Of the Irregularities in the Planetary Motions, Caused by the Mutual Attraction of the Planets: In a Letter to Charles Morton, M. D. Secretary to the Royal Society, by Charles Walmesley, F. R. S. and Member of the Royal Academy of Sciences at Berlin, and of the Institute at Bologna

ventricle, and the extravasation covering the fissure in the aorta, exactly marked, as they appeared to, My Lord, Your Lordship's most obedient and most humble servant, Frank Nicholls. LII. Of the Irregularities in the planetary Motions, caused by the mutual Attraction of the Planets: In a Letter to Charles Morton, M. D. Secretary to the Royal Society, by Charles Walmesley, F. R. S. and Member of the Royal Academy of Sciences at Berlin, and of the Institute at Bologna. S I R, Read Dec. 10, 1761. Finding that the influence, which the primary planets have upon one another, to disturb mutually their motions, had been but little considered, I thought it a subject worthy of examination. The force of the sun, to disturb the moon's motion, flows from the general principle of gravitation, and has been fully ascertained, both by theory and observation; and it follows, from the same principle, that all the planets must act upon one another, proportionally to the quantities of matter contained in their bulk, and inverse ratio of the squares of their mutual distances; but as the quantity of matter contained in each of them, is but small when compared to that of the sun, so their action upon one another, is not so sensible as that of the sun upon the moon. Astronomers generally contented themselves with solely considering those inequalities of the planetary motions, that arise from the elliptical figure of their orbits; but as they have been enabled, of late years, by the perfection of their instruments, to make observations with much more accuracy than before, they have discovered other variations, which they have not, indeed, been able yet to settle, but which seem to be owing to no other cause, but the mutual attraction of those celestial bodies. In order, therefore, to assist the astronomers in distinguishing and fixing these variations, I shall endeavour to calculate their quantity, from the general law of gravitation, and reduce the result into tables, that may be consulted, whenever observations are made. I offer to you, at present, the first part of such a theory, in which I have chiefly considered the effects produced by the actions of the earth and Venus upon each other. But the same propositions will likewise give, by proper substitutions, the effects of the other planets upon these two, or of these two upon the others. To obviate, in part, the difficulty of such intricate calculations, I have supposed the orbits of the earth and Venus to be originally circular, and to suffer no other alteration, but what is occasioned by their mutual attraction, and the attraction of the other planets. planets. Where the forces of two planets are considerable, with respect to each other, as in the case of Jupiter and Saturn, it may be necessary, in such computations, to have regard to the eccentricity of their orbits; and this may be reserved for a subject of future scrutiny. But the supposing the orbits of the earth and Venus to be circular, may, in the present case, be admitted, without difficulty, as the forces of these two planets are so small, and the eccentricity of their orbits not considerable. On these grounds, therefore, I have computed the variations, which are the effects of the earth's action: first, the variation of Venus's distance from the sun; secondly, that of its place in the ecliptic; thirdly, the retrograde motion of Venus's nodes; and, fourthly, the variation of inclination of its orbit to the plane of the ecliptic. The similar irregularities in the motion of the earth, occasioned by its gravitation to Venus, are here likewise computed: but it is to be observed, that the absolute quantity of these irregularities is not here given, it being impossible, at present, to do it; because the absolute force of Venus is not known to us. I have, therefore, stated that planet's force by supposition, and have, accordingly, computed the effects it must produce; with the view, that the astronomers may compare their observations with the motions so calculated, and, from thence, discover how much the real force differs from that which has been supposed. But the exact determination of the force of Venus must be obtained, by observations made on the sun's place, at such times, when the effect of the other planets is either null or known. The influence of Venus upon the earth being thus computed, that of the other planets upon the same, may likewise, hereafter, be considered: by which means, the different equations, that are to enter into the settling of the sun's apparent place, will be determined; the change of the position of the plane of the earth's orbit will also be known; and, consequently, the alteration that thence arises in the obliquity of the ecliptic, and in the longitude and latitude of the fixed stars. These matters of speculation are reserved for another occasion, in case what is here offered should deserve approbation. I am glad to have it in my power to present you with this testimony of my gratitude for past favours, and of my respect for your distinguished merit; and it is with sincerity, I subscribe myself, SIR, Your very humble servant, Bath, Nov. 21, 1761. Cha. Walmesley. De Inequalitatibus quas in motibus Planetarum generant ipsorum in se invicem actiones. Quoniam in theoriae hujus decursu frequens erit usus fluentium quae arcubus circuli, vel eorum finibus, cosinibus, et finibus versis, exprimuntur, idcirco lemma sequens, quod alibi olim tradidi, lubet hic apponere. LEMMA. Lemma. Dato cosinu arcûs cujuvis, invenire cosinum et sinum arcûs alterius qui sit ad priorem in ratione \( \lambda \) ad 1. Detur \( c \) cosinus arcûs A ad radium 1, et sit arcus B = \( \lambda \) A, cujus cosinus dicatur \( t \); eritque, ut notum est, \( \dot{A} = \frac{-c}{\sqrt{1 - cc}} \), atque \( \dot{B} = \lambda \dot{A} = \frac{-i}{\sqrt{1 - tt}} \). Ponatur \( c = \frac{1 + xx}{2x} \), et \( t = \frac{1 + yy}{2y} \), fietque \( \dot{A} = \frac{\dot{x}}{x \sqrt{-1}} \), \( B = \frac{\dot{y}}{y \sqrt{-1}} \): sed est \( \dot{A} \cdot \dot{B} :: 1 \cdot \lambda \), adeoque \( \frac{\dot{x}}{x} = \frac{\dot{y}}{y} \); unde log. \( x^2 = \log. y \), et \( x^3 = y \). Verùm æquationes \( c = \frac{1 + xx}{2x} \) et \( t = \frac{1 + yy}{2y} \) dant \( x = c + \sqrt{cc - 1} \), \( x = c - \sqrt{cc - 1} \), et \( y = t + \sqrt{tt - 1} \), \( y = t - \sqrt{tt - 1} \); unde est \( x^2 = t + \sqrt{tt - 1} = c + \sqrt{cc - 1} \), atque inde \( 2t = c + \sqrt{cc - 1} + c - \sqrt{cc - 1} \). Fiat igitur \( c + \sqrt{cc - 1} = l \), et \( c - \sqrt{cc - 1} = m \), eritque \( lm = 1 \), et \( c = \cos. A = \frac{l + m}{2} \), et sin. \( A = \frac{l - m}{2} \sqrt{-1} \); atque inde \( t = \cos. B = \frac{l + m}{2} \), et sin. \( B = \frac{l - m}{2} \sqrt{-1} \). Itaque in circulo, cujus radius est 1, si duorum arcuum vel angulorum A et B alteruter B sit ad alterum A ut numerus quilibet \( \lambda \) ad 1, et ponatur \( \cos. A = \frac{l + m}{2} \), existente \( lm = 1 \), erit sin. \( A = \) \[ \frac{l-m}{2} \sqrt{-1}, \text{ atque } \cos B = \cos \lambda A = \frac{l^2 + m^2}{2}, \] et \[ \sin B = \sin \lambda A = \frac{l^2 - m^2}{2} \sqrt{-1}. \quad Q.E.I. \] **Coroll. I.** Hinc habetur \[ \cos A \times \cos B = \frac{l^2 + m^2}{2} \times \frac{l^2 + m^2}{2} = \frac{l^{2+1} + m^{2+1}}{4} + \frac{l^{2-1} + m^{2-1}}{4}; \] sed, quemadmodum per hoc lemma est \[ \frac{l^2 + m^2}{2} = \cos \lambda A, \] erit \[ \frac{l^{2+1} + m^{2+1}}{2} = \cos \lambda + 1 \times A = \cos A + B, \] atque \[ \frac{l^{2-1} + m^{2-1}}{2} = \cos \lambda - 1 \times A = \cos B - A, \] adeoque \[ \cos A \times \cos B = \frac{1}{2} \cos A + B + \frac{1}{2} \cos B - A. \] Atque hoc calculi methodo facile eruuntur sequentes formulæ pro duobus angulis \(A\) et \(B\), adverterendo esse \[ \cos B - A = \cos A - B, \] \[ \sin B - A = -\sin A - B, \] et \[ \cos o = 1. \] 1°. \[ \cos A \times \cos B = \frac{1}{2} \cos A + B + \frac{1}{2} \cos A - B. \] 2°. \[ \sin A \times \sin B = -\frac{1}{2} \cos A + B + \frac{1}{2} \cos A - B. \] 3°. \[ \sin A \times \cos B = \frac{1}{2} \sin A + B + \frac{1}{2} \sin A - B. \] Atque ex illis hæ sequentes eliciuntur, 4°. \[ \cos A + B = \cos A \times \cos B - \sin A \times \sin B. \] 5°. \[ \cos A - B = \sin A \times \sin B + \cos A \times \cos B. \] 6°. \[ \sin A + B = \sin A \times \cos B + \cos A \times \sin B. \] 7°. \[ \sin A - B = \sin A \times \cos B - \cos A \times \sin B. \] Tùm ex his valores tangentium haud ægrè derivantur, Quippe Quippe cum sit generatim pro quovis angulo A, \[ \tan(A) = \frac{\sin(A)}{\cos(A)} \] erit \( \tan(A + B) = \frac{\sin(A + B)}{\cos(A + B)} = \frac{\sin(A)\cos(B) + \cos(A)\sin(B)}{\cos(A)\cos(B) - \sin(A)\sin(B)} \) \[ \times \frac{1}{\tan(A)} = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}. \] Simili calculo prodit \( \tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}. \) Unde statui possunt, 1°. \( \tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}. \) 2°. \( \tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}. \) 3°. \( \tan(A)\tan(B) = \frac{\tan(A) + \tan(B) - \tan(A)\tan(B)}{\tan(A) + \tan(B)}, \) vel \( \tan(A)\tan(B) = \frac{\tan(A) - \tan(B) - \tan(A)\tan(B)}{\tan(A) - \tan(B)}. \) Coroll. II. Erat in lemmate \( \dot{A} = \frac{x}{x^{\sqrt{-1}}}, \) unde est \( A = \sqrt{-1} = \log(x). \) Denotet igitur E numerum cujus logarithmus hyperbolicus est 1, eritque \( E^{\sqrt{-1}} = x, \) et cum fit \( x = c + \sqrt{cc - 1}, \) inde obtinetur \( c = \cos(A) = \frac{E^{\sqrt{-1}} + E^{-\sqrt{-1}}}{2}, \) atque \( \sin(A) = \frac{E^{\sqrt{-1}} - E^{-\sqrt{-1}}}{2}. \) Vol. LII. Sunt qui his sinuum et cosinuum valoribus potius utuntur; verum ii valores, quos exhibet corollarium praecedens, simpliciores sunt et calculo plerumque aptiores. Coroll. III. Quoniam est $2 \times \cos A = l + m$, erit $$2^\lambda \times \cos A^\lambda = \begin{cases} l^\lambda + \lambda l^{\lambda-1}m + \lambda \times \frac{\lambda - 1}{2} l^{\lambda-2}m^2 + \lambda \\ \times \frac{\lambda - 1}{2} \times \frac{\lambda - 2}{3} l^{\lambda-3}m^3 + \text{etc.} \\ m^\lambda + \lambda m^{\lambda-1}l + \lambda \times \frac{\lambda - 1}{2} m^{\lambda-2}l^2 + \lambda \\ \times \frac{\lambda - 1}{2} \times \frac{\lambda - 2}{3} m^{\lambda-3}l^3 + \text{etc.} \end{cases}$$ assumendo scilicet primos et ultimos terminos homologos seriei exprimentis quantitatem $\frac{l+m}{2}$: unde, propter $lm = 1$, provenit $$2^{\lambda-1} \times \cos A^\lambda = \frac{l^\lambda + m^\lambda}{2} + \lambda \times \frac{l^{\lambda-2} + m^{\lambda-2}}{2} + \lambda \times \frac{\lambda - 1}{2}$$ $$\times \frac{l^{\lambda-4} + m^{\lambda-4}}{2} + \lambda \times \frac{\lambda - 1}{2} \times \frac{\lambda - 2}{3} \times \frac{l^{\lambda-6} + m^{\lambda-6}}{2} + \text{etc.