_Vires determinare quibus perturbatur motus Satellitis circa Primarium
suum revolventis._
Exhibeat jam sphærois prædicta planetam quemvis figurâ hac donatum,
et corpus S satellitem circa planetam tanquàm primarium gyrantem.
Quantitas materiæ globo sphæroidis interiori incumbentis æqualis est
4_bbc_D ⁄ 3_a_ sive 4_bc_D ⁄ 3 proximè, et si materia illa locaretur
in centro sphæroidis C, attraheret satellitem S secundum SC vi 4_bc_D
⁄ 3_l_², quæ reducta ad directionem SD fit 4_bck_D ⁄ 3_l_³, et ad
directionem DC fit 4_bch_D ⁄ 3_l_³. Cum igitur vis 4_bc_D ⁄ 3_l_² non
turbat motum satellitis, utpote quæ tendat ad centrum motûs et quadrato
distantiæ ab eodem centro sit reciprocè proportionalis, vires illæ
4_bck_D ⁄ 3_l_³, 4_bch_D ⁄ 3_l_³, in quas resolvitur, etiam motum non
turbabunt. Itaque ex vi D × _c_ × (4_kb_ ⁄ 3_l_³ - 4_kb_³ ⁄ 5_l_⁵ +
2_khhb_³ ⁄ _l_⁷) auferatur vis 4_bck_D ⁄ 3_l_³, et ex vi D × _c_ ×
(4_hb_ ⁄ 3_l_³ + 2_hb_³ ⁄ 5_l_⁵ - 2_hkkb_³ ⁄ _l_⁷) auferatur 4_bch_D
⁄ 3_l_³, et remanebunt vires D × _c_ × - (4_kb_³ ⁄ 5_l_⁵ + 2_khhb_³ ⁄
_l_⁷), D × _c_ × (2_hb_³ ⁄ 5_l_⁵ - 2_hkkb_³ ⁄ _l_⁷), motuum satellitis
S perturbatrices. Designetur vis D × _c_ × (2_hb_³ ⁄ 5_l_⁵ - 2_hhkb_³ ⁄
_l_⁷) per rectam S_r_ (_Fig. 2._) ac resolvatur in vim S_q_ tendentem
ad centrum planetæ primarii C et ob triangula similia S_rq_, SDC,
æqualem D × _c_ × (2_b_³ ⁄ 5_l_⁴ - 2_kkb_³ ⁄ _l_⁶), existentibus ut
priùs, SD = _k_, DC = _h_, SC = _l_; et in vim _rq_ rectæ SD parallelam
et æqualem D × _c_ × (2_kb_³ ⁄ 5_l_⁵ - 2_k_³_b_³ ⁄ _l_⁷); atque hæc vis
posterior subducta ex vi D × _c_ × - (4_kb_³ ⁄ 5_l_⁵ + 2_khhb_³⁄_l_⁷)
relinquet D × _c_ × 4_kb_³ ⁄ 5_l_⁵ pro vi perturbatrice in directione
SD. Unde cum massa tota planetæ sit 2_ab_D ⁄ 3, gravitas satellitis
tota in planetam erit 2_ab_D ⁄ 3_l_² proximé, vel etiam 2_bb_D ⁄ 3_l_²,
et hæc gravitas est ad vim D × _c_ × 4_kb_³ ⁄ 5_l_⁵ ut 1 ad 6_kbc_ ⁄
5_l_³.
Deinde vis illius D × _c_ × 4_kb_³ ⁄ 5_l_⁵ secundum SD pars ea quæ agit
in directione SC est D × _c_ × 4_kkb_³ ⁄ 5_l_⁶, quæ addita vi Sq dat D
× _c_ × (2_b_³ ⁄ 5_l_⁴ - 6_kkb_³ ⁄ 5_l_⁶) vim perturbatricem tendentem
ad centrum planetæ primarii, atque hæc vis est ad satellitis gravitatem
2_bb_D ⁄ 3_l_² in primarium ut 3_bc_ ⁄ 5_l_² - 9_kkbc_ ⁄ 5_l_⁴ ad 1.
_Q. E. I._