TAUTOCHRONIS IN VACUO. 55 



cujus circuli gcncratoris diamctcr cft d. Defcendat 

 corpus in hac cycloide cx pundo E acque ako ac C 

 TclD-, erit velocitas cjus in L \t v\A-.v). Ergo 

 tcmpufculum pcr L/ c(t dv : V(h-x). Id quod igi- 

 tur acqualc ell fummae tempulculorum per clemen- 

 ta Mm, Nn. Qinre totum tempus defcenfus per 

 ELA, acquale erit fummae tcmporum per arcus CA 

 ct DA. Ofcillatio ergo pcr CAD contemporancii 

 c(l dimidiac ofcillationi penduli longitudinis 2^, fcu 

 intcgrac ofciliationi pcnduli long. -^a. 



§. II. Ex his jam facile apparet, quomo- 

 do data altera curua AC inueniri dcbeat altera 

 AD. Sit quaefuae AD applicata PNzz s •, crit 

 Nfi—dt—V(/x-^^z-. Erit igitur (Js-\-V(dx'-\-(iz-) 

 —dxV{.i : .V). Vnde V((lx--^dz-)—dxV(a : x)-ds. 

 Dcnique dz—V(adx- : x-\-ds--dx'^-zdsdxV(a : .r) ). 

 Ciim curua A.MC fit data, dabitur ds iu .v ct dx; 

 ponitur i^itur dsrzzpdx. Erit dz—d.xV(a:x-\-p-—i 

 — ipl'(ii : .V) ). Quae acquatio, cum p in .v dari po- 

 natur, exprimct naturam curuae AND quacfitac. 

 Hinc intclligitur, cum a non a curiui pendcat , ct 

 idco pro Uibitu accipi poffit, infinitas inueniri pofle 

 curuas loco quacfitae AND, (juac cum AAIC jundae 

 tautochronas pracbeant. Notandum tamcn accide- 

 rc cafus, quibus, fi rt quantitate quadam minor ac- 

 cipiatur, curua quacfita fiat immnginaria. 



§. i£. Sit curua data AC liuca recfla, cum Fig. 2, 

 vcrticali AB anguhmi qucmcunqucBAC couilitucns, 

 cnt ds:z-ndx j n dcnocaiuc oumcrum ci angulo con- 



uc- 



