Ax*+ Bx + Cl’ = xo px’ + gx* rx! + sx + tx 4-- v (= 0); which, by in- volution and tranfpofition, will give FA a eas — A’x*- 2ABy' -+-

33. 5hy — w = ; 4° EYY Re — te Now, as all the other terms, that would arife in thefe equati- ons (befides thofe put down) are affected with , and are there- fore divifible thereby, it is manifeft that the four quantities ipy + iaB—t, tay+iPB—v, iby —w, and tyy — z, here brought. out, muft likewife be, all of them, divifible by the fame common divifor x, when the equation given is capa~ ble of being reduced. If, therefore, no fuch common divifor (under the reftrictions {pecified in the preceding Cafes, depend- ing on p, r, ¢, or w being an odd number) can be difcovered (which will moft commonly happen) the work will then be at an end. From the fame method of operation, which may be looked upon as a fort of examination, whether the equation be Te cible 117 i18 Of the Refolution of Algebraic Equations, cible or not, we may find all the quantities to which 2 ought to be a common divifor, when the equation given is of 10, 12, or a greater number of dimenfions. Thus, let there be given x +} px%—"- gx 4- rx¥—3 + sxve—4 +L tx?—5 Fe. == 0, and let there be aflumed xt tpn 4Ont ta a tt Rg tt B x xo 3A Sn ty Kx ee, — nx hx Ld? bmx 3 Bc.) = x pt Lh gx? Be. then, by fquaring x«* + tpx°' + Qin + taxx* Ge. and tranfpofing x -+- px**—* 4- gx*—? Ge, it will appear, that the terms of this equation, in which z enters not, will be ot B y } 0 € 2 = I e-" I 1 1 ) GP* ¢XH A Spa ¢Xx* ee Lx ret oP yen eo 4 r aa (Xp tay Sxxts = —? 2A | ae &c. From the former half of which terms, all the quantities a, 6, y, &c. will be determined, by affuming the coefficients equal to nothing: thus we have aq — pp, B—=r— ipa, y == 5 — 1pB — laa, 0 t— i py—taB, Sc. And then, thefe quantities being known, the coefficients of the remaining terms will likewife be known; which ought, all of them, to be divifible by 7, in order that the reduction may fucceed ; that is, they ought to be fuch, as to admit of a common divifor (7) under the reftrictions before fpecified. For example, if the equation given were to be of twelve di- menfions, as x%* +- pxt? + gxt? + rx? 4 sx8 + tx? + ox? + ax’ bx" + cx’ + dx + éx + fo, we fhould have «==g— +f, Ee oe SPR, eS — ipB— tan, d = t— ipy — ia, and e== VU — ipd — Lay —-+@; and the coefficients of the other fix terms (whereof 7 ought to be a common divifor) would be ape 200+ {Ry—a, tae+ 100+ i yy—b, ihe -+iyd—e, aye oe = dd — , t0¢—eé, and te —f. Thefe operations, for finding of 2, as this fort of reduction is feldom poffible in high equations, will moft commonly end the work. If fuch a value, however, fhould be found for z, as to anfwer all the conditions above fpecified, it is not by pur- fuing the fame method of divifors, laid down in the refolution of by the Method of Surd Divifors. IQ of the preceding Cafes, that the values of 2, 1, &c. can from thence be determined, without a prodigious deal of trouble. There are indeed various other means of trying thefe quantities, by affuming fome of them, and finding the others from thence ; and fo proceeding on, changing the values continually, till all the conditions of the feveral equations, arifing from the com- parifon of the homologous terms, are fulfilled. But as this.is exceedingly laborious, and fecing after all, the ufe of fo great reduétions (as the fagacious Author himfelf obferves) is very little, there not being, perhaps, one cafe in a thoufand in which they can fucceed; I fhall, therefore, defift here. TR ey Fig. 24. ¢ 1 ; 5 7 SUZ SY WZ YZ eh S3Z ‘] A250 ee GU WA ZNO? Se ONO OK OK OK RHO KOKA OR OHO HOE aN /aD UN aD og vase Nao oCs Nas UMEDA 20 Caen PED Gwin gi} RPO eR ee ee aoe Yea OF Some GENERAL Prosiems in Mecuanics, and PuysicaL AsTRONOMY, Saree eT | Suppofe a fyfiem of bodies A, B, C (conneéted together) to re- volve about a center, or axis (P), with a given angular celerity ; it 1s propofed to find the momentum (k) which, atting at a given diftance QP from the center, fhall be juft fufficient to flop, or take away the whole motion of the Sifem. oer the given angular celerity of the fyftem, at any di- #4 1 48 ftance PG from the axis, be denoted by v, the cele- BSE Soe rities of the feveral bodies A, B, and C will be truly exprefled by nee Vv BE xv and Ea re{pectively. Hence PG > PG . PG ; (by the property of the Lever) it will be, as PQ is to AP, fo : AP AP* is Cre xv xX 4) the momentum of the body A, to OPxPa * vx, the momentum, which acting at Q, is a juft counter- poife to the action of 4. And, in the very fame manner, the momentum, acting at Q, fufficient to take away the motion of BPS ge, 7 OPxPG XX B; and fo on. Whence it is manifeft, that the fum of all zhefe, mutt be the true momen- : A X AP? +. B x BP? + C x CP? tum required ; or that & —= hal iS de ea oo, tesla phere a PG x QP 9. EB, I. B, appears to be C-O-R:O LU AR Yo. If the motion of the fyftem is that which might be pro- duced by any given momentums a, 4, ¢ (or forces capable of producing thofe momentums) ating on the bodies A, B, C, in | directions The Refolution of Jome General Problems. directions perpendicular to AP, BP, and CP; then (by the property of the lever) the force a irae, acting at Q, having the fame effect to turn the fyftem about its axis, as the force a, acting at the diftance AP, &c. it follows that the force, which, by acting at Q, is fufficient to deftroy the whole moti- 4 AP BRosi CP on of the fyftem, will here be ax OP \-bx opt ¢ x op: which being fubftituted in the room of 4, our general equation, axAP-+ 4x BP -+-¢xCP Ax AP?+-Bx BP? CxCP” fhewing the angular celerity at the diftance PG, produced in the fyftem by the action of the given forces; which celerity is, in the laft article, will become v—= PG x therefore, in proportion to the celerity ( uy that the given force (or momentum) a is capable of producing in the fingle body AxPG ax AP + 5x BP +--x CP A, as 7 — & a AP-PBKBP = CxCP (© Unity. CO ROL LoAR YY, oil. If the momentum & be given equal to that of the whole fyftem (A, B, C) in a direction perpendicular to the line PGQ paffing through the common center of gravity G; then the length of the lever (PQ) by which & acts, may be determin- ed from hence. For, the celerity of the point G being repre- fented by v, the momentum laft named will, by the property of the center of gravity, be rightly defined by vu x A+B+C; which being fubftituted in the room of 4, we thence get Ope Ax AP* + Bx BP?-++ Cx CP A+B+CxGP the center of percuffion (Q.) at which an immovable obftacle receives the whole force of the ftroke. ; exhibiting the diftance of COROLLARY. IT If a fingle body S, equal to the fum of all the bodies A, B, C, be fuppofed to revolve (independent of the others) about the fame center, with the common angular celerity of R the 12k 122 The Refolution of Jome General Problems : SP Sp Sa cee the fyftem, its momentum 57, Xv x S, or Ba XU A+B+C, will be in proportion to the momentum & (given by the Pro- AP? 2 2 pofition) as QP x SP to suohith Be BE de Ler Ok® By mak- ing thefe two quantities equal to each other, we have SP == bon BR for the diftance of the body $ from A+B+C x QP the axis of motion, when its momentum is equal to the mo- mentum 4, or when equal forces, applied to the fingle body at S, and to the fyftem at Q, can take away, or produce equal angular celerities in both, about the common axis of motion P. COR O hale wR OY si; Hence, if the point Q_be fuppofed to coincide with S, our laft equation will become SP = ah 2 as, fhewing the diftance of the center of gyration, or the place of the body S, where the fame force can take away, or pre- duce the whole motion of the fyftem A, B, C, as can take away, or produce the motion of the fingle body 5, equal to the fum of all the former, and revolving with the fame angular celerity. CO RO1 A RY. V. But if the point Q be taken in the common center of gravity G of the fyftem, and Gg and Ss be drawn perpendicular to the horizontal line TP; then, the force of gravity-by which the whole fyftem is urged in the direction Gg perpendicular to the horizon, being the fum 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 = But the force whereby the weight S, in a direction perpendicular to the fame PS, ‘is accelerated, is equal toSx i = A+B+C x =£ (becanfe S=A-+B 3 Ps iy Pe Z ev E . +C, and $5 =p A Therefore, feeing the forces acting at S and in Mechanics and Phyfical Aftronomy. 123 and Q are here equal, it is evident, from Corol. II. that the diftance SP, fo that the fame angular celerity may be produced in the fingle body as in the fyftem, will be truly exhibited by AX AP AB BEES Cxele there A+B+CxQP derived; the point S thus determined being the center of ofcil- lation, and the fame with the center of percuffion, found in Cor. If. having its diftance from the axis, equal to a third pro- portional. to the diftance of the center of gravity and that of gyration, determined in Corol. IV. the general equation SP = COROLLARY... Vi. Hence, alfo, the preffure on the axis of fufpenfion P may be deduced: for, fince the angular celerities, produced in the fyftem, and in the fingle body 5, by the equal forces a Ta Pg Ps A+B+C X pa and Sx sp that the abfolute celerity produced in G, during any given time, will be but the ae only the i part of the gravity of the fyitem is employed in are the fame, it is manifeft part of that produced in S; fo that accelerating its motion, the other part = being loft on the axis of fufpenfion ; which axis will therefore, in a direction perpendicular to PG, fuftain a force exprefied by A+B-+C ghey pe PG PS is affected ; fince, befides the other part of the force of gravity (Apa ones acting in the fame direction, is to be taken into the confidera- tion; whereof the quantity will be the fame, as if the whole mafs of the fyftem was to be placed in its common center of gravityG. For, if upon PS the perpendiculars Aq, Bd, Ce be Fig: 25. let fall; then, the centrifugal forces of the feveral bodies A, B, C being as the maffes drawn into the refpective diftances from the center P, the effet of thofe forces in the dire¢tion PG, R 2 will But this is not the only force by which the axis } in the direction GP, the centrifugal force, : : a ea Da) Fria Rn aka ae. 12 ~ The Refolution of ‘Jome General Problems will therefore be exprefied by Ax Pa+BxP5+CxPr, which (by the property of the center of gravity) is known to be equal to A+ B-+Cx PG. But the preffure on the axis may be otherwife deduced, in- dependent of the center of ofcillation : for the angular celerity generated in the fy{tem about its center of gravity G (which is the fame with the angular celerity about the point of fufpenfi- on P) is intirely the effect of the action on the point of fut penfion ; and the momentum, or force, fufficient to produce that celerity, is found (by the Propofition) to be A x AG*+ Bx BG’* + C x CG* PG mentum v X A+ B+ C, generated in the fyftem, as SRO Re ee PG. Therefore the force aét- A+B+CxPG ing on the axis of fufpenfion, in a direction perpendicular to PG, mutt be to the force employed in accelerating the motion of the fyftem (in the like dire@ion), in the fame proportion above fpecified ; fo that, to have the true meafure of each, the force of gravity muft be divided in that ratio: whence (taking Gs ax te ee) it will be, as GS + PG A+B+4CxPG (PS) is to GS, fo is the force of gravity, in a direction perpen- dicular to PG, to the force a¢ting on the axis of fufpenfion, in the like diretion. | That the proportion here determined is the fame with that found above, and the point S, the center of ofcillation, is thus made to appear. Since AP* = GP* + GA* — 2GP x Ga, BP’ = GP* + GB* + 2GP x Ga, as CP’ = GP* + GC* — 2GP x Ge, it is evident that Ax AP* + Bx BP* + Cx CP” (as given above) will be = A+B+CxGP’+AxGA’*+B x GB’ + C x GC’ — 2GP x Ax Ga—BxGsh+Cx Ge = A-+-B-+-Cx GP*+AxGA’+B x GB'+C x GC’, barely; becaufe (from the property of the center of gravity) all the quan- ; which is to the abfolute mo- UX . tites AxGa—BxGé -++ Cx Ge deftroy one another. Hence, by in Mechanics and Phyfical Afironomy. by fubftituting the quantity here found, inftead of its equal (2 Cor. V7.) we get sp LAA BEE AGE bait Bae BS CxGC A+B-+CxGP == Gra seee 4+ Bx GB?