IY2

CAPUT FI II.

arcum nullus circulus poiTit transire, feu is minori radio fO, feu majorigO describi cogitetur (431. 432. §.): ob hanc causam dicetur circulus cur-vedinis datae curvae in punfto O verfus X osculari arcum OX; hinc ap-pellabitur is circulus osculator-, ejusque radius tO radius osculi, vel radiuscirculi osculatoris.

434. Corollarium 6.

Circulus radio tO descriptus arcumque OXFdatae curvae in 0 oscu-lans tangit eundem arcum in punfto O interne vel externe (433. §.).

435. Theorema.

Si in data curva MOX fuerit fumma quadratorum ordinatae m s etabfcffae Os divifa per duplam abfcijfam Os aequalis Juturnae Z + ^ ve ^differentiae Z — \f/ duarum quantitatum Z, i/s quarum prior independensfitab abfcffa Os, pofierior vero ita ab ea pendeat, ut decrefcente abfcffa Ospojfit fieri minor data quavis quantitate; erit Z —tO radius circuli cur -vedinis in punfto O, qui arcum O X tanget in punfto O interne cafu primo,et externe cafu fecundo.

D e m 011 ft r a t i o.

tn s 2 4* u 2

1. Per hypothefim erit——-— Z -j- hinc ms 2 — 2Zu— u 2

+ 2yu pro abfciffa Osr=u.

rn s 2 u 2 ,

2. Vel —— — Z—hinc ms 2 — 2 uZ—u 2 —2^u.

2 u

3. Quodfi jam radio tO;=Z cogitetur circulus effe descriptus, inquo abfciffae u —Os respondeat ordinata y — rs, vel y —ks; erity 2 :=2 Zu — u 2 .

4. Hinc jam patet evidenter, femper efle ms>y in prima hypothefi(i), et ms<y in fecunda hypothefi (2): in prima hypothefi jacebit ergo•arcus circularis radio tO—Z descriptus alicubi in 0 « intra arcum OXdatae curvae; in fecunda vero hypothefi jacebit is alicubi in 0/3 extra ar-cum OX.

5. Ponamus jam in prima hypothefi (1) praeter circulum 0 « radiotO = Z descriptum describi circulum 0/3 radio gO —Z-fo>, pro diffe-rentia w = gt quantumcunque parva; respondebit in hoc circulo abfciffaeOs = u quadratum ordinatae k s 2 ;=2Z u + 2« u—u 2 .

6. Cum