}$$ atque adeò per lemma $$\cos A^\lambda = \frac{1}{2^{\lambda-1}} \text{ in } \cos \lambda A + \lambda \cos \lambda - 2 \times A + \lambda$$ $$\times \frac{\lambda - 1}{2} \cos \lambda - 4 \times A + \lambda \times \frac{\lambda - 1}{2} \times \frac{\lambda - 2}{3} \cos \lambda - 6$$ $$\times A + \text{etc.}$$ Ubi $\lambda$ est numerus impar, terminus ultimus seriei erit ille in quo numerus $\lambda$, vel $\lambda - 2$, vel $\lambda - 4$, etc. qui multiplicat angulum $A$, evadit æqualis 1. Ubi vero $\lambda$ est numerus par, terminus ultimus seriei erit ille in quo numerus praedictus evadit æqualis 0, quo in casu semissis tantum ultimi termini sumenda est; cum enim series haec colligatur ex numero pari terminorum homologorum, quae tamen, ubi \( \lambda \) est numerus par, constare debet ex terminorum numero impari, ideò duplum exhibet terminum ultimum. Simili modo cum sit \( 2 \times \sin A = \sqrt{-1} \times \left\{ \begin{array}{c} \frac{\lambda - 1}{2} \times \frac{\lambda - 2}{3} \times \cdots \\ \pm m^{\lambda - 1} + \lambda \times \frac{\lambda - 1}{2} \times \cdots \\ \pm \lambda \times \frac{\lambda - 1}{2} \times \frac{\lambda - 2}{3} \times \cdots \end{array} \right. \) Terminis inferioribus hujus seriei praesignuntur alternativim signa \( \pm \) ubi \( \lambda \) est numerus par, et signa \( \mp \) ubi \( \lambda \) est numerus impar, adeoque in priore casu est \( 2^{\lambda - 1} \times \sin A = \sqrt{-1} \times \left\{ \begin{array}{c} \frac{\lambda + m^{\lambda}}{2} - \lambda \times \frac{\lambda - 2 + m^{\lambda - 2}}{2} + \lambda \\ \pm \frac{\lambda - 1}{2} \times \frac{\lambda - 4 + m^{\lambda - 4}}{2} - \lambda \times \frac{\lambda - 1}{2} \times \frac{\lambda - 2}{3} \times \frac{\lambda - 6 + m^{\lambda - 6}}{2} + \cdots \end{array} \right. \) et in casu posteriori \( 2^{\lambda - 1} \times \sin A = \sqrt{-1} \times \left\{ \begin{array}{c} \frac{\lambda - m^{\lambda}}{2} - \lambda \times \frac{\lambda - 2 - m^{\lambda - 2}}{2} + \lambda \\ \pm \frac{\lambda - 1}{2} \times \frac{\lambda - 4 - m^{\lambda - 4}}{2} - \lambda \times \frac{\lambda - 1}{2} \times \frac{\lambda - 2}{3} \times \frac{\lambda - 6 - m^{\lambda - 6}}{2} + \cdots \end{array} \right. \) Adeoque si \( \lambda \) fit numerus par, erit \( \sin A = \frac{1}{2^{\lambda - 1}} \times \cos \lambda A + \lambda \cos \lambda - 2 \times A + \lambda \) \( \times \frac{\lambda - 1}{2} \cos \lambda - 4 \times A + \lambda \times \frac{\lambda - 1}{2} \times \frac{\lambda - 2}{3} \cos \lambda - 6 \) \( \times A + \cdots \) Signa Signa hæc alternatim mutantur, et superiora sunt adhibenda, ubi \( \lambda \) exprimit unum ex numeris 4, 8, 12, 16, &c. quia tunc est \( \sqrt{-1}^{\lambda} = i \); inferiora autem adhibenda, ubi \( \lambda \) exprimit unum ex numeris 2, 6, 10, 14, &c. quia tunc est \( \sqrt{-1}^{\lambda} = -i \). Si \( \lambda \) sit numerus impar, cum per lemma sit \[ \frac{l^{\lambda} - m^{\lambda}}{2} \sqrt{-1} = \sin. \lambda A, \quad \text{et} \quad \frac{l^{\lambda-2} - m^{\lambda-2}}{2} \sqrt{-1} = \sin. \lambda - 2 \times A, \&c. \] habetur \[ \sin. A^{\lambda} = \frac{1}{2^{\lambda-1}} \sin. \lambda A + \lambda \times \sin. \lambda - 2 \times A + \lambda \times \frac{\lambda - 1}{2} \sin. \lambda - 4 \times A + \lambda \times \frac{\lambda - 1}{2} \times \frac{\lambda - 2}{3} \sin. \lambda - 6 \times A + \&c. \] ubi signa superiora sunt usurpanda, cum \( \lambda \) exprimit unum ex numeris 1, 5, 9, 13, &c. quia tunc est \( \sqrt{-1}^{\lambda} = \sqrt{-1} \); et signa inferiora, cum \( \lambda \) fuerit unus ex numeris 3, 7, 11, 15, &c. quia tunc est \( \sqrt{-1}^{\lambda} = \sqrt{-1} \). Notandum autem, seriei ultimum terminum esse illum in quo numerus \( \lambda \), vel \( \lambda - 2 \), vel \( \lambda - 4 \), &c. est æqualis 1 ubi \( \lambda \) est numerus impar; atque terminum ultimum esse illum in quo praedictus numerus est æqualis 0 ubi \( \lambda \) est numerus par, quo in casu semifiss tantumulti termini assumenda est ob rationem superiùs datam. Ex his sinuum et cosinuum expressionibus alia hujusmodi theoremata deducere liceret, sed quæ hæ traduntur ad praesens institutum sufficiunt. Coroll. Coroll. IV. Notum est fluentem fluxionis \( \dot{A} \cos A \) esse fin. \( A \), atque fluentem fluxionis \( \dot{A} \sin A \) esse fin. vers. \( A \). Pariter si sumatur arcus \( \lambda A \) qui sit ad arcum \( A \) ut numerus quilibet \( \lambda \) ad 1, cum sit \( \lambda \dot{A} \cos \lambda A \) æqualis fluxioni finûs arcûs \( \lambda A \), erit flu. \( \dot{A} \cos \lambda A = \frac{\text{fin. } \lambda A}{\lambda} \), et flu. \( \dot{A} \sin \lambda A = \frac{\text{fin. vers. } \lambda A}{\lambda} \). Itemque, si ad arcum \( \lambda A \) adjungatur arcus datus \( d \), cum fluxio arcûs \( \lambda A + d \) sit æqualis \( \dot{A} \), erit flu. \( \dot{A} \cos \lambda A + d = \frac{\text{fin. } \lambda A + d}{\lambda} \), et flu. \( \dot{A} \sin \lambda A + d = \frac{\text{fin. vers. } \lambda A + d}{\lambda} \). Sumantur jam duo anguli, vel duo arcus \( \lambda A \) et \( \mu A \), qui sint ad angulum, vel arcum \( A \) respectivè, ut \( \lambda \) et \( \mu \) ad 1, atque per Coroll. II. habetur \( \cos \lambda A \cos \mu A = \frac{1}{2} \cos \lambda + \mu \times A + \frac{1}{2} \cos \lambda - \mu \times A \); unde erit fluens fluxionis \( \dot{A} \cos \lambda A \times \cos \mu A \) æqualis \( \frac{\text{fin. } \lambda + \mu \times A}{2 \times \lambda + \mu} + \frac{\text{fin. } \lambda - \mu \times A}{2 \times \lambda - \mu} \). Atque hoc methodo prodeunt sequentes formulæ 1°. Flu. \( \dot{A} \cos \lambda A \times \cos \mu A = \frac{\text{fin. } \lambda + \mu \times A}{2 \times \lambda + \mu} + \frac{\text{fin. } \lambda - \mu \times A}{2 \times \lambda - \mu} \). 2°. Flu. \( \dot{A} \sin \lambda A \times \sin \mu A = - \frac{\text{fin. } \lambda + \mu \times A}{2 \times \lambda + \mu} + \frac{\text{fin. } \lambda - \mu \times A}{2 \times \lambda - \mu} \). 3°. Flu. 3°. Flu. \( \dot{A} \sin \lambda A \times \cos \mu A = \frac{\sin \text{verf. } \lambda + \mu \times A}{2 \times \lambda + \mu} \) \[ + \frac{\sin \text{verf. } \lambda - \mu \times A}{2 \times \lambda - \mu}. \] Advertendum autem est, ubi \( \lambda = \mu \), tunc esse \[ \cos \lambda A \times \cos \mu A = \frac{1}{2} \cos 2\lambda A + \frac{1}{2}, \quad \sin \lambda A \times \sin \mu A = \frac{1}{2} \sin 2\lambda A; \quad \text{adeoque in hoc casu formulæ praecedentes evadunt} \] 1°. Flu. \( \dot{A} \times \cos \lambda A \) \(^2 = \frac{\sin 2\lambda A}{4\lambda} + \frac{A}{2} \). 2°. Flu. \( \dot{A} \times \sin \lambda A \) \(^2 = -\frac{\sin 2\lambda A}{4\lambda} + \frac{A}{2} \). 3°. Flu. \( \dot{A} \times \sin \lambda A \times \cos \lambda A = \frac{\sin \text{verf. } 2\lambda A}{4\lambda} \). Si angulo \( \lambda A \) addatur angulus datus \( d \), erit \( \cos \lambda A + d \times \cos \mu A = \frac{1}{2} \cos (\lambda + \mu \times A + d + \frac{1}{2} \cos \lambda - \mu \times A + d \), atque inde 1°. Flu. \( \dot{A} \cos \lambda A + d \times \cos \mu A = \frac{\sin \lambda + \mu \times A + d}{2 \times \lambda + \mu} \) \[ + \frac{\sin \lambda - \mu \times A + d}{2 \times \lambda - \mu}. \] 2°. Flu. \( \dot{A} \sin \lambda A + d \times \sin \mu A = -\frac{\sin \lambda + \mu \times A + d}{2 \times \lambda + \mu} \) \[ + \frac{\sin \lambda - \mu \times A + d}{2 \times \lambda - \mu}. \] 3°. Flu. \( \dot{A} \sin \lambda A + d \times \cos \mu A = \frac{\sin \text{verf. } \lambda + \mu \times A + d}{2 \times \lambda + \mu} \) \[ + \frac{\sin \text{verf. } \lambda - \mu \times A + d}{2 \times \lambda - \mu}. \] 4°. Flu. 4°. Flu. \( \dot{A} \cos \lambda A + d \times \sin \mu A = \frac{\sin \text{vers} \lambda + \mu \times A + d}{2 \times \lambda + \mu} \) \[- \frac{\sin \text{vers} \lambda - \mu \times A + d}{2 \times \lambda - \mu}.\] Si fuerit \( \lambda = \mu \), erit \( \cos \lambda A + d \times \cos \lambda A = \frac{1}{4} \cos 2 \lambda A + d + \frac{1}{2} \cos d, \&c. \) adeoque formulae præcedentes in has abeunt, 1°. Flu. \( \dot{A} \cos \lambda A + d \times \cos \lambda A = \frac{\sin 2 \lambda A + d}{4 \lambda} \) \[+ \frac{\cos d}{2} A.\] 2°. Flu. \( \dot{A} \sin \lambda A + d \times \sin \lambda A = \frac{\sin 2 \lambda A + d}{4 \lambda} \) \[+ \frac{\cos d}{2} A.\] 3°. Flu. \( \dot{A} \sin \lambda A + d \times \cos \lambda A = \frac{\sin \text{vers} 2 \lambda A + d}{4 \lambda} \) \[+ \frac{\sin d}{2} A.\] 4°. Flu. \( \dot{A} \cos \lambda A + d \times \sin \lambda A = \frac{\sin \text{vers} 2 \lambda A + d}{4 \lambda} \) \[- \frac{\sin d}{2} A.\] **Propositio I. Problema.** In systemate duorum planetarum circa Solem in orbibus penè circularibus revolventium, requiratur vis planetæ exterioris ad perturbandum motum interioris. Revolvantur planetæ duo \( P \) et \( Q \) (Fig. 1.) in eodem plano circa Solem in \( S \), et jungantur \( SP, SQ, PQ \). Orbis Orbis planetæ interioris Q, cu- jus motus hîc investigamus, circu- laris supponitur nisi quatenûs mu- tatur ejus figura vi planetæ P; or- bem verò planetæ P ut accuratè circularem habemus. Poétâ ergò unitate pro distantiâ corporis Q à Sole ubi ambo planetæ versantur in conjunctione cum Sole, fiant $SQ = x$, $PQ = z$, $SP = k$; tum- que vis attractionis Solis in distan- tiâ æquali i sit ad vim attractionis planetæ P in eâdem distantiâ ut i ad $\phi$, eritque $\frac{\phi}{z^2}$ gravitas planetæ Q in planetam P. Producatur jam, si opus est, $PQ$ ad O ut fit $PO = \frac{\phi}{z^3}$, et ducâtâ $OI$ parallelâ ipsi $QS$ occurrente rectæ $PS$ productæ in I, propter triangula similia $PQS$, $POI$, erit $PQ \cdot PS :: PO \cdot PI$, hoc est, $PI = \frac{\phi k}{z^3}$, atque $PQ \cdot QS :: PO \cdot OI$, hoc est, $OI = \frac{\phi x}{z^3}$. Sed, quia parûm differt $x$ ab unitate et admodûm exigua est vis $\phi$, pro $x$ scribi potest i in omnibus iis terminis qui ducuntur in $\phi$, adeoque $OI = \frac{\phi}{z^3}$. Ex vi $PI$ aufe- ratur vis $\frac{\phi}{k^2}$ qua gravitat Sol in planetam P, et vis re- sidua $\frac{\phi k}{z^3} - \frac{\phi}{k^2}$ est ea qua perturbatur motus planetæ Q in directione parallelâ rectæ $PS$: nàm cùm motus planetarum planetarum referantur ad Solem spectatum tanquam immotum, vis $\frac{\phi k}{z^3}$ pars ea $\frac{\phi}{k^2}$, qua simul urgentur Sol et planeta Q versus P secundum lineas parallelas, non mutat corporum S et Q situm ad se invicem, idoque differentia virium sola perturbationem inducit. Quare differentia illa, nimirùm $\frac{\phi k}{z^3} - \frac{\phi}{k^2}$, exponatur per lineam QT parallelam rectae PS, et in SQ demisso perpendiculo TR, vis QT resolvetur in vires TR, QR, eritque vis QT ad vim TR ut radius r ad finum anguli QSP, adeoque vis TR = $\frac{\phi k}{z^3} - \frac{\phi}{k^2}$ × sin. QSP, et vis QR = $\frac{\phi k}{z^3} - \frac{\phi}{k^2}$ × cos. QSP. Ex vi autem QR tollatur vis OI utpotè in contrarium agens, et manebit vis $\frac{\phi k}{z^3} - \frac{\phi}{k^2}$ × cos. QSP = $\frac{\phi}{z^3}$. Vires igitur, quibus planeta P perturbat motum planetæ Q quatenus in eodem