Miscellanea Curio fa. 109 

 | is exactly limitted from the foregoing 



Equation y/ddd =a fc 3 +* i ? co £ bp.\ vizi 

 when the Circle touches the Parabola. There- 

 fore i q ought to be lefs than - 6 bp — £ 7 b 3 -\- 

 \ld i - 0 but if p be greater than alibi # 

 ought to be bigger than '% bp ~* 2 ± b 3 \/d 3 y 

 that the Center may not fall in the little 

 Space NUW. And with thefe Conditions 

 the Equation will always be explicable by 

 three Roots - 7 otherwife but by one. But 

 whether there be three or one, they are al- 

 ways Affirmative ones, becaufe of the por- 

 tion of the Center R to the Right Hand of 

 £he Line DP. 



And this is the molt difficult Cafe \ fo that 

 thofe that well underftand what has gone be- 

 fore, will without any trouble take what 

 comes after. Now let the Equation x} — < h 

 z. 2, 'V pa + q^ a^ be given. Here (that there 

 may be three Roots had) the Center of the 

 Circle ought to be found fomewhere within 

 the Space PNa, determin'd by the right 

 Lines PN, Pa, and the Curve of the Para- 

 bola Na; wherefore fince EF is ca | bb r p 

 ought to be lefs than \ bb. Now for the de- 

 termination of the quantity q y d being s=s f 

 b 2 f p as before, *Jdl Hr 2 j b 2 — i bp ought 

 always to be greater than % q, that fo the 

 Center of the Circle may be pofited in the 

 forementioned Space PNa ; which when 'tis 

 jfb, fueh an Equation has two Affirmative 

 Roots, and one Negative. But if p be greater 

 than f bb % or i q greater than Vd 3 -\- 2 j b % i- 

 i bp \ it is explicable but by one (aad that a 

 Negative) Root, 



