Page:Philosophical Transactions - Volume 003.djvu/245

 construendarum ratio: nempe, ut qua ratione augetur Pondus, eadem minuatur Celeritas; quo fiat, ut Factum ex Celeritate & Pondere, eadce Vi movendo, idem sit: puta $$\mathrm V. \mathrm{P C} :: \mathrm V. \mathrm P \times \frac{1}{m} \mathrm C = \mathrm{P C}.$$

9. Si Pondus $$\mathrm P$$, Vi $$\mathrm V$$, Celeritate $$\mathrm C$$, latum, in pondus Quiescens (non impeditum) $$m\mathrm P$$ directe impingat; ferentur utraque Celeritate $$\frac{1}{1 + m} \mathrm C$$. Nam, propter eandem Vim, majori Ponderi movendo adhibitam, eadem ratione minuetur aucti Celeritas: nempe $$\mathrm V. \mathrm{PC} :: \mathrm V. \frac{1 + m}{1} \mathrm P \times \frac{1}{1 + m} \mathrm C = \mathrm{PC}$$. Adeoque Alterius Impetus (intellige factum ex Pondere & Celeritate) fiet $$\frac{1}{1+m} \mathrm{PC}$$; Reliqui $$\frac{1}{1+ m} m \mathrm{PC}$$.

10. Si in Pondus $$\mathrm P$$, (Vi $$\mathrm V$$) Celeritate $$\mathrm C$$ latum, directe impingat aliud, eadem via, majori Celeritate infequens; puta Pondus $$m \mathrm P$$, Celeritate $$n \mathrm C$$, (adeoque Vi $$mn \mathrm V$$ latum; ferentur ambo Celeritate $$\frac{1 + mn}{1 +m} \mathrm C$$. Nam $$\mathrm V. \mathrm {PC} :: mn \mathrm V. mn \mathrm{PC} : \mathrm V + mn \mathrm V = \frac{1+mn}{1} \mathrm V. \frac{1 + mn}{1} \mathrm{PC} = \frac{1+m}{1} \mathrm P \times \frac{1 + mn}{1 + m} \mathrm C.$$ Adeoque præcedentis Impetus fiet $$\frac{1 + mn}{1+ m} \mathrm{PC}$$; subsequentis, $$\frac{1+mn}{1+m} m \mathrm{PC}$$.

11. Si Pondera contrariis Viis lata, sibi directe occurrant sive impingant mutuo, puta, Pondus $$\mathrm P$$ (Vi $$\mathrm V$$) Celeritate $$C$$, dextrorsum; & Pondus $$m \mathrm P$$, Celeritate $$n \mathrm C$$ (adeoque Vi $$m n \mathrm V$$) sinistrorsum: Utriusque Celeritas, Impetus, & directio, sic colliguntur. Pondus dextrorsum latum, reliquo si quiesceret, inferret Celeritatem $$\frac{1}{1 + m}\mathrm C$$, adeoque Impetum $$\frac{1}{1+m} m \mathrm{P C}$$, dextrorsum, sibique retineret hanc eandem Celeritatem, adeoque Impetum $$\frac{1}{1+m} \mathrm{P C}$$ dextrorsum (per Sect. 9.) Pondusque sinistrorsum latum (simili raione) reliquo si quiesceret, inferret Celeritatem $$\frac {mn}{1+m} \mathrm C$$, adeoque Impetum $$\frac{mn}{1+m} \mathrm{PC}$$ sinistrorsum; sibique retineret hanc eandem Celeritatem, adeoque Impetum $$\frac{mn}{1 + m} m \mathrm{PC}$$ sinistrorsum. Cum itaque motus utrinque fiac; Impetus dextrorsum prius lati, jam aggregatus erit ex $$\frac{1}{1+m} \mathrm{PC}$$ dextrorsum, & $$\frac{mn}{1+m} \mathrm{PC}$$ sinistrorsum; adeoque re adse vel dextrorsum vel sinistrorsum, prout ille vel hic major fuerit, eo impetu qui est duorum differentia: h.e. (posito $${}+{}$$ signo dextrorsum, & $$ {} - {} $$ sinistrorsum significante,) Impetus erit Rh