404 PHILOSOPHICAL TRANSACTIONS. [aNNO J 687. 



px' — qz + r = O; make BD, fig. 11, = 46 ; AB = ^i^ ; BK = i, or ^ 

 the parameter; KC = 2AB = -^bb; KE — ib b — i-p ; AE — 4. = -r^bb 

 — 4^,- FE = tV^^ — Tbp ; and EG = ^f^b^ — ^b p + xq ; which done, a 

 circle from the centre G with the radius ^ GD^ — r will intersect the parabola, 

 either in none, or 2, or 4 points; from whence perpendiculars let fall on DH, 

 will ^ive all the roots z. But that they may be 4, it is evident that the centre 

 of the circle should be found somewhere within a space, from any point of 

 which, three perpendiculars may be let fall upon the curve of the parabola ; and 

 also that the radius be less than the greatest of those perpendiculars, and greater 

 than the middle one. But if the centre be posited without this space, so that 

 only one perpendicular can be let fall on the parabola, and the radius be greater 

 than it ; or if it be less than the middle one of the three perpendiculars, but 

 greater than the least of them, then there can be only two roots ; but there is 

 none at all, when the radius v^G^D^ — r is less than the least of the three, or 

 one, as often as there is but one. 



Now it remains to inquire, of what kind this space is, by what limits it is 

 distinguished, and under what conditions the radius of the circle is less or 

 greater than the above-mentioned perpendiculars. And first of all we must 

 show how a perpendicular is to be let fall upon the parabola. Let ABC, fig. J 2, 

 be a parabola ; AE its axis ; AV half the parameter ; G the point from whence 

 the perpendicular is to be let fall ; let GE be drawn perpendicular to the axis, 

 and let VE be bisected in F, and erecting the perpendicular FH on the same 

 side of the axis, make FH = ^-GE ; then the circle described on the centre H, 

 with the radius HA, will intersect the parabola either in three points, or one, 

 Z, to which the right lines GZ being drawn, will be perpendicular to the curve 

 of the parabola. 



But that there may be three such intersections, the centre H, fig. 10, should 

 be so posited, as that it may be within the space included by the paraboloids, 

 that is, that FH may be less than ■/^^j-VP, or Fff less than the cube of 

 fVF; and so GE = 4FH will be less than 4v^^»^VF^, or4/-5VVF/; that 

 is, the square of GE will be less than 4-fVE^. Therefore these limits coin- 

 cide with two paraboloids of the same kind with those which were used in cubic 

 equations, but whose parameter is twice" less; viz. -f-J^ of the" parameter of the 

 parabola, that is V of A V ; and therefore it is that very curve, by whose evolu- 

 tion the parabola is described, as Huygens has demonstrated ; and which is 

 always touched by the line DF, fig. 11, a perpendicular to the parabola in the 

 point D. And the point P, in which the right line DF touches the paraboloid, 

 is the centre of a circle, which being described with the radius DP, coincides 