+Cx GC A+B+€xGP (SG) the diftance of the center of ofcillation, or percuffion from A x GA* + BOR a eee ss Ack. +B+CxGP wery fame as above. Hence it alfo appears, that, if the plane of the motion remains unchanged, the rectangle under SG and GP will be a conftant quantity ; and that, if S be made the point of fufpenfion, then P will become the center of ofcil- lation; and, laftly, that the ofcillations will be performed in the fhorteft time poffible, when SG and GP are are equal to one another, and equal, each, to ees ee a tt ee 3 being well known, that the fum of two lines, whofe rectangle is given, will be a minimum when the lines themfelves are equal to each other. ; and confequently the center of gravity —= The fame method laid down above, for finding the preffure upon the axis of fufpenfion at reft, anfwers equally when that axis is fuppofed to have a motion, or when the fyftem, or body, has a progreffive motion, as well as an angular one (as is the cafe of a cylinder, which, in its defcent, a made to re- volve about its axis, by means of a rope wrapped about it, whereof one end is made faft 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 Gir producing it, drawn into the diftance os 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 rett. Another thing it may be proper to take notice of, which is, that in the foregoing confiderations the bodies A, B, C are fup- pofed to be very Gait: fo as to have all their parts, nearly, at the fame diftance from ie axis of motion. But, to have the con- clufion accurately true, every particle of matter in the fyftem ought saci i i a 125 | | Hes h spit) | Ae | | ay | ea HN WV | | i iW an REDE ( i et: Hyatt Hid ; ay ah 126 Fig. 26. The Refolution of fome General Problems ought to be confidered, and treated, as a diftinét body: from whence, by means of the method of fluxtons, the fum of all the momenta will be truly found: but this relating merely to mat- ters of calculation, I have no defign to touch upon it here. I {hall only add, that the center of ofcillation may be otherwife, very readily, computed, /vom Coro/. F. even in cafes where the forces acting on the bodies A, B, C have any given relation to each other. For, if a, 4, ¢ be taken to reprefent the, re{pec- tive, meafures of the faid 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 fyftem (at the dif tance 1, from the center, during the fame time) will be truly ex- ate : which, in cafe of a AX AP +BXBP +CxCP fingle body S, acted on by the force s, becomes oan < (or ; x prefled by sa): Therefore, by putting this laft value equal to the s. Ax AP +BxBP +CxCP- Se x AP BBP oc OP. * fhewing at what diftance from the point of fufpenfion the fingle body S muft be placed, to acquire, by means of the foree s, the fame angular celerity as the fyftem itfelf acquires, from the action of all the other forces given. former, we have SP = LEMMA. If a given angle AOB be divided into two parts AOC, BOC, the product (or folid) contained under the fquare 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 tue parts, 7s one-third of the fine of the whole given angle. For, if the fine (CD) of the former part be denoted by w, and that (CE) of the latter by y, it is well known that (x) the celerity of x’s increafe (fuppofing C to move from A to B) will be in proportion to the celerity (— y) of y’s decreafe, as the co-fine in Mechanics and Phyfical Aftronomy. x co-fine of FCD to the co-fine of GCE; that is, as oa But, when x*y is a maximum, we have 2xxy + x’*y == 0, and ue to GC confequently «:—y::«:2y. Hence, by equality, ac : ne :: x: 2y3 and therefore FC—2GC. Let, now, OH be drawn to bifeét FC in H, and let HM and GN be perpendicular to FO; then, fince it is proved that FC == 2GC, it follows that FH = +FG, and that HM, by fimilar triangles, mutt likewife 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 alfo manifeft. PROBLEM IL. Suppofe that a plane ABC, moving with a velocity and diretiion reprefented by bB, ts acted om by a medium, or fluid, whofe particles move with a velocity reprefented by DB, and in directions parallel thereto; to determine the effect of the fluid on the plane, in the direction of its motion BH, and alfo what the angle of inclination ABD muft be, that the effec? may be the greateft poffible. Becaufe a particle, impinging on the plane at B, moves thro’ the {pace DB in the time that the plane itfelf, from abc, arrives at the pofition ABC, it is evident that the diftance (De) of the faid particle from the plane (produced), at the beginning of that time, will be the meafure of the relative celerity where- with the particles of the fluid approach the plane in a direction perpendicular thereto; and, confequently, that the force of the ftream in that direction, will be as Da (it being well known that the force of a ftream upon any plane-furface, 1s always as the fquare of the relative celerity with which the particles ap- proach it, ina perpendicular direction). Hence, by the refolu- tion of forces, it will be, as the radius, is to the fine of the an- ele ABH (or abH)), fo is the force Del, to its required efficacy in the propofed direction BH. More- 127 Fig. 27. The Refolution of [ome General Problems Moreover, with regard to the latter part of the Problem, the angle BD, which the directions of the two motions make with each other, being given, as well as the fides Bb, BD con- taining it, the remaining angle Bé4D will from thence be known, as likewife Dd: and fo De being the fine of the angle Dée, to the given radius Dé, the effect (Del x fin. adH) will therefore be a maximum, when fin. Dée] x fin. abH is.a maxi- mum ; that is (by the Lemma), when the fine of the difference of the angles Dée, abH, is equal to + part of the fine of the whole given angle B/D: from whence the difference being given, the angles themfelves will be known.—The geometri- cal conftruction from hence, is extremely eafy ; for, having from the center 4, with any radius, defcribed the arch mr, on rb produced (if neceffary) let fall the perpendicular mp ; take py = ~ of mp, and draw gs parallel to fr, cutting the circle ins; then bifect the arch ms by the line dae, and the thing is done: for the fine su of Sr (or of the difference of the angles Dée, abH) is by conftruction (= pg) = + of mp the fine of the whole given angle BJD; as it ought to be, by the Lemma. But, if you had rather have a general Theorem exprefled in algebraic terms, then let the velocity (6B) of the plane be put = a, and that (DB) of the fluid = 4; and let the fine, and co-fine of the given angle DBd (to the radius 1) be denoted by m, and n, refpectively ; alfo, having drawn BFL perpendicular to 4Fe, put 6F = x, and BF = y; then, fince FB and FL are tangents of the angles FB and FéL, to the common radius 6F, it appears, by the Lemma, that FL is (= 2BF) = 2y; whence (fuppofing LR and DQ to be perpendicular to B/Q, we have (by fimilar triangles) as B4 (2) : BF (y) :: BL (39) aed yee © 22, and therefore BR (BL — Bd) = oe Al- fo, Bb (a) : OF (x) :: BL(3y) : LR — #9, But the value of DQ being mé, and that of 4Q = nb — a, we have again, by fim. triang.) mb:nb—a:: 3%. a a a and confequently nb nb By) — 4a == —T" x 3x9, OF 3 — yy — xx = BE x 3H (be- in Mechanics and Phyfical Aftronomy. (becaufe yy + xx = aa); whence ss 1. Sixt a oet { 72 a and from thence, by completing the {quare and extracting 5 lh pd hice ah 2. nh—a U4 the root, te See — 2x s, equal to mb 2 m the tangent of the angle dBF, the complement of the re- quired angle aH, or ABH. Q,E. I. COR OFF EA Rayo. If the given angle DB be a right-one (which is the cafe when regard is had to the wind ftriking againft the fails of a windmill); then, being = 1, and z=0, our expreffion for the tangent of BF (which here is equal to the angle of inclination ABD) will become a 2 huans + a ; and this, if a be taken = 0, or the plane be fuppofed at reft, will be = v2, barely; anfwering to an angle of 54° 44’. But if the velocity of the plane be fuppofed 4, 4, or 4 of the velocity of the medium or ftream, then the angle of inclination ABD will be found from hence equal to 58°14’, 61°27’, or. 66° 58’, refpectively ; fo that, the greater the velocity of the plane is, the greater alfo will be the angle of inclination. Hence it ap- pears that the fails of a windmill, that the effect may be the greate{t, ought to be more turned towards the wind in the ex- tream parts where the motion is fwifteft, than in the parts nearer to the axis of motion; in fuch fort, that the tan- gent of the angle formed by the direétion of the wind and the fail, may be, every-where, equal to a 2+ a, + 2 ; the velocity a being proportional to the diftance from the axis of motion. S COROL- 129 130 The Refolution of fome General Problems COROLLARY Il. If, inftead of the angle DBH (or DB@), the angle DBA, ‘which the direction of the {tream makes with the plane, be given ; then it will appear, tliat the effect will, in this cafe, 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 faid* given angle DBA, in the given proportion of {BD to Bd. For, .. the force in the perpendicular direction FB being exprefied by Del , its effect in the direction BH will, therefore, be defined Del el (fappofing BA ; produced to meet DeE in E).. Now DB and the angle DBE (as well as Bé4) being fuppofed given, DE is given from thence. But it is well known, that the fquare 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. Confequently De muft here be the by De. x ae or its equal Gouble of Ee; which laft, or its equal BF, will therefore be «|:-: = SDE, But, fn, Be: - radius +: BF (DE), } Bos: And, radius : fin. DBA :: BD aoe? > = by compounding of which, we have the proportion above laid down. But that proportion, it may be obferved, can only take place when Bd is equal to, or greater than + of .DE: for, when Bé is lefs than + of DE, Ee (which is always lefs than Bd) cannot be equal to + of DE; but will approach the neareft to it, when BF coincides with Bd, that is, when the angle FOH, or ABH is a right-one; and in this cafe, the effect will be a maximum, when the direction of the motion is per- pendicular to the plane.—If the given angle DBA be a right- one (which pofition appears from hence to be the moft ad- vantageous, becaufe DE then becomes —= DB) it follows that the fine of the angle ABH, which the required direction makes with the plane, will be to the radius, as +*part of the velocity of the ftream 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 thip, greater than + part of its own celerity, it is evident that the fhip may run fwifter up- on in Mechanics and Phyfical Afironomy. 135 7 on an oblique. eourfe, than when the fails directly before the wind *, PROBLE MTT. Suppofe that a thread ACnC AS having two equal weights . A, fy [pended at the ends thereof, 1s hung over two tacks C Gik the Jame horizontal line ; and that to the middle point of the aa (1 e) pel diftant from th the tacks, another given weight B 1s fixed, . i é bid j oO which 1s pern Aud to defcend by its own gravity, fo as to caufe the Be two Weg US, at the fame time, to afcend : it 7s propofed to find the law of the « velocity by which the faid weights afcend and defcend ; abftratiing from the 1 efifpance of the air, the weight of the thread, and the friction on the tacks. that might be uniformly gone over in one fecond of time), and let 2 (== 32: feet) =: the meafure of the’ velocity which gravity can generate in a falling body, in one fecond ; putting CE — a, En =x, Ca= y, and the tenfion of the Let v denote the velocity of B (meafured by the diftance Fig. 28. thread — w: then = being the time in which B would, uni- VU formly, defcribe the diftance x, we fhall have, as 1 (fecond) is to ” fo is (the velocity generated by gravity in one fecond) to, —, the velocity generated (or deftroyed) by gravity in the . x U Moreover it will be, as BC (y) : Ex (/yy—aa) :: 0: wv Wnt ihe velocity with which the weights A, A afcend; J whofe fluxion Wy aax yo + aay} (oom sex sok att ) is there- yy yy — aa oa -fore the increafe. of that velocity, in-the-time =: but were ve) * In the above confiderations the velocity of the plane (or fail) is, all along, treated as a given quantity ; becaufe the fame direction that gives the effective force the greateft, when the ve elocity is given, mutt neceflarily give the velocity the greateft poffible, when the force, alone, -is.given. SR not iq? The Refolution of fome General Problems not the {tring to act on the faid weights, their velocity (inftead of being increafed) would be diminifhed, and that by the quan- tity 2 (as is found above). Therefore the whole alteration of ae aa arifing from the tenfion of the {tring 1 is to that arifing from the aeNos of gravity, in the proportion of bi ee eax ral as to =; and, confequently, the tenfion of U yyx UV the ftring (w) will be to the weight of the body A, in the fame y —aaxX YUV aavu proportion: whence we havew—Ax1 epee eee byyxx Again, it will be (by the refclution of dotees) as Ci (y) is to En (x x), fo is 2w (the double of the tenfion of the Re to or the effect of that tenfion to retard the defcent of the weight B; which being fubtracted from the gravity (B), the remainder B— ay will be the force by which B's motion is > accelerated. Hence we finn. as is to Be 2 40 Is = 2; the velocity that would be generated by the gravity in the time =; to that (v) generated in the fame time, by the force B— —— From whence, by multiplying extreams and means, = * * d \: A ee 17 an 2a)? in we get vb — by — 2 __ hy 2bAKe ON yD aa X yuu + a’v'y | By oy Bas {by fubftituting the value of w) = 4x — ant as — +t. = x Be 2 ie (becaufe yy = xx) = bx — = mie a + = x ie Le and, ae by taking the fluent, . oe = se oy a ate at d (d being the necef- {ary correction); which equation, if a She put =m, will be reduced to v = oe whaling 2bms — aby Fain a the true ve- m4a-l.yy—aa locity in Mechanics and Phyfical Aftronomy. locity of the body B; whence that of the: body A (== fobs ah oe will alfo be known. Q. E. I. m—+L.yy—aa COR OL is hehe If the firft values of x and y, when the motion commences, be exprefied by f and g, refpectively; then, w being =o, when x =f, and y=g, we fhall have o = 2bmf— 2bg + 2dm, and confequently 2d” = — 2bmf- 2g; fo that the general va- : ‘abhmx ~ 2by—2bm ‘abe he s'h Pe foa h oe lue of v will bey / a i RE ee eee m--1.