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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

Retentis iis quæ sunt in lemmate superiori demonstrata; esto C centrum
sphæroidis, cujus æquatori parallelus sit circulus IMK. Sphæroidis
hujus semiaxis major sit _a_, semiaxis minor _b_, eorum differentia
_c_, quam exiguam esse suppono; et dicatur D circumferentia æquatoris.
Centro C et radio æquali semiaxi minori describi concipiatur
circulus qui secet IK in _i_, eritque gravitas in directione SD, qua
urgetur corpus S versus materiam sitam inter circumferentiam IMKN et
circumferentiam centro X et radio X_i_ descriptam, æqualis gravitati
in lemmate præcedenti definitæ ductæ in rectam I_i_. Sed est I_i_.
_c_∷ IX. _a_, atque _d_. D∷ IX. _a_; unde I_i_ × _d_. D × _c_∷ (IX)².
_aa_, hoc est, ex naturâ ellipseos, ob CX = _z_, et IX = _r_, I_i_
× _d_. D × _c_∷ _bb_ - _zz_. _bb_, adeoque I_i_ × _d_ = (D × _c_) ⁄
(_bb_) × (_bb_ - _zz_), atque _rr_ = _aa_ - (_aazz_) ⁄ (_bb_); scribi
autem potest in sequenti calculo _bb_ - _zz_ pro _rr_ ob parvitatem
differentiæ semiaxium in quam omnes termini ducuntur. Gravitas igitur
corporis S in materiam inter circumferentias supradictas consistentem
exprimetur per (D × _c_) ⁄ (_bb_) × (_bb_ - _zz_) × (_k_ - _z_) in 1
⁄ _l_³ + (3_kz_) ⁄ _l_⁵ - (3_bb_) ⁄ (2_l_⁵) - (15_zz_) ⁄ (4_l_⁵) +
(15_bbhh_) ⁄ (4_l_⁷) + (45_kkzz_) ⁄ (4_l_⁷). Et si addatur gravitas
in similem materiam ex alterâ parte centri C ad æqualem à centro
distantiam, quia tunc CX sive _z_ evadit negativa, gravitas corporis S
in hanc duplicem materiam erit (D × _c_) ⁄ _bb_ × (_bb_ - _zz_) in 2_k_
⁄ _l_³ - 6_kzz_ ⁄ _l_⁵ - 3_kbb_ ⁄ _l_⁵ + 15_k_³_zz_ ⁄ _l_⁷ + 15_hhkbb_
⁄ 2_l_⁷ - 15_hhkzz_ ⁄ 2_l_⁷. Ducatur jam gravitas hæc in _ż_, et sumptâ
gravitatum omnium summâ, factâ _z_ = _b_, gravitatio tota corporis S
in totam materiam globo interiori superiorem secundum directionem SD
æquatori perpendicularem prodit (D × _c_) × (4_kb_ ⁄ 3_l_³ - 4_kb_³ ⁄
5_l_⁵ + 2_khhb_³ ⁄ _l_⁷). Simili ratiocinio gravitatio corporis S in
eamdem materiam secundum directionem SR æquatori parallelam invenitur
æqualis D × _c_ × (4_hb_ ⁄ 3_l_³ + 2_hb_³ ⁄ 5_l_⁵ - 2_hkkb_³ ⁄ _l_⁷).
Tum si addatur gravitatio corporis S in globum interiorem, ex unâ parte
scilicet 2_b_³_k_D ⁄ 3_al_³, et ex alterâ 2_b_³_h_D ⁄ 3_al_³, habebitur
gravitas corporis S in totum sphæroidem. _Q. E. I._


COROLL.

Igitur gravitas corporis S secundum SD est ad ejusdem gravitatem
secundum SR sive DC in materiam sphæroidis globo interiori incumbentem
ut 2_k_ ⁄ 3 - 2_kb_² ⁄ 5_l_² + _khhb_² ⁄ _l_⁴ ad 2_h_ ⁄ 3 + _hb_² ⁄
5_l_² - _hkkb_² ⁄ _l_⁴, adeoque si gravitas prior exponatur per _k_,
posterior exprimetur per _h_ - 3_hb_² ⁄ 5_l_² quamproximè. Unde cum
sit DC = _h_, patet gravitatem corporis S in sphæroidem oblatam non
tendere ad centrum C, sed ad punctum _c_ rectæ DC in plano æquatoris
jacentis vicinius puncto D.


PROPOSITIO I.

PROBLEMA.

