VOL. XVI.] PHII/JSOPHICAL TBANSACTIONS. 405 



with the parabola in the point D, or has the same curvature with it, as is 

 manifest. 



Having therefore described such paraboloids VXP, VNa, on each side of 

 the axis, it is plain, that unless the centre of the circle be placed within these 

 limits, it cannot intersect the parabola in more than two points ; from whence 

 we may determine, under what conditions, the coefficients of the intermediate 

 terms are restrained in biquadratic equations, so that there may be 4 roots. And 

 at first sight it is plain, that p cannot be greater than \bb (viz. in those for- 

 mulas wherein it is + p) nor q than -rV^*- But in general, ~^^'+ -^pb +" 

 4^9, that is E G, the distance of the centre from the axis, should be less than 

 E H = 4v^-JV VE^ that is, {hecaaae V E = -^b b + ^p) than -kbb + ^p 

 ^tV^^ + or — -^p; the signs + and — being left doubtful, that they may 

 be varied according to the nature of any equation, as was shown above in 

 cubits. 



As for the limitation of the last term r, it cannot be found with the same 

 ease ; because the letting fall a perpendicular on the curve of a parabola is a 

 solid problem, which cannot be resolved without the solution of a cubic equa- 

 tion. Therefore, in the first place, let the second term be wanting, or if it be 

 in the equation, let it be taken away, so that it may have this form z*. ^. pz'. 

 qz. r =z o. Now if it be — r,^it is always explicable by two or four roots ; and 

 that there may be four, the centre of the circle should be posited within the 

 above-mentioned paraboloids, or that it may be — p, and qq may be less than 

 ■^p^, or the cube of ^p ; then let the roots of this equation y^. :4j- ipy- 4? = O 

 be found, the quantities p and q having the same signs as in the biquadratic ; 

 and these roots are found expeditiously enough by the table of sines. But having 

 found those three y, (which are ordinates to the axis of the parabola, from the 

 points where perpendiculars fall on its curve, viz. YZ, fig. 12,) then j&yy — 

 3y* from the lesser y, will denote the greatest quantity r, if it be — r, than 

 which if r be less, the equation will have four roots, otherwise only two. But 

 if it be -|- r, it should be less than 3y* — pyy from the middle y ; for if it be 

 greater, it can have but two roots ; at least if r be less than 3y* — pyy from 

 the greatest y. But if it be greater than this, the equation is not explicable by 

 any true root at all. And these same limits are otherwise expressed by the 

 quantity q ; viz. ^^qy — y* in the first case, y* — \qy in the second, and y* + 

 \qy in the third. 



But it may happen, that the two lesser quantities y may not be far distant 

 from each other, and thence both the perpendiculars greater than the right line 

 GA, viz. when ^y is greater than ^VP'j but less than -jVP'*? the centre falling 

 within the space contained between the paraboloids of both the figures 10 and 



