VOL. XVI.J PHILOSOPHICAL TRANSACTIONS. 379 



last term r ; therefore taking away the 2d term, and finding the centre E, the 

 line E O is the radius of the circle ; viz. since « r := 0, the whole 5th term, in 

 the new equation, arises from the taking away of the 2d term. As for 



Exam. 2. U z^ — b z^ + apz + a^ q = O. This multiplied by z, it be- 

 comes z* — bz^ -{- apz^ -\- a^ qz = O. To take away the 2d term, let x + 

 ■J- /j = z ; then it will be 



X* + bx> + ^b^x^ + ^b'x + r^b* = + z\ 

 — bx' — ib'^x^ — ^b'x — ^b^ = — bz\ 

 4- a/)z*-f ^abpx+ -^apb* = + a/)z% 

 + a^ qx -{• ^ a^ qb = + a*5'Z. 



In this new equation, the half co-efficient of the 3d term divided by a, viz. — 



-g- -\- 4- /J, is to be substituted for :^p; and the half co-efficient of the 4th 



term divided by a^, the square of the parameter, viz. — -g-^ + ^ + 4- ?» 



instead of \ q, in Descartes's construction ; from hence the centre E is deter- 

 mined. Then drawing a parallel to the axis, at the distance of 4- ^ on the left 

 hand side, because ar + -j^ 6 = z, and let O be its intersection with the para- 

 bola ; then a circle described with the centre E and radius E O, will either cut 

 or touch the parabola in as many points as the equation has true roots : which 

 roots, or z, are perpendiculars drawn from these points to the parallel to the 

 axis, the affirmative on the right side, and the negative on the left. 



If the 3d or 4th term, or both, be wanting in the equation, in investigating 

 the central rule, there is no manner of difference at all to be observed ; but the 

 quantity p or q being wanting, those parts of the lines C D, and D E, somehow 

 deduced from that quantity, will be wanting too ; and we are to proceed with 

 the remaining co-efficients of the third and fourth terms in the new equation, 

 according to the method prescribed in the preceding examples. 



Hitherto Mr. Baker's general method has been considered, than which none 

 more easy and expeditious can be expected, assuming for the construction 

 either a parabola, or any other curve, viz. when the equation rises to a biqua- 

 dratic : but while I was writing these things, I hit upon a geometric effection 

 of the central rule, which is expeditious beyond what can be hoped for, and 

 will sufficiently satisfy the curious in those matters. Having described the 

 parabola N AM, fig. 8, pi. 10, whose vertex is A, axis ABC, and parameter a, 

 let the equation be reduced to this form z*. bz". apzz. aaqz. ar = O; or to this, 

 if it be only a cubic, z'. bz^. apz. aaq = O; then at the distance B D = ^ /; 

 let the line D H, meeting the parabola in D, be drawn parallel to the axis to 

 the left if it be — b, and to the right i( -{• b', and from D let fall the perpen- 



3 c 2 



