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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

Designet CK (_Fig._ 3.) lineam intersectionis planorum æquatoris
planetæ et orbitæ satellitis, et resolvatur vis SD = 6_kbc_ ⁄
5_l_³, quæ agit perpendiculariter ad planum æquatoris, in vim DR
perpendicularem ad planum orbitæ satellitis, et in vim SR jacentem
in eodem plano. Producatur SR donec occurrat CK in K, eritque SK
normalis ad CK, et planum SDK normale ad planum orbis satellitis;
ac proptereà ob similia triangula SDK, SRD, si _m_ denotet sinum ad
radium 1 et _n_ cosinum anguli SKD, inclinationis scilicet orbitæ
satellitis ad æquatorem planetæ, erit DR = SD × _n_ = 6_kbcn_ ⁄
5_l_³, et SR = SD × _m_ = 6_kbcm_ ⁄ 5_l_³, existente 1 gravitate totâ
satellitis in primarium suum. Jam quoniam vis SR jacet in plano orbitæ
satellitis, hujus plani situm non mutat; accelerat quidem vel retardat
motum satellitis revolventis, sed hæc acceleratio vel retardatio ob
brevitatem temporis ad quantitatem sensibilem non exurgit: vis DR eidem
plano perpendicularis continuò mutat ejus situm, et motum nodi generat,
quem sequenti propositione definiemus.


PROPOSITIO II.

PROBLEMA.

_Invenire motum nodi ex prædictâ causâ oriundum._

Per motum nodi in hac propositione intelligo motum intersectionis
planorum æquatoris planetæ et orbitæ satellitis; orbitam autem
satellitis quamproximé circularem suppono. Esto S locus satellitis in
orbe suo SN cujus centrum C, (_Fig._ 4.) SF arcus centro C descriptus
perpendicularis in circulum æquatoris planetæ FN; SB arcus eodem
centro descriptus perpendicularis ad orbem SN, atque in SB sumatur
lineola S_r_ æqualis duplo spatio, quod satelles percurrere posset
impellente vi DR in Coroll. præced. determinatâ, quo tempore in
orbe suo describeret arcum quàm minimum _p_S: per puncta _r_, _p_,
describatur centro C circulus _rpn_ secans equatorem in _n_, qui
exhibebit situm orbitæ satellitis post illam particulam temporis, nodo
N translato in _n_. Agantur SC, CN, et SH perpendicularis in lineam
nodorum CN, et N_m_ perpendicularis in _rpn_. Jam cum sint lineolæ
S_r_, N_m_, ut sinus arcuum S_p_, SN, erit S_p_. S_r_∷ SH. N_m_; deinde
in triangulo rectangulo N_mn_ habetur _m_. 1∷ N_m_. N_n_; unde per
compositionem rationum S_p_ × _m_. S_r_∷ SH. N_n_ = ((S_r_ × SH) ⁄
(S_p_ × _m_)): dato igitur arcu S_p_, est N_n_ sive motus nodi ut S_r_
× SH. In triangulo sphærico rectangulo SFN est sinus anguli N, hoc est,
anguli inclinationis orbitæ satellitis ad æquatorem planetæ, ad sinum
arcûs SF, ut radius ad sinum arcûs SN, id est, _m_. (_k ⁄ l_)∷ 1. SH,
adeoque _k ⁄ l_ = _m_ × SH; est igitur _k ⁄ l_ ut SH. Vis autem S_r_
per Coroll. Prop. præced. est ut _k ⁄ l_, adeoque ut SH; quamobrem
est S_r_ × SH, proindeque et N_n_, ut (SH)², hoc est, motus horarius
nodi vi præfatâ genitus est in duplicatâ ratione distantiæ satellitis
à nodo. Et quoniam summa omnium (SH)², quo tempore satelles periodum
suam absolvit, est dimidium summæ totidem (SC)², ideò motus periodicus
est subduplus ejus qui, si satelles in declinatione suâ maximâ ab