¥y— aa mt1I. yy aa From whence the ¢gréateft diftance through which the weight B can defcend, before its whole motion is deftroyed by the other weights A, A, may be eafily determined: for, fince the velocity, at the loweft point of the defcent, vanifhes, or be- comes equal to nothing, we fhall, in that circumftance, have obaxx—f—2bxy—g=—0, rmxx—f/+sg(H=y=e J xx aa) = SV xx + gg — ffs which, {quared, gives m xx—f\ + 2mgxx—f = xx — ff: whence, dividing by « —f, we have m* x x — ft 2mg—=x-+/f; and con- fequently x = ames tert ; exhibiting the diftance of the point 7 below the horizontal line CC, when the whole motion is deftroyed, and all the weights begin to move the contrary way. But it muft be obferved, that this can only happen when m is lefs than unity, or when the weight B is lefs than the fum of the other two: for, if 7 be equal to unity, « will be infinite; and, if m be greater than unity, the value of x will come out negative ;. which fhews the thing to be impoffible, or that the weight B muft continually defcend; except when mz is lefs than unity, or B lefs than 2A: in which laft cafe, it appears that the bodies will ofcillate, backwards and forwards, continu- ally; in fuch fort, that the two extream diftances from the horizontal line CC will be expreffed by f and ee sl 1 — 717 2ma whereof the latter, when f= 0, will become Ls 7 I:2%2 Ju The Refolution of fome General Problems ——"_ x CC ; fhewing the loweft defcent of 2, from the line Ei ere CC, when the motion commences from that line.—By mak- : 2mg—i-+-nm. f ing f and — equal to each other, we cet f—= me oe qual to eacl » we get f= me B < ; ™ e ° eee Mie, OFS og et: B >: 2A.° Prom'which it appears, that, Bix if the frft pofition of the weight B be fuch, that Ez is to Cr in the given proportion of B to 2A, no motion at all will en- fue, but the weights remain in eguilibrio. Whence it is evi- dent, that, if the motion commences from any point below ‘hat here determined, the weight B will firft of all afcend, 2mg —i+-mm.f an after which it will = 7 again defcend, to its firft diftance f; and fo on, backwards and forwards, continually. till the diftance from CC is COR OL LORY. “iH: If CnC, in the firft pofition of B, be fuppofed to coincide with the horizontal-line CEC; and the body B be impelled fron thence with any given celerity ¢ (meafured, as above, by the {pace that would be uniformly gone over in one fecond of time): then, 0 beiio = when “ ='o and y = a, we fhall, by fubftituting thefe values in the general equation 2bmx — 2b; 2d . —— SF d (v Oe Eom 2) obtain c = ators ee — m—+1.yy — aa rb cpa ries ‘adm — 2b J) a -, and confequently . 2dm == mc* + 2ba ; fo that v m sj ; 2bmx — 2b 2ba + me? : is bere —= igi Ye BAe oe ; which, when /—=0, or, 1 m-+-1.yy—aea when the bodies are not aéted on by gravity, will become v = ¥ m*c ‘ ‘ : : . 2. And in this cafe, the time of moving thro’ Ex m-+- 1 yy — aa / “ : a/ iy 1 a. : A bs 2 (whereof the fluxion is * —" —"F ror SE ne ) VU us mcy as meV xx + aa may be readily found, by means of «an hyperbola, whole tran{- verfe in Mechanics and Phyfical- Afronomy. 135 : ° 2a . . a verfe and conjugate axes are —= and 2a; 7¢ being in propor- 3 Vm tion to the time of moving uniformly over the fame diftance (x) Mies with the given celerity at E, as the arch of the hyperbola is to its ordinate x. PRORT EM iy, Suppofing. a Jpherical body of 1ce, or any other matter, revolving about its axis, to be reduced to a fiate of fluidity ; to determine the change of figure thence arifing. ‘Tt is demonftrable, that the figure of an homogeneous fluid, Fig. revolving about an axis (PS), having all its particles qui- {cent with regard to each other, muft be ‘that of an oblate {pheroid OAPES (fee Art.395 of my Dottrine of Fluxtons); and that the particular {pecies of fuch fpheroid, anfwering to any gi- ven time of revolution f, will be truly defined by the equation Ga N OD eb 2 P= a7 aoe ; where 1: 1+7¢¢ :: PS*: AE’; PS be- 3fuUXA=3t a ing the axis, and AE the equatoreal diameter; alfo A = the circular arch, whofe radius is unity, and tangent ¢; and g = : the time wherein a folid fphere, of the fame magnitude and am denfity with the fpheroid, mutt revolve, fo that the centrifugal force at the equator thereof, may be exactly equal to the at- traction, or gravity. Now it is evident, that, whatfoever figure a fluid, revolving about an axis, at any time hath, the niomentum of rotation about the axis will be no-ways changed, with the figure, by the action of the particles on each other : fo that thé momentum of our propofed fluid, arifing from the {phere of-i¢e; will, at all times, be the very fame with that of the {phere itfelf. From whence it may be eafily proved, that the time wherein one intire revolution of the fluid, con- fidered as a fpheroid, might be uniformly performed, mutt be always as AE’: therefore, if we make e = ALF, and put d = the diameter of the fphere (or of the fluid, when AE = PS) it follows that the faid time * will be truly exprefied Toe here * At Art. 399. of my Fluxions, this time is, by miftake, put down et 7 ; é e* pee : . ‘ t j S ee = x5 (inftead of =X 5)5 whereby the remaining part of that Article is by rendered erroneous, 135 equiltbrio, we {hall thence have The Refolution of fome General Problems Fé ‘ ¢ j + ¢ i } te by aes (fuppofing s to denote the given time of revolu tion of the body, when under the form of a fphere): 2g? 2x Oe as in the revolution is performed, when.the particles are in SEX Aa 1 2e c d* pee : the time where- / which being put (= 9/ : 35. et But, Wecaufe PS; (s= _ — —, the mafs of the fphe- Vite 1+ tt roid will therefore be as ae (AE x PS); and ther of I tt the fphere, as d’: which two quantities ‘being made equal to d* I : ; each other, we have — == ———. And, this value being é i cH? . Se tix A — 3¢ wrote in the room’ of its equal, we have — = rs From the ee i ce) ae refolution of which equation the value of ¢, and the fpheroid itfelf, will be known. ‘The fpheroid thus determined, is that under which the fluid might remain in eqguz/zbrio, were the par- ticles to be, once, quiefcent with refpect to each other: but the particles, in their recefs from the axis, do, through the cen- trifugal force, acquire a motion from the axis, which is not immediately deftroyed, on the fluid’s affuming the figure, or degree of oblatenefs above determined ; the equatoreal parts ftill continuing to recede from the axis, till the gravitati- on, by degrees, prevails, and in the end quite overcomes the faid motion. After which the equatoreal parts will begin to fubfide, and again approach the axis, in the very fame manner they before receded therefrom: and fo will continue ofcillating, backwards and forwards, ad infinitum. But if the fluid is fup- pofed to have fome degree of tenacity, the ofcillations will be, every time, contracted, and the parts of the fluid will then converge to an equilibrium, under the form. above deter- mined. se Da Vig in Mechanics and Phyfical Aftronomy. 137 4 LEMMA. | _ Suppofng a body to move with an uniform celerity, in a right- line AD ; to determine the rate of increafe 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 = «; and Fig 30. let the meafure of the body’s celerity, or the fpace gone over in a given time g, be denoted by c: then will 2 exprefs the : c time of defcribing x (or BS); and it is well known, that ’ AB ° 1 CX eG Se atl will be the true meafure of the cele- WV xx + 4a BC: rity with which CB increafes; whofe fluxion, ao -, is XM + aa * therefore the (uniform) increafe of that celerity, in the time =": hence it will be, as: g (the time given) :: —“* is : xx aay* 24? ao ae pa (a gC), the required increafe, that would uni- ‘xx +b baz CB)’ formly arife in the given time g: which increafe, fince ‘ is reprefents the paracentric velocity of the body (in a direction perpendicular to CB) will be, always, expreffed by the {quare of the meafure of the body’s paracentric velocity, ap- plied to the diftance (BC) from the given point, or center. 2. bed COR GT LAR Y. It is evident from hence, that if a force, which in the given time g is fufficient to generate the faid increafe of velocity, be fuppofed to urge the body towards the center C, and there- by deflect it from its rectilineal motion, the celerity with which CB increafes will then be uniform ; becaufe the, force applied, each moment of time, is juft fufficient to deftroy the increafe that would arife, in the fame moment, from the body’s being futfered to continue its motion uniformly in a right-line—If the dire€tion (B4) of the motion is perpendicular to CB, the . body, thus acted on (as no celerity is generated in the dire@tion CB), will move in the circumference of a circle. Confequently mp ie i ths 138 The Refolution of fome General Problems the force above determined is the fame with the centrifugal force in a circle, when the diftance from the center, and the angular celerity are the fame. But all this may be made to appear in a different manner, by ftippofing Bd exceeding {mall: for, if BE be made perpen- dicular to CB (produced), BE will then exprefs the length whereby CB would be uniformly augmented, in the time ( g ) of defcribing. Bd; and therefore eb, the excefs of Cd above CE, will be the {pace through which the force muft caufe the body to deftend, in order that the increafe of the diftance from the center C may be the fame as would uniformly arife with the firft celerity, at B. But it is evident that this excefs e (which, by the property of the circle, is = iy or alfo exprefles the effect of the force, neceflary to caufe a body to revolve in the circumference of a circle Ee/,; with the fame angular celerity——-To determine, from hence, the velocity which this force would generate in the given time g, we have, é sa (the fquare of the time of defcribing B4, or Be) is to go Ea ( ex EA” | g’, fo is —= to ICR x the fpace through which the as 2CB ball might fall, by means of the faid force, in the given c x El : x ACY CB x Bd] CBx CB] 7 is, therefore, the true meafure of the velocity fought ; becaufe the diftance gone ever by a falling body is but the half of that which might be defcribed in the fame time, with the velocity ac- quired at the end of the defcent.—The quantity here determin- ed (as has been before obferved) is the meafure of the force by which the body is made to recede from C with an uniform ce- lerity: if a force, lefs or greater than this, be fuppofed to act, the difference will caufe an increafe or decreafe of celerity in the line CB, proportional to the faid difference. time g; the double of which, (or its equal PR O- in Mechanics and Phyfical Aftronomy. PROBLEM V. % Suppofe that a body, let go from a given place A, in a given direétion, with a given celerity, 1s continually folicited towards a given point C, by a given centripetal force; to determine the path ABP in which the body will move. 139 From the center C, through A, let the circumference of a Fig. 31. circle ADK be defcribed ; and, fuppofing B to reprefent the place of the body, (the radius CD ofthe circle .°.. 5... -- er ee the rags Geter CBS ce Ce ae | the arch AD, meafuring theaneles ACT. he put the time of defcribing the angle ACB ...... a. | the meaf. of the celerity with which the lineCB incr. = 4, =— ai, the meaf. of the celer. with which the area ACB incr. ihe meafure of the centripetal force ....... ; where, by the meafure of a celerity, I mean the fpace that would be uniformly defcribed with that celerity in a given time g; and by the meafure of a force, I underftand the meafure of the celerity that might be uniformly generated by the force, in the fame given time. Since the celerity with which the area ACB increafes is ex- preffed by aw, it is evident that the paracentric velocity of the radius vector CB, at the middle point (4), will be exprefled by =) and that of the body itfelf by a the fquare of which laft, divided by x, will give we for the true meafure of the centrifugal force (by the Lemma); whence it appears that ( > —Q) the excefs thereof above the centripetal force 2, is that force whereby the celerity v is accelerated: therefore we have g:¢:: sla — Q (the meafure of the celerity gene- rated in the given time g) : v. - But, becaufe the paracentric velocity of the body is =, that of the point D (defcribing the pee circular ; : ee ee ae == Sat hy by iN Wa Hea Ne. Hi} At Hi) aie | Hd: | ty Alt a i ye i iN, A 4 ae) i; Ma, Hii ti ey ih ia i ie ih i NK: MN | ‘Hh i. | i tl Y Hl i" Ne t lh. ny i) Hi My) al fia i Ma! NW He ah Mr Ae qe y i) : i —— aa = eS Se SS eS 140 The Refolurion of. [ome General Problems : : 204: a 2a°u circular arch AD) will be-—~ x*—, or ; we x“ xx ; and fo we have g:t:1 22% (the diftance defcribed in the time g) : & (the di- nx {tance defcribed in the time ¢) : whence, by equality, —— Q £02: 7@%:%; and confequently do — ““—Qx ke asp ice ie Again, the fpaces 2 and x (defcribed in the fame time) be- ere “ : ° ie 2a7u ° ing in the fame proportion as the celerities “— and v, with xx 26AUe. To ex- which they -are defcribed, we alfo have v = MZ terminate v and wv out of ¢his, and the preceding equation, make = = w, (or w= —+-+1); then v = xX x & 2auw ; ‘and 2auw +- 2auw . 4a°u- ae acess. [0 Micmrme x x Qz ooo 2uxI—w) . w w I uw Q. equation may be reduced to = += a ae fecat exprefling the general relation of w and z, or of x and z, ac- cording to any value of w. But in the cafe propounded, where- in no force is fuppofed to act, befides that tending to the center C, the celerity aw with which the area ACB increafes, will be a conftant quantity ; and therefore, z being here = 0, our . 