JSiifcellanea, Curtoja. 125 



gative and which Affirmative, and that the 

 Limits of thefe Roots are deriv'd from the 

 found Quantities y>. But for an Example 

 fwhich any one may imitate in the reft of 

 of the Cafes) let it be propos'd to difcover 

 the Limits or Conditions, under which, there 

 may be four Affirmative Roots in a Biqua- 

 dratic^ Equation. Now this will be as of- 

 ten as the Center of the Circle G is polited 

 in th© fpace UPK (Eg. 2.) and alfo ~br or 

 the Radius of the Circle is lefs than GD. 

 Whence 'tis plain, that the Equation here 

 concern'd is of this Form, *,4 — +/?*2 

 €jz, -f- r and that p cannot be greater 

 than f bbj nor | pb (in this cafe) than r 6 bz 

 ~\~ % °li again, 'tis neceflfary that %bb — %p 



in Vr 6 bb — i p fhould be greater than r 6 h 

 ^ i % h and from thefe Limits, it will 

 be manifelt that the Center is contained 

 within the fpace UPK. But in order to 

 the determination of the Quantity r, this 

 Cubick E quat ion muft be firft folv'd, yl.*., 



-h b 2 - %py^ h bi ~Y \q—ipb ; and fo 

 will be obtain'd the Points upon which fall 

 the Perpendiculars from the Center to the 

 Curve of the Parabola. Now having found 

 the three Values of this y, the Quantity r 

 ought to be lefs than 2 \ 6 k + % —r 6 bbp 

 4? 3jy4 — § b2y2 -\~pyy of the middle J, but 

 greater than 2 j 6 U % hq — r 6 bbp rf 374 — 

 $b*y2-\-pyy of the leaft y. But if r exceed 

 thefe Limits, there can be but two Roots 

 obtain'd, Laftly, if af 6 £4 1 bp-~~- 6 bbp ~b 

 3y4 — 8 Wy + pjyy of t he great eft y 7 be grea- 

 ter than V r then the Equation proposed is 



