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  "text": "I. De Pressionibus Ponderum in Machinis motis.\n\nRead at the Royal Society, June 27, 1754.\n\nNimus erat aliquando in legem resistentiæ, quam patiuntur corpora in superficie aquæ mota, inquirendi. Suasit hoc cura, quam dudum mihi imposuit officii ratio, scientiarum navallium, quarum pleræque vel ipsi resistentiæ theoriæ innituntur, vel ita sunt cum eadem connexæ, ut resistentiam ipsam supponant cognitam. Ratiociniis et calculo interdum insuperabili fidere nolui, experimenta licet plurima pro omni casu non sufficere vidi; experimenta cum ratiociniis ea propter conjungere statui, modumque quæsivi per experimenta in leges resistentiæ inquirendi.\n\nRepresentavi mihi corpus specifice levius aquæ stagnanti ad certam immersum profunditatem, idemque filo duas ambiente trocleas potentiæ ita junctum, ut, hâc suo præpondio verticaliter descendentе, motu illud horizontali donaretur, quæsivi ex tempore spatio et ponderibus datis, resistentiam pro quovis corpore determinare.\n\nNon licuit heic, ut moris est, abstrahere à fricationibus,\ntionibus, à rigiditate fili, ab inertia materiæ: introducenda erant hæc omnia in expressione vis acceleratricis, si ducta illa in elementum temporis monstraret verum celeritatis incrementum.\n\nAt nemo erat, quantum constitit, qui ita dilucidavit theoriam frictionis ex incremento pressionis in machinis motis oriundæ, ut nexus cum primis mechanicæ principiis dilucide pateret; quod me invitavit ad indagandam solutionem, quam non ut omni numero perfectam, sed potius ab eruditis et meliora edoctis corrigendam et ulterius perficiendam, Illusterrimæ Societati proponam.\n\nRepræsentat adjecta figura axem in peritrochio. Sit potentia movens \\( A \\), distantia ipsius à centro motus \\( a \\). Sit quoque pondus \\( B \\), ejusque à centro distantia \\( b \\). Sit radius axis, in quem frictio cadit = \\( c \\). Pondus machinæ = \\( M \\) distantia centri virium a centro gravitatis = \\( d \\). Quæritur jam pressio in axem, cum potentia descendens \\( A \\) machinam agit.\n\nSi jam pressio oriunda ex descendente potentia \\( A \\), seu illa qua filum tenditur ad latus \\( \\alpha \\), appelletur \\( \\pi \\), erit ob actionis et reactionis æqualitatem pressio seu tensio ad alterum latus \\( \\beta = \\frac{\\pi a}{b} \\), unde integra pressio, excluso pondere machinæ funisque, = \\( \\pi + \\frac{\\pi a}{b} = (1 + \\frac{a}{b}) \\pi \\). Sit jam constans ratio pressionis ad frictionem ut \\( 1 : \\mu \\); erit frictio \\( = (1 + \\frac{a}{b}) \\pi \\mu \\); et momentum hujus frictionis \\( = (1 + \\frac{a}{b}) c \\pi \\mu \\): momentum vero frictionis ex pondere machinæ \\( = M c \\mu \\); quod priori momento adjectum dat \\( ((1 + \\frac{a}{b}) \\pi + M) c \\mu \\); unde momentum potentiae moventis\nmoventis = Aa - Bb - \\left(1 + \\frac{a}{b}\\right) \\pi + M)c \\mu.\n\nCum vero momentum inertiae sit \\(Aa^2 + Bb^2 + Md^2\\),\nerit vis acceleratrix \\(= \\frac{Aa - Bb - \\left(1 + \\frac{a}{b}\\right) \\pi + M)c \\mu}{Aa^2 + Bb^2 + Md^2}\\)\n\net pro acceleratione puncti \\(a\\), seu potentiæ moventis \\(A\\), habetur per principia mechanicæ:\n\n\\[\n\\left(\\frac{Aa^2 - Ba b - \\left(1 + \\frac{a}{b}\\right) \\pi + M)a c \\mu}{Aa^2 + Bb^2 + Md^2}\\right) dt = dc; \\text{ ubi } dt \\text{ significat elementum temporis; } dc \\text{ vero incrementum velocitatis. Si autem } A \\text{ liberé cecidifset, fuisset } \\frac{A}{A} dt = dt'. \\text{ Cum autem incrementa vel decrementa velocitatum eadem temporis particula in eodem corpore genita sint ut vires generantes, licebit inferre ut } dt': dt' - dc = A: \\text{ ad vim generantem decrementum celeritatis } dt' - dc, \\text{ quæ eadem vis est, quæ lapsum corporis retardat, filum tendit, et ad latus } a \\text{ premit; unde substitutis valoribus habetur analogia sequens}\n\n\\]\n\n\\[\n1:1 - \\frac{Aa^2 - Ba b - \\left(1 + \\frac{a}{b}\\right) \\pi + M)a c \\mu}{Aa^2 + Bb^2 + Md^2} = A: \\pi\n\\]\n\nideoque \\(\\pi = \\frac{ABb^2 + AMd^2 + ABab + \\left(1 + \\frac{a}{b}\\right) \\pi + M)Aac \\mu}{Aa^2 + Bb^2 + Md^2}\\)\n\nex qua æquatione invenitur \\(\\pi = \\frac{ABb^2 + AMd^2 + ABab + AMac \\mu}{Aa^2 + Bb^2 + Md^2 - \\left(1 + \\frac{a}{b}\\right)Aac \\mu}\\)\n\nB 2\net \\(\\frac{\\pi e}{b} = A B a b + A M d^2 a^2 + A B a^2 + A M a^2 c \\mu\\)\n\n\\[\n\\frac{A a^2 + B b^2 + M d^2 - (1 + \\frac{a}{b}) A a c \\mu}{A a^2 + B b^2 + M d^2 - (1 + \\frac{a}{b}) A a c \\mu}\n\\]\n\net pressio integra \\(= \\pi + \\frac{\\pi a}{b} =\\)\n\n\\[\n\\frac{A B (a + b)^2 + A M (d^2 + a c \\mu) (1 + \\frac{a}{b})}{A a^2 + B b^2 + M d^2 - (1 + \\frac{a}{b}) A a c \\mu}\n\\]\n\nSi jam frictionem et pondus machinæ excludere placuerit, habetur pressio integra \\(= \\frac{A B (a + b)^2}{A a^2 + B b^2}\\): et si,\n\nut in troclea evenit, supponatur \\(a = b\\), erit pressio integra \\(= \\frac{A B (a + a)^2}{(A + B) a^2} = 4 A B\\)\n\n\\[\n\\frac{A B (a + a)^2}{(A + B) a^2} = 4 A B\n\\]\n\nChristianus Hée,\nProfessor Mathef. et Phys. 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    "identifier": "jstor-104894",
    "title": "De Pressionibus Ponderum in Machinis Motis",
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    "year": 1755,
    "volume": "49",
    "journal": "Philosophical Transactions (1683-1775)",
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