406 PHILOSOPHICAL TRANSACTIONS. [aNNO 1687. 



11. In this case, if it be + r, there can be only two roots, y* + -^gy from 

 the greatest y, being greater than r, otherwise none. But if 4.9^ — y'* from 

 the least y, will be greater than r marked with the sign — ; but r greater than 

 ■^qy — y* from the middle y, then there will be four roots ; but two only, if r 

 be found greater than the former, or less than the latter. 



But if in the equation it be + p or — p, and qqhe greater than -fiyp^, the 

 equation y^. >(;. -i-pz/. 4-y is explicable by only one rooty; that is, there can be 

 only one perpendicular let fall from the centre of the circle ; whence it may be 

 certainly concluded, that there can be only two roots in the given equation, 

 whose sum, if it be — r, is increased with the quantity r ; but if it be + r, 

 having obtained the quantity y, that quantity r should be less than y* -f iqy ; 

 for if it be greater, the equation proposed is absurd and impossible. 



It would be both tedious and needless to run over all the equations of this 

 kind, since it is evident from what has been already said, which are negative and 

 which affirmative roots; and that the limits of these roots are derived from the 

 quantities y found. But for an example, which any one may imitate in other 

 cases ; let it be proposed to discover the limits or conditions, under which, there 

 may be four affirmative roots in a biquadratic equation. Now this will be the 

 case as often as the centre of the circle G falls within the space VPK, fig. 11, 

 and also as often as it is + r, or the radius of the circle is less than GD : whence, 

 it is plain that the equation, here designed, is of this formula, z* — bz^ -{• pz' 

 — ^z + 7 = O; and that/> cannot be greater than fbb, nor -^pb in this case 

 greater than -^b^ + -^9; again, it is necessary that -^bb — ^pV ^bb — -^p 

 should be greater than -^b^ -\. :^q — ^bp ; and from these limits it will be 

 plain, that the centre is contained within the space VPK. But to determine 

 the quantity r, this cubic equation must be first solved, y^. sjc. — ^b'^ — ■^. py 

 = -jV^' + -T? — ■f/'^ ; and we shall have the points upon which perpendicu- 

 lars, from the centre, fall on the curve of the parabola. 



Now having found the three values of this y, the quantity r should be less 

 than -TTs-b* + t^9 — -rV^^P + ^y* ~ \^'^yy + pyy from the middle y, but 

 greater than -^^b* -^ \bq — -t^bbp + 3y* — -f-i^y'^ + Pyy from the least y. 

 But if r exceed these limits, there can be only two roots obtained. Lastly, 



if tI-b-^'' + -tI^P — -TT^l^P -\- 3^* — f^^yy + pyy from the greatest y, be 

 less than r, then the equation proposed is impossible. It may also happen that 

 there are four affirmative roots, when the centre G falls in the little space VTS ; 

 viz. drawing RTS perpendicular on the middle of the supposed line AD ; but 

 this happens when p is greater than -^bb, and-^bb — ip^xV^^ — iP greater 

 than -^pb — -r^b bb — \q; in which case always two, sometimes three of the 

 roots are greater than \b. 



