CAPUT IX.

227

determinato cafu, quo fuerit u — 0 vel u = -£a dabit cofmr=o vel Sin n — i :nimirum tangens ad verticem axeos conjugati DE etl huic axi perpendicu-laris, et parallela axi transverso AB; tangens vero ad verticem axeostransverti AB eft eidem perpendicularis, et parallela axi conjugato DE.

2 b u

Cofm — Sinn

/(a 4 —4U 2 (a 2 — b*)/

535. Corollarium 10,

sUi/Ya 4 — 4u 2 (a 2 —b 2 )) „ . _ 4

Pro /i=rDQerit — -p«r(;Z2. 469. §.) ;

quodfi ergo integrale hujus exponentis per (359- 2 38'§0 ita determinetur,ut pro u —o fiat quoque ,u.”0; obtinebitur arcus ,it~DQ debitus ab-fciflae u —Lp; adeoque pro u:==CB;=Jla obtinebitur quadrans ellip-feos DB.

536. Corollarium 11.

Et pro S=DCpQ erit per (488- 53°-§0 «8—^-6 u /'(a 2 — 4U 2 ) r

hinc, quia debet fieri S = DCpQ=o pro u=:Cp:=o, inveniemus per

(289. 238. §) S^=DCpQ = —/(a 2 —4U 2 ) + ~ ArcSin^- Quamob-

4 a o a

2 u

rem pro ur=|a, quo cafu fit ArcSin-^— = ArcSin 1 ~\x, obtinebiturarea quadrantis elliptici DDB — ^ ^ - area igitur totius ellipfeosADBEA^='ab7r efl ad aream circuli, qui fuper axe AB —a trans-

verso tanquam diametro descriptus cogitetur, ficut axis conjugatus b adtransversum a.

537. Corollariu m 1 2.

Si pro Ba—x, aP = y arcus ellipseos BP(i8.Fig.) revolvatur'circa ^ Igaxem transverfum, generabitis segmentum S— BPp fpb.aercidis elliptici,cujus genefis revolutione totius ellipseos circa axem transversum abfulvi*tur: igitur per (490. 525. §.) repelletur sS, unde integrando per (249.

7r t > 2

238- %■) obtinebimus soliditatem segmenti S = B Pp = (3 ax 2 —2x 3 ).

Hinc pro x — ^a invenietur foliditas dimidii (phaeroidis elliptici —,

.. . . . x a b z

et fouaitas integri fphaeroidis —— -r —-

Ff 2

538.C0-