plano moventur, sunt 1°. Vis TR ad radium QS perpendicularis, qua augetur vel minuitur area tempore dato descripta, estque æqualis $\frac{\phi k}{z^3} - \frac{\phi}{k^2}$ × fin. QSP. 2°. Vis $\frac{\phi}{z^3}$ × k cos. QSP = 1 − $\frac{\phi}{k^2}$ cos. QSP, qua retrahitur planeta Q à Sole in directione radii SQ. Ut autem harumce virium expressiones formam induant calculo accommodam, ope trianguli PSQ habebitur $PQ^2 = zz = kk + xx - 2kx \times \text{cos. QSP}$, sive, positâ $x = 1$ ob rationem dictam, $zz = kk$ Vol. LII. Pp \[ \frac{1}{2} k \times \cos QSP. \] Assumatur jama angulus \( s \) qui semper fit ad angulum QSP in ratione \( n \) ad 1, eritque QSP = \( \frac{1}{n} s \), et positio \( kk + 1 = tt \), et \( \frac{2k}{t} = b \), erit \( x^2 = t^2 \times 1 - b \cos \frac{1}{n} s \), hincque \( \frac{1}{x^3} = \frac{1}{t^3} \times 1 - b \cos \frac{1}{n} s \). Si \( b \) fuerit unitati ferè æqualis, et evolvatur quantitas \( 1 - b \cos \frac{1}{n} s \) in seriem modo solito, series illa parùm convergit, estque ad operationes analyticas minus commoda. Series igitur alia investiganda est, et quia ex lemmate patet hujusmodi quantitatem \( \cos A \) exprimi posse aggregato terminorum, quorum singuli ducuntur in cosinus angularum qui sunt anguli A multiplices, generatim supponemus \( 1 - b \cos \frac{1}{n} s \) \( = R + S \cos \frac{1}{n} s + T \cos \frac{2}{n} s + V \cos \frac{3}{n} s + W \cos \frac{4}{n} s + \ldots \). Atque ut inveniantur valores coefficientium \( R, S, T, \ldots \) sumatur utrinque fluxio, nempe \( \frac{mb}{n} s \times \sin \frac{1}{n} s \times 1 - b \cos \frac{1}{n} s \) \( = -S \times \frac{1}{n} s \times \sin \frac{1}{n} s - T \times \frac{2}{n} s \times \sin \frac{2}{n} s - V \times \frac{3}{n} s \times \sin \frac{3}{n} s - W \times \frac{4}{n} s \times \sin \frac{4}{n} s - \ldots \), &c. atque ducatur æquatio hæc in \( 1 - b \cos \frac{1}{n} s \), et substituto pro \( 1 - b \cos \frac{1}{n} s \) ipfius valore \( R + S \cos \frac{1}{n} s + T \cos \frac{2}{n} s + \ldots \), &c., fiet \( mb \times \sin \frac{1}{n} s \times \ldots \). \[ R + S \cos \frac{1}{n} s + T \cos \frac{2}{n} s + V \cos \frac{3}{n} s + \ldots = 1 - b \cos \frac{1}{n} s \] \[ - S \times \sin \frac{1}{n} s + 2 T \times \sin \frac{2}{n} s \times 3 V \times \sin \frac{3}{n} s + W \times \sin \frac{4}{n} s + \ldots \] et factâ multiplicatione, cum sit (per Coroll. I. Lem.) \[ \sin \frac{1}{n} s \times \cos \frac{r}{n} s = \frac{1}{2} \sin \frac{r+1}{n} s - \frac{1}{2} \sin \frac{r-1}{n} s, \text{ ac} \] \[ \sin \frac{r}{n} s \times \cos \frac{1}{n} s = \frac{1}{2} \sin \frac{r+1}{n} s + \frac{1}{2} \sin \frac{r-1}{n} s, \text{ emergit} \] \[ + 2mbR \quad + 2S \quad - 2bT \quad - mbT \] \[ \times \sin \frac{1}{n} s \quad \left\{ \begin{array}{c} +mbS \\ -bS \\ +4T \\ -3bV \\ -mbV \end{array} \right\} \times \sin \frac{2}{n} s \quad \left\{ \begin{array}{c} +mbT \\ -2bT \\ +6V \\ -4bW \\ -mbW \end{array} \right\} \times \sin \frac{3}{n} s, \&c. = 0 \] Deinde nihilo æquando singulos terminos, prodeunt \[ T = \frac{2S + 2mbR}{m + 2 \times b}, \quad V = \frac{4T + m - 1 \times bS}{m + 3 \times b}, \quad W = \frac{6V + m - 2 \times bT}{m + 4 \times b}, \&c. \text{ quorum valorum progressus satis manifestus est.} \] Datis igitur primis duobus coefficientibus \( R \) et \( S \), dabuntur et reliqui: \( R \) et \( S \) autem sic inveniuntur. \[ \text{Est } 1 - b \cos \frac{1}{n} s = 1 - mb \cos \frac{1}{n} s + m \] \[ \times \frac{m-1}{2} b^2 \cos \frac{1}{n} s^2 - m \times \frac{m-1}{2} \times \frac{m-2}{3} b^3 \cos \frac{1}{n} s^3, \&c. = R + S \cos \frac{1}{n} s + T \cos \frac{2}{n} s + V \cos \frac{3}{n} s + \&c. \] \[ \text{Evolvantur termini } \cos \frac{1}{n} s^2, \cos \frac{1}{n} s^4, \cos \frac{1}{n} s^6, \&c. \text{ per methodum traditam in Coroll. III. Lem. ac, collectis simul omnibus terminis qui nullo cosinu afficiuntur, prodibit} \] \[ Pp_2 \quad R = \] \[ R = 1 + \frac{m}{2} \times \frac{m-1}{2} b^2 + \frac{m}{2} \times \frac{m-1}{2} \times \frac{m-2}{4} \times \frac{m-3}{4} b^4 + \ldots \] \[ + \frac{m}{2} \times \frac{m-1}{2} \times \frac{m-2}{4} \times \frac{m-3}{4} \times \frac{m-4}{6} \times \frac{m-5}{6} b^6 + \ldots \] cujus seriei progressio satis patet; atque adeò, cum sit in hoc nostro problemate \( m = -\frac{3}{2} \), erit \[ R = 1 + \frac{3 \times 5}{4 \times 4} b^2 + \frac{3 \times 5}{4 \times 4} \times \frac{7 \times 9}{8 \times 8} b^4 + \frac{3 \times 5}{4 \times 4} \times \frac{7 \times 9}{8 \times 8} \times \frac{11 \times 13}{12 \times 12} b^6 + \frac{3 \times 5}{4 \times 4} \times \frac{7 \times 9}{8 \times 8} \times \frac{11 \times 13}{12 \times 12} \times \frac{15 \times 17}{16 \times 16} b^8 + \ldots \] Inspicienti indolem hujus seriei patebit terminum quemlibet æquari termino antecedenti ducto in \[ r + i \times r - 1 b^2, \text{ sive } \frac{r^2 - 1}{r^2} b^2, \text{ } r \text{ existente æquali numero quadruplicato terminorum præcedentium: sic terminus sextus, quia habetur in hoc casu } r = 5 \times 4 = 20, \text{ æqualis est termino quinto } \frac{3 \times 5}{4 \times 4} \times \frac{15 \times 17}{16 \times 16} b^3 \] ducto in \( \frac{19 \times 21}{20 \times 20} b^2 \). Termino igitur quovis hujus seriei dicto B, terminus subsequens erit \( Bb^2 \times \frac{r^2 - 1}{r^2} : \) et manente deinceps eodem, quem in hoc termino habet, numeri \( r \) valore, termini subsequentes erunt, \( Bb^4 \times \frac{r^2 - 1}{r^2} \times \frac{(r+4)^2 - 1}{(r+4)^2}, Bb^6 \times \frac{r^2 - 1}{r^2} \times \frac{(r+4)^2 - 1}{(r+4)^2} \times \frac{(r+8)^2 - 1}{(r+8)^2}, \) \( Bb^8 \times \frac{r^2 - 1}{r^2} \times \frac{(r+12)^2 - 1}{(r+12)^2}, \ldots \), &c. Sed est \( \frac{r^2 - 1}{r^2} = 1 - \frac{1}{r^2}, \frac{(r+4)^2 - 1}{(r+4)^2} = 1 - \frac{1}{(r+4)^2}, \ldots \), &c., et si fuerit \( r \) numerus numerus aliquantum magnus, erit \( \frac{r^2 - 1}{r^2} \times \frac{(r+4)^2 - 1}{(r+4)^2} \) \( = 1 - \frac{1}{r^2} - \frac{1}{r+4} \), et \( \frac{r^2 - 1}{r^2} \times \frac{(r+4)^2 - 1}{(r+4)^2} \times \frac{(r+7)^2 - 1}{(r+8)^2} \) \( = 1 - \frac{1}{r^2} - \frac{1}{r+4} - \frac{1}{r+8} \), atque ita porrò, rejiciendo fractiones hujus generis \( \frac{1}{r^2 \times r+4} \) et alias his minores. Unde termini omnes praedicti, incipiendo à termino B, erunt \[ B + Bb^2 + Bb^4 + Bb^6 + Bb^8 + \ldots \] \[ = B \times \frac{1}{1-b^2} \] \[ = \frac{Bb^2}{r^2} - \frac{Bb^4}{r^2} - \frac{Bb^6}{r^2} - \frac{Bb^8}{r^2}, \ldots \] \[ = \frac{B}{r^2} \times \frac{b^2}{1-b^2} \] \[ = \frac{Bb^4}{r+4^2} - \frac{Bb^6}{r+4^2} - \frac{Bb^8}{r+4^2}, \ldots \] \[ = \frac{B}{r+4^2} \times \frac{b^4}{1-b^2} \] \[ = \frac{Bb^6}{r+8^2} - \frac{Bb^8}{r+8^2}, \ldots \] \[ = \frac{B}{r+8^2} \times \frac{b^6}{1-b^2} \] \[ = \frac{Bb^8}{r+12^2}, \ldots \] \[ = \frac{B}{r+12^2} \times \frac{b^8}{1-b^2} \] \[ \ldots \] ac proinde tandem fit \[ R = 1 + \frac{3 \times 5}{4 \times 4} b^2 + \frac{3 \times 5}{4 \times 4} \times \frac{7 \times 9}{8 \times 8} b^4 + \frac{3 \times 5}{4 \times 4} \times \frac{7 \times 9}{8 \times 8} \] \[ \times \frac{11 \times 13}{12 \times 12} b^6 \times \frac{3 \times 5}{4 \times 4} \ldots \frac{15 \times 17}{16 \times 16} b^8 + \ldots \] \[ = \frac{B}{1-b^2} \] \[ \times 1 - \frac{b^2}{r^2} - \frac{b^4}{r+4^2} - \frac{b^6}{r+8^2} - \frac{b^8}{r+12^2} - \frac{b^{10}}{r+16^2}, \ldots \] Unde si, computatis, exempli gratia, decem terminis, undecimus designetur per B, erit \( r = 10 \times 4 = 40 \), et summa illorum decem terminorum addita summæ summæ seriei \( \frac{B}{1 - b^2} \times 1 - \frac{b^2}{r^2} - \frac{b^4}{r \times 4^2} \), &c. dabit valorem ipsius R. Simili modo si in æquatione praedictâ \( 1 - mb \cos \frac{t}{n} s + m \times \frac{m-1}{2} b^2 \times \cos \frac{1}{n} s^3 + m \times \frac{m-2}{3} b^3 \times \cos \frac{1}{n} s^3 + \), &c. \( = R + S \cos \frac{t}{n} s + T \cos \frac{2}{n} s + V \cos \frac{3}{n} s + \), &c. evolvantur quantitates \( \cos \frac{1}{n} s^3 \), \( \cos \frac{1}{n} s^7 \), &c. in finos valores, prout in Coroll. III. Lem. edoctum est, et colligantur omnes termini qui ducuntur in \( \cos \frac{1}{n} s \) exurget \[ S = -mb - m \times \frac{m-1}{2} \times \frac{m-2}{4} b^3 - m \times \frac{m-1}{2} \times \frac{m-2}{4} \times \frac{m-3}{6} \times \frac{m-4}{6} b^5 - m \times \frac{m-1}{2} \times \frac{m-2}{4} \times \frac{m-3}{6} \times \frac{m-4}{6} \times \frac{m-5}{8} \times \frac{m-6}{8} b^7 - \), &c. sive, posito \( m = -\frac{3}{2} \), \[ S = \frac{3}{4} b + \frac{3}{4} \times \frac{5 \times 7}{4 \times 8} b^3 + \frac{3}{4} \times \frac{5 \times 7}{4 \times 8} \times \frac{9 \times 11}{8 \times 12} b^5 + \frac{3}{4} \times \frac{5 \times 7}{4 \times 8} \times \frac{9 \times 11}{8 \times 12} \times \frac{13 \times 15}{12 \times 16} b^7 + \frac{3}{4} \times \frac{13 \times 15}{12 \times 16} \times \frac{17 \times 19}{16 \times 20} b^9 + \), &c. Patet autem terminam quemlibet hujus seriei æquari termino antecedenti ducto in \( \frac{r+1 \times r+3}{r \times r+4} b^2 \), existente \( r \) æquali numero terminorum praecedentium quadruplicato: sic terminus sextus, quia tunc \( r = 5 \times 4 = 20 \), est est æqualis termino quinto $\frac{17 \times 19}{16 \times 20} b^p$ ducto in $\frac{21 \times 23}{20 \times 24} b^2$. Quamobrem termino quovis hujus seriei dicto $B$, terminus subsequens erit $Bb^2 \times \frac{r+1 \times r+3}{r \times r+4}$, sive $Bb^2 \times 1 + \frac{3}{r \times r+4}$, et manente jam eodem valore numeri $r$, termini reliqui erunt, $Bb^4 \times 1 + \frac{3}{r \times r+4} \times 1 + \frac{3}{r+4 \times r+8}$, $Bb^6 \times 1 + \frac{3}{r \times r+4} \times 1 + \frac{3}{r+4 \times r+8} \times 1 + \frac{3}{r+8 \times r+12}$, &c. Sed si fuerit $r$ numerus aliquantum magnus, erit $1 + \frac{3}{r \times r+4} \times 1 + \frac{3}{r+4 \times r+8} = 1 + \frac{3}{r \times r+4} + \frac{3}{r+4 \times r+8}$ quamproximè, et $1 + \frac{3}{r \times r+4} \times 1 + \frac{3}{r+4 \times r+8} \times 1 + \frac{3}{r+8 \times r+12} = 1 + \frac{3}{r \times r+4} + \frac{3}{r+4 \times r+8} + \frac{3}{r+8 \times r+12}$, &c. Unde termini omnes prædicti incipientes à termino $B$ erunt $$B + Bb^2 + Bb^4 + Bb^6 + Bb^8 + \ldots = \frac{B}{1 - b^2} + \frac{3Bb^2}{r \times r+4} + \frac{3Bb^4}{r \times r+4} + \frac{3Bb^6}{r \times r+4} + \frac{3Bb^8}{r \times r+4} + \ldots = \frac{3B}{r \times r+4} \times \frac{b^2}{1 - b^2} + \frac{3Bb^4}{r+4 \times r+8} + \frac{3Bb^6}{r+4 \times r+8} + \frac{3Bb^8}{r+4 \times r+8} + \ldots = \frac{3B}{r+4 \times r+8} \times \frac{b^4}{1 - b^2} + \frac{3Bb^6}{r+8 \times r+12} + \frac{3Bb^8}{r+8 \times r+12} + \ldots = \frac{3B}{r+8 \times r+12} \times \frac{b^6}{1 - b^2} + \frac{3Bb^8}{r+12 \times r+16} + \ldots = \frac{3B}{r+12 \times r+16} \times \frac{b^8}{1 - b^2} + \ldots$$ Ac proinde erit \[ S = \frac{3}{2} b + \frac{3}{2} \times \frac{5 \times 7}{4 \times 8} b^3 + \frac{3}{2} \times \frac{5 \times 7}{4 \times 8} \times \frac{9 \times 11}{8 \times 12} b^5 + \frac{3}{2} \times \frac{5 \times 7}{4 \times 8} \times \frac{9 \times 11}{8 \times 12} \times \frac{13 \times 15}{12 \times 16} b^7 + \ldots + \frac{B}{1 - b^2} \] \[ \times 1 + \frac{3b^2}{r \times r + 4} + \frac{2b^4}{r + 4 \times r + 8} + \frac{2b^6}{r + 0 \times r + 14} + \frac{3b^8}{r + 1 \times r + 16} + \ldots + \text{&c.