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Philosophical transactions, Vol. L. Part II. For the year 1758.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.Various

History

Philosophical transactions, Vol. L. Part II. For the year 1758.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

Name: Philosophical transactions, Vol. L. Part II. For the year 1758.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.
Author: Various

Various

Science -- Periodicals

in integrâ nodi revolutione vi prædictâ genitus. Excessu igitur hoc
minuatur motus nodi lunaris periodicus 360°, et remanebit motus ille
quem generat vis solis.


PROPOSITIO IV.

PROBLEMA.

_Variationem inclinationis orbis lunaris ad planum eclipticæ ex figurâ
terræ spheroidicâ ortam determinare._

Esto ANH (_Fig._ 6.) æquator, AG ecliptica, et A punctum æquinoctii
autumnalis: fit NGRM orbis lunæ secans eclipticam in G et æquatorem in
N, in quo sumantur arcus NL, GR, æquales quadrantibus circuli. Jam
si nodus æquatorius N per temporis particulam vi prædictâ transferri
intelligatur in _n_, et per punctum L describatur circulus _n_L_r_,
exhibebit hic situm orbis lunæ post tempus elapsum, et si in eumdem
demittantur perpendicula N_m_ et R_r_, posterius R_r_ designabit
variationem inclinationis orbitæ lunaris ad eclipticam eodem tempore
genitam. Est autem N_n_: N_m_∷ 1: _m_, itemque N_m_: R_r_∷ 1: sin. LR;
sed ob NL = GR, est NG = LR; unde conjunctis rationibus est N_n_: R_r_∷
1: _m_ × sin. NG; ex quo patet variationem inclinationis momentaneam
esse proportionalem sinui distantiæ nodi lunaris ecliptici à nodo
æquatorio. Ad diametrum NM demittatur perpendiculum GK, et existente
G_h_ decremento arcûs NG facto quo tempore nodus æquatorius N describit
arcum N_n_, agatur _hk_ parallela ipsi GK, eritque 1: GK sive sin.
NG∷ G_h_. K_k_; proindeque jam erit N_n_: R_r_∷ G_h_: _m_ × K_k_,
adeoque summa omnium variationum R_r_, quo tempore nodus eclipticus
G descripsit arcum MG, genitarum erit ad summam totidem motuum N_n_,
hoc est, ad motum nodi æquatorii N eodem tempore factum, ut summa
omnium K_k_ ducta in _m_, ad summam totidem arcuum G_h_, id est, ut
_m_ × MK ad MG. Sit NH motus nodi N tempore revolutionis nodi G ab uno
equinoctio ad alterum, eritque variatio inclinationis eodem tempore
genita, hoc est, variatio tota æqualis ((2_m_ × NH) ⁄ MGN). Unde cùm NH
⁄ MGN exprimat rationem motûs nodi æquatorii ad motum nodi ecliptici,
prodit theorema sequens: _Est motus nodi lunaris ecliptici ad motum
nodi æquatorii, ut sinus duplicatus inclinationis mediocris orbitæ
lunaris ad æquatorem, ad sinum variationis totius inclinationis ejusdem
orbitæ ad eclipticam._

Public-domain text, read in full here on John Shaqi.

Opening the text

History

Philosophical transactions, Vol. L. Part II. For the year 1758.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

Various

Science -- Periodicals

in integrâ nodi revolutione vi prædictâ genitus. Excessu igitur hoc
minuatur motus nodi lunaris periodicus 360°, et remanebit motus ille
quem generat vis solis.


PROPOSITIO IV.

PROBLEMA.

_Variationem inclinationis orbis lunaris ad planum eclipticæ ex figurâ
terræ spheroidicâ ortam determinare._

Esto ANH (_Fig._ 6.) æquator, AG ecliptica, et A punctum æquinoctii
autumnalis: fit NGRM orbis lunæ secans eclipticam in G et æquatorem in
N, in quo sumantur arcus NL, GR, æquales quadrantibus circuli. Jam
si nodus æquatorius N per temporis particulam vi prædictâ transferri
intelligatur in _n_, et per punctum L describatur circulus _n_L_r_,
exhibebit hic situm orbis lunæ post tempus elapsum, et si in eumdem
demittantur perpendicula N_m_ et R_r_, posterius R_r_ designabit
variationem inclinationis orbitæ lunaris ad eclipticam eodem tempore
genitam. Est autem N_n_: N_m_∷ 1: _m_, itemque N_m_: R_r_∷ 1: sin. LR;
sed ob NL = GR, est NG = LR; unde conjunctis rationibus est N_n_: R_r_∷
1: _m_ × sin. NG; ex quo patet variationem inclinationis momentaneam
esse proportionalem sinui distantiæ nodi lunaris ecliptici à nodo
æquatorio. Ad diametrum NM demittatur perpendiculum GK, et existente
G_h_ decremento arcûs NG facto quo tempore nodus æquatorius N describit
arcum N_n_, agatur _hk_ parallela ipsi GK, eritque 1: GK sive sin.
NG∷ G_h_. K_k_; proindeque jam erit N_n_: R_r_∷ G_h_: _m_ × K_k_,
adeoque summa omnium variationum R_r_, quo tempore nodus eclipticus
G descripsit arcum MG, genitarum erit ad summam totidem motuum N_n_,
hoc est, ad motum nodi æquatorii N eodem tempore factum, ut summa
omnium K_k_ ducta in _m_, ad summam totidem arcuum G_h_, id est, ut
_m_ × MK ad MG. Sit NH motus nodi N tempore revolutionis nodi G ab uno
equinoctio ad alterum, eritque variatio inclinationis eodem tempore
genita, hoc est, variatio tota æqualis ((2_m_ × NH) ⁄ MGN). Unde cùm NH
⁄ MGN exprimat rationem motûs nodi æquatorii ad motum nodi ecliptici,
prodit theorema sequens: _Est motus nodi lunaris ecliptici ad motum
nodi æquatorii, ut sinus duplicatus inclinationis mediocris orbitæ
lunaris ad æquatorem, ad sinum variationis totius inclinationis ejusdem
orbitæ ad eclipticam._

Public-domain text, read in full here on John Shaqi.

Philosophical transactions, Vol. L. Part II. For the year 1758.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

Philosophical transactions, Vol. L. Part II. For the year 1758.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

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