2 a equation becomes —— = 1—w— & z: from whence, RR ; 4u*X 1— w) when Q is given in terms of x (or w), the relation of w and % may be determined. COR OL LAR YUE. Hence, if the centripetal force, by which a body defcribes a given orbit at reft, be known, the increafe of that force, when the orbit itfelf is {uppofed to have a motion round the center of force, may be eafily deduced: for, let the angular motion of the orbit, be to that of the body in the orbit, in the conftant ratio of mto 13; then, the whole angular celerity of the Q x wire Quz Qx*z 2uzxI-w 2a°u Os 2a°u a a (by writing vie for its equal x); which ’ . ee in Mechanics and Phyfical Aflronomy. the body, here, being in proportion to the angular celerity when the orbit is quiefcent, as m--1 to 1, the centrifugal force here, will therefore be to that (44) in the quiefcent or- = q bit, in the duplicate ratio of mt 1 to 1 (dy the Lemma), and 5 fo will be truly exprefled by m-+-1 | x oo . From whence it appears, that mm-+-2m x ee is the increafe of the centri- x fugal force arifing from the motion of the orbit : which quan- tity, therefore, muft be that whereby the centripetal force ought to be likewife increafed, in the moveable orbit ; fo that the difference of the two forces, whereby the motion of the body in the line CB is accelerated, may be the fame here, as in the quiefcent orbit ; in which cafe the value of CB itfelf, in all contemporay pofitions, muft neceffarily be-the fame. Hence it appears, that the increafe of the centripetal force, in order to the defcription of a moveable orbit, will be always inverfely as the cube of the diftance ; and will, moreover, be to the centrifugal force in the quiefcent orbit (in all contemporary po- fitions), in the conftant ratio of mm - 2m to 1. CHR OL LARRY (Mi If the centripetal force (Q) be fuppofed, inverfely, as the fquare of the diftance, and the given value thereof, at the lower apfe A, be to the centrifugal force there, in any given ratio of 1 —e 4a°u* ; ) of the centrifugal to 1; then, as the general value ( force, will, at A, become = a... the centripetal force there a will be expreffed by a x 1—e; and confequently that at B eo. x 2 2 eee NG 2 h i by - ~. ba xs or its equal ; “x i— x . Which a value being fubftituted for Q, our equation here becomes aw é : é ; = =o: whence, multiplying by w and taking the aw w* » fluent, we get —— — ew — — (where, the angle CAB being 22% 2 fuppofed _ IAI 142 The Refolution of fome General Problems Fig. 32. fuppofed a right-one, no correction is neceflary); fo that we ° LU é ew havees — + gee et : but the laft of 2ew — ww a. W2w—ww thefe quantities is known. to exprefs the fluxion of a circular ° 2 ez arch (4), whofe verfed-fine is w and radius e; therefore be- a ing== 4, or a:e:: 2: A, it follows that the arches z and 4 (which are in the fame proportion with their radii a, ¢) mutt be fimilar, and confequently their verfed-fines, AF and w, in the fame proportion above fpecified, or as a to e: whence we have ¢ Re AR AF ACC eet ee ee es “E (fuppofing BE perpendicular to AC) =o =I—G (fuppofing perpendicular to ; AC CE as oa therefore 1 — BC =! I-—-Gp and confequently BC — AC (= BD) =exCB— CE; from which equal quantities take away e x BD, fo fhall BD —e x BD —ex AE, and therefore I—e:¢:: AE: BD; which is a.known property of the conic fections, with re{pect to lines drawn from the focii. Hence it appears, that the trajectory will be an ellipfe, parabola, or by- perbola, according as the antecedent 1 — e is greater, equal to, or lefs than the confequent e; or according, as the centripetal force at A, is greater, equal to, or lefs than half the force fuffi- cient to retain the body in the circular orbit ADK.—As to the particular {pecies of the curve, corre{ponding to any given va- lue of e¢, it is, from hence, very eafily determined : for, if AO be made to reprefent the femi-tranfverfe axis, then will AO :OC (:: AE: BD, p. conics) +: 1—-e: e;_ therefore, by divifion, AO : AC :: 1—¢ : 1—2e; whence-AO is known. COROLL AMY III. If to the foregoing force, varying in the inverfe ratio of the: {quare of the diftance, another force, which is inverfely as the cube of the diftance, be joined (which, at A, is to the former part in any given ratio of s to 1—e ), the place of a body, thus acted on, may be found. in the fame conic fection Fig. 33. APRS, above determined, fuppofing it to have a motion about Se : ’ ' : r \ in Mechanics and Phyfical Aftronomy. about the focus C, which is to that of the body in the feGtion (referred to the fame point C) in the conftant ratio of m to 1; the value of m being = /1 + 5 —1, or fuch, that mm—+t.2m:1::5: 1 (y Corol. I.) —If the centripetal force be barely as the cube of the diftance inverfely, 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 itfelf BA’ (al- ways touching the circle in A’) is fo carried along by the mo- tion of the radius CA’, that the angle ACA’ fhall be to the angle A’CB, in the conftant proportion above {pecified; the ra- tio of the centripetal, and centrifugal forces at A (and confe- quently in every other pofition) being exprefied by that of sto1+s. COROLT-AR ¥. IV; If the centripetal force to be as any power of the dif- tance, whofe exponent is 7, and the given value thereof, at A, be in proportion to the centrifugal force (=), as r tol; rue x rue I we fhall then have Q = 4“ x5 —#* x —_,; and here 7 - a” g aw 4 a’w ; aQ_ : our equation, —- == 1 — w — =, will become es 4u-XI1—wW a aw r a—— IT wo RB T—u\""* and the fluent taken, we thence get on = w— — — 2 Fie t = na-+iI oe : which being multiplied by w, ~- sn and confequently eed ; from whence the J ob — up — 2F I + a } 3 n+ n-+1 value of 2, by infinite feries, or the quadrature of curves, may be found. But when r differs but little from unity, and the orbit is Sa: (becaufe of the fmallnefs of w) nearly circular, wil 143 144 ert mais Ga The Refolution of fome General Problems will. be nearly equal to rx 1+n-+2.w; and there- 2 fore —- = I — wWwo—yr—nt2.nw=>Iier— r-—T ‘| n—+-3.w, nearly; and confequently —— ae — Ww: n+ 3. RR eT . Pat ee a and A= n+ 3x2, that is, let 4 re- prefent an arch A’D of the circle ADK, which is always .to the arch AD (or 2), in the conftant ratio of /a-+ 3 to 1: then AA being = nae x 2B, our equation, by fubftituting thefe values, will become = = f— W5 _ which differs in no- thing from that (2= =e — Ww) refolved in Corol. IT, excepting only, that 4 and f are here uifed, inftead of z and e: whence it is manifeft, that the value of w (there reprefented by e x ver!- ed-fine of z) will here be tr uly" ‘exprefied by fx verfed-fine of A; from whence and what is ¢here.demonttrated, it alfo ap- pears, that the place of the’ body ‘will.be in the periphery of a given ellipfe A’BR, revolving about its focus C, with an angular celerity, which is to that of the body in the ellipfe, in the ¢ con- {tant ratio of the arch AA’ to the arch A’D, or as-1 — /n 3 to /nu+2 +3. And it is evident, that the motion of the apfides will be to the motion of the body in the ellipfe, referred to the focus C, in the fame given ratio; and that the angle defcribed by the body in moving done one. apfide to the other (becaufe AD is always = A’D x7 = ave be ==. 180°.x Vn +2 E80° ae All which conclufions, as well as thofe derived in Hi 3 the preceding Corollaries, exa@tly agree with what Sir Isaac Newron has demonttrated, by a very different method, in the third and ninth Seé#ions of the firtt book of his Principia.—As to the motion of the apfides of the lunar orbit, with the other inequalities depending on the fun’s aétion, thele require the ufe of other principles, and the {olution of the following PR O- ood ee 2-3 in Mechanics and Phyfical Aftronomy. PROBLEM VI. The fame being fuppofed as in the laf Problem, and that, befides the force tending to the center C, another force, whofe meafure 7s R, acts continually on the body, in a direétion perpendicular to the radius-vector BC; it is propofedto determine the curve ABP which the body, fo acted on, will deferibe. 145 Every thing in the preceding Problem being retained, Fig. 31. we have nothing more to do here, than to get an equa- tion for ~, by means of the new force, whereon the increafe or decreafe of uw intirely depends. In order to this, we have, as ¢ (the given time) is to ¢, fo is R, the velocity gene- rated in the time g, to Re the velocity generated in the time t, in a direction perpendicular to BC: whence the cor- refponding increafe of the celerity au, with which the area : ; : eRe ACB is generated, will be exprefled by x4, that is, = x= 2a°u will be = az. But, as it has been proved that g : fi: ae NS: xs t ; ‘ Rx3z “ we :have — = 3; and. therefore az = —"*, or 2uaz —= g 24aaUu 4.aau IRyz LRz : pad =——; (becaufe x = ae Hence, by taking the a I—w I—w Re : fluent, we have v2? — ¢ + flu.—2".. (c* being put for the A rece 0 neceflary correction, or the value of w* when z==0). From : - w Ww I uw Q. which equation, and that ( = --—=——-—- — : pe aa, (aa Uxz 4au*x 1—w| derived by the preceding Problem, the relation of u, w, and z may be determined, when the law of the forces Q and R is affigned. QF. TI. Ly C O- 146 The Refolution of fome General Problems COROLLARY If the forces 2 and R are fuppofed to be in proportion to (#4) the centrifugal force at A, as A tol, and Il tol, re- a fpectively (A and Il being any variable quantities whatever), and if the celerity (az) with which the area ACB increafes be fuppofed in proportion to (ac) the firft value thereof at A, as S* to 1; then, Q being = “ee, R ee and u=c =, our two equations, by fubftituting thefe values, will become az 1 Ww I yw A 5 —1-- Au. ——— and 4 -- — = — — 2x re + axi—w) Le aa aa oe Yes a@=xi—w\ i 2 Tz or > ow A or S=—1-+- flu. =, and = +woI— yx — —— ah 1—w\ ‘ at 22, 2% =x I—w » by making AC unity; which laft equation will be rendered ftill arc - 2Ilz more commodious, by writing for © its equal rs whence 3 ? 3 : uu Tw A will be had % + w (= 1 — ——=—; — ee SN al = iF Eexi—al =xi—wi = into E— A x jw) —x “y wl °: from which the = values of w and © may be found, when thofe of A and Ml are afligned: by means whereof the time (t) of defcribing the an- e é t De S ele z, will alfo be known, from the equation > = = “ta g 2aau bove derived): which by = ee c=: and === for their XK os : renee Sv equals w and —» gives f == oo To exemplify the ufe of the equations here derived, by the refolution of a cafe on which the determination of the lunar or- bit depends, let the force A, whereby the body is folicited towards the center, be confidered, as compofed of two parts; whereof the principal (4% 1—7") is in the inverfe duplicate-ratio of the diftance; the other part, which is fuppofed {mall in comparifon e . TI e . of the former, being as the diftance (=) direQtly, drawn in- Will here become oe i £S= 1+ 2 fluent PZ fin. pz + Qz fin. gz &e. x 1—wl *, and Ww +w= > into 5 —4—P’cof. pz +Q col. gz &c. x I—w act — x P fin. pz+Q fin. gz &e. x I—wr * Now, the orbit being fuppofed nearly circular, we may, in order to a firft approximation, neglect w in both the factors i—w\ 3 and 1—wl*, as being very {mall in refpect of unity; by which means © will become = 1-+2 flu. pz fin. pz+Qz fin. gz &e. = 1-+-d— = cof, pz— x cof. gz &c. (fee p. 82.) where d, reprefenting the neceffary correction to the flu- ent, muft be taken == = + “= &c. fo that & may be ==1, when z=—=0. This value of © being now fubftituted in the fe- cond equation — + w= = x E—b—P' cof. pze—Q col. gu Ke. (where “ x P fin. pz-+Q fin. gz &c. on account of the {mall- * This affumption is not the lefs general by the multiples of z being taken the fame here as inthe value of A; becaufe, if any multiple of z, in the one value, enters not into the other, it is but fuppofing the correfponding coefficient in this laff, to va- nifh or become equal to nothing. . uw 2 nefs 147 148 The Refolution of fome General Problems nefs of =) there cometh out x& w I 2P ? % 20% , “ Ses la dah pz— — xcof.gz&c. 2 i =xi+d b 3p TP xeo pz— +OQ'xcofig where the general multiplicator = may be alfo omitted, as differ- ing very little from unity: this being done, and the fluent being taken, according to the method on f. g2,we thence find w==1-++-d P" cof. pz Q”" cof. gz pe 2P yy —b+acofiz— ee Bcc wancre.-P" oe: : +P’, Ce aS + Q’, &c. and where the term a cof. z, by which the fluent is corrected, muft have its coefficient fo taken, that w and z may have their origin together, that is, # muft be made Pp’ Oe == b6— 1—d-+ ——. bec: aE Aoooe le i Having thus found a value nearly equal to w, we may by help thereof, proceed now to a fecond approximation, by {ub- ftituting that value for w, in the fators 1—wl * and I os : wherein it was before neglected ; and,to facilitate the computation, the terms in the value of 1—w (whofe reciprocal is the diftance of the body from the center of force) may be expreffed by the general feries of cofines, ex 1—B cof. @z—C cof.yz—D cof. dz &c. (as it appears from above, that the value of w will confift of fuch): by which means the fame terms before determined will be again brought out, together with a number of others, ferving as a farther correction. But, fince the former opera- tion is made, more with a view to difcover the form of the feri- es, than to be regarded for its exactnefs, I fhall have no far- ther reference thereto, but proceed to determine the value of the feveral quantities e, B, C &c. de nove, by a method fome- thing different from that ufed above. Firft, then, from the equation 1— w==ex 1—Bcof. Bz—C cof. yz— &c. will be had ——\__ I 9 QZ eee eee fl = = XB cof. @ 2-+-C col. yz &c. + &e. a an cd in Mechanics and Phyfical Aftronomy. and ;—wl *= 42 x B cof. Bz+C cof. yz &c. + &c. which laft value being multiplied by Pz x fin. pz (accord- ing to the purport of our firft equation, 21 + 2 fluent Pz fin. pz+Qz fin.gz &e. X i—w!