_Vires determinare quibus perturbatur motus Satellitis circa Primarium
suum revolventis._

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

Retentis iis quæ sunt in lemmate superiori demonstrata; esto C centrum
sphæroidis, cujus æquatori parallelus sit circulus IMK. Sphæroidis
hujus semiaxis major sit _a_, semiaxis minor _b_, eorum differentia
_c_, quam exiguam esse suppono; et dicatur D circumferentia æquatoris.
Centro C et radio æquali semiaxi minori describi concipiatur
circulus qui secet IK in _i_, eritque gravitas in directione SD, qua
urgetur corpus S versus materiam sitam inter circumferentiam IMKN et
circumferentiam centro X et radio X_i_ descriptam, æqualis gravitati
in lemmate præcedenti definitæ ductæ in rectam I_i_. Sed est I_i_.
_c_∷ IX. _a_, atque _d_. D∷ IX. _a_; unde I_i_ × _d_. D × _c_∷ (IX)².
_aa_, hoc est, ex naturâ ellipseos, ob CX = _z_, et IX = _r_, I_i_
× _d_. D × _c_∷ _bb_ - _zz_. _bb_, adeoque I_i_ × _d_ = (D × _c_) ⁄
(_bb_) × (_bb_ - _zz_), atque _rr_ = _aa_ - (_aazz_) ⁄ (_bb_); scribi
autem potest in sequenti calculo _bb_ - _zz_ pro _rr_ ob parvitatem
differentiæ semiaxium in quam omnes termini ducuntur. Gravitas igitur
corporis S in materiam inter circumferentias supradictas consistentem
exprimetur per (D × _c_) ⁄ (_bb_) × (_bb_ - _zz_) × (_k_ - _z_) in 1
⁄ _l_³ + (3_kz_) ⁄ _l_⁵ - (3_bb_) ⁄ (2_l_⁵) - (15_zz_) ⁄ (4_l_⁵) +
(15_bbhh_) ⁄ (4_l_⁷) + (45_kkzz_) ⁄ (4_l_⁷). Et si addatur gravitas
in similem materiam ex alterâ parte centri C ad æqualem à centro
distantiam, quia tunc CX sive _z_ evadit negativa, gravitas corporis S
in hanc duplicem materiam erit (D × _c_) ⁄ _bb_ × (_bb_ - _zz_) in 2_k_
⁄ _l_³ - 6_kzz_ ⁄ _l_⁵ - 3_kbb_ ⁄ _l_⁵ + 15_k_³_zz_ ⁄ _l_⁷ + 15_hhkbb_
⁄ 2_l_⁷ - 15_hhkzz_ ⁄ 2_l_⁷. Ducatur jam gravitas hæc in _ż_, et sumptâ
gravitatum omnium summâ, factâ _z_ = _b_, gravitatio tota corporis S
in totam materiam globo interiori superiorem secundum directionem SD
æquatori perpendicularem prodit (D × _c_) × (4_kb_ ⁄ 3_l_³ - 4_kb_³ ⁄
5_l_⁵ + 2_khhb_³ ⁄ _l_⁷). Simili ratiocinio gravitatio corporis S in
eamdem materiam secundum directionem SR æquatori parallelam invenitur
æqualis D × _c_ × (4_hb_ ⁄ 3_l_³ + 2_hb_³ ⁄ 5_l_⁵ - 2_hkkb_³ ⁄ _l_⁷).
Tum si addatur gravitatio corporis S in globum interiorem, ex unâ parte
scilicet 2_b_³_k_D ⁄ 3_al_³, et ex alterâ 2_b_³_h_D ⁄ 3_al_³, habebitur
gravitas corporis S in totum sphæroidem. _Q. E. I._


COROLL.

Igitur gravitas corporis S secundum SD est ad ejusdem gravitatem
secundum SR sive DC in materiam sphæroidis globo interiori incumbentem
ut 2_k_ ⁄ 3 - 2_kb_² ⁄ 5_l_² + _khhb_² ⁄ _l_⁴ ad 2_h_ ⁄ 3 + _hb_² ⁄
5_l_² - _hkkb_² ⁄ _l_⁴, adeoque si gravitas prior exponatur per _k_,
posterior exprimetur per _h_ - 3_hb_² ⁄ 5_l_² quamproximè. Unde cum
sit DC = _h_, patet gravitatem corporis S in sphæroidem oblatam non
tendere ad centrum C, sed ad punctum _c_ rectæ DC in plano æquatoris
jacentis vicinius puncto D.


PROPOSITIO I.

PROBLEMA.

_Vires determinare quibus perturbatur motus Satellitis circa Primarium
suum revolventis._

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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