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

Designet CK (_Fig._ 3.) lineam intersectionis planorum æquatoris
planetæ et orbitæ satellitis, et resolvatur vis SD = 6_kbc_ ⁄
5_l_³, quæ agit perpendiculariter ad planum æquatoris, in vim DR
perpendicularem ad planum orbitæ satellitis, et in vim SR jacentem
in eodem plano. Producatur SR donec occurrat CK in K, eritque SK
normalis ad CK, et planum SDK normale ad planum orbis satellitis;
ac proptereà ob similia triangula SDK, SRD, si _m_ denotet sinum ad
radium 1 et _n_ cosinum anguli SKD, inclinationis scilicet orbitæ
satellitis ad æquatorem planetæ, erit DR = SD × _n_ = 6_kbcn_ ⁄
5_l_³, et SR = SD × _m_ = 6_kbcm_ ⁄ 5_l_³, existente 1 gravitate totâ
satellitis in primarium suum. Jam quoniam vis SR jacet in plano orbitæ
satellitis, hujus plani situm non mutat; accelerat quidem vel retardat
motum satellitis revolventis, sed hæc acceleratio vel retardatio ob
brevitatem temporis ad quantitatem sensibilem non exurgit: vis DR eidem
plano perpendicularis continuò mutat ejus situm, et motum nodi generat,
quem sequenti propositione definiemus.


PROPOSITIO II.

PROBLEMA.

_Invenire motum nodi ex prædictâ causâ oriundum._

Per motum nodi in hac propositione intelligo motum intersectionis
planorum æquatoris planetæ et orbitæ satellitis; orbitam autem
satellitis quamproximé circularem suppono. Esto S locus satellitis in
orbe suo SN cujus centrum C, (_Fig._ 4.) SF arcus centro C descriptus
perpendicularis in circulum æquatoris planetæ FN; SB arcus eodem
centro descriptus perpendicularis ad orbem SN, atque in SB sumatur
lineola S_r_ æqualis duplo spatio, quod satelles percurrere posset
impellente vi DR in Coroll. præced. determinatâ, quo tempore in
orbe suo describeret arcum quàm minimum _p_S: per puncta _r_, _p_,
describatur centro C circulus _rpn_ secans equatorem in _n_, qui
exhibebit situm orbitæ satellitis post illam particulam temporis, nodo
N translato in _n_. Agantur SC, CN, et SH perpendicularis in lineam
nodorum CN, et N_m_ perpendicularis in _rpn_. Jam cum sint lineolæ
S_r_, N_m_, ut sinus arcuum S_p_, SN, erit S_p_. S_r_∷ SH. N_m_; deinde
in triangulo rectangulo N_mn_ habetur _m_. 1∷ N_m_. N_n_; unde per
compositionem rationum S_p_ × _m_. S_r_∷ SH. N_n_ = ((S_r_ × SH) ⁄
(S_p_ × _m_)): dato igitur arcu S_p_, est N_n_ sive motus nodi ut S_r_
× SH. In triangulo sphærico rectangulo SFN est sinus anguli N, hoc est,
anguli inclinationis orbitæ satellitis ad æquatorem planetæ, ad sinum
arcûs SF, ut radius ad sinum arcûs SN, id est, _m_. (_k ⁄ l_)∷ 1. SH,
adeoque _k ⁄ l_ = _m_ × SH; est igitur _k ⁄ l_ ut SH. Vis autem S_r_
per Coroll. Prop. præced. est ut _k ⁄ l_, adeoque ut SH; quamobrem
est S_r_ × SH, proindeque et N_n_, ut (SH)², hoc est, motus horarius
nodi vi præfatâ genitus est in duplicatâ ratione distantiæ satellitis
à nodo. Et quoniam summa omnium (SH)², quo tempore satelles periodum
suam absolvit, est dimidium summæ totidem (SC)², ideò motus periodicus
est subduplus ejus qui, si satelles in declinatione suâ maximâ ab

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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