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.

by Various; Royal Society

I have added at the end a proposition concerning the diurnal motion of the earth. This motion has been generally esteemed to be exactly uniform; but as there is a cause that must necessarily somewhat alter it, I was glad to examine what that alteration could amount to. If we first suppose the globe of the earth to be exactly spherical, revolving about its axis in a given time, and afterwards conceive that by the force of the sun or moon raising the waters its figure be changed into that of a spheroid, then according as the axis of revolution becomes a different diameter of the spheroid, the velocity of the revolution must increase or diminish: for, since some parts of the terraqueous globe are removed from the axis of revolution and others depressed towards it, and that in a different proportion as the sun or moon approaches to or recedes from the equator, when the whole quantity of motion which always remains the same is distributed through the spheroid, the velocity of the diurnal rotation cannot be constantly the same. This variation however will scarce be observable, but as it is real, it may not be thought amiss to determine what its precise quantity is.

I am sensible the following theory, as far as it relates to the motion of Jupiter’s satellites, is imperfect and might be prosecuted further; but being hindered at present from such pursuit by want of health and other occupations, I thought I might send it you in the condition it has lain by me for some time. You can best judge how far it may be of use, and what advantage might arise from further improvements in it. I am glad to have this opportunity of giving a fresh testimony of that regard which is due to your distinguished merit, and of professing myself with the highest esteem,

Reverend Sir,
Your very humble Servant,
C. Walmesley.

Bath, Oct. 21. 1758.

Lemma I.

Invenire gravitatem corporis longinqui ad circumferentiam circuli ex particulis materiæ in duplicatâ ratione distantiarum inversè attrahentibus constantem.

ESto NIK (Vid. Tab. [xxxiii.] Fig. 1.) circumferentia circuli, in cujus puncta omnia gravitet corpus longinquum S locatum extra planum circuli. In hoc planum agatur linea perpendicularis SH, et per circuli centrum X ducatur recta HXK secans circulum in I et K, et SR parallela ad HX: producatur autem SH ad distantiam datam SD, et agantur rectæ DC, XC, ipsis HX, SD, parallelæ. Tum ductâ chordâ quavis MN ad diametrum IK normali eamque secante in L, ex punctis M, N, demittantur in SR perpendiculares MR, NR, concurrentes in R; junctisque SM, SN, erit SM = SN, MR = NR, SR = HL. Dicantur jam SD, k; HX sive DC, h; XL, x; CX, z; XI, r; eritque HL = h - x, et SH = k - z. Est autem SM ad SH ut attractio 1 ⁄ (SM)² corporis S versus particulam M in directione SM ad ejusdem corporis attractionem in directione SH, quæ proinde erit SH ⁄ (SM)³: sed est SR = HL, et (SM)² = (SR)² + (MR)² = (SR)² + (SH)² + (ML)²; unde sit SH ⁄ (SM)³ = SH ⁄ ((HL)² + (SH)² + (ML)²)³⁄ ², et ductâ mn parallelâ ad MN, vis qua corpus S attrahitur ad arcus quàm minimos Mm, Nn, exponitur per SH × 2Mm ⁄ (SM)³ = SH × 2Mm × ((HL)² + (SH)² + (ML)²) ⁻³⁄ ². Est autem (HL)² + (SH)² + (ML)² = kk - 2kz + zz + hh - 2hx + rr, hincque ponendo kk + hh = ll, ((HL)² + (SH)² = (ML)²)⁻³⁄² = 1 ⁄ l³ + 3kz ⁄ l⁵ + 3hx ⁄ l⁵ - 3rr ⁄ 2l⁵ - 3zz ⁄ 2l⁵ + 15kkzz ⁄ 2l⁷ + 15khzx ⁄ 2l⁷ + 15hhxx ⁄ 2l⁷, neglectis terminis ulterioribus ob longinquitatem quam supponimus corporis S. Quarè, si scribatur d pro circumferentiâ IMKN, gravitas corporis S ad totam illam circumferentiam secundum SH, sive fluens fluxionis SH × 2Mm × ((HL)² + (SH)² + (ML)²) ⁻³⁄ ² evadit (k - z) × d in 1 ⁄ l³ + 3kz ⁄ l⁵ - 3rr ⁄ 2l⁵ - 3zz ⁄ 2l⁵ + 15kkzz ⁄ 2l⁷ + 15hhrr ⁄ 4l⁷. Simili modo obtinebitur gravitas ejusdem corporis S secundum SR. Q. E. I.

Lemma II.

Corporis longinqui gravitatem ad Sphæroidem oblatam determinare.

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 Xi descriptam, æqualis gravitati in lemmate præcedenti definitæ ductæ in rectam Ii. Sed est Ii. c∷ IX. a, atque d. D∷ IX. a; unde Ii × d. D × c∷ (IX)². aa, hoc est, ex naturâ ellipseos, ob CX = z, et IX = r, Ii × d. D × c∷ bb - zz. bb, adeoque Ii × 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 ⁄ l³ + 3kz ⁄ l⁵ - 3bb ⁄ 2l⁵ - 15zz ⁄ 4l⁵ + 15bbhh ⁄ 4l⁷ + 45kkzz ⁄ 4l⁷. 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 2k ⁄ l³ - 6kzz ⁄ l⁵ - 3kbb ⁄ l⁵ + 15k³zz ⁄ l⁷ + 15hhkbb ⁄ 2l⁷ - 15hhkzz ⁄ 2l⁷. 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) × (4kb ⁄ 3l³ - 4kb³ ⁄ 5l⁵ + 2khhb³ ⁄ l⁷). Simili ratiocinio gravitatio corporis S in eamdem materiam secundum directionem SR æquatori parallelam invenitur æqualis D × c × (4hb ⁄ 3l³ + 2hb³ ⁄ 5l⁵ - 2hkkb³ ⁄ l⁷). Tum si addatur gravitatio corporis S in globum interiorem, ex unâ parte scilicet 2b³kD ⁄ 3al³, et ex alterâ 2b³hD ⁄ 3al³, habebitur gravitas corporis S in totum sphæroidem. Q. E. I.