TAB.LXI,Fig. 7r.

TAB.LXI.Fig, IX,

458 _N°.CXLIX. LecTio XVII. INVEÌÏTIQ CENTRI

generatur altéra ; affcremus exemplum Parabolse supra allatum ,ubi videbitur , quod eadem a?quatio proveniat pro natura cur-va?, qua? per suam evolutionein describit Parabolam» quX pro-venît pro natura curva: centrorum osculantium.

Sit itaque A B Parabola, A E x, B E ZZZ y ZZ vGx ìquia generaliter in omnibus curvis B E)ZZZ(à* ) d(dx t•L'dy 1 ):—— ddydx, substituendus est valor ipsius dy & ddy. Estautem ìn Parabola dy == adx : 2 d ax^ proinde dy l = adx*'. 4X,& ddy = — adx* : 4X d a x ; ergo ( dx* -f. dy x ) : — d dy =( a- 3 r^x') s/ax : a , & /(c/x* ) : dx=z d (<* 4 * 4X) : ^4x4

îdeoque B D z=(^+4-v) V' (a-h 4X): / 4^. Sed ut natu-ra habeatur curvae in qua punctum D , juxta modum Car-THSII, sit AL = í, LD = t ; quia E H === A a , erit B H= V ( ax+’^aa'); ob BH: BE = BD: BE + L D , inve-nitur LD = 4.x dax: a = t, & ob BH:EH=BD:EL>rcperitur EL = ia + zx , proinde AL = 1*4- 3x1= s j sit»ut supra posuimus j s=:±a + r, erit r= 3X, velx = ir, & sic4 xs/ax: a =z/p d \ar : ^a=^=f } reducta œquatione habetur 1 6r ì— 2 yatt , quae eadem est .rquatio, quam supra invenimus. Ergo» &c.

Sit A B Parabola cubicalis prima, parameter = a » A Ë

BE

d a a x erit d y = adx: 3 $ a x x »

-.aadx i : 9 ^ 4 ax? , & ddy = — 2 a d x*: 9 ^ ^x 5 —

_a/ï+9x fy aax _ 2 a _

x a a x ' 9 xtf axx

2 adx*:$x y' < *xx:ergo(Wx í '4-tì[)' i ):—

^ (/x _j_9x^á z x 2 ); 2 ^ ^ x, & \/ (dx* -\-dy í d x = d (aa+?x y aax)-. < 9 *y xxx, IdeoqueBD = ^+?*f««

a a + 9 5C^ aax,

9 x aax. /

2 a a

* V(

Si AB sitHyperboIa» diameter ==a ì parameter =£, AE*=x, E B = y = / (4^x4- <*xx v'erit dy= (aa-\- zax)dx :3 d ( aabx 4- ab x x ) y d y 1 = Qt ì 4- a x 4- 4 a x x') dx z : (j^abx4- 4^xx ) & ddy = — a} d x* : (q.ax-{-xx) d (aab x-\-ab x x~)kErgo (dx* -{-dy* ) : - ddy==8cc.

Sit À B Cyclois , AGH circulus genitor» AE = x, EBz==y = EG 4-AG = V (a a x — XX ) 4- § ; erit ergo d y