} \] Itaque si, computatis, exempli gratia, quindecim terminis, decimus sextus designetur per B, erit \( r = 15 \times 4 = 60 \), et summa terminorum quindecim illorum addita summæ seriei \( \frac{B}{1 - b^2} \times 1 + \frac{3b^2}{r \times r + 4} + \frac{3b^4}{r + 4 \times r + 8} + \ldots + \text{&c.} \) dabit valorem coefficientis S. Determinatis hoc pacto quantitatibus assumptis R, S, T, &c. jam ut ad expressiones virium revertamur, vis TR ad radium QS perpendicularis erat \( \frac{\phi k}{x^3} - \frac{\phi}{k^2} \times \sin. QSP \); sed posuimus angulum QSP \( = \frac{1}{n} s \), estque \( \frac{x}{x^3} = \frac{1}{t^3} \) in R \( + S \cos. \frac{1}{n} s + T \cos. \frac{2}{n} s + V \cos. \frac{3}{n} s + W \cos. \frac{4}{n} s + \ldots + \text{&c.} \) Unde vis TR \( = \frac{\phi k}{t^3} \) in R \( - \frac{t^3}{k^3} - \frac{T}{2} \times \sin. \frac{1}{n} s \) \[ + \frac{S - V}{2} \sin. \frac{2}{n} s + \frac{T - W}{2} \sin. \frac{3}{n} s + \frac{V - X}{2} \sin. \frac{4}{n} s + \ldots + \text{&c.} \] Et vis quae planetam Q distrahit à Sole in directione radii QS erat \( \frac{\phi}{x^3} \times k \cos. QSP = 1 - \frac{\phi}{t^3} \cos. QSP \), hoc est, \( \frac{\phi}{t^3} \) in \( \frac{kS}{2} - R + kR + \frac{kT}{2} - \frac{t^3}{k^3} - S \times \cos. \) \[ \times \cos \frac{1}{n} s + \frac{kS + kV - 2T}{2} \times \cos \frac{2}{n} s + \frac{kT + kW - 2V}{2} \] \[ \cos \frac{3}{n} s + \frac{kV + kX - 2W}{3} \cos \frac{4}{n} s + \text{&c. Q.E.I.} \] **Propositio II. Problema.** Inaequalitates motûs planetæ interioris ex viribus praedictis ortas investigare. Exeant simul planetæ \( P, Q \) (Fig. 2.) de locis \( D, C \), ubi jacebant in eâdem rectâ cum Sole posito in \( S \), et post aliquod temporis spatium reperiantur in \( P \) et \( Q \), et jungantur \( SP, SQ, PQ \). Esto \( CS = 1 \), et arcus circularis \( CQ \) sive angulus \( CSQ = s \); denotent præterea \( P \) et \( Q \) respectivè tempora periodica planetarum \( P \) et \( Q \), eritque ang. \( QSC : \text{ang. } PSD :: P : Q \), adeoque angulus \( QSP : \text{ang. } QSC :: P - Q : P \), unde ang. \( QSP = \frac{1}{n} s \), posito \( n = \frac{P}{P-Q} \). Vis attractionis Solis ad distantiam \( QS \), et tempus quo corpus, eâdem vi uniformiter agente, impulsum acquirere possit eam velocitatem, qua planeta \( Q \) in circulo \( CQ \) revolvitur, tum illa ipsa velocitas, exponantur figillatim per unitatem; et si, sumpto arcu \( CH = CS = 1 \), \( CH \) exprimat tempus illud unitati æquale, arcus quilibet quàm minimus \( Qq \) exprimet tempus quo uniformi illâ velocitate describitur. Vol. LII. Unde, Unde, cum velocitates viribus quibusvis constantibus genitae sint ut ipsae vires et tempora, quibus haec velocitates generantur, conjunctim; erit velocitas \( i \) planetae \( Q \) in circulo \( CQ \) revolventis ad incrementum vel decrementum velocitatis vi \( Z \) genitum (scripto nempe \( Z \) pro vi planetae \( P \) normaliter ad radium \( QS \) agente, prout est in propositione praecedente definita) quo tempore planeta \( Q \) describit arcum quam minimum \( Qq \), ut vis attractionis Solis \( i \) ducta in tempus \( CH \) sive \( i \), ad vim \( Z \) ductam in tempus descriptionis arcus \( Qq \) sive in ipsum in arcum \( Qq \): adeoque incrementum vel decrementum velocitatis vi \( Z \) genitum, quo tempore describitur arcus \( Qq \), exprimetur per \( Z \times Qq \) sive \( Z \times i \). Est autem \( Z = \frac{\phi k}{t^3} \) in \( R - \frac{t^3}{k^3} - \frac{T}{2} \times \sin. \frac{1}{n} s + \frac{S-V}{2} \sin. \frac{2}{n} s + \frac{T-W}{2} \sin. \frac{3}{n} s + \ldots \), &c. et hac quantitate ducta in \( i \), tum sumpta fluente, prodit velocitatis acceleratio sive retardatio, quam voco \( U \), genita quo tempore describitur a planeta \( Q \) arcus \( CQ \), æqualis \( \frac{\phi kn}{t^3} \) in \( R - \frac{t^3}{k^3} - \frac{T}{2} \times \sin. \text{vers}. \frac{1}{n} s + \frac{S-V}{4} \sin. \text{vers}. \frac{2}{n} s + \frac{T-W}{6} \sin. \text{vers}. \frac{3}{n} s + \frac{V-X}{8} \sin. \text{vers}. \frac{4}{n} s + \ldots \), &c. sive posito \( b = R - \frac{t^3}{k^3} - \frac{T}{2} + \frac{S-V}{4} + \frac{T-W}{6} + \frac{V-X}{8} + \ldots \), &c. \( U = \frac{\phi kn}{t^3} \) in \( b - R - \frac{t^3}{k^3} - \frac{T}{2} \times \cos. \frac{1}{n} s - \frac{S-V}{4} \cos. \frac{2}{n} s - \frac{T-W}{6} \cos. \frac{3}{n} s - \frac{V-X}{8} \cos. \frac{4}{n} s - \ldots \), &c. Hoc Hoc pacto obtinetur variatio velocitatis in hypothesi quod revolvatur planeta Q semper ad eamdem distantiam à Sole, quod in praecedenti calculo supponi potest, cum tantillum varietur distantia S Q actione planetae P. Hoc facto, ut investigetur variatio distantiae planetae Q à Sole, fingamus planetam descriptisse, non arcum circularem CQ, sed arcum curvae Cr (Fig. 3.) et reperiri in puncto r ubi radius S Q productus fecat curvam. Ducatur recta S t vicinissima ipsi S Q occurrens circulo et curvae q et t; tum centro S et radio Sr describatur arcus rp, sitque Sr = x. Si planeta Q urgeretur solâ vi tendente ad centrum S, describeret areas temporibus proportionales, atque adeò, cum ipsius velocitas angularis in loco C supponatur esse 1, in loco r foret æqualis \(\frac{1}{x}\); sed in illo quem exhibet schema situ planetarum minuitur hæc velocitas quantitate U suprà definitâ, unde velocitas angularis in loco r erit \(\frac{1}{x} - U\); et tempus, quo describeretur arcus Qq velocitate 1, est ad tempus quo describitur arcus rp velocitate \(\frac{1}{x} - U\), ut Qq ad \(\frac{rp}{\frac{1}{x} - U}\), hoc est, ut s ad \(\frac{x^5}{\frac{1}{x} - U}\); unde, cum s exprimat ex jam dictis tempus descriptionis arcûs Qq velocitate 1, exprimet quantitas titas $\frac{x^j}{\frac{1}{x} - U}$ tempus quo describitur arcus $r \rho$ velocitate. $\frac{1}{x} - U$. His positis, quoniam planetae $Q$, recessus à centro vel ad idem accessus pendet ex differentiâ virium, centrifugae scilicet et centripetae, quibus urgetur in $Q$; si hæc differentia virium dicatur $P$, et $v$ denotet velocitatem ascensûs vel descensûs planetae $Q$ secundum radium $SQ$, per idem planè ratiocinium, quod mox usurpavimus in investigatione velocitatis $U$, habebitur $v = P \times \frac{x^j}{\frac{1}{x} - U}$. Quoniam ex hypothesi planetae $Q$, sepositâ actione planetae $P$, descripteret circulum, vires (centripeta et centrifuga) sibi invicem et unitati forent æquales: existente autem planetæ $Q$ in $r$, ipsius attractio in Solem est $\frac{I}{x^2}$, ex qua auferenda est vis ea qua juxta propositionem præcedentem distrahitur à Sole, nimirum $\frac{\phi}{t^3}$ in $A + B \cos \frac{I}{n} s + C \cos \frac{2}{n} s + D \cos \frac{3}{n} s + E \cos \frac{4}{n} s + \&c.$ positis $A = \frac{kS}{2} - R$, $B = kR + \frac{kT}{2} - \frac{t^3}{k^2} - S$, $C = \frac{kS + kV - 2T}{2}$, $D = \frac{kT + kW - 2V}{2}$, $E = \frac{kV + kX - 2W}{2}$, &c. atque harum virium differentia componit vim centripetam. Vis autem centrifuga est semper in ratione duplicata areae temporis momento descriptæ directe et triplicata distantiae inversè; unde si hæc vis fuerit æqualis 1, ubi incepit planeta movere in C, erit æqualis \( x^2 \times \frac{1}{x} - U \) \( \times \frac{1}{x^3} = \frac{1}{x} \times \frac{1}{x} - U \) \( \times \frac{1}{x^3} \) ubi movetur in r. Differentia igitur inter viam centrifugam et centripetam, qua urgetur planeta in r supra designata per P, est \( \frac{1}{x} \times \frac{1}{x} - U \) \( \times \frac{1}{x^2} + \frac{\phi}{t^3} \) \( \times A + B \cos \frac{1}{n} s + C \cos \frac{2}{n} s + D \cos \frac{3}{n} s + \ldots \) hincque habetur \( \dot{v} = j \times \frac{1}{x} - U - \frac{s}{x \times \frac{1}{x} - U} + \frac{\phi}{t^3} \) \( \times \frac{x^5}{\frac{1}{x} - U} \times A + B \cos \frac{1}{n} s + C \cos \frac{2}{n} s + \ldots \) Vires, quibus perturbatur motus planetæ Q, cum exprimantur seriebus quorum termini ducuntur in sinum vel cosinum anguli \( \frac{1}{n} s \), vel anguli hujus multiplicitis, fingemus differentiam inter distantias S Q et S r exprimi serie simili, ac propterea ponemus \( x = 1 - Q + K \cos \frac{1}{n} s + L \cos \frac{2}{n} s + M \cos \frac{3}{n} s + N \cos \frac{4}{n} s + \ldots \) existente \( Q = K + L + M + N + \ldots \) ut sit S r, sive \( x = 1 \), ubi planetæ Q et P incipiunt movere à linea conjunctionis SCD. Quantitates autem assumpta K, L, M, &c. sunt exiguae, ideoque erit \( \frac{1}{x} = 1 + Q - K \cos \frac{1}{n} s - L \cos \frac{2}{n} s - M \cos \frac{3}{n} s - N \cos \frac{4}{n} s - \ldots \), &c. quamproximè. proximè. Substituantur ergò in æquatione suprà traditâ valores quantitatum \( x, \frac{1}{x}, \) et \( U; \) et sumptâ fluente, rejectis iis terminis qui ducuntur in altiorem quàm unam dimensionem quantitatum \( \phi, Q, K, L, \) &c. prodit \( v = -\frac{2\phi kn}{t^3} - \frac{\phi}{t^3} A - Q \times s \) \[ + \frac{2\phi kn}{t^3} \times R - \frac{t^3}{k^3} - \frac{T}{2} + \frac{\phi}{t^3} B - K \times n \times \sin. \frac{1}{n} s \] \[ + \frac{\phi kn}{t^3} \times \frac{S - V}{4} + \frac{\phi}{t^3} \times \frac{C}{2} - \frac{L}{2} \times n \times \sin. \frac{2}{n} s \] \[ + \frac{\phi kn}{t^3} \times \frac{T - W}{9} + \frac{\phi}{t^3} \times \frac{D}{3} - \frac{M}{3} \times n \times \sin. \frac{3}{n} s \] \[ + \frac{\phi kn}{t^3} \times \frac{V - X}{16} + \frac{\phi}{t^3} \times \frac{E}{4} - \frac{N}{4} \times n \times \sin. \frac{4}{n} s +, \text{ &c.