*), and a proper regard being, at the fame time, had to the Theorems on p. 80, the ae will as thus, = ~= xfin pe —= x —Bfin. B—Pp. 2+B fin. B+p.z &e. or thus, ———————Se zs x fin. pz + oe XBL p-p.2+Bipte z+ Cf p-y.z4+ Clhp+y.z&c. ° whereof the fluent will be a ere — ie sie Reine ee BoolpB.x , Bool pe.x , Ceokp-yx » Coolp+y-® g pe e+ pee ?+8 ; p—y ga Cy In the very fame manner, the terms arifing from the multi- — plication of Q& x fin. gz will be exhibited; and we fhall there- fore ane Id ae x cof, pz -4- ax cof. darko x cof, rz &e. | ne __ 4? Beof. 5k % Boot. i+6- epee: Cool. p—y.% Cook phy on SiN eRe OB oe ee Pe p-Y P+y 5 s Bcof.g—8 Bol. g—B.% Bool. q+8.2 Ccof.g—y.z . Ccokg+y.z a par ——Y g+y &c. In Lae te quantity d, aflumed to denote the neceflary cor- rection, muft be fo taken, that 2 =. be Saf when z==0, ; P that is, 7d muft be equal to 2 x B ae ; ce oe &e. + QP By BB Se ee ey Be pe T pte py a a ae E 4Q. B B C se a oF + air 4. ; =: &e. + &c. ees now to fubftitute in our other equation C oS er WXE— r+ b + P’ cof. pz+-Q cof gz&e. x 1 + wl 8 + 149 ae re ne oe Fi op ae SE ae eee ae Fa Ree i i i \ i { i 2 150 The Refolution of fome General Problems w x Piin.pz+Qfin. gz &c. x 1—wl +o) we fhall, in the firft place, by taking the Auxion of w= I—e-+-exB cof.6z-+-Ceof-v2&e, have = == — exPB fin. Gz+yC fin. yz &e. (vid. p. 82 and 83.) whereof the fluxion being, again, taken, we get = —=—ex BB fin. Bz--y* Cfin. yz &c. and confequently rt zy twa 1—e-+ex 1-88 x Beof. Bz -+ Ivy x Cool. yz &e. Hence 2 +- w X Lamm Bae — ey Se eX 1—@Bx Beof. Pei yy XC cof-ye &e. X 1fd—— x = coGpet x cof. gx &c, which (by an actual multiplication of terms of the two feries’s into each other, and a proper application of Lem. 1, on p. 76, neg- lecting at the fame time, all terms wherein two, or more di- menfions of the quantities B, C, D &c. would arife) will be reduced to —l+titds —88.eBcol.pzt 1 +d. 1—yy.eCcof.yz+-1 +d.1—33.eDeof.dz &c. Se ee ee ee —I—B8. pg > x cof.p-—8.2-+-cof.p+ 8.2 . x COf. g-—8.z-L-cof. 7-+-B.x &e, lee ae re cof.p—y.z--cof.p+-y.-+ = Xcof. g—y.z-L-cof. g+y.x &c, &c. i &c, ° In like manner we have P’cof.px+- Qcof.gzk&e. x 1—wlh3— P’cofpz-+Qoofgz bic. x 4. a = x B cof. 6z4-Ccof- yz &c. = x P’cof. pz+Q‘ cof. gz+-R’ col. rz &ec._ é fr | Sr, ST DSSS ge se “ 3X Beof. p—8. 2-+B cof. p+. 2+-C cof.p—y.z-+C cof. p+-y.z &c, 8Q0 sermon + = x Beof.g—@.%-+-B cof, 9+8-%-4+-Ccof. gq—y.z-+-C cof. g+y.z &c. Ll &c. &e. Laftly becaufe = = —ex@Bhin. Czy Cfin. yz &c. it fol- low S in Mechanics and Phyfical Afironomy. 151 lows that “ x P fin. pz + Q fin. gz &c. X eT —ex BBfin.Bz-+-yC fin. yz &c. x Pfin. pz-+-Qfin. gz &c, x = x S xB cof.Bz+-C col.ye&c. whence, by proceeding as a- bove (neglecting the latter part, =x Beof, Bz--C cof. yz &c. of the laft factor, as producing terms involving two dimenfions of the quantities B, C, D, &c.) will be had Pree ee eee =e X Bx cof. p—6.z—BB cof. p+6.z+yCcof. p—y.2z—yC col. py. % 8c. See ee net a BB cof. q—B.%—6B cof, gb. 24+-yC cof. g—y.z—yCcol.g-+y.% &c, &c. &c. Now let the three values, thus determined, be collected to- gether, taking inftead of X, its equal, as given by the firft equa- tion, putting f—=1-+d, and dividing the whole by 1---d.e; by which means our equation is, at length, reduced to 1—88. B cof. bz 1—yy.C cof. yz 1—0. Dcof. dz Src 7} b Fe oP ! wD: ’ — I-- i: + ae x o +-P’xcof.pz—- a +. Q’ x cof. gz 8c. AP. RCE aE Pos ; i Js +i. % x af cof. p—B.% | OP FP ae eB cata fe + oer ee Re ae ee is PLP ae Py | ot a 2 —I—yy. p x g ecoigae AP. Sy — icy cA ae —I yy if x caf cof. py. % Sen); Se. J But as the coefficients of the terms in this equation are much compounded, it will be proper to make a fubftitution for them: ih De (8 ey dle os Thos, lee = 3 et Q= hg aX x “ PBi= —~* gy Te eee aia 2P 2 ? ee 4 4 3h 4 Bey BB PC ¥52 The Refolution of fome General Problems 4 Pr ¥ I—yy PC ee 2 ee ou xX ef? a a e ——— 4 3P y _ I—-£€8 PC_ RC = Top a 2 X = &e fuppofing thefe fubftitutions to be continued on, to take in the terms affected by the other given quantities Q, hy i oc. (which are had from thofe above, by barely writing Qand 4 &c. in the place of P and p &c.) . This being done, our equation will ftand thus 1—GG. Bool. Bz-+1—yy, C cof. yz+-1—d9.D cof. me &e. —r +t 4 +-Px cof. pz +Qx cof. qz &e. | + PBixcofp—e.z+PB2xcoflp+B.z a O -+-QBrx cof. q—B.% -- QB2 xcof.g+-B.z &c. | J +PCri x cof p—y.z + PC2 x cof.pty.z A ORg x col. g—y.z + QC2 xcof.gty.z &c. &c. &c. From whence, by comparing the multiples of the arch *, in the firft and fecond lines, we have y=p, d=9, and fo on, to as many values (”) as there are quantities Pe 9,7; CC. And, by equating the correfponding coefficients “of thof equi-multiples, we alfo have C— — . De Bi I—pp’ I—7” E> — oe and {o on, toz terms. Then, the value of the firit —rr m terms of the feries B cof G@z-+ C cof yz + Deolde &c. (exclufive of B cof. @z, of which more hereafter ) being thus known, the terms in the 3d, 4th, sth, &c. lines of our equa- tion, being compounded of them and the given quantities P, p> Q, 4, &c. will alfo become known; from whence, by continuing the comparifon with the terms of the upper line, after thofe already taken, a new fet of terms, involving two dimenfions of the quantities B, C, D, &c. P, P’, Q, QO &e. will be determined: by means whereof, ftill continuing the operation in the fame manner, a third fet of terms may be ob- tained ; and fo on, at pleafure. As in Mechanics and Phyfical Aftronomy. 153 As to the firft term 1—@8. Boof. 6z, whereof no ufe has been yet made, it is referved to take off, or deftroy any other term, or terms, of the fame fpecies, that may arife in the ge- neral equation. If no fuch term fhould occur, it is but making the coefficient 1—@@.B==o, and every thing will be right: But that fuch terms do actually arife, will appear in the fol- lowing illuftration of the general method of proceeding, applied to a particular cafe, whereon the determination of the lunar or- bit depends. Let P=P, U=+P, and Q, R &c. all equal to nothing *; then our equation will become 1—26.Beof. gz +. I—yy.Ccof.y z-1-1—90.D cof. dees. E cof.ex &c. } Diy Lee ere Jee me ate F +Px cof. pz-+ Q+- PB 1xcof. p—@.z-+ PB 2xcof.p+-8.z + QBr + QB2x cof. gz + PCr X cof. p—y. z+ PC2xXcof.p+y.z a + Qb1 + QC2 x cof. yz-4+ PDr xcof. p—F. z 4. PD2 xcof. p+o.z + QD1 4+ QD2x cof. dz PET xcof. p—z. z + PE2x cof. p+e. z aor QE2 x cof.ez-+ &c, J Make now, y=), d= —, and = shen a : will ftand thus P P P+8 en the equation I—£B. ooo Bz-+ 1—yy. Coof. yz+- 1—9. D cof. dz-+- &c. —1+ of +-P xcof. pz+ Q+PBi xcof.p—B. 2+ PB2xcof.p+2.z +-QBi+ QB2x cof. ez4+ PCr + PC2 xcof.2p2-+ QCr + QC2 x cof-pz +PD1xcof. 62+ PD2xcof.29-6.2+ QD1+QD2 xcof. p—8. Zs SC. Put B’=QBi+QB2 + PDi + &c. . | ts C= P+ QCr+ Cz + &c D’=PBr + QDr + Qh2 + &e E’ = B2 + &ce. F’ = PC2 aH &e, G’ =PD2 + &c. Then, expunging the terms which have no multiple-cofine in them (as alfo deftroying each other) we, at length have * That thefe affumptions are fo made, as to exprefs (nearly) the forces ° whereby the fun difturbs the moon’s motion about the earth, will be fhewn hereafter. X i ie — a a oe ee am 154 The Refolution of fome General Problems 7 e a —es. E cof.ex &c. AA: Beol.gz-+ 1—yy. Ccof.yz-+- 1—09. D cof.dz-}- Ise. cofex Se. J : ifs B'cof. 6z-+- C'cof.pz-t- D’cof.p-6.z-+- E’cof.p+6.% p=? 4-F’cof.2pz- G'cof2p—8.2-- H’cof.2p+8.2-+- &c. I—(GB: F o D po whence 1—66 x B—=—B’, C=— oo E’ F’ G ee ek ae diag wali = oe Ne pe eer tari op He a ink. HY’ =—;, &c. and confequentl 1—2p-+-6) q y I—w (==ex 1—B cof. Bz— C cof. yz &c.) C’xcof. pz D' cof. p—g. z E’ cof.p-+6.% I—B cof. 8 z+ 4 + ~ I—pp 1—j—4) I—p+48] = F' cof. 2p% G’ cof. 2p—8.% H’cof.2p+-8.z +. &c 1—4pp I—2p—A) Ss 1— 2p +A “ In deriving the equation here brought out, all terms involvin e two, Or more dimenfions of the quantities B, C, D, &c. are neglected; it will, therefore, be neceflary to fhew now, how the effect of thofe terms may be computed, and the approxi- mation carried on, to any farther degree of exactnefs defired. Previous to which, it will be proper to obferve, that, in the preceding calculations, the quantity 1—wl? was taken —= - + 3 x Beofl. Bz + Ccoflyz &c. barely, and 1—wl* = = = x Beof Bz -+-C cof. y = &c. whereas the true value of j—wl3, orex1— Book Rz—C cof, yzbcc.b ? IS = ———_—— oo —,Ci : a 4 x Beof.pz+ Ceol. yz &c. + Gi ¥ Bool, Bz4-Cookyzke} +L &c. ReLaaNe I 3. eee and that of 1—w!* == — + a x Beof. Bz -- C cof. yz &e. é é 4. “= x Beof. Bz-+- Ccof. yzkc.) Now Beof. 6z-\-~-Ccof. 7 0CC) (before neglected) is evidently equal to B* x cof. G2 + 2B Cx cof, Bx x C'eof, y 2% &e, —-B*xy -- cof. 20 BCxcof. y—B. z-+-cof. y+. z + &c. all which cofines, to- gether with their coefficients, will be known, feeing GB, B, y, C &c. Se ee ee Se ee ee. Ll in Mechanics and Phyfical Aftronomy. bcc. are given (or nearly fo) ) by the preceding operation. And, in the fame manner, a feries of cofines expreffing the value of Beof. @ z+- Ccof. yz &c.)’ (or of the moft confiderable terms thereof) may be found, fhould it be neceflary to continue the approximation fo far. To find what alteration thefe quantities will produce in the value of 1—w (before found) let ie new term thus arifing in the value of 1—w)~°, be denoted by 5 — x cof. az, andthe term ihileiens to it in the value of 1—wi*, by ®xcof. az: then the correfponding increafe in the value of — e£ ae —e—2ex flu. P zfin. pz+Q 2 fin. gz. &e.x 1—wl*,) willbe — 2¢ x flu. N P xfin. pz pe + Qa hin, gz. &e. x = xcokaz—=— into flu. Pix fin. p—o. e+ fin.p--a.2z-+-Qéexfin.g—a.2-+-fin.g--a.z &c. any Poof. paz Pcofiptaz , Qcolig—az , Qeof. g+a.% Se LE tee nts he ee aa es Ae ie p—a pe q—a qa ect In like manner, the increafe of the fecond member (P’ cof. pz -}-Q’ cof. gz &c. x 1—w]’) of our general equati- on (/ee p.147.) appears to be P’ cofpz4+-Q’cofigz &e.x = x cof. a 2 r= 5p xPcol.p—a.z-+-P cof.p+a.z-+ Qcol.qg—a. z+ Q'cofg a. xz &e. And the increafe of the laft term, or member, P fin.pz +- Qfin. gz be. x * x flux. 1—wh3, is had — P- fins pz + Q fin. gz &c. x ix—"F x fin. & % eee erg ee pa ee Ma = P fin. pz + Q fin.gz &c. x— Zax fina Zo ee Zaz xP cof, pa. z—P cof. paz + Q cof. ‘gong. Qyeol 7g Ita &e. bet thefe three quantities be now collected together; from whence the new terms entering into the general qe when the whole is divided by oy will ftand thus 2 P a a ee Soe 155 156 The Refolution of fome General Problems Pot Nowra) 4) BIEN MP: gM’ trodes pu Gx pane are pa “i 2P =o 6 xcof, pta-% + &c. So that, from the general method of operation before laid down, it appears that the proper correction for the value of A M N i—w, arifing from the terms, =e cof. «zand 23% cof. az, be- fore neglected in the refpective values of 1—w) 3 and 1—w\*> : : Pp "N , MP aM. cof.p—a.z will be exprefled by e zuto Beet ee. 6 See pur — PN) Me aM cpa Feet ae oh eee N MO” aM .. cof. g—a. z af q—a + oo ee Q N MQ aM cof. q+. % Be ee In refpect to which it may be obferved, that no regard has been had to fuch quantities as would arife from the multiplica- tion of the new terms in £, by the feries 1—€@.B cof. Gz ae I yy. C cof. yz + &c. (as ought, in ftrictnefs, to have been done); becaufe thefe terms being very fmall, they will, after multiplication into the fmall quantities 1—PBxB, 1—yyxC &c. be fo far reduced, as to become quite inconfiderable. And it _ may be obferved farther, that even the whole produd arifing from the multiplication of the value of © into the faid feries, except the terms — e © +-1—£6.e/Beof Bz-+ 1—yy. ef C cof. y z &c. might be alfo rejected, without producing an error of more than a few feconds, in finding the place of the moon in her orbit. Having fhewn what alteration will be caufed in the value of 1—w (the reciprocal of the diftance of the body from the cen- ter of force) from the taking in of any {mall, affigned terms in the values of 1—wl3 and 1—w!4, it may be a pro- per place here to fhew (as it is of great importance to be known) what change will arife in the faid value of 1—w from the addition of any {mall, new terms, A fin. 72, and in Mechanics and Phy fical Afironomy. and A’ cof. rz, to the refpective forces. P fin. pz + Qfin. gz &c. and P’ cof. pz Q col. gz &c. whereby the motion of the body is difturbed. What this alteration, or correction ought to be, is eafily difcovered from the general equation on p. 151; from whence, by fubftituting 7, A, and A’ inftead of p, P, and P’, refpectively, the new terms affected by 7, en- tering into the faid equation, will appear to be ARS * + 2 x cof. rz of 4 3A B 1-68 | AB ——— + 4. lyn Ea Od “fase eas ' 4 A’ B 1-@8 | AB = and confequently the increafe in the value of 1—w arifing cof, 7z : A 2 A therefrom — e into at ate ak oe 4 3A’ 8 1-66 . AB cof.a—B.