User:Wwoods/Principia149

''Exempl. 3. Assumentes m & n'' pro quibusvis indicibus dignitatum Altudinis, & b, c pro numeris quibusvis datis, ponamus vim centripetam esse ut $$ \frac{bA^m + cA^n}{A\,cub.} $$, id est ut $$ \frac{b\mathrm{ in }\overline{T-X}^m+ c\mathrm{ in }\overline{T-X}^n} {A\,cub.} $$ seu (per eandem Methodum nostram Serierum convergentium) ut $$ \frac{bT^m - mbXT^{m-1} + \frac{mm-m}{2}bX^2T^{m-2} + cT^n - ncXT^{n-1} + \frac{nn-n}{2}cX^2T^{n-2} \;[et]c.}{A\,Cub.} $$ [** For some reason, it chokes on the & in the fraction.] & collatis numeratorum terminis, fiet ''RGq. - RFq. + TFq. ad $$ bT^m + cT^n $$, ut -Fq.'' ad $$ -mbT^{m-1} - ncT^{n-1} + \frac{mm-m}{2}XT^{m-2} + \frac{nn-n}{2}XT^{n-2} $$ &c. Et Sumendo rationes ultimas quæ prodeunt ubi orbes ad formam circularem addedunt, fit Cq. ad $$ bT^{m-1} + cT^{n-1} $$, ut Fq. ad $$ mbT^{m-1} + ncT^{n-1} $$ & vicissim Gq. ad Fq. ut $$ bT^{m-1} + cT^{n-1} $$ ad $$ mbT^{m-1} + ncT^{n-1} $$ Quæ proportio, exponendo altitudinem maximam CV seu T Arithmetice per unitatem, fit Gq. ad Fq. ut b + c ad mb + nc, adeoq; ut 1 ad $$ \frac{mb+nc}{b+c} $$. Unde est G ad F, is est angulus VCp ad angulum VCP, ut 1 ad $$ \sqrt{\frac{mb+nc}{b+c}} $$. Et propterea cum angulus VCP inter Apsidem summam & Apsidem imam in Ellipsi immobili sit 180 gr. erit angulus VCp inter easdem Apsides, in Orbe quem corpus vi centripeta quantitati $$ \frac{bA^m + cA^n}{A\, cub.} $$ proportionali describit, æqualis angulo graduum $$ 180\sqrt{\frac{b+c}{mb+nc}} $$. Et eodem argumento si vis centripeta sit ut $$ \frac{bA^m - cA^n}{A\, cub.} $$, angulus inter Apsides invenietur $$ 180\sqrt{\frac{b-c}{mb-nc}} $$ graduum. Nec secus resolvetur Problema in casibus