} \] \[ + Z, \] designante \( Z \) quantitatem idoneam qua compleatur fluens. At, quoniam velocitas \( v \) supponitur nulla evadere, non solum ubi \( s, \) five arcus \( CQ = 0, \) id est, ubi planetæ versantur in primâ illâ conjunctione, sed etiam in omnibus aliis conjunctionibus subsequen- tibus, hoc est, ubi est angulus \( \frac{1}{n} s, \) seu \( PSQ = 0, \) vel \( = r \times 180^\circ, \) scripto scilicet \( r \) pro quovis ex numeris naturalibus \( 1, 2, 3, 4, \) &c. fiet. \( Z = \frac{2\phi kn}{t^3} - \frac{\phi}{t^3} A - Q \times s \) adeoque \[ v = \frac{2\phi kn}{t^3} \times R - \frac{t^3}{k^3} - \frac{T}{2} + \frac{\phi}{t^3} B - K \times n \times \sin. \frac{1}{n} s \] \[ + \frac{\phi kn}{t^3} \times \frac{S - V}{4} + \frac{\phi}{t^3} \times \frac{C}{2} - \frac{L}{2} \times n \times \sin. \frac{2}{n} s \] \[ + \frac{\phi kn}{t^3} \times \frac{T - W}{9} + \frac{\phi}{t^3} \times \frac{D}{3} - \frac{M}{3} \times n \times \sin. \frac{3}{n} s \] \[ \frac{\phi k n}{t^3} \times \frac{V - X}{16} + \frac{\phi}{t^3} \times \frac{E}{4} - \frac{N}{4} \times u \times \sin. \frac{4}{n} s + \ldots \] Deinde, cum fit \( tp \), five \( \dot{x} \) ad \( rp \), five \( x \dot{s} \), ut velocitas \( v \) qua describitur \( tp \) ad velocitatem \( \frac{1}{x} - U \) qua describitur \( rp \), erit \( \dot{x} = v \times \frac{x \dot{s}}{\frac{1}{x} - U} \), five, quia valor velocitatis \( v \) componitur ex quantitatibus exiguis, \( \dot{x} = v \dot{s} \) quamproximè, et \( \frac{\dot{x}}{\dot{s}} = v \). Verùm etiam æquatio assumpta \( x = 1 - Q + K \cos. \frac{1}{n} s + L \cos. \frac{2}{n} s + M \cos. \frac{3}{n} s + \ldots \), &c. dat \( \frac{\dot{x}}{\dot{s}} = -K \times \frac{1}{n} \sin. \frac{1}{n} s - L \times \frac{2}{n} \sin. \frac{2}{n} s - M \times \frac{3}{n} \sin. \frac{3}{n} s - N \times \frac{4}{n} \sin. \frac{4}{n} s, \ldots \) Habitis igitur duobus velocitatis \( v \) valoribus, eorum termini homologi statuantur æquales, atque inde obtinebuntur quantitates assumptæ, nempe \[ K = \frac{\phi}{t^3} \times \frac{n^2}{n^2 - 1} \times 2kR - \frac{2t^3}{k^2} \times n + \frac{1}{2} - kT \times n - \frac{1}{2} - S \] \[ L = \frac{\phi}{2t^3} \times \frac{n^2}{n^2 - 4} \times kS \times n + 1 - kV \times n - 1 - 2T \] \[ M = \frac{\phi}{3t^3} \times \frac{n^2}{n^2 - 9} \times kT \times n + \frac{3}{2} - kW \times n - \frac{1}{2} - 3V \] \[ N = \frac{\phi}{4t^3} \times \frac{n^2}{n^2 - 16} \times kV \times n + 2 - kX \times n - 2 - 4W \] &c. indeque manifesta fit harum quantitatum progressio: atque atque hoc pacto habetur semper distantia \( x \) planetæ Q à Sole. Jam ut definiatur planetæ Q motus verus qui designatur per \( s \), dicatur \( w \) motus medius, sive, quod perinde est, tempus quo planeta descripterit arcum quemlibet \( C r \); atque ex demonstratis est \( w = \frac{x^5}{x - U} \); unde, substitutis valoribus quantitatum, \( x \), \[ w = 1 - 2Q + \frac{\phi kbn}{3} \times s + 2nK - \frac{\phi kn^2}{t^3} \times R - \frac{t^3}{k^3} - \frac{T}{2} \] \( \times \sin. \frac{1}{n} s + nL - \frac{\phi kn^2}{8 t^3} \times S - V \times \sin. \frac{2}{n} s \) \[ + \frac{2nM}{3} - \frac{\phi kn^2}{18 t^3} \times T - W \times \sin. \frac{3}{n} s \] \[ + \frac{nN}{2} - \frac{\phi kn^2}{32 t^3} \times V - X \times \sin. \frac{4}{n} s + , \&c. + Z \] denotante \( Z \) quantitatem idoneam ut compleatur fluens. Sed, quia motus verus medio æqualis evadere supponitur in qualibet planetarum P et Q conjunctione cum Sole, id est, ubi angulus PSQ sive \( \frac{1}{n} s \) æquatur, vel nihilo, vel angulo \( r \times 180^\circ \), exhibente \( r \) quemvis ex numeris naturalibus 1, 2, 3, 4, &c. erit \( Z = 2Q - \frac{\phi kbn}{t^3} \times s \). Ponantur igitur \( F = -2nK + \frac{\phi kn^2}{t^3} \times R - \frac{t^3}{k^3} - \frac{T}{2}, G = -nL + \frac{\phi kn^2}{8 t^3} \times S - V, H = -\frac{2nM}{3} + \frac{\phi kn^2}{18 t^3} \times T - W, I = -\frac{nN}{2} + \frac{\phi kn^2}{32 t^3} \times X \) \[ V - X, \text{&c. eritque motus verus, } s = w + F \sin. \frac{1}{n} s + G \times \sin. \frac{2}{n} s + H \times \sin. \frac{3}{n} s + I \times \sin. \frac{4}{n} s + \text{&c. vel, quia parum admodum differt motus verus à motu medio } s = w + F \times \sin. \frac{1}{n} w + G \times \sin. \frac{2}{n} w + H \times \sin. \frac{3}{n} w + I \times \sin. \frac{4}{n} w + \text{&c. Q.E.I.} \] **Coroll. I.** His ita generatim definitis, ut specialis eliciatur in motu cujuspiam planetæ inæqualitatum mensura, determinandæ sunt quantitates assumptæ. Itaque planeta \( P \) designet Terram, planeta \( Q \) Venerem, et quoniam est distantia Terræ ad distantiam Veneris à Sole ut 100000 ad 72333, hæc erit ratio \( k \) ad 1, adeoque \( k = \frac{100000}{72333} \), \( kk + 1 = tt = 2.91129 \), \( b = \frac{2k}{t^3} = 0.94975 \); atque inde per methodum in Prop. I. expositam prodibunt \[ \begin{align*} R &= 9.3925 \\ V &= 11.1964 \\ Y &= 5.3380 \\ S &= 16.6782 \\ W &= 8.8504 \\ Z &= 4.1029 \\ T &= 13.8877 \\ X &= 6.9045 \\ \end{align*} \] &c. Tum, existente periodo Terræ annuâ dierum 365.2565, et periodo Veneris dierum 224.701, est ex jam dictis \( n = \frac{365.2565}{365.2565 - 224.701} = 2.59866 \); et cum gravitas in Solem sit juxta Newtonum ad gravitatem in Terram, paribus distantìis, ut 1 ad \( \frac{1}{72333} \) erit \( \phi = \frac{1}{72333} \). Vol. LII. R r Unde Unde, redactis in numeros formulis in hac propositione datis, emergunt \[ K = 0.0000103 \] \[ L = 0.0000444 \] \[ M = 0.0000377 \] \[ N = -0.0000065 \] \[ O = -0.0000024 \] \[ O' = -0.0000011, \text{ &c.} \] Atque ex his tandem deducuntur \[ F = -0.0000473 \] \[ G = -0.0001078 \] \[ H = -0.0000684 \] \[ I = 0.0000100 \] \[ I' = 0.0000033 \] \[ \text{&c.} \] Hinc ergo habentur valores coefficientium aequationis \( s = w + F \times \sin. \frac{1}{n} w + G \times \sin. \frac{2}{n} w + H \times \sin. \frac{3}{n} w + \ldots, \text{ &c.} \) ubi \( s \) denotat motum Veneris verum, \( w \) motum medium, et \( \frac{1}{n} w \) angulum PSQ five differentiam longitudinum heliocentricarum Terræ et Veneris; vel, reductis quantitatibus \( F, G, H, \text{ &c.} \) ad exprimendas more astronomico circuli partes, fit \[ s = w - 9''.76 \times \sin. \frac{1}{n} w - 22''.24 \times \sin. \frac{2}{n} w \] \[ + 14''.11 \times \sin. \frac{3}{n} w + 2''.06 \times \sin. \frac{4}{n} w + 0''.68 \times \sin. \frac{5}{n} w + \ldots, \text{ &c.} \] Ut exemplum apponam, esto angulus PSQ five \( \frac{1}{n} w = 40^\circ \), ac prodibit \( s = w - 15''.5 \); motus igitur medius superat verum, eorumque differentia est 15''.5. Computatâ hoc pacto differentiâ inter motum Veneris verum et medium respectu Solis, sequenti modo innotescet quanta evadat cum e Terrâ spectatur. Esto PSQ PSQ (Fig. 4.) angulus exhibens, ut prius, differentiam longitudinum planetarum P et Q tempore quovis dato, et in circulo RQ exhibente portionem orbitae planetae Q, summatur arcus Qq æqualis differentiae motuum praedictæ, et ductis Sq, Pg, centro P et radio Pg describatur arcus qr secans PQ in r; atque, ob parvitudinem arcuum Qq, qr, erit Qq : qr :: rad. : fin. PQq; deinde $\frac{Qq}{QS} : \frac{qr}{PQ} ::$ ang. QSq : ang. QPq; adeoque $\frac{\text{rad.}}{QS} : \frac{\text{fin. PQq}}{PQ} ::$ ang. QSq : ang. QPq, unde ang. QPq = ang. QSq × $\frac{QS}{PQ}$ × $\frac{\text{fin. PQq}}{\text{rad.}}$. Datis igitur angulo PSQ et distantiai PS, QS, dabitur distantia PQ, et angulus PQS, adeoque et angulus PQq: unde innotescet angulus quaestus QPP, hoc est æquatio motûs, prout appareat spectatori in centro Terræ locato. Hincque, quamvis sit modica motûs Veneris inæqualitas telluris actione genita, qualis tamen sit ut pateat, libet eam in sequenti tabulâ oculis subjicere. Hujus tabulae columna prima exhibet angulum QPS, sive elongationem Veneris à Sole mediam; secunda indicat correctionem hujus elongationis, à conjunctione Veneris inferiore usque ad maximam ejus elongationem quae in orbe circulari est $46° 19' 50''$ circiter. Tertia et quarta columna eodem modo exhibent elongationem Veneris, ejusque correctionem, à tempore elongationis maximæ usque ad conjunctionem superiorem. Rr2 Elonga- | Elongatio Ven. à Sole | Correctio. | Elongatio Ven. à Sole | Correctio. | |----------------------|-----------|----------------------|-----------| | ° ' " | " | ° ' " | " | | 0 | 0 | 46 19 50 | 0 | | 5 | 0 | 46 | + 2.3 | | 10 | 0 | 45 | 5.1 | | 15 | 0 | 40 | 9.5 | | 20 | - 0.5 | 35 | 7.3 | | 25 | 0.8 | 30 | 1.8 | | 30 | 1.5 | 25 | - 4.4 | | 35 | 2.8 | 20 | 9.2 | | 40 | 2.9 | 15 | 11.2 | | 45 | 2.7 | 10 | 10.2 | | 46 | 1.7 | 5 | 6.0 | | 46 19 50 | 0 | 0 | 0 | Exempli gratiâ, si Venus à conjunctione inferiore digressa motu suo medio discesserit à Sole angulo elongationis $40^\circ$, erit vera Veneris elongatio $40^\circ - 2''$.9 = $39^\circ 59' 57''$.1: pariter, si ulterius delata Venus pervenerit ad eamdem elongationem $40^\circ$, erit tunc vera Veneris elongatio $40^\circ 0' 9''$.5. Eadem omnino sunt correctiones et cum iisdem signis adhibendae, ubi post conjunctionem superiorem eadem eveniunt elongationes. Coroll. II. Ex praecedentibus etiam deducitur distantia Veneris à Sole pro quolibet ejus cum Terrâ et Sole aspectu, in in hypothesi quod, seclusâ Terræ attractione, in orbitâ circulari revolvet. Sic, si angulus $\frac{1}{n} s$, seu $PSQ$ fit $90^\circ$, vel $270^\circ$, æquatio $x = 1 - Q + K \cos \frac{1}{n} s + L \cos \frac{2}{n} s + M \cos \frac{3}{n} s + N \cos \frac{4}{n} s + \ldots$, &c. fit $x = 0.9999437$ circiter; et si fit $PSQ = 180^\circ$, fit $x = 1.0000607$. Unde, si distantia Veneris à Sole in conjunctione inferiore ponatur In quadraturis cum Terrâ erit ipsius distantia In conjunctione superiore erit Item innotescit differentia inter tempus periodicum Veneris, quale nunc est, et tempus illud periodicum, quale foret, si unicâ Solis attractione in orbe circulari moveretur. Siquidem, cum Venus post discessum suum à conjunctione ad eamdem redierit, æquatio generalis in propositione tradita, quæ exprimit relationem inter motum Veneris verum et medium, evadit: $$w = 1 - 2Q + \frac{\phi k h n}{t^3} x s,$$ five $w = 1.0000066 x s$ circiter: unde tempus periodicum Veneris est ad tempus illud alterum periodicum, ut $1.0000066$ ad $1$; adeoque, si nulla foret gravitatio Veneris in Terram, revolutionem suam circa Solem minutis duobus horæ primis citius perageret. PROPO- Propositio III. Problema. In systemate duorum planetarum in orbitis circularibus circa Solem revolventium, motum nodorum orbitae planetae interioris, quatenus ex vi planetae exterioris oritur, investigare. Per motum nodorum hic intelligendus est motus intersectionis plani orbis planetae interioris cum plano orbis planetae exterioris spectato ut immoto. Itaque esto Sol in S (Fig. 5.) et centro S atque radio SQ de- scribantur in superficie sphærae duo circuli QN, PN, se se interfecantes in N, quorum prior QN designet situm plani orbis planetae interioris Q, et posterior PN situm plani orbis planetae exterioris, cujus locus sit in rectâ SP producâta. Eodem centro S et radio SP describatur circulus PK, cujus planum sit plano SQN SQN perpendicularis, secetque circulum QN in K, et in SK demittatur perpendicularum PH: tum ductâ QT parallelâ rectae SP et TB in planum SQN normali, si linea QT exhibeat vim qua trahitur planeta Q in directione QT, seu SP, TB exhibebit vim qua distrahitur perpendiculariter à plano suae orbitae; erit- que triangulum QTB simile triangulo SPH, atque adeò, TB : QT :: PH : SP :: fin. PK : 1; deinde in triangulo sphærico rectangulo PKN habetur, 1 : fin. PN :: fin. PNK : fin. PK; unde, conjunctis rationibus, et scripto c pro sinu anguli PNK ad ra- dium 1, hoc est, pro sinu inclinationis orbis QN ad orbem PN, provenit TB = QT x c x fin. PN. Sumatur jam arcus quam minimus Qq, ad quem erigitur lineola perpendicularis qr, æqualis duplo spatio quod planeta Q percurrere posset impellente vi TB quo tempore in orbe suo describeret arcum illum Qq, et centro S descriptus circulus rQn secans circulum PN in n exhibebit situm orbis planetæ Q post tem- pus illud, nodo N translato in n; atque in QN de- misso perpendicularo nm, et in Sq perpendicularo Qs, erit angulus qQr, sive NQn ad duplum angulum qQs, id est, ad angulum QSq, ut vis TB ad gra- vitatem (nempe 1) planetæ Q in Solem; hoc est, \[ \frac{nm}{\text{fin. } QN} : Qq :: TB : 1; \] in triangulo autem rectan- gulo Nmn, est Nn : nm :: 1 : c; quarè conjunctis his rationibus, prodit Nn = \[ \frac{TB \times \text{fin. } QN \times Qq}{c}; \] sed suprà invenimus TB = QT x c x fin. PN, unde fit Nn = QT x