% 5 rc ae an B 1—@8 AB __ cof. a+8.z Sr Tok aa a eee ew ae where, in many cafes, thefirft term alone, will be fufficient. In making the different corrections above pointed out, it will be found neceflary to have a particular regard to fuch multiples of the arch z, as are, either, very fmall, or nearly equal to unity (of which two kinds, thofe whofe exponents are p—2, and p—@, will be found the moft confiderable.) For, though the coefficients of fuch terms fhould appear, at firft, to be fmall, they ought not, therefore, to be immediately rejected; becaufe the divifors which they afterwards receive (the former in obtaining the value of 1—w, and the latter, in finding the anomaly from thence) are fuch as may render the effect, or quotient, too confiderable to be intirely difregarded. And it may be eafily conceived, without the help of cal- culation, that a term, or force of the former kind, exprefied by the fine or co-fine of a very {mall multiple of the longitude z, muuft neceflarily have a much greater effect, than another (having the fame coefficient) which is proportional to the fine or co-fine of a large multiple of the fame angle : becaufe, ee e TDi 158 The Refolution of fome General Problems the index, or multiple is avery {mall one, the term itfelf, while % increafes, will continue, for a confiderable time, neatly of the fame value; and confequently, will have its whole effe¢ exerted in the fame direction; but when the multiple is a large one, the changes from fofifive to negative, and from thence to poftive again, are fo quick, that fufficient time is not allowed for producing any confiderable inequality in the body's motion, before that effect is again deftroyed by the fame force, acting equally in the oppofite direction. The value of 1—w (the reciprocal of the body’s diftance from the center of force) being, by the formule laid down above, approximated to a fufficient degree of exactnes, we may from thence, and the equation = - x xa (given at page 146.) proceed to compute the time (¢) of defcribing the angle 2; whereby the difference between the ¢rue and mean anoma- lies will alfo be known; which, in the azar theory, is the great point in queftion, and is befides, abfolutely neceffary in -or- der to introduce the proper quantities of the forces whereby the moon’s motion is difturbed. Let, therefore, the value of 2 (as given by the firft equa- tion, at p. 149,) be here reprefented by J x 1—B cof. Rz—Ccof. yz—Deof.dz&c. fo thal a= xf z x1~B cof.6z— Ccofiyz bse: 3 xe x 1B cof. gz &c\-* But 1~Beof Bz—Ceolyzdcel —1 +2xB cof bz &. +L **Boof. Bz +C cof. yz 8c. | Ye I+ x Beol,Bz+Ceofyz Sze. -|- +X BB x 1 cof. 2z-+BC x cof y—B.z + cohy + B.z&e, Andi—Beof,Gz—Ccofiyzéc.)-* = 1 +2 x B cof. Bz &c. -+ 3XiBBx1+cof2@z4+ BC x cof.9¢—f. z+ cofy-LB.z 8c. Which values being multiplied together, we thence get ¢ = 2 x Boot. Gz--Ceokyz &c. ST +X B cof. P2-+C cof » z &e. Bers act into +4 in Mechamies and Phyfical Aftronomy. ———[—$<==————E_ - BBxi-}-cof262-+BCxcof.y—B.2-+ cof. y- Box &C. ee eee + <3 y+BBxi-tcof. 28z-1-BCxcof. y—B. y—Bz+cofy +P. z s &ec. | £xBBx1-tcof. "1 -beol20z-+-BC-+BCxcoly—6. pak cisaede Bisedecs Put 6=14+-3xBB+CC+DD &c. + +xBB+CC+DD &e, +z xBB4+CC-+DD &c. (B) ==2B-+- 3 B, (BB). == 3 BB-}- 3 BB+ + BB, (BC) =3BC+3 BC+ 1BC+- 4 BC, &c. conti- nuing on thefe fubftitutions, fo as to take in the terms affected by the other quantities D, E, F, &c. (which are had from thofe above, by barely writing one letter for another); by means whereof our equation is reduced to (4 +-(B) xcof.Bz-+(C (C)xcoly2z4(D)xcofdz &e. pi se “J (BB) x cof. 2@z4+(BC)xcof.y—B. z-tcoty y+B.2 z 2ce'f* | (CC) x cof. egg Stee eras a s rats &e. From whence, by taking the eG &c. we have (B) xfin. Cz (C)xfin.yz a oz [2p ee eee te + &c. pe (BB)xfin. fin.26% (BC). finy—@2z fin. fin. 78. 2he roe ye ae + &e. |(CC)xfin.zyz , (CD) finy—dz | finy3x & L 2by i eee yo “i . In : ‘efpedt to which it will be needful to obferve, that the firft term, me ° xz, will, when z = the whole circumference, be the true meafure of the mean periodic time; becaufe all the other terms being c ted of thefines of multiples of the arch z, they will, ¥ yhile mk keeps increafing, change from pofitive to nega- tive, andfrom a to esate again, and fo on continually ; and hee can have nothing to do in the mean motion; being themfelves no other than the proper equations whereby the mean and ¢rue motions differ from each other; fo that, the true motion pete oe J by z, the mean motion will be exprefied (C)fin.yz by 2 ho ~ -|- 7p ——= &c. as above determined. {in De oP. From 160 Fig. 31. Pig; 34 The Refolution of fome General Problems From the expreflion ae xz, here found, the proportion between the mean periodic time of the body in its orbit ABP &cc. and the periodic time in the circular orbit ADK, that might be defcribed, independent of the perturbating forces, by means of a centripetal force fufficient to caufe the body to move there- in, will be known: for the quantities e, f, and 4 being here, each, equal to unity, the faid expreffion will, in this cafe, be- come = = yz: whence it is evident, that the periodic time k c in the circle, will be in proportion to the periodic time in the orbit ABP &c. as unity is to ras Apprticarion to the Lunar Orbit. In order to apply the conclufions derived in the preceding pages, to the determination of the lunar orbit and the different inequalities of the motion therein, it will be neceflary, firft of all, to inveftigate the fun’s force to difturb the motion of the moon about the center of the earth ; from whence all thofe ze- qualities, except that arifing from the excentricity, are pro- duced. Let C, $, and B reprefent any three cotemporary places of the earth, fun, and moon, refpectively; and, upon the dia- gonal BS, let the parallelogram BCSH be conftituted ; mak- ing BF perpendicular to CS. If & be aflumed to denote accelerative force of the earth to the fun, the accelerative force of the moon to the fun will be truly reprefented by & x = ; which force may be refolved into two others, the one in the direction BC, exprefied by kx <= x aS ; and the other in the direction BH, exprefied SC? SC ; : : by &X = x gp; from which laft, let the force & in the pa- rallel diretion CS be fubtracted; fo fhall the remainder kX LS se le in Mechanics and Phyfical Aftronomy. SC}—5B? ° ° qe ° kX — be that part of the force, acting in the dire@ion BH, whereby the motion of the moon about the earth is dif- SCs 8B ae. turbed: But this quantity, 4x —==—, is evidently equal to SB’ kx SC—SBx eee 3 which, as SB (by reafon of the great diftance of the fun) is nearly equal to SF, will be alfo SC*-.8CxSB-+45B° CF equal to x CF x —— aa ,0r to3&X =p eq ° And the forces £x ex EF cof2SCB, pal te x i fin. 2 SCB, whereby the motion of the moon about the earth is difturbed, will therefore be truly defin- | gaitehf CB me tae CB ed by => X qq xcof2SCB +=, and aE GE & fin. 25CR. If the fun and moon be fuppofed to move from the line CA at the fame time, fo that the angle ACS may be the fun’s ap- parent motion about the earth, whilft the moon in her orbit moves from A to B, it will be, as1:m::2 (= the angle ACB): mz == the angle ACS, nearly; which would be ftrictly true, were the frue motions of the fun and moon to be exactly in the fame proportion with the mean motions. Hence SCB (—ACB —ACS) will be had = z—mz, nearly; and confequently 28CB=pz (by making p=1—m x2): and fo the forces found above, by fubftituting this value, and writing — in the sath) . CB : room of its equal ca Will, by means thereof, be reduced to I ame*f . met Bt Sat Be 1—w 2bb x cof, pz-- ahh » and r—w <> opp fin. pz: which quantities, with contrary figns (becaufe they diminith the centri- petal force to the earth, and the area defcribed, inftead of in- creafing them) being compared with the two generak ex- preffions — x P’ cof. pat-Q cof. gz-+-R’ cok rz Sc. and r Rescues a fp * P fin. pz-+-Q fin, gz-ER fin rz &c. as given on p. 147, * In deriving this conclufian, regard is had to the moon’s rehative motion about the earth’s center 5 but if we confider the motion, as performed about the common center of gravity of the earth and moon, the refult will be exaéily the fame. we in Mechanics and Phyfcal Aftronomy. we fhall, in this cafe, haveg==0, r==0, &c. P’ = — ar ef etf ME OOS: i e X75 == 0,0084 Xx pQ=— Ze == —0,0028 x5 I / ous 37m” ef. es Ey, ea Bas (25 342) Pe — 2 xe (=+P*),-Q=n0,;- Reso. &c! Bit here, inftead of 4, or its equal, 1 +ix BB+CC+DD &c. (given at ~. 159,) it will be fufficient to make uf of the firft term of the feries only (till a more exact value, by means of the fubfequent calculations, can be known); becaufe all the quan- tities B, C, D, &c. being fmall in comparifon of wnity, the iquares of them, which are here neglected, will be ftill {maller, and of lefs confequence. If, now, the cafe under confideration be compared with chat laid down, and refolved, at g. 153, they will appear to be the fame; fo 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 mult be taken notice of, refpecting the-principal (B cof. Gz) of thofe values, on which the great elliptical equation, arifing from the eccentricity, depends; which cannot be known but from the obfervations of Affronomers; fince it is owing to the projeCtile-velocity which the moon, jirft, received, more than to the perturbating force of the fun, whofe effec we are about to calculate. But, though the term B x cof. Gz, cannot be deter- mined by theory alone, yet the value of its exponent 6, on which the motion of the apogee depends, may from hence be deduced. ‘This value, in a former operation (fee page 148,) was found to be an wnt (as the circumftatice of the problem, when the apogee is at reft, ab{olutely requires); the true value cannot, therefore, differ much from an wit, which may be ufed inftead of B, till a more exaét value, by means of a fecond operation, 1s known. Making, therefore, B tat g “ = 0,0084; P’=P; Q=0; Ui P= 1P; m= 0,0748, p (=1—m xX 2) = 18504 y=p, d—p—P, -=f4+0, &c. (as before determined) and {ubftituting in the equations on p. 151, r52, 153, we have : y3 iP The Refolution of fome General Problems = oP P 2x—0,0084 = P (=—— + =) es ae —0,084==— 0,001748 ; pef ag , 0,0084 Q OQ (= 23 + Sl H— Ft = — 010028: Pay (a te ob ee ee) oR PBi (= 5-3 - 2 p ef = — 0,040D; ae 4 @ 1—@8 ., PB PBs (erg eet 5 mee pe a) — — 0,0285B; Se 4 ys ayy PC PC2 (apa saat eo et pnt a7) = 0,0404Cs 4 gone S i—.. PD\ __ : Ei a = ae =) —— 0,0414D; 9 Bes 3 = eo — —_ = e PDa(=4y +t +5— 5X ap) =m 02733 Se WR gO) EQ eee” OB PB 0,0042 B (becaufe Q=o, and Q’==!P); and in the very fame manner, QB2== —0,0042B ; QC1=—o0,0042C ; QC2—= — 9,0042C; QDi1= —0,0042D, QD2=-—0,0042D, &c. &e. *. Whence %* In thefe calculations, all terms whofe divifors are found equal to no- thing, will themfelves be nothing, and not infinite, asmight at firft be. ima- col.p—y.z ee ae Saad gined: thus the term —, by having its divifor p—y==0, intirely va- nithes. ‘The reafon whereof will appear evident, if it be confidered, that this term, oul aue a arifes by taking rhe fluent of —z X fin. p—y.%, Or—z X02 which is, manifeftly, equal to nothing. But an objection may, perhaps, be brought from hence, againft the truth and univerfality of the /uxionary, calculus; feeing thereby an expreffion is here derived, which, though aQually equal to nothing, appears neverthelefs under the form of an infinite quantity. fip—y.x To clear up this point, it muft be obferved, that See er is not the complete fluent of —z x fin. p—y.z, but the variable part of it only; the corrected flu- : I cof. p—y.% z Serine ent bein a ee as > or att 5 8 ~—y moka Gat boi gine X —i--cof.p—y.z. But, whatever value p—y is fuppofed to have, the cofine of p—y.z will, it is known, be ex: prefied in Mechanics and Phyfical Aftronomy. Whence it ts evident, that B’(= QB1+QB2-+PD1 &c.) = —0,0084B —0,0414D, C (=P + QCi 4 QC2 &c.) =—0,01748—0,008 4C, D’ (= PBi+QD1 + QD2 &e.) = —0,048B—0,0084D, E’ (= PB2 &c.) == —0,0285B, F’ (= PC2 &c.) = —0,0404C, G’ (=PDz &c.) = —0,0273D, &c. But B’ being = — 1—60.B, C—=—1—yy.C, D'=—1—2.D, &c. the fecond and third of thefe equations, will by fubfti- tution, be changed to 1—yy.C==0,01748-+0,0084C, and 3—60. D = 0,048 B +0,0084D; whence C is given = 0,01 748 001748 I—yy—0,0084 1~1,8504/'—0,0084 5048 B . ? : ose yc etha eh A = 0,178 B: which values being fub- I—0,8504 '—0,0084 {tituted in the other equations,they will become B= —0,0158B, E’ — —o,0285B, FY = 0,0003, G’ = — 0,005 B; whence E’ A E (= —- — —0,004B, ES — =. ( aa sik ae 0,000023, G (= — ==) —=-—0,0008B; and confe- oe quently 1—w= (ex 1—Bcof.@z—Ccof. yz—D cof.dz &c.) — ¢ into. 1—Bcof. Bz-+-0,007186 cof.pz—0,178 Bcof. p—C.z -\.0,004Bcofp-++B.z—0,000023cof.2pz-|-0,0008Bcol.2p—.z &c. expreffing the reciprocal of the moon’s diftance from the earth. As to the quantity B, which enters into the greater part of the terms of the feries here found, it cannot, as has been al- ae Cee preffed byx—~*—#-— = p—y Vx 4 &c. So that, the fluent in queftion will be: p—y N72” p—y. V4 Tas truly reprefented by Pana = &c. or its equal —- oy 1.2 1.2. 3.4 —y.2? —y |Px4 : 7 : ‘ —— + oe &c. which entirely vanifhes, as it ought to do, whem p—-y=0.. ready, — = —0,007186; and D== 165 The Refolution of fome General Problems ready intimated, be otherwife determined than from the ob- fervations of Afronomers: nor will the equation above exprefling the relation of B’ and B, afford us the leaftshelp therein: For, by fubftituting —o,01 58 B inftead of its equal B’, that equation will become —o,0158 B ==— 1—£8.B, or 1—@G—=0,0158; where, B intirely vanifhing, nothing in relation to it can, therefore, be determined. We have here, indeed, an equa- tion for finding the value of @; which from thence is given ==\/1—0,01 58==0,99206; bymeans whereof the motion of the apogee will be known: for it will appear, 4y cor.1 and 3 toprob.V1.) that 1—w==e x 1—B cof. Gz is the equation correfponding toa moveable ellipfe, turning about the focus, or center of force, with an angular celerity which is to thatof the body in the ellipfe, every-where in the conftant proportion of 1—6 to 8: 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 0,008 455 to 1) as given from obfervation, by about ‘, part of the whole value: nor ought this to feem ftrange, as a number of ({mall) terms yet remain to be introduced into the value of 1—w, by the corrections pointed out on-f.156.and.157: befides which, the difference arifing in. the co-efficients of the terms already found, by fubftituting this, zew, value for 8, will amount to fomething confiderable,. from whence, alone, near half the error would be taken away. But, to avoid the trouble of repeating the fame operation, againand-again, with the new values of @, thus found, I shall here, at once, take @ equal to the true value (03991 54:5) as given from: obfervation; but wre icine tric cue ltt Ua el heiie bocag sal If the eccentricity B, be fuppofed to vanifh, 1—w will then become = ex 1-+0,007186 cof. px—0,000023c0/. 