VOh. XVI.] PHILOSOPHrCAL TRANSACTIONS. 3Qf 



a circle passing through the vertex, in the same manner as if the second term 

 were wanting ; and therefore to determine the number of roots, it is sufficient 

 to know the properties of the locus or curve, which distinguishes the space, 

 wherein, if the centre of the circle, which passes through the vertex of the 

 parabola, be placed, its circumference will intersect the parabola, either in one 

 or three other points ; that is, to define the nature of that curve, on which 

 the centres of all the circles, passing through the vertex and then touching the 

 parabola, do fall. 



Now that locus is the paraboloid, which, with Dr. Wallis, we may call the 

 semicubical, or in which the cubes of the ordinates are as the squares of the 

 corresponding abscisses ; whose latus rectum is y of the latus rectum of the 

 given parabola, and its vertex the point V, the line AV, being half the latus 

 rectum of the same parabola. That is, if we put unity for the latus rectum of 

 the given parabola, then -J^ of the cube of the ordinate will be equal to the 

 square of the absciss ; or the cube of •§■ VH = the square of HR, viz. if R b6 

 the centre of the circle which passes through the vertex of the parabola, and 

 touches the same afterwards. This is that curve which our countryman Mr. 

 Neil first of all demonstrated to be equal to a given right line ; and which de- 

 scribed on both sides of the axis of the parabola, as VNL, VPX, comprehends 

 a space, in which, if the centre of the circle, which passes through the vertex 

 A, be placed, it will cut the parabola in three other points ; but the spaces 

 more remote from the axis afibrd centres for circles, which will cut the parabola 

 only in one point besides the vertex. 



These things being well understood, we shall now proceed to determine the 

 number of the roots. And first, let the second term be wanting, and let the 

 latus rectum be 1, or AV = -^; in the construction VH is = ^p, and RH = iq; 

 and since, if it be + /> in the equation, i/j is to be set off from V towards the 

 upper parts, the centre of the circle is always without the space LVX ; and 

 therefore explicable by one root only, which is affimative if it be — g, negative 

 if + 9j which indeed are the roots found by Cardan's rule ; but if it be — p, 

 then VH = ij& is set off towards the lower parts ; but it may be that HR may 

 fall between the axis and the curve VX or VL, viz. if the cube of ■*- VH, or of 

 ■J-/), be greater than the square of 4-q, or -^'-^p^ be greater than i^qq; in which 

 case there are three roots, two negative if it be — q, and one affirmative equal 

 to the sum of the others ; but if it be -|- q, then there are two affirmative roots 

 and one negative; but if -^ppp be less than -^qq, then there is but one root, 

 affirmative if it be — q, negative '\i -\- q. 



Now let all the terms be complete, and first let there be proposed, for ex- 

 ample, this equation, 7? — bz^ -{• pz — q = O, for which fig. 10 serves. In the 

 construction of this, BC = ^i, VG = 4.AC =^bb,VE=zib\VH = ibb — j.p, 

 GH = 4-6^ — i/>, orip — -^bi hence HO = Vt^* — -s-^/'> ot -^bp—^l^, and 