fin. PN x fin. QN x Qq. Esto SC linea conjunctionis planetarum, fiatque, ut in propositione præcedente, arcus CQ = s, Qq = s, SQ = 1; et, quia inclinatio orbis QN ad orbem PN exigua supponitur, erit etiam hic ang. PSQ = \frac{1}{n} s quamproximè; proindeque, posito arcu CN = a, erit QN = s + a et PN = s - \frac{1}{n} s + a quamproximè. Porrò, cum lentissimè moveantur nodi, arcus CN spectari potest quasi invariabilis per multarum planetæ revolutionum seriem, atque adeò fluxio arcus QN eadem erit cum fluxione arcûs QC. His positis, habebitur fin. PN x sin. QN = \frac{1}{2} \cos. \frac{1}{n} s - \frac{1}{2} \cos. 2s - \frac{1}{n} s + 2a, estque per propositionem primam QT = \frac{\phi k}{z^3} - \frac{\phi}{k^2} = \frac{\phi k}{t^3} in R - \frac{t^3}{k^3} + S \cos. \frac{1}{n} s + T \cos. \frac{2}{n} s + V \cos. \frac{3}{n} s + W \cos. \frac{4}{n} s +, &c. unde substitutis his valoribus in æquatione Nn = QT x sin. PN x sin. QN x Qq, et sumptâ fluente per methodum in Coroll. IV. lemmatis edoctam, prodibit summa omnium Nn, five motus nodi, quo tempore planeta Q à loco conjunctionis C procedens in orbe suo descriperit arcum CQ, æqualis \frac{\phi kn}{2 t^3} in \frac{s}{2n} s + R - \frac{t^3}{k^3} + \frac{T}{2} x \sin. \frac{1}{n} s + \frac{S + V}{4} \sin. \frac{2}{n} s + \frac{T + W}{6} \sin. \frac{3}{n} s +, &c. + \frac{\phi kn}{2 t^3} in Z x sin. 2a - R - \frac{t^3}{k^3} x \frac{1}{2n - 1} \sin. 2s - \frac{1}{n} s + 2a - \frac{S}{2} x \frac{1}{2n} \sin. 2s + 2a - \frac{S}{2} x \frac{1}{2n - 2} \sin. 2s - \frac{2}{n} s + 2a - \frac{T}{2} x \[ \frac{1}{2n-3} \sin_{2s} - \frac{3}{n} s + 2a - \frac{T}{2} \times \frac{1}{2n+1} \sin_{2s} + \frac{1}{n} s + 2a - \frac{V}{2} \times \frac{1}{2n-4} \sin_{2s} - \frac{4}{n} s + 2a - \frac{V}{2} \times \frac{1}{2n+2} \sin_{2s} + \frac{2}{n} s + 2a, \text{ &c. existente} \] \[ Z = 2n - 1 \text{ in } R - \frac{t^3}{k^3} \times \frac{1}{2n-1} + \frac{s}{2n-2+2n} + \frac{T}{2n-3 \times 2n+1} + \frac{V}{2n-4 \times 2n+2} + \frac{W}{2n-5 \times 2n+3} + \text{ &c. atque in his seriebus patet terminorum progressio. Q.E.I.} \] **Coroll. I.** Hic liquet multas oriri in motu nodorum æquationes; sed quia minutæ sunt, et locum planetæ \( Q \) ferè nihil mutant, idè satis erit rationem habere motûs nodorum medii et æquationis solius periodicæ, qui sic ex præcedentibus deducuntur. Cum in planis parùm ad se inclinatis moveri supponantur planetæ \( P \) et \( Q \), quoties revertentur ad conjunctionem, angulus \( PSQ \), sive \( \frac{1}{n} s \), qui metitur eorum distantiam à se invicem, evadet \( = 360^\circ \) vel \( = r \times 360^\circ \), existente \( r \) numero integro: et quia, sumpto arcu quolibet \( A \), est semper \( \sin_{r \times 360^\circ} + A = \sin_{A} \); hinc, si computatur motus nodi pro tempore conjunctionum, expressio illa generalis et prolixa in propositione tradita in hanc simplicem abit \( \frac{\phi k}{2t^3} \times \frac{s}{2} - nZ \times \sin_{2s+2a} - \sin_{2a}, \) sive per Coroll. I. lemmatis \[ \frac{\phi k}{2t^3} \times \frac{s}{2} - 2nZ \times \sin_{s} \times \cos_{s} + 2a. \] Vol. LII. SS Hic Hic est igitur motus nodorum factus, quo tempore planetae P et Q à conjunctione proiecti post quotlibet- cunque revolutiones ad conjunctionem quamvis aliam pervenerint, exhibente s arcum à planetâ Q in suâ or- bitâ interea descriptum. Terminus \( \frac{\phi k}{2t^3} \times 2nZ \times \sin.s \) \( \times \cos.s + 2a \) indicat æquationem periodicam et fa- cillimè computatur: cumque hæc æquatio modò sit additiva, modò subtractiva, patet termino altero \( \frac{\phi k}{2t^3} \times \frac{s}{2} \) exprimi generatim motum nodi medium. Coroll. II. Esto planeta P Terra, Q Venus, et revolutionem Veneris ab unâ conjunctione inferiore cum Terrâ ad alteram vocemus, brevitatis gratiâ, revolutionem syn- odicam; eritque post unam revolutionem synodicam \( \frac{1}{n}s = 360° \), proindeque \( s = n \times 360° = 935° 31' \); hic igitur est arcus descriptus à Venere inter duas ejusdem generis conjunctiones. Hinc motus nodi medius tempore revolutionis unius synodiciæ, qui juxta corollarium praecedens est \( \frac{\phi ks}{4t^3} \) fit \( \frac{\phi kns}{4t^3} = 360° = 23''.087 \); atque hic motus imminutus in ratione tem- poris periodici Terræ circa Solem ad revolutionem Veneris synodicam, id est, in ratione 1 ad \( n - 1 \), evadit 14''.44, motus scilicet annuus nodorum Ve- neris regressive, qui spatio centum annorum fit 24' 4''. Æquatio periodica \( \frac{\phi knZ}{t^3} \times \sin.s \times \cos.s + 2a \) ut adhuc simplicior evadat, ponamus arcum \( a \) five CN perexiguum perexiguum esse vel nullum, id est, supponamus conjunctionem Terrae et Veneris fieri proximè in nodo, quemadmodum contingit hoc anno 1761, eritque æquatio periodica \( \frac{\phi k n Z}{t^3} \times \sin. s \times \cos. s = \frac{\phi k n Z}{2 t^3} \times \sin. 2s \). Cum igitur sit \( Z = 32.33 \) circiter, formula \( \frac{\phi k S}{4 t^3} - \frac{\phi k n Z}{2 t^3} \sin. 2s \), quæ per corollarium praecedens exprimit generatìm motum nodi in qualibet serie revolutionum synodiarum consecutum, sit \( 0.000006855 \times s - 14''.2 \times \sin. 2s \). Æquatio igitur periodica \( 14''.2 \times \sin. 2s \), quam generalem voco, est ut sinus dupli arcûs à Venere descripti in datâ serie revolutionum synodiarum, nec ultra \( 14''.2 \) ascendit. Jam, si pro \( s \) substituatur \( 935° 31' \), erit \( \sin. 2s = \sin. 71° 2' \), et regredientur nodi, in primâ revolutione synodicâ post conjunctionem factam in nodo, per arcum \( 23'' - 14''.2 \times \sin. 71° 2' = 10'' \): et, si \( r \) denotet numerum quemcumque revolutionum synodiarum, motus nodi, peractis illis revolutionibus, erit \( r \times 23'' - 14''.2 \times \sin. r \times 71° 2' \); pariterque, peractis revolutionibus quarum numerus est \( r - 1 \), idem motus erit \( r - 1 \times 23'' - 14''.2 \times \sin. r - 1 \times 71° 2' \); posterior motus ex priore auferatur, et remanet \( 23'' - 14''.2 \times \sin. r \times 71° 2' - \sin. r - 1 \times 71° 2' = 23'' - 14''.2 \times 2 \sin. 35° 31' \times \cos. r \times 71° 2' - 35° 31' = 23'' - 16''.5 \times \cos. 2r - 1 \times 35° 31' \) pro motu nodi facto, tempore illius revolutionis synodiciæ, cujus locum in serie revolutionum indicat numerus \( r \). Exempli gratiâ, si desideretur motus nodi tempore revolutionis quartæ synodiciæ post conjunctionem factam in nodo, erit \( r = 4 \), et regressus nodi erit \( 23'' \). \[23'' - 16''.5 \times \cos. 7 \times 35^\circ 31' = 29''.\] Sic ope hujus formulæ \(23'' - 16''.5 \times \cos. 2r - 1 \times 35^\circ 31'\) facile computatur sequens tabula, quæ exhibet regressum nodi Veneris in plano eclipticæ, pro duodecim sigillatim revolutionibus synodiciis quæ proximè sequuntur conjunctionem Terræ et Veneris factam in nodo vel proximè ad nodum. | In revol. Ven. synod. | Regressus nodi Ven. | In revol. Ven. synod. | Regressus nodi Ven. | |----------------------|---------------------|----------------------|---------------------| | | '' | | '' | | 1ᵃ. | 10 | 7ᵃ. | 26 | | 2ᵃ. | 28 | 8ᵃ. | 39 | | 3ᵃ. | 39 | 9ᵃ. | 30 | | 4ᵃ. | 29 | 10ᵃ. | 11 | | 5ᵃ. | 10 | 11ᵃ. | 8 | | 6ᵃ. | 9 | 12ᵃ. | 25 | Qui motus potest, cum libuerit, ad annos communes reduci. Denique patet æquationem periodicam, nempe \(16''.5 \times \cos. 2r - 1 \times 35^\circ 31'\), quam specialem appello, ubi maxima est, evadere \(16''.\); ac proinde regressum nodi in unâ revolutione synodiciâ nusquam superare \(39''.\), nec minorem esse \(6''.\). Propo- PROPOSITIO IV. PROBLEMA. Iisdem positis, variationem inclinationis orbis planetæ interioris ad planum orbis planetæ exterioris determinare. Est NQV (Fig. 5.) quadrans circuli, cui erigatur perpendicularis Vt occurrens arcui n Q r producto in t, eritque Vt mensura variationis inclinationis orbis NQV factæ quo tempore nodus N transfertur in n. Est autem Vt : nm :: sin. QV sive cos. QN : sin. QN, atque nm : Nn :: c : 1, c denotante sinum inclinationis orbis QN ad orbem PN, adeoque Vt : Nn :: c x cos. QN : sin. QN; unde Vt = Nn x \(\frac{c \times \cos. QN}{\sin. QN}\), sive, quia per propositionem superiori habetur Nn = QT x sin. PN x sin. QN x Qq, Vt = c x QT x sin. PN x cos. QN x Qq. Hinc, cum fit sin. PN x cos. QN = \(\frac{1}{2} \sin. 2s - \frac{1}{n} s + 2a - \frac{1}{2} \sin. \frac{1}{n} s\), sumptâ fluente prodit variatio inclinationis, quo tempore planeta Q à loco conjunctionis C movetur per arcum CQ, æqualis \(-\frac{\phi c kn}{2t^3}\) in R \(-\frac{t^3}{k^3}\) \(-\frac{T}{2}\) x sin. vers. \(\frac{1}{n} s + \frac{s - V}{4}\) sin. vers. \(\frac{2}{n} s + \frac{T - W}{6}\) sin. vers. \(\frac{3}{n} s + \frac{V - X}{8}\) sin. vers. \(\frac{4}{n} s + \ldots\) &c. \(+\frac{\phi c kn}{2t^3}\) in \(-Z\) x sin. vers. \(2a + R - \frac{t^3}{k^3}\) x \(\frac{1}{2n - 1}\) sin. vers. \(2s - \frac{1}{n} s + 2a + \frac{s}{2} \times \frac{1}{2n - 2}\) sin. vers. \(2s\). \[ \frac{2s - \frac{2}{n}s + 2a + \frac{T}{2} \times \frac{1}{2n+1} \text{fin. vers. } 2s + \frac{1}{n}s + 2a}{\text{fin. vers. } 2s - \frac{3}{n}s + 2a + \frac{V}{2} \times \frac{1}{2n+2}} \] \[ \frac{\text{fin. vers. } 2s + \frac{2}{n}s + 2a + \frac{V}{2} \times \frac{1}{2n-4} \text{fin. vers. } 2s - \frac{4}{n}s + 2a + \frac{W}{2} \times \frac{1}{2n+3} \text{fin. vers. } 2s + \frac{3}{n}s + 2a}{\text{fin. vers. } 2s - \frac{5}{n}s + 2a, \&c.} \] Existente hic eodem valore quantitatis \( Z \) ac in propositione praecedente. Q. E. I. **Coroll.** Si computetur variatio inclinationis pro tempore conjunctionum, facile obtinebitur; hæc enim per formulam in propositione traditam evadit \( \frac{\phi c kn}{2t^3} \times Z \times \text{fin. vers. } 2s + 2a - \text{fin. vers. } 2a \) quæ item, si prima conjunctionum, à qua fumitur motûs exordium, statuatur in nodo, fit \( \frac{\phi c kn}{1t^3} \times Z \times \text{fin. vers. } 2s \). Hoc est igitur decrementum inclinationis orbis planetæ \( Q \) factum in qualibet serie revolutionum ad conjunctionem, designante \( s \) arcum interea à planetâ circa Solem descriptum. Conferatur hæc inclinationis variatio cum æquatione nodi periodicâ eodem tempore genitâ, prout in propositione superiori definitur, et patebit priorem esse ad posteriorem ut \( c \times \text{fin. vers. } 2s \) ad fin. \( 2s \). Ut ad orbem Veneris hæc transferantur, quem si inclinari ad orbem Terræ supponatur angulo \( 3° 23' 20'' \), erit erit \(\frac{\phi ckn}{2t^3} \times Z \times \sin.\) vers. \(2s = o''.84 \times \sin.\) vers. \(2s\). Unde palàm fit: 1°. in quacumque serie revolutionum synodicarum, post conjunctionem factam in nodo, decrementum inclinationis orbitæ Veneris ad eclipticam non superare \(2 \times o''.84 = 1''.68\), quod è Terrà spectatum evadit \(4''.4\): 2°. cum, peractâ unâ revolutione synodicâ, sit fin. vers. \(2s = \sin.\) vers. \(71^\circ 2'\), inclinationis decrementum pro qualibet serie revolutionum synodicarum quarum numerus est \(r\), esse \(o''.84 \times \sin.\) vers. \(r \times 71^\circ 2'\), et pro serie revolutionum quarum numerus est \(r - 1\), esse \(o''.84 \times \sin.