2p &Fc. which, when p20, or the moon is the fyzigy, will be =e x 14-0,007186—0,000023 = 1,007163xe; but when ap2=90 degrees, or the moon is in the quadrature, it will then be = ex i—0,007 186—0, 00002 3=0,992791 xe. Therefore, when the orbitis with- out excentricity, the diftance of the moon:from the earth, in the fyzigy is in pro- . ‘ ‘ i portion to the diftance in the quadrature, as —— to ———, or 1,007163 xe 0.992791 xe 0,99279f ta. 1,007163, that 185 asOQ 10705 very near; thefameas is found by Sir Tfaac Newton in kis Princip. B. TM. prop. 28, a fhall, as in Mechanics and Phyfical Aftronomy. fhall, at the fame time, put down the feveral terms arifing in the equation for the value of 8; by means whereof it will ap= pear, in cafe both fides are found to be equal, that the va- lue of the root 6 has been rightly affumed; and that the mo- tion of the moon’s apogee (which has been the fubjeét of fo much fpeculation and controverfy) is intirely confiftent with the general laws of gravitation. Now, of the quantities above determined, PBi and PD) are thofe that are the moft affected by altering the value of ~: thefe being, therefore, computed a-new (making @ equal to 0991545, inftead 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=o,1869 B, and 1—06 (= — 8 ==0,01619. As to the values of E, F, G, &c. they are fo fmall, in themfelves, and fo little affected by @, that to compute them a-new, would be quite unnecef- fary; the difference not producing an error of a fingle fecond, in the place of the moon. To apply, now, the obfervations laid down at page 156, in order to obtain from thence a farther corre¢tion of the value 1—Beof.0z-+-0,007186cof.pz—o,1869B cofp—f.z (ex hse? 004Beof.p 1-B.2-0 000023 cof.29z-+ 0,0008 Beof.2p-G.: rvs PP: ’ : > {.2p-6.z) above found, the feveral powers of the feries B cof Gz — 0,007186 cof. pz-+-o,1869 B cof. p—B.z &c. (before omitted) muft be now taken, or fuch terms thereof, at leaft, as are of confequence enough to merit regard. Thus, in the fquare, or fecond power, the terms which appear confiderable enough to merit an examination, at leaft, will be thofe arifing from the {quares and the double rectangles of the three firft terms of the root, which are vaftly larger than the others. Thefe will be + B* + +B’ cof. 282 — ,007186 B cof, p—G.z — ,007 186 cof. p-+ B.z-+- 0, 1869 B* cof. p—28.z + 0,1869 B’ cof. pe ,00002,5 -} ,00002§ cofine 2p2 — ,00135 B cofine Bz — 00135 B cof. 2p—@.2 + ,0175 B’+- ,0175B* cof. 2p—2f. x. But, in the value of 1—w/! 3, thefe terms are affected with the = abd oes ‘ common tmultiplicator = (vide p. 154); and, in the value of 567 168 The Refolution of fome General Problems odes Pa 10 : r—w, *, by the common multiplicator =: fo that, in order to find the effect of the firft of them (4B’) from the formula.at weigO TOAiat p. 156, we muft compare + B xB ahd = B* x, with x N e* N= 58 ;.and ao: and confequently, P. “N , MP’. caM. cof p—a.z cof. az,and-~ x cof. az, refpectively ; whence M = 3B, er ee ee or gt P Ne aM cof. ptaz ee ee oe ee In the fame manner, with refpect to the fecond term, + B’ cof, 28z, we have M=3B*, N==sB’ (as before), and a—2PB; from whence the correétion, anfwering thereto (exclufive of the general faétor e) will come out, 0,3178 B’ cof. p—28.z -}- 0023 B’cof. p+28.z +,0028 cof. 2Bz. Again, in relation to —,007186Bcof. p—B.z, we have M=~—.,431B, N=—,0718B, and a=p—B; from which will be found ,000735 B x ae —,000073 Beof. 2p—B.z +,000464Bcofp—.z; where the firft term is of that fpecies on which the motion of the apogee depends, and where the /e- cond is too {mall to be farther regarded. In like fort, from —,007186 Bcof.p+2.z, we fhall get &c.=,029 B? cof. pz. od {, Bz : —,00060 Bx ee befides two other terms; which, on ac- count of their extreme fmallnefs, may be intirely neglected. By proceeding on, in this manner, two, or three {mall terms, more, (producing equations of a few feconds, each) will be found; which being joined to thofe above, we fhall from thence have, -+ ,03 B* cof. pz + ,00055 B cof. p—B.z + 0,3028B'cofp—26.2-1,0072B’cof 2Bz-+,0070B' cof.2p—28.z cof. Bz ; 1—6e ? tore, will give the whole increafe of 1—w, by taking in the fquare of the feries aforefaid. As to the increafe arifing from the +,00016 Bx which, drawn into the general multiplica- in Mechanics and Phyfical Afronomy. 169 the cube thereof, few of the terms will be found confiderable enough to deferve notice, the only ones of any confequence to be collected from thence, being —,o0004B-+0,7B' 0,7B’ x cof p—f.z z —o,11B’ xcofine p —3 bey z = 68 B x cof, 2p—3. % cof.ez —Sp * —,055B’ x : But, from the dzquadrate, we fhall only have one term (1, 1B cof. p—28. z) that can produce an equation of more than a fecond, or two; though the effect of this one term will, alone, amount to near a minute; which is owing to the {mallnefs of the exponent p—2, confonant 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 {pecified. If, now, the feveral quantities above brought out, be collected into one fum, we fhall have 0,03B*cof. pe-+,00051B 51B—o,7B’ x cofine p—B. zt 0,3028B’ + 1,1B* x x cofine p—28. Be: -+,0072B’cof.2@z-+,0070B’cof.2p—26.z—0,11B’ cof p—3h.z ——,00b Col. emda Was 16B—,055B’x —— of all which terms, 0,3028B° -+- 1,1B*x cof. p—26.z is by far the greateft, and fo confiderable as to produce other terms, or correcti- ons, of confequence enough to be regarded here. The new terms arifing from the entering of this term into the firt power of the feries, for the value ff: I—w, will appear to b be —,0032B'cof.20z—,0047B*cof.2p—28.2-+,0026B’col.p—2@. 3*colp—2e.z (which are found as above, by making M3 x —0,3028B’, N=4 xX —0,3028B", a=p—2@, and negle@ing 1,1 B*): And by the entering thereof into the fquare of that fe- ries, from the double rectangle under it ar it and the firft term, there will be had -++,088B’ Fists 2p—38.% — ,019B' cof.p—3B.2-+-,03B° me ee which, together ae os quantities before found, bee Goined to the former value of 1—w, we haveat length Z The Refolution of fome General Problems | 1-Beol.fz4 5007 186-+,03B"xcof.pz—0,1864B-+0,7B'xcol p—2.z +-,004Bcof.p-+6-2-+0,3054B*--1,1B*xcof.p—28.z—,000023cof.2pz -}-,coo8B cof. 29p—6.2-4-,004B* cof. 262-+-,0023B? cof. 2p —255.% {+-0,13B3cof p—38.x In which equation the terms of the fpecies, cof.@z, are not put down; becaufe the original term—B cof. £z,which is equivalent to them, is ftill retained. Thofe terms, however, though of no ufe in this equation, are not to be intirely difregarded; fince on them the motion of the apogee depends. By collecting them, and the former value T—-w—éx : : 01619B of —B, into one fum, we get the equation —B=— 2 ,00016B—,05 5B? o2B3 01603B+,0255° 4 OER Toe a at —_ — 01b03B 2.025 : from whence, I-38 I—38 I—33 when B isaffigned, the value of 6 will be found, but /7// fome- thing fhort of the true value, as given from obfervatzon. But it ought to be now remembered, that all the equations hitherto brought out, are derived on the hypothefis, that the true motion of the fan is to that of the moon, ima conftant pro- portion: which is near the truth, only; fince the diftance of the fun from the moon, whereon the perturbating forces de- pend, is fometimes greater, or lefs by two degrees, than ac- cording to that hypothefis: which difference muft, of confe- quence, render other corrections neceflary. How rhefe may be introduced I fhall here inftance, by farft confidering the moti- on of the earth about the fun, as uniformly performed in acir- cular orbit; leaving, to be determined by a future operation, the other equations arifing from the eccentricity and parallax, which will be obtained in the fame manner. Tt is found at page 159, that the moon’s mean motion, an- fwering to the true motion z, is tightly reprefented by z + (B) xlin.gz (C)xfin.yz (D) xfin.dz Gan Get ege re Mert pe &c; from whence the mean, or true motion of the fun (for, in acircular orbit, they are the ip aos {> , (B)xfin.gz , (C)xfin. yz , fame) will be had = mx 24----> : ~- Fr a B)-xfin.€z C) fin. = mz—-+-mz’, fuppofing 2’ = — -|- Ce fc, Hence the in Mechanics and Phyfrcal Aflronomy. the true diftance of the moon from the fun, will be 2 —#12z—= mz, and the double thereof equal to I—MX23—2MZ, or to pz—2mz': of which the fine is —fin.pz x cof. 2mz—col. pz x fin. 2mz’; but the arch 2mz! being exceeding fmall, the fine thereof will be nearly equal to the arch itfelf, and the co- fine vety nearly equal to the radius, or unity; and confe- quently the fine of the double diftance of the moon from the 2m(B)xfin.6z boas m (C) — fun == fin. pe—cof. pz x 2mz! = fin. pz— cof. pz x B ——————_—_— errr &c. == fin. pz +- ue x fin. pe fin, p4t-B.2 — x fin. p —y.2—fin. pty. z oe. 2 In like manner, the co-fine thereof will be had (= cof. pz x cof. 2mz'4- fin. pz x fin. ame! col pz--fin.pz x 2miz’ m(B) nearly) == cof.pz.-+ —j3- X cof. pha cof. p+6.% m(C) We eee Te ae ae 3 Which two quantities being multiplied by P, it will ap- pear, that the new terms arifing in the two expreffions of the forces (befides thofe, P fin.pz and Pof.pz. in the former hypothefis) are vail: lit, p—B.z— fin.p+-6.2-+ me ed i we P (RB oe ey. x finp—y.z—p+ y 2. and x cof. p—B.2—col.p+f.z mP c le “P ee) x cof. p—=y.z—py.z &e. Now it will appear, by a comparifon with the formula on p. 157, that the effect of the two firft, corréfponding terms here. mP(B) ~ 2 d-1%2 cof, 9% 8 og 6 2 ar 1 : earn ag exhibited, making s=p—P, will be eer me ee 4s B 28 mPB(B) cof. BB 4 — ee WA Wee eee x hgetf Iam! te, Ey, APBB) SOME peu ephich; bes wee 2 ae fae pe &e. which, be Pr ‘ eaufe m=,0748, w= —,0084, andd==1, will be equal to r7! 172 The Refolution of fome General Problems 00063(B) . 2 cof. 7% 4 B I—98 See Ot x a testo I—77 m—B T Beof. t—32.% 4 gp I—28 Beof.7+3.z x 1—7—4] a a+B or ote phe ok ee +- enna) I—7+2," Ccof. r—y.z = + &c. = —,0081 (B) I—7—y¥!} x cofine p—G.z +-,0187B (B) x cof. p—28.z -+- ,0011 B(B) x cof. pz &c. where all the terms, after the two firft, may, on account of their extreme {fmallnefs, be intirely neglected. In the very fame manner, expounding # by p+, p—y (=0), py (=2p): P—8 (=8), and p-+3 (=2p—8,) faccet- fively, and fubftituting -+ —- a sa gi iad ,— es) 4 refpectively, in the room of te ,00063(D ,00063(D) ee Soe Me and -{- ey ee coe) ; there will, 22 the jfirff cafe, be found the term —,o0015 (B) x cof.p+6.2; being the only one producing an equation of more than a fingle fecond; 2 the fecond and third cafes, no term at all, worth notice, will be found: but, zz the fourth (where =f) two pretty confiderable ones ,o0222 (D) x SS and— 0,0233 D(D) x cof. p—2/3.2, do arife ; whereof the former is of that fpecies on which the motion of the apogee depends, and gives, an increafe of that motion, of about =*- part. By purfuing the fame method, the effect of the fecond order ofterms, oo ee : b.2B : b.y—B ed; but here it will be fufficient to make ufe of the firft term of the general formula alone: from whence are obtained the quantities +-,0045 (BB) x cof. p—28.z, and — ,oo59 (DD) x cof. p—26.2; which are both of that kind, fpecified at p. 157, rendered confiderable by the fmallnefs of their expo- nents,and are the only ones 4ere, that merit regard, Thefe being, therefore, joined to thofe found above, the whole correction fought will be, —0,0081(B) x cof. p-—3.2—0,0001 5 (B) xcof. p+6.z I &c. may alfo be comput- in Mechanics and Phyfical Aftronomy. ee enna -- 0,0187B(B) —0,233D (D) 40,0045 (BB) —0,0059 (DD) oe eer x cof.p—2@.z. But (B)being 2B, (D) =2D(BB)—=? B’, and (DD)=:D*, nearly, wd. p. 159 (B, C, D, &c. being neglected, as too inconfiderable to be regarded here) our expreili- on may therefore be changed to -— 0,0162 Bcol.p—£.z —0,0003 Beof. p+ B.z-+0,0441 BY —o,05 54D’ xcof.p—26.% which being added to the value of 1—w, before found, we. thence have 1—Bcof.6z-+4-0,007 86-4-0,03 B® xcof_pz—o,20264 +-0.78 xcof.p— 2.% [-wa=tex +-0,0037Bcof.p+8.2-++-0,03495B*-0,0554D* + 1,1B*xcof.p—28.% —,000023 cof. 262 + ,0008 B cof. 2p-—8.z + ,004B* cof. 28 z + 30023 B*cof. 2 —2,8.%-4-0,1 3B* cof. p—36.z 0,00222(D)xcof.gz ne ieee in the equation for the motion of the apogee, it will here be- 0,01603B-+4 0,025B 0,00222(D) __ I—88 Pe i Rese BB Ge And, by writing the coefficient of the term — ee come —B= — 0,01693B-+-0,028B? a whence 1—6G = 0,01693-++00028B’—=0,017015§ (fuppofing B—=0,05505); and confequently 1—@—=0,00854; which value is, now, about ~~ part greater than the true value, giv- en from obfervation, and is near enough to fhew, that the force of the fun is fufficient to produce all the motion of the moon’s apogee, without fuppofing a change in the general law of gravitation, from the inverfe ratio of the fquares of the di- ftances. Several very {mall terms belonging to this equation, have been omitted, befides thofe arifing from the fun’s excentri= city, and the inclination of the lunar orbit ; which together, may very well be fuppofed, fufficient to caufe a difference equal to that abovementioned. If, now, the value of, B expreffing the mean excentricity, in Sir Ifaac Newton's lunar theory, be expounded by 0,05505, according to that author, the general equation of the orbit, found above, will here become TE TST s pecaule (D) 2D 2x o; 2026B-+0,7B’: | 173 The Refolution of fome General Preoblents Ne ,05505¢of. 624+-0,007276cof, pz— 0,01127 cof. p—P.x + needed 000204.cof. p-+B.%-4-0,0010623 cof. s-28.z- 0,000023 cof. 2p% ry +-0,000044cof. 2p—B. Z+-0,00001 2cof, 26%-+-0,000007 cof. ap —2.2.% { -[-0,000021 cof. p—26.% But this: value muft now be corrected by the difference arifing frotn our having, in all the preceding calculations, taken the di- vifor b—=1, inftead of the true value 1-+-2 Cor a (wid. p. 159.) which value, becaufe B—o,o5505, 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 ea in whofe exponents the quantity p is, fugly, concerned, ught to be diminifhed in the ratio of 1 to 1,0097 ; and that ail “thofes where 2p is in like manner concerned, ought to be di- minifhed, in the duplicate of that ratio: by which means our equation is, at leneth, reduced to 10,055 05 cof.62-+0,007206cof. pz—0,01 116 cof. p—B.z -++0,000202 cof. p+3.2-4+-0,001052 cof, p—28.2—0,000022 cof. 2p% } } }-0,000043¢0f, 2p—B.2- %-4-0,00001 2cof. 28%+0,0¢0007cof. 2p—28.% ( +-0,00002 I cof. p==38.%. From whence all the great equations of the moon’s motion, and ail the {maller ones, except thofe depending on the fun’s excentricity &c. ace obtained, within leis than half a mznute Priv ste Kis of the truth; fuppofing the mean excentricity (B) to be here truly affigned. Ifit fhould be found neceffary to augment, or diminifh the value thereof, then the term —o0,01116cof.p —6.z (producing the equation, called the evecfion) mutt be alio aug- mented or diminifhed, in the fame ratio; and the east -+0,001062cof. p—28.z (which is the next confiderable of thofe wherein @ enters) muft be augmented or diminithed in the duplicate of that ratio. As to the reft of the terms, they are fo {mall, that a little alteration in the value of B will produce no difference in them worth notice. In the fame manner, the inequalities caufed in the moon’s motion by the fun’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 fun, being in pro- portion thereto as m to 1, will be alfo known; from hens, and the excentricity, the fun’s true anomaly, and diftance from the in Mechanics and Phyfical Aftronomy. I7vg / the earth will be had, in a feries of fines and co-fines of the multiples of the arch z; whofe exponents and coefficients are all intirely known: by meansof which, the fun’s true difiance from the moon being (nearly) obtained, the mew terms arifing from the eccentricity, in the general expreffions for the perturbating forces, will be had, in confequence thereof; and, laftly, the effects themfelves, 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 diftance from the earth) is thus deter- mined to a fufficient degree of exactnefs (by repeating the ope- ration if neceflary), the difference between the struc, and mean motions of the moon, mutt, from thence, be found, in terms of the latter: this may be done by firft finding the megan mo- tion in terms of the grue (as is fhewn at fp. 159) and then reverting the feries; or, otherwife, without Hading the meaz motion at all, by the refolution of the fluxionary equation > ~S e mA : cr —_ — os WY > met af ae al according to the method made ufe of of in a former paper, in this collection. The procefs (whict is more laborious than difficult) I {hall not infert here, as my defign, in this mzfcellaneous work, is not to exhibit every opera- tion neceflary to the forming of a complete’ theory of the mcon’s motion; which, for its importance, and the great variety and intricacy of calculations arifing therein, may very well merit to be the fubject of a volume, by itfelf. It may fuffice, in this.place, to have pointed out (by a me- thod not very perplexed) how the different znegualities of that motion may be determined, and the moon’s true place, accerd- ing to gravity, afcertained. Atanother time, if health permits, I may, not only give, at large, the application of the cguations and precepts here delivered, but alfo a new fet of /unar tables, deduced therefrom; which (though a work of much Jabour) I fhall the more chearfully undertake, as Dr. Bradley has ery obligingly offered to affift me: with any obfervations, that may be wanting in order to the compleating of the defign, and eftablifhing the theory on a proper dafs. And I have fome reafon to hope, that, when that part of the daca, which can be only known from obfervations, is truly fettled, 2 the ~ 176 The Refolution of Jome General Problems the place of the moon may be a/ways obtained, within about a minute of the truth, even, without the ufe of a multitude of ta- bles, by means of proper contrivances, refpecting the {maller equations, Or the lefs confiderable terms of the general {eries. I thali conclude what I have to fay on this fubject, at prefent, with obviating a difficulty in the application of the equations abovementioned, when the effect in the moon’s motion, de- pending on the fun’s eccentricity and parallax *, are computed thereby. In this cafe, a new {pecies of (very {mall) terms, af- fe&ted with the co-fine of the arch z, will be found to enter into the general equation (07 page 151) whereof the effect can- not be determined in the fame manner with that of the other terms, affected with the co-fines of the multiples of that arch: For if, according to the method of proceeding there laid down, we aflume a term (as 1—az.Icof. zz) in the upper line, 1—(G.B cof. Bz-1—yy. Ccof. yz-+- 1—09. Dcof. dz 8c. in or- der to compare it with aterm, gcof. z,. of the aforefaid fpecies, we {un’s eccentricity and parallax ; \et a=CS, x=CB, andy=CF; then BS be- sige aa Ce? CB. e's ; ing=\/aa—2ay+-*x, the forces x a6 1a in the direétion CB (vid. p. hx Dye welts © ye 3y css 160) will be= 7 XI—7 eS AE + ~ &c. and that (Ax ong 2 |? —h) in the dire&tion BH, equal tox I— = ao — -k=kX x J = — = 2 &c. which, becaufe yx xX cof. BCS, will be reduced to ce at —_—_— 2 ce = x1 += x cof, BCS &c. and aX = cof. BCS +041 scof. 2BCS&c. refpeCtively. “But the laft of thefe forces may, again, be refolved into two o- thers; the one in the direétion BC, exprefled by & cof. BCS x 3x xe kx Sx cof, BCS += x94 15 cof. 2BCS, (= 3 4¢ 1+ cof. 2BCS + Pa ern ge ene ee : xX 11 cof. BCS+-5cof. 3BCS); and the other, in a dire€tion perpendicular 3% ae siti es a en thereto, expreffed by éfin. BCS x i. x cof, BCS + qaa*9+ 15 cof. 2BCS is: in Mechanics and Phyfical Aftronomy. we fhall then have r—=1, and confequently I—ar=o; and fo, the whole (1—zz.Icof. rz) intitely vanifhing, we have nothing left to compare with the term propofed (g cof. z) where- by its effect can be known. Neverthelefs, it is by means of a term (Ticofl wz) of the fame {pecies, enterin g into the feries for the value of 1—w, that the effect in queftion muft be exhibit- ed: for, though indeed the term I cof. zz (or cof. z), by re- ceiving the co-efficient 1—2z (=o) intirely vanifhes, in the firft line of the general equation, yet, that will not be the cafe in the other parts of the equation; wherein it will be found, affected in the fame mannerwith the term Beof Gz (whereof the determination hath been given, in computing the motion of the apogee); which muft neceflarily appear to be the cafe, if it be confidered, that the terms TI cof. zz and B cof. Bz are alike concerned in the original feries 1— 1 cof.zz—B cof. z-—-C cof yz Sc. from whence the equation itfelf is derived. Hence we not only have a proper term to compare with the given one (g cof. z), but alfo an eafy way to difcover what the refult will Oe iinet tte eid ee RS ps AL hE kx HO emer or So <— x fin. 2BCS + e x fin. BCS + 5 fin. 3BCS)... From the former kx 3a é of which, let the force |G *%it ZX cof. BCS. in the’ oppofite direGion be MRE Pi cs is fubtracted, and the remainder, = x If 3cohaBCS 4 ee S 3 cof. BCS+-5 cof. 3BCS, will be the force whereby the gravity of the moon tothe earth, isdiminifhed. But the quantity 4 (which muft be exterminated ) is, at the mean diftance (d) of the fun and earth, found to. be equal to dx ' mer f id. 6 3 h . 2. qt. -d met a met — tl bbxCA (vid. p12 )3 whence, a” :d*:: XTPXCA* ga * FhxCA Srre general value of 4; which being fubftituted inftead thereof, the force in the di- rection of the radius vedtor will become Af d’ CB OE ie ited ae re XGA X 3+ 3c0f.2BCS Fos x 2 cof. BCS + . cof.3BCS ; and: that in the perpendicular direéion, : me*f. d¥ "CB CB cae ic = TO eS KR x3 fin. 2BCS + Cp * 4 fin. BCS +- a fin. 3 BCS: 1 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 fubftitution is. made in the general formula (on p, 157.) intirely vanifhes. Ava be s. ae ure — The Refolution of fome General Problems be; for feeing that M1 cof. rz and B cof. Bz have there the fame coefficients, and that the fum of thofe belonging to Bcof. Gz has been already found, in computing the motion of the apogee, to be ——1—f8, it follows, that —-1—@@.1 cof. rz (or 1 — 6.11 cof.z) will exprefs the value cf all the terms of this {pecies, arifing in the equation from the introduction of Iicof.z into the original, or affumed feries: whence, by compar- ing this quantity with the given one, gcof.z (that is, by making {Pe €Ol, <-- 2 Cone o)< we get TL = ; and confequently I cof. g== — xcof. z; whereby the I—Sp” 1—f tequired effect of the quantity g cof. z, 1s known. >is true, indeed, that the new term II cof. z, thus deter- mined, will introduce an infinity of others; but, of thefe, none will be confiderable enough to merit the leaft degree of atten- tion, except (perhaps) fuch as are exprefled by the co-fine of 1—(.2; which, becaufe of the fmallnefs of 1—, may, for reafons before fpecified, produce an effect not to be neglected without a proper examination. To give here the quantity of this effect, it will be neceflary to obferve, that, of all the terms in the values of 1—w\? and 1—w! *, zbat which arifes by the seks : IOll . multiplication of => x cof.z into 2D xCof. dz, and comes from the part ao xTicof. z-+-Beof. Bz+-C cof. yz--D cof. dz &c.\ will fo much exceed, in its effect, the others wherein I is con- cerned, as to render them,.in a manner, inconfiderable. But this roll rolID —— z wars: zx2Deof.dz (or —7—. cof. z x 2cofp—fP.z) Is 10lID a 1olID OL Sey =— X cof. p—B-+-1.z,and —[— xcolp—B—1.25 term -refolved into whereof the former part only, in which the difference of @ and 1 is concerned, is for our purpofe: from whence (as appears by the obfervations at ~. 156) a new term expreffed by eP 10olID cof.p—t1.z eP—- yoriD — ; af (or ap * ioe * cof. B—1.z, nearly) will B—1 “ 1—B—1]" arife in the required value of 1—w. Of the two terms here found, the former gives a very fmall equation depend- ing o in Mechanics and Phyfical Afironomy. 879 ing on the moon’s diftance from the fun’s apogee; and the latter, an equation fomething more confiderable, whofe ar- gument is the diftance of the moon’s apogee from that of the fun. By having fhewn above, that the effe& of fuch terms, or forces, as are proportional to the co-fine of the arch z, is expli- cable by means of the co-fines of that arch, and of its multiples, (no lefs 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, fince it appears thereby, that no terms enter into the equation of the orbit but what by a regular increafe and decreafe, do after a certain time return again to their former values, it is evident from thence, that the mean motion, and the greateft quantities of the feveral equations, undergo no change from gravity. TIPE ND, ERRATA. Page 5 line 18, for R read R’; 1. 21, for R’ r. Rs p. 30. 1.9, for ha-+-dh r- ba+b; 1. 13, fortr. hs p. 39.1, 14, put the comma before YO; p. 58. I. 11, for PSO r. PSC; p. 59. 1. 15, for ry 1. 225 for For pres, a a 1. 32. r. powers of; p. 127, 1. 9, for AGO r. AOG; 0.125.190, for bl, BR; p. 131.123, for BCr, mC; p. 138. 1.15, for ef r. Ee; p. 143, 1. 15 dele to; p. 152, 1. 2, for BR, r. yy; p. 160. 1. 27, for = r. = In fig. 18, at the interfeétion of the circumference of the circle and the right line Fm, place ane. SR AIO TILL LT aE 9ST SLE TO ET — mnted Nourse the following Books, all written by Mr. aa Tuomas Simpson, Ff. R. S. 4 SSAYS on SEVERAL CURIOUS AND USEFUL SUBJECTS, in fpeculative and mixed Mathematicks ; An which are explain- ed the moft difficult Problems of the firft and fecond Books of Sir Ifaac Newton’s Principia; being an ufeful Introdu@tion to Learners for the Underftanding that illuftrious Author , Quarto, 1740. Price 6s. few’d, Il. 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To which is added, The Geometrical Conftruction of a great Number of linear and plane Problems, with the Method of refolving the fame numerical] The Second Edition, with large Additions; OGavo, 1755. y- VI. Erements of Prane GeomMerry. To which are added, an Effay on the Maxima and Minima of Geometrical Quantities, and a brief Treatife of regular Selids; alfo the Menfuration of both Superficies and Solids, together with a Conftruction of a large Variety of Geometrical Problems ; defigned for the Ute of Schools. Odtavo, 1747. 7 VU. Taz Docrrine anp APPLICATION oF FLUxIoNs, containing (befides what is common on the Subjeét) a Number of new Im- provements in the Theory, and the Solution of a Variety of new and very interefting Problems in different Branches of the Mathema- ticks; 2 Vols. Odtavo, 1750. . Vill. Triconometry PLain anp SpuHeERIcat, with the Conftruc- tion and Application of Logarithms, O&avo, 1749. IX. 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