VOh. XVI.] i-HILOSOPHieAI. TBAlif3ACTION'S. 40)1 



the difference between b and the sum of the affirmative roots. But the centre 

 being within the space ENGa, either of the affirmative is greater than -^b, 

 but less than ^4-66 + ^b; and the negative always less than i-b. But the 

 limits of the greatest affirmative root become nearer from the given quantity py 

 being always less than ^^ \bb ^ p + 4-^> and greater than ^ -{bb — i-p + 

 ■^b ; but the other affirmative root, which diminishes with the quantity q, is less 

 than this limit, also the negative root is always less than ^^^bb + ^p — -yb, 

 and it vanishes when the quantity q is wanting. 



In the third formula, there are two negative roots and one affirmative root. 

 In this, as in the fourth, the roots are not limited by the quantity b : but 

 the affirmative root is ever less than '^ ^b b -\- ^p -\- -^b, yet greater than 

 V p ^ ^bb -\- i^b; and the greatest of the negative is always greater than 

 V ^bb + ip — i-b, but less than Vp + ^bb — 46 ; and the lesser of the nega- 

 tives is always lessened with the diminished quantity q. 



In the fourth formula, the centre falling within the space LaPD; if there, 

 be two affirmative roots and one negative, the greatest of the affirmative cannot. 

 be greater than Vp + -^bb + -^b, nor less than ^-^bb + ■§■/> + -^b ; but 

 the lesser root is diminished by these limits, the quantity q being decreased. 

 A lso the negative root is less than ^^^^bb + ^p — ^b, and greater than 

 Vp + ±.bb — -yb. 



But here it is to be noted, that the negative roots are every where marked' 



with the affirmative sign, because these are the affirmative roots of those four 



equations, wherein -j- b is found, and q affected with the contrary sign, as was 



hinted above. The demonstration of all this follows from hence, that wherever 



the centre R of the circle falls on the curves VPX or VaL, its circumference 



touches the parabola in a point, whose distance from the axis is t/-^ VH, and 



cuts it on the other side of the axis at the distance of 2 v'-J- VH ; but when the; 



centre falls on the line DPD, one of the roots is = 0, and consequently the 



cubic equation is reduced to a quadratic, or to z^ — bz + p = O, whose roots, 



give the limits, where the quantity q vanishes: and by how much the less y 



becomes, by so much the nearer the roots approach to those limits. The^ 



equation is also a quadratic, when the centre falls in the axis ; that is, when ^g 



= ^bp — -,\b^ in the first formula, or -^q = -^bbb — ^bp in the second; 



in the third it is impossible ; but in the fourth, when \q = -^^b^ + ^bp, in 



which case, the lesser of the affirmative roots is \b, and the greater V-^bb -f- p 



-f- -ig^b ; but the negative '^■^bb + p — ^b. In the first formula, the roots 



are '^b, and ^b ±_ V^ bb — p. But in the second, -^b and >/ -^bb — p + -i^b 



are affimative, and »/ -^bb — p — l-i '\s negative. 



VOL. in. 3 F 



