Fvndamenta Nova Investigationis Orbitae Verae Qvam Lvna Perlvstrat, Qvibvs Annexa Est Solvtio Problematis Qvatvor Corporvm Breviter Exposita

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Fvndamenta Nova Investigationis Orbitae Verae Qvam Lvna Perlvstrat, Qvibvs Annexa Est Solvtio Problematis Qvatvor Corporvm Breviter Exposita
Peter Andreas Hansen
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Secundum hypothesin statutam ponere licet — iGz=2Fcos(at4-|5) , |^^H=2;W^8in (a^-f^) u)ide 24 Digitized by Google 186 r = A — I e€? 4- 2 Slr^ «(«> cos [xy4- «f-f fl ubi in SDinmatione efiam valor x= o excludendus est, quia termini omnes, i(i quibus x = e est, extra summationis signum appositi sunt, et ubi *(x) pj^4.fr»iiw Positis deinceps in hac aequatione et x-|-l, et — «, et — x — 1 ioco x, elicitur „(-x) _ yj^_jf^^m) de a' .(-'«-I) r^t!l_fF(^+i)B(x4..) ubi F et n^ ubique eaedem quantitates s...unt, quoniam e tc non pendent.
Eliminatis (jleinceps V ei fV ope primae et tertiae harum quatuor aequa- tioQum, nanciscimur F = W = a(x) _ ^(-x) 2xjR(;) q«ae in seeunda et quarta aequatione substitutae subministrant ubi X absolute, hoc est sine respectu signi sui algebraici sumenda est, et ubi M ^ | L de J (x-4-1) JJ(x-+i) K^') x-S»» r =i- Habemus igitor hoc IV de J (x-»-l) J2(''+'>l ^de J *m Digitized by Google 18?
Theorema.
Quotiea F est functid huius farmae r — Zr 4- G ^ cos 9 4- H ^ sin 9 ubi Ly G atque H indolis supra designatae sunt: ope aequationum (^)^ quarum prior ad terminos ipsius JT, m quihus % positiuus^ pBsterior vero ad terminos^ m quibus x negatitms, adhibenda esty termini, in quibus x sine respectu signi sui algebraici maior est quam 1 , ex iis^ m quibus x=l rfx= — 1, faciWma opera computantur, unde in evolutione ipsius r ceteroquin instituenda non nisi ad terminos, in quibus x = o , % = 1 e* X = — 1 est^ respicere opus est Facile quoque reperitur theorema idem locum habere, quoties G for- mae Vsin J^at^P) et H formae W cos (a* + ^) sunt.


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