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Mechan. Pars I. Caput VI. Art. III.

451. Problema IL Si medium refifiat in ratione duplicata celeri-tatum, idque ponatur uniforme, & vis sollicitans P conftans, uti prius, di-rectionibus ad AP perpendicularibus agat, determinare curvam projefto-riam.

ResoL. Sit P = £, ceterae denominationes, ut ante. Fietgdx zv zvdxddy . . ds

— - . & gds •+■ 2 vidy ss o; item dv ss — gdy —

R

O dat v — — — r —: quare hoc valo-

ds

vds ..... —

~Y~’ Ex his aequationibus eliminata g jam invenimus (450)«* v —

Sed aequatio gds 2 *+■ 21 ddy = o dat v ss — —-da: 5 a o ± 2dH

. € k pds* cds* -i-

re fubftituto prior abit in — —= —, five f * gdx 4 — — zaddy } &dudla aequatione in dy, e k gdx 2 dyss — zadyddy. Sed sumpta dx conflan-te fit (449) dyddy ss dsdds; igitur e k gdx 2 dy ss — 2 adsids. Ponaturdx

~ - = p, five dx ss pds; erit ob dx conftans dpds + pdds=s o, & dds ss —dpds .

—item dx 2 ■+■ dy* == ds 2 9 dy 2 — d§' —> dx' --- d§ 4 — p a d/’, dy —

/~- .... — 2fldp

dsy 1 — p*. Hifce fubftitutis obtinetur aequatio e e k ds ss ——-jr= ,

ptyi—pp

quae ad conftru&ionem fatis eft. Integra!is vero aequatio proveniet gke * =

ha-

dx

Reddito valore p — ^,

—, quae eft differentiatis primi gra-

c __ a V} —J2. __ aL 1 1/ 1 — P P

p* p

, , S „ adyds ds -i-dy

betur gh > = C - -^r - »L —-

dus, neque fimplicior reddi poteft.

452. CorOLL. I. ^Equatio differentialis tertii gradus illico obtinetur,

fi in aequatione dv ss — gdy — fubftituatur pro dv & v valor ex ae-gis‘ , * gdLrfry

quatione v ss —- Jjjy Haec enim praebet dv = — gdy -t- 5 &

l . , gds 2 d ! y

hmc - giy 4 - ^r- = - S iy

M$’ vel ^

453 - CoROLL. II. Sit finus anguli, quem in A curva cum axe APconftituit, =s m, & altitudo ibidem debita celeritati = b. Pofita s ss o,

fiet t’ as b . Sit dein sumpto finu toto s= 1, 11 = |/i — mm, erit ds : dx