\) vers. \(r - 1 \times 71^\circ 2'\); unde horum decrementorum differentia \(o''.84 \times \sin.\) vers. \(r \times 71^\circ 2' - \sin.\) vers. \(r - 1 \times 71^\circ 2'\) \(= o''.84 \times 2 \sin. 35^\circ 31' \times \sin. 2r - 1 \times 35^\circ 31' = o''.98 \times \sin. 2r - 1 \times 35^\circ 31'\), exprimit variationem inclinationis genitam tempore revolutionis synodicæ illius, cujus locum in serie revolutionum denotat numerus \(r\): atque hæc variatio, ut patet, nusquam excedit \(o''.98\) è Sole conspecta, quæ spectatori in centro Terræ collocato sub angulo \(2''.\frac{1}{2}\) apparebit. Cum igitur tantilla sit orbitæ Veneris inclinationis variatio, non videtur operæ pretium de eâ ulteriùs exquirere. Demonstratis, quæ ad perturbationem motûs planetæ interioris spectant, supereft ut, quibus perturbationibus afficiatur motus planetæ exterioris, vicissim expendamus. Propo- Propositio V. Problema. In systemate duorum planetarum circa Solem in orbibus penè circularibus revolventium, determinare vim planetæ interioris ad perturbandum motum exterioris. Simili ratiocinio ei, quod in propositione primâ usurpavimus, etiam hoc problema solvitur. Itaque posita unitate pro distantiâ planetæ P à Sole, ubi ambo planetæ P et Q conjunguntur cum Sole, (Fig. 1.) fiat \( SP = x \), \( SQ = k \), \( PQ = z \). Sit \( I \) ad \( \phi \) ut gravitatio planetæ P in Solem in distantiâ \( I \) ad ejusdem planetæ P gravitationem in planetam Q in eadem distantiâ, eritque \( \frac{\phi}{z^2} \) gravitas planetæ P in planetam Q in distantiâ PQ. Productâ, si opus est, PQ ad O ut fit PO = \( \frac{\phi}{z^2} \), et ductâ OI parallelâ rectæ QS occurrente PS productæ in I, resolvatur vis PO in vires PI et OI, eritque propter similia triangula PQS, POI, vis OI = \( \frac{PO \times QS}{PQ} = \frac{\phi k}{z^3} \), atque vis PI = \( \frac{PO \times PS}{PQ} = \frac{\phi x}{z^3} \) five vis PI = \( \frac{\phi}{z^3} \) quamproximè. Vis OI impellit planetam P in directione parallelâ rectæ SQ, et in eundem sensum urgetur Sol vi \( \frac{\phi}{k^2} \) qua gravitat in planetam Q: exceffu igitur solo vis prioris supra posteriorem, nempe \( \frac{\phi k}{z^3} - \frac{\phi}{k^2} \), censendus est urgeri planeta P in directione parallelâ rectæ SQ. Porrò Porrò vis $\frac{\phi k}{z^3} - \frac{\phi}{k^2}$ ea pars, quae agit perpendiculariter ad radius PS, est $\frac{\phi k}{z^3} - \frac{\phi}{k^2} \times \sin. PSQ$, atque altera pars, quae amovet planetam P à Sole secundum PS, est $\frac{\phi k}{z^3} - \frac{\phi}{k^2} \times \cos. PSQ$. Auferatur hæc posterior vis ex vi PI, et manebit vis $\frac{\phi}{z^3} + \frac{\phi k}{z^3} \times \cos. PSQ$, qua planeta P urgetur versus Solem. Esto DCS (Fig. 2.) linea conjunctionis planetarum, et arcus DP, sive angulus DSP vocetur $s$, denotentque P et Q respectivè tempora periodica planetarum P et Q, eritque, posito $n = \frac{Q}{P-Q}$, ang. PSQ = $\frac{1}{n}s$. Tum, si fiat $t^2 = 1 + kk$, et $b = \frac{2k}{t^2}$, erit uti in Prop. I. exposuimus, $x^2 = t^2 \times 1 - b \cos. \frac{1}{n}s$, atque $\frac{1}{z^3} = \frac{1}{t^3} \times R + S \cos. \frac{1}{n}s + T \cos. \frac{2}{n}s + V \cos. \frac{3}{n}s + \&c.$ et quemadmodum ibi erat $b = \frac{2PS \times SQ}{PS^2 + SQ^2}$, hic item est $b = \frac{2PS \times SQ}{PS^2 + SQ^2}$, adeoque valores quantitatum assumptarum R, S, T, &c. iidem hic sunt ac in propositione primâ. Unde vis $\frac{\phi k}{z^3} - \frac{\phi}{k^2} \times \sin. PSQ$, qua sollicitatur planeta P in directione ad radius PS perpendiculari, sic exprimetur $\frac{\phi k}{t^3}$ in $R - \frac{T}{2} \times \sin. \frac{1}{n}s + \frac{S-V}{2}$ Vol. LII. Tt fin. \[ \sin \frac{2}{n} s + \frac{T-W}{2} \sin \frac{3}{n} s + \frac{V-X}{2} \sin \frac{4}{n} s + \ldots \] Et vis \( \frac{\phi}{x^3} + \frac{\phi k}{k^2} - \frac{\phi k}{x^3} \times \cos \) PSQ, qua urgetur planeta P in Solem secundum radium PS, fiet \( \frac{\phi}{x^3} \text{ in } R - \frac{kS}{2} - kR + \frac{kT}{2} - \frac{t^3}{k^2} - S \times \cos \frac{1}{n} s \) \( - \frac{kS + kV - 2T}{2} \cos \frac{2}{n} s - \frac{kT + kW - 2V}{2} \cos \frac{3}{n} s \) \( - \frac{kV + kX - 2W}{2} \cos \frac{4}{n} s + \ldots \) Q. E. I. **Propositio VI. Problema.** Inaequalitates motûs planetæ exterioris ex viribus praedictis ortas investigare. Per analysim in propositione secundâ institutam vis ad radium PS perpendicularis generabit acceleratiōnem, vel retardationem velocitatis, dum arcus quilibet DP describitur à planeta P, æqualem \( \frac{\phi k n}{t^3} \) in \( b = R - \frac{t^3}{k^3} - \frac{T}{2} \times \cos \frac{1}{n} s - \frac{S-V}{4} \cos \frac{2}{n} s \) \( - \frac{T-W}{6} \cos \frac{3}{n} s - \frac{V-X}{8} \cos \frac{4}{n} s + \ldots \) existente \( b = R - \frac{t^3}{k^3} - \frac{T}{2} + \frac{S-V}{4} + \frac{T-W}{6} \) \( + \frac{V-X}{8} + \ldots \) Deinde si scribatur \( p \) pro vi illâ planetæ Q, qua urgetur planeta P in Solem, prout in propositione præcedente definita est, et \( v \) pro velocitate ascensûs vel descensûs planetæ P secundum radium PS, et jam supponatur supponatur \( SP = x = 1 - Q + K \cos \frac{1}{n} s + L \cos \frac{2}{n} s + M \cos \frac{3}{n} s + N \cos \frac{4}{n} s + \ldots \), &c. existente \( Q = K + L + M + N + \ldots \), &c. erit \( \frac{1}{x^2} + p \) vis centripeta planetæ \( P \), et \( \frac{1}{x} \times \frac{1}{x} + U^2 \) ejusdem vis centrifuga, atque inde habebitur \( v = \frac{1}{x^2} + p - \frac{1}{x} \times \frac{1}{x} - U^2 \times \frac{x^5}{x^2} + U \). Tum restitutis valoribus quantitatum \( U, p, x \), et prosequendo calculum prout in Prop. II. positis \[ A = Kn + \frac{2\phi kn^2}{t^3} \times R - \frac{T}{k^2} - \frac{\phi n}{t^3} \times kR - \frac{T}{k^2} - S + \frac{kT}{2} \] \[ B = L \times \frac{n}{2} + \frac{\phi kn^2}{4t^3} \times S - V - \frac{\phi n}{4t^3} \times kS + kV - 2T \] \[ C = M \times \frac{n}{3} + \frac{\phi kn^2}{9t^3} \times T - W - \frac{\phi n}{6t^3} \times kT + kW - 2V \] \[ D = N \times \frac{n}{4} + \frac{\phi kn^2}{16t^3} \times V - X - \frac{\phi n}{8t^3} \times kV + kX - 2W \] &c. prodibit \( v = \frac{\phi}{t^3} \times R - \frac{kS}{2} - \frac{2\phi kn}{t^3} - Q \times s \) \(+ A \times \sin \frac{1}{n} s + B \times \sin \frac{2}{n} s + C \times \sin \frac{3}{n} s + D \times \sin \frac{4}{n} s + \ldots \), &c. + Z, et factâ hypothesi quòd sit \( v = 0 \) ubi angulus \( PSQ = 0 \), vel \( r \times 180^\circ \), exprimente \( r \) unum ex numeris naturalibus \( 1, 2, 3, 4, \ldots \), &c. erit \( Z = -\frac{\phi}{t^3} \times R - \frac{kS}{2} - \frac{2\phi kn}{t^3} - Q \times s \), \[ T t_2 \] ac proinde \( v = A \times \sin. \frac{1}{n} s + B \times \sin. \frac{2}{n} s + C \times \sin. \frac{3}{n} s + D \times \frac{4}{n} s + \ldots \) Tùm, quia vis centripeta hîc excedere supponitur vim centrifugam, cùm contrarium suppositum fuerit in propositione secundâ, habetur — \( \dot{x} = v \times \frac{\dot{x}}{\dot{x}} + U \) sive — \( \dot{x} = v \dot{s} \) proximè, et — \( \dot{\dot{x}} = v = K \times \frac{1}{n} \sin. \frac{1}{n} s + L \times \frac{2}{n} \sin. \frac{2}{n} s + M \times \frac{3}{n} \sin. \frac{3}{n} s + N \times \frac{4}{n} \sin. \frac{4}{n} s + \ldots \) Unde factâ collatione terminorum hujus valoris velocitatis \( v \) cum terminis homologis valoris supra inventi, emergent \[ K = -\frac{\phi}{t^3} \times \frac{n^2}{n^2-1} \times 2kR - \frac{2t^3}{k^2} \times n - \frac{i}{2} - kT \times n + \frac{i}{2} + S \] \[ L = -\frac{\phi}{2t^3} \times \frac{n^2}{n^2-4} \times kS \times n - i - kV \times n + i + 2T \] \[ M = -\frac{\phi}{3t^3} \times \frac{n^2}{n^2-9} \times kT \times n - \frac{i}{2} - kW \times n + \frac{i}{2} + 3V \] \[ N = -\frac{\phi}{4t^3} \times \frac{n^2}{n^2-16} \times kV \times n - 2 - kX \times n + 2 + 4W \] &c. atque ità patet hujusmodi quantitatum progressio. Innotescet igitur \( x \), seu distantia planetæ \( P \) à Sole in quovis ejus cum planetâ \( Q \) aspectu. Ut obtineatur planetæ \( P \) motus verus \( s \), designet \( w \) motum medium, et cùm sit \( \dot{w} = \frac{\dot{x}s}{\dot{x}} + U \), substi- tuantur quantur valores quantitatum \( x \), \( U \), et sumptâ fluente, positis \[ F = 2nK + \frac{\phi kn^2}{t^3} \times R - \frac{t^3}{k^3} - \frac{T}{2} \] \[ G = nL + \frac{\phi kn^2}{8t^3} \times S - V \] \[ H = \frac{2nM}{3} + \frac{\phi kn^2}{18t^3} \times T - W \] \[ I = \frac{nN}{2} + \frac{\phi kn^2}{32t^3} \times V - X \] \&c. proveniet \( w = 1 - 2Q - \frac{\phi kbn}{t^3} \times s + F \times \sin. \frac{1}{n}s \) \(+ G \times \sin. \frac{2}{n}s + H \times \sin. \frac{3}{n}s + I \times \sin. \frac{4}{n}s + \&c. + Z. \) Et factâ hypothesi quod motus verus coincidat cum medio ubi est \( \frac{1}{n}s \), seu angulus PSQ = o, vel \( r \times 180^\circ \), exhibente \( r \) quemvis ex numeris 1, 2, 3, 4, \&c. erit \( Z = 2Q + \frac{\phi kbn}{t^3} \times s; \) ac proinde, scriptis \[ \frac{1}{n}w, \frac{2}{n}w, \&c. \text{ pro. } \frac{1}{n}s, \frac{2}{n}s, \&c. \text{ quia parùm admodùm differt motus verus à medio, habetur motus verus, five } \] \( s = w - F \times \sin. \frac{1}{n}w - G \times \sin. \frac{2}{n}w - H \times \sin. \frac{3}{n}w \) \(- I \times \sin. \frac{4}{n}w - , \&c. \text{ Q.E.I. } \) Coroll. I. Designet jam planeta P Terram, Q Venerem, et quia posuimus esse distantiam mediocrem Terræ à Sole Sole ad distantiam mediocrem Veneris à Sole ut 1 ad \( k \), erit híc \( k = 0.72333 \), atque \( t = \sqrt{1 + kk} = 1.234182 \). Item est \( n = \frac{Q}{P - Q} = \frac{224.701}{365.2565 - 224.701} = 1.59866 \). Quantitates \( b, R, S, T, \) &c. eosdem híc retinent valores quos habebant in Coroll. I. Prop. II. Verùm, ut motuum Terrestrium accurata institueretur computatio, dignoscere necesse esset effectus aliquos ab actione Veneris provenientes, ex quibus derivare liceret vim attractivam istius planetæ, sed quia specialis hujusmodi effectus nulli, quantum noverimus, observationibus astronomicis explorati habentur, propertèa vim Veneris nunc conjecturâ definiemus, ut inde inæqualitates in motu Telluris computatæ, atque cum observationibus astronomicis collatae inservire posthac possint ad eamdem vim certius determinandam. Itaque supponemus gravitatem in Solem esse ad gravitatem in Venerem, paribus distantiis, ut 400000 ad 1, hoc est, esse \( \phi = \frac{400000}{1} \). Qui tamen valor vis \( \phi \) si major vel minor postèa deprehensus fuerit, in eàdem ratione sequentes omnes determinationes augendæ sunt, vel minuendæ, adeoque ad justam mensuram facillimè reducentur. Erunt igitur \[ \begin{align*} K &= -0.00000575 \\ L &= 0.00001643 \\ M &= 0.00000259 \\ N &= 0.00000090 \\ O &= 0.00000039 \\ O' &= 0.00000022, \text{ &c.} \end{align*} \] Indeque colliguntur \[ \begin{align*} F &= -0.00002459 \\ G &= 0.00002795 \\ H &= 0.00000345 \\ I &= 0.00000105 \\ I' &= 0.00000042 \\ &\text{&c.} \end{align*} \] atque reductis quantitatibus \( F, G, H, \) &c. in partes circuli, circuli, tandem habetur $s = w + 5''\cdot07 \times \text{fin.} \frac{1}{n}w - 5''\cdot76 \times \text{fin.} \frac{2}{n}w - 0''\cdot71 \times \text{fin.} \frac{3}{n}w - 0''\cdot22 \times \text{fin.} \frac{4}{n}w$, &c. ubi $s$ denotat motum Terræ verum, $w$ motum medium, et $\frac{1}{n}w$ angulum PSQ, sive differentiam longitudinum heliocentricarum Terræ et Veneris. Inde computatur sequens tabula exhibens æquationem motûs Solis pro variâ distantiâ Veneris à Terrâ quam metitur angulus PSQ, sive pro variâ differentiâ longitudinum heliocentricarum Terræ et Veneris quam metitur arcus circuli maximi inter Terram et Venerem interjectus et secundum feriem signorum à loco Terræ computatus. Diff. | Diff. long. hel. Terræ et Ven. | Æquatio motûs Solis. | |-------------------------------|---------------------| | Sig. o. | " | | 0° | — 0 | | 10° | 1.6 | | 20° | 2.8 | | 30° | 3.4 | | Diff. long. hel. Terræ et Ven. | Æquatio motûs Solis. | |-------------------------------|---------------------| | Sig. VI. | " | | 0° | — 0 | | 10° | 2.6 | | 20° | 5.0 | | 30° | 7.0 | | Sig. I. | 3.1 | | 20° | 2.1 | | 30° | 0.4 | | Sig. VII. | 8.4 | | 20° | 9.1 | | 30° | 9.2 | | Sig. II. | + 1.6 | | 20° | 3.8 | | 30° | 5.8 | | Sig.VIII. | 8.6 | | 20° | 7.5 | | 30° | 5.8 | | Sig. III. | 7.5 | | 20° | 8.6 | | 30° | 9.2 | | Sig. IX. | 3.8 | | 20° | 1.6 | | 30° | + 0.4 | | Sig. IV. | 9.1 | | 20° | 8.4 | | 30° | 7.0 | | Sig. X. | 2.1 | | 20° | 3.1 | | 30° | 3.4 | | Sig. V. | 5.0 | | 20° | 2.6 | | 30° | 0. | | Sig. XI. | 2.8 | | 20° | 0.6 | | 30° | 0. | Coroll Coroll. II. Si tellus gravitate suâ in Solem in circulo revolvi posse supponatur, adveniente Veneris actione variari debere distantiam ejus à Sole patet ex hac propositione. Esto angulus $\frac{1}{n} s$, seu $PSQ = 90^\circ$, vel $270^\circ$, atque æquatio generalis $x = 1 - Q + K \cos \frac{1}{n} s + L \cos \frac{2}{n} s + M \cos \frac{3}{n} s + \ldots$, &c. in hanc abit $x = 0.9999693$; et si sit $PSQ = 180^\circ$, fit $x = 1.0000053$. Unde si distantia Terræ à Sole, ubi versatur in conjunctione cum Venere, ponatur In quadraturis cum Venere erit ipsius distantia Atque in oppositione Propositio VII. Problema. In systemate duorum planetarum in circulis circa Solem revolventium, motum nodorum orbis planetæ exterioris in plano orbis planetæ interioris investigare. Esto $P$ locus planetæ exterioris (Fig. 5.) in orbe suo $PN$, $SQ$ recta conjungens Solem et planetam interiorem, et dicatur $\phi$ sinus inclinationis duorum orbium ad se invicem ad radium $r$, atque per propositionem quintam est $\frac{\phi k}{z^2} - \frac{\phi}{k^2}$ vis qua planeta $P$ amovetur ab orbe suo secundum directionem parallelam rectæ $SQ$, hujusque vis ea pars quæ perpendiculariter agit agit in planum orbis PN, per simile ratiocinium quo- ut si sumus in Prop. III. prodit æqualis \( c \times \sin. QN \) \( \times \frac{\phi k}{z^3} - \frac{\phi}{k^2} \), et motus intersectionis plani orbis PN cum plano orbis QN fit \( \frac{\phi k}{z^3} - \frac{\phi}{k^2} \times \sin. PN \times \sin. QN \) \( \times Pp \) quo tempore planetæ P describit in orbe suo arcum quam minimum \( Pp \). Deinde si designaverit D locum planetæ P ubi ver- satur in conjunctione cum planetâ interiore, et ponan- tur DP = s, PP = \( \dot{s} \), DN = a, erit PN = \( s + a \), QN = \( s + \frac{1}{n} s + a \) quamproximè, atque sin. PN \( \times \sin. QN = \frac{1}{2} \cos. \frac{1}{n} s - \frac{1}{2} \cos. 2s + \frac{1}{n} s + 2a \). Unde, calculum prosequendo uti in propositione tertiâ, motus nodorum factus, quo tempore planetæ P à loco conjunctionis D discedens descripterit in orbe suo arcum quemlibet \( DP \), exprimetur per \( \frac{\phi kn}{2t^3} \) in \( \frac{S}{2n} s + R - \frac{t^3}{k^3} + \frac{T}{2} \times \sin. \frac{1}{n} s + \frac{S+V}{4} \sin. \frac{2}{n} s \) \( + \frac{T+W}{6} \sin. \frac{3}{n} s + \frac{V+X}{8} \sin. \frac{4}{n} s + \ldots \), &c. \( + \frac{\phi kn}{2t^3} \) in \( Z \times \sin. 2a - R - \frac{t^3}{k^3} \times \frac{1}{2n+1} \sin. 2s + \frac{1}{n} s + 2a \) \( - \frac{S}{2} \times \frac{1}{2n} \sin. 2s + 2a - \frac{S}{2} \times \frac{1}{2n+2} \sin. 2s + \frac{2}{n} s + 2a \) \( - \frac{T}{2} \times \frac{1}{2n-1} \sin. 2s - \frac{1}{n} s + 2a - \frac{T}{2} \times \frac{1}{2n+3} \sin. \) \( 2s + \frac{3}{n} s + 2a - \frac{V}{2} \times \frac{1}{2n-2} \sin. 2s - \frac{2}{n} s + 2a \) \[ \frac{V}{2} \times \frac{1}{2n+4} \text{ fin. } 2s + \frac{4}{n}s + 2a - \frac{W}{2} \times \frac{1}{2n-3} \text{ fin. } 2s - \frac{3}{n}s + 2a - \frac{W}{2} \times \frac{1}{2n+5} \text{ fin. } 2s + \frac{5}{n}s + 2a, \&c. \] existente \( Z = 2n + 1 \) in \( R = \frac{t^3}{k} \times \frac{1}{2n+1} + \frac{s}{2n \times 2n+2} + \frac{T}{2n-1 \times 2n+3} + \frac{V}{2n-2 \times 2n+4} + \frac{W}{2n-3 \times 2n+5} + \&c. \) In quibus seriebus manifesta est terminorum progressio. Q. E. I. **Coroll.** Hinc in conjunctionibus expressio motûs nodi evadit \[ \frac{\phi k}{2t^3} \times \frac{s}{2} s - nZ \times \text{ fin. } 2s + 2a - \text{ fin. } 2a. \] Hicque est motus nodi factus quo tempore planetæ \( P \) et \( Q \) à conjunctione procedentes ad conjunctionem quamvis aliam pervenerint, exhibente \( s \) arcum à planetâ \( P \) in suâ orbitâ interea descriptum. Terminus \[ \frac{\phi k}{2t^3} \times \frac{s}{2} s \] exprimit motum nodi medium, et terminus alter \[ \frac{\phi kn}{2t^3} Z \times \text{ fin. } 2s + 2a - \text{ fin. } 2a \] indicat æquationem periodicam generalem; vel etiam, si conjunction illa à qua desumitur computationis initium, fieri supponatur in nodo, vel propè ad nodum, æquatio periodica generalis fit \[ \frac{\phi kn}{2t^3} Z \times \text{ sin } 2s. \] Designet jam planetæ \( P \) Terram, \( Q \) Venerem, eritque post unam revolutionem synodicam, id est, post revolutionem Veneris ad Terram, \( \frac{1}{n}s = 360^\circ, \) U u 2 proindeque proindeque \( s = n \times 360^\circ = 575^\circ 31' \). Quare motus nodi medius huic temporis spatio congruens fit \( \frac{\phi k n}{4t^3} S \times 360^\circ \), qui imminutus in ratione revolutionis Terræ circa Solem ad ejusdem revolutionem ad Veneream, hoc est, in ratione 1 ad \( n \), evadit \( \frac{\phi k}{4t^3} S \times 360^\circ = 5''.20 \), motus scilicet nodi medius annuus quo regreditur intersecctio planorum orbium Terræ ac Venereis; atque hic motus spatio centum annorum fit 8' 40''. In computo æquationis periodicæ generalis \( \frac{\phi k n}{2t^3} Z \times \sin. 2s \), advertendum est omnes terminos, ex quibus componitur valor quantitatis \( Z \), eodem hinc esse ac in Prop. III. praeter terminum primum \( R - \frac{t^3}{k^3} \times \frac{1}{2n+1} \) qui ob diversum valorem quantitatum \( t \) et \( k \) diversus est. Hic igitur provenit \( Z = 31.59 \), adeoque \( \frac{\phi k n}{2t^3} Z \times \sin. 2s = 5'' \times \sin. 2s \); unde patet æquationem hanc nunquam superare 5''. Motus igitur nodi verus, nimirùm \( \frac{\phi k}{2t^3} \times \frac{S}{2} s - nZ \times \sin. 2s \), peractâ unâ revolutione synodica post conjunctionem factam in nodo, evadit 8'.3 — 5'' × sin. 71°.2', quia tunc est sin. 2s = sin. 2 × 575°.31' = sin. 71°.2'; et per ratiocinium simile ei, quod in Coroll. II. Prop. III. usurpatum est, constabit 8''.3 — 5''.8 × cos. 2r — 1 × 35°.31' exprimere regressum nodi factum tempore illius revolutionis synodicae, cujus locum cum in serie revolutionum indicat numerus \( r \). Hinc computatur tabula sequens quae exhibet regressum nodi orbitae Terrestris in plano orbis Veneris pro duo- decim figillatium revolutionibus synodicis quae proximè sequuntur conjunctionem Terrae et Veneris factam in nodo, vel proximè ad nodum. | In revol. synod. | Regressus nodi Ter. | In revol. synod. | Regressus nodi Ter. | |-----------------|--------------------|-----------------|--------------------| | | " | | " | | 1 | 4 | 7 | 9 | | 2 | 10 | 8 | 14 | | 3 | 14 | 9 | 11 | | 4 | 10 | 10 | 4 | | 5 | 4 | 11 | 3 | | 6 | 3 | 12 | 9 | Patet autem æquationem periodicam specialem, nempe \( 5''.8 \times \cos(2r - 1 \times 35^\circ.31') \), ubi maxima est, evadere \( 5''.8 \), et regressum nodi in quavis revo- lutione Terrae ad Venerem non assurgere ultra \( 14'' \), nec minui citra \( 2''\frac{1}{2} \). **Propositio VIII. Problema.** Iisdem positis, variationem inclinationis orbis planetæ exterioris ad planum orbis planetæ interioris deter- minare. Designet \( I \) variationem inclinationis factam quo tempore planeta \( P \) describit arcum quàm minimum \( Pp \), Pp, et N motum nodi eodem tempore confectum, ac per ratiocinium omnino simile ei quod adhibitum est in propositione quartâ habetur \( I = N \times \frac{c \times \cos. PN}{\sin. PN} \): sed per propositionem præcedentem est \( N = \frac{\varphi k}{x^3} - \frac{\varphi}{k^2} \) \( \times \sin. PN \times \sin. QN \times Pp \), adeoque fit \( I = \frac{\varphi k}{x^3} - \frac{\varphi}{k^2} \) \( \times c \times \cos. PN \times \sin. QN \times Pp \). Unde, cum hic sit \( PN = s + a \), \( QN = s + \frac{1}{n} s + a \), proindeque \( \cos. PN \times \sin. QN = \) \( \frac{1}{2} \sin. \frac{1}{n} s + \frac{1}{2} \sin. 2s + \frac{1}{n} s + 2a \), sumptâ fluente prodit variatio inclinationis genita, quo tempore plau- neta descripsit in orbe suo arcum quemlibet DP à loco conjunctionis D, æqualis \( \frac{\varphi c kn}{2t^2} \) in \( R - \frac{t^3}{k^3} - \frac{T}{2} \) \( \times \sin. \text{vers}. \frac{1}{n} s + \frac{S - V}{4} \sin. \text{vers}. \frac{2}{n} s + \frac{T - W}{6} \sin. \) \text{vers}. \frac{3}{n} s + \frac{V - X}{8} \sin. \text{vers}. \frac{4}{n} s + , &c. + \frac{\varphi c kn}{2t^2} \) in \(- Z \times \sin. \text{vers}. 2a + R - \frac{t^3}{k^3} \times \frac{1}{2n + 1} \sin. \text{vers}. \) \( 2s + \frac{1}{n} s + 2a + \frac{S}{2} \times \frac{1}{2n} \sin. \text{vers}. 2s + 2a + \frac{S}{2} \) \(\times \frac{1}{2n + 2} \sin. \text{vers}. 2s + \frac{2}{n} s + 2a + \frac{T}{2} \times \frac{1}{2n - 1} \) \(\sin. \text{vers}. 2s - \frac{1}{n} s + 2a + \frac{T}{2} \times \frac{1}{2n + 3} \sin. \text{vers}. \) \(\frac{1}{s} + \frac{3}{n} s + 2a + \frac{V}{2} \times \frac{1}{2n - 2} \sin. \text{vers}. 2s - \frac{2}{n} s + 2a \) + \[ + \frac{V}{2} \times \frac{1}{2n+4} \text{fin. vers. } 2s + \frac{4}{n}s + 2a, \text{ &c. Eum-} \] \[ \text{dem hic habet valorem quantitas } Z \text{ ac in propositione præcedente. Q. E. I.} \] **Coroll.** Ubi angulus PSQ est nullus, vel multiplex anguli \(360^\circ\), id est, ubi planetæ versantur in conjunctione, variatio inclinationis genita generatim est \(\frac{\phi ck n}{2t^3} Z\) \(x\) fin. vers. \(2s + 2a - \text{fin. vers. } 2a\) quæ, si ponatur arcus DN \(= a = 0\), fit \(\frac{\phi ck n}{2t^3} Z x \text{ fin. vers. } 2s\). Atque hoc est decrementum inclinationis orbis planetæ P ad orbem planetæ Q factum in qualibet serie revolutionum ad conjunctionem, initio sumpto à conjunctione factâ in nodo, vel prope ad nodum, et designante \(s\) arcum interea à planetâ P in orbe suo descriptum. Si inde computetur decrementum inclinationis orbis Terrestris supra planum orbitæ Veneris factum post quotcumque revolutiones Veneris ad Terram, fiet \(\frac{\phi ck n}{2t^3} Z x \text{ fin. vers. } 2s = 0''.3 x \text{ fin. vers. } 2s\), adeoque hoc decrementum, ubi maximum evadit, non superat \(0''.6\), ac proinde in omni casu